Frontiers in Relativistic Celestial Mech

Frontiers of Relativistic Celestial Mechanics.
Volume 1
Editor: Sergei M. Kopeikin
February 15, 2014

List of Contributing Authors
Prof. Dr. Thibault Damour
Institut des Hautes Études Scientifiques (IHÉS)
35 route de Chartres
Bures sur Yvette, F-91440
France
E-mail: damour@ihes.fr
Prof. Dr. Toshifumi Futamase

Tohoku University
Astronomical Institute
Sendai 980-8578
Japan
E-mail: tof@astr.tohoku.ac.jp
Prof. Dr. Sergei Kopeikin

University of Missouri
Department of Physics & Astronomy
223 Physics Bldg.
Columbia, MO 65211
USA
E-mail: kopeikins@missouri.edu
Prof. Dr. Gerhard Schäfer

Friedrich-Schiller-Universität Jena
Theoretisch-Physikalisches Institut
Max-Wien-Platz 1
D-07743 Jena
Germany
E-mail: gos@tpi.uni-jena.de
Prof. Dr. Michael Soffel

Dresden University of Technology
Lohrmann Observatory TU Dresden
Planetary Geodesy
Mommsenstr 13
01062 Dresden
Germany
E-mail: soffel@rcs.urz.tu-dresden.de
ii
Dr. Pavel Korobkov

Solovetsky Township
Arkhangelsk Region 164070
Russia
Dr. Alexander Petrov

Moscow Lomonosov State University
Sternberg Astronomical Institute
Universitetskiy Prospect 13
Moscow 119992
Russia
E-mail: alex.petrov55@gmail.com
Dr. Xie Yi
Nanjing University
Department of Astronomy
22 Hankou Road
Nanjing Jiangsu 210093
P. R. China
E-mail: yixie@nju.edu.cn
Contents
Preface v
1 The General Relativistic Two Body Problem 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Multi-chart approach to the N -body problem . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 EOB description of the conservative dynamics of two body systems . . . . . . . . 5
1.4 EOB description of radiation reaction and of the emitted waveform during
inspiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 EOB description of the merger of binary black holes and of the ringdown of
the final black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 EOB vs NR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6.1 EOB[NR] waveforms vs NR ones . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6.2 EOB[3PN] dynamics vs NR one . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7 Other developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.1 EOB with spinning bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.2 EOB with tidally deformed bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.3 EOB and GSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
References 25
2 Hamiltonian Dynamics of Spinning Compact Binaries 37
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Hamiltonian formulation of general relativity . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.1 Point particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.2 Spinning particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.3 Introducing the Routhian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 The Poincaré algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4 Post-Newtonian binary Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.1 Spinless binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.2 Spinning binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
iv Contents
2.5 Binary motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.5.1 Spinless two-body systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5.2 Particle motion in Kerr geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.5.3 Two-body systems with spinning components . . . . . . . . . . . . . . . . . . 56
References 58
3 Covariant Theory of the Post-Newtonian Equations of Motion of Extended
Bodies 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 A Theory of Gravity for Post-Newtonian Celestial Mechanics . . . . . . . . . . . . . 68
3.2.1 The Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2.2 The Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3 Parametrized Post-Newtonian Celestial Mechanics . . . . . . . . . . . . . . . . . . . . . 70
3.3.1 External and Internal Problems of Motion . . . . . . . . . . . . . . . . . . . . . 70
3.3.2 Solving the Field Equations by Post-Newtonian Approximations . . . . . 72
3.3.3 The Post-Newtonian Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.3.4 Conformal Harmonic Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4 Parametrized Post-Newtonian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.4.1 The Global Post-Newtonian Coordinates . . . . . . . . . . . . . . . . . . . . . . 79
3.4.2 The Local Post-Newtonian Coordinates . . . . . . . . . . . . . . . . . . . . . . . 82
3.5 Post-Newtonian Coordinate Transformations by Asymptotic Matching . . . . . . 90
3.5.1 General Structure of the Transformation . . . . . . . . . . . . . . . . . . . . . . 90
3.5.2 Matching Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.6 Post-Newtonian Equations of Motion of Extended Bodies in Local
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.6.1 Microscopic post-Newtonian Equations of Motion . . . . . . . . . . . . . . . 98
3.6.2 Post-Newtonian Mass of an Extended Body . . . . . . . . . . . . . . . . . . . . 99
3.6.3 Post-Newtonian Center of Mass and Linear Momentum of an
Extended Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.6.4 Translational Equation of Motion in the Local Coordinates . . . . . . . . . 102
3.7 Post-Newtonian Equations of Motion of Extended Bodies in Global
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104
3.7.1 STF Expansions of the External Gravitational Potentials in Terms of
the Internal Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.7.2 Translational Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.8 Covariant Equations of Translational Motion of Extended Bodies . . . . . . . . . . 117
3.8.1 Effective Background Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.8.2 Geodesic Motion and 4-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119
3.8.3 Four-dimensional Form of Multipole Moments . . . . . . . . . . . . . . . . . . 121
3.8.4 Covariant Translational Equations of Motion . . . . . . . . . . . . . . . . . . . 123
3.8.5 Comparison with Dixon’s Translational Equations of Motion . . . . . . .127
Contents v
References 128
4 On the DSX-framework 139
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.2 The post-Newtonian formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.2.1 The general form of the metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.3 Field equations and the gauge problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.4 The gravitational field of a body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.4.1 Post-Newtonian multipole moments . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.5 Geodesic motion in the PN-Schwarzschild field . . . . . . . . . . . . . . . . . . . . . . . 151
4.6 Astronomical Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.6.1 Transformation between global and local systems: first results . . . . . . . 155
4.6.2 Split of local potentials, multipole-moments . . . . . . . . . . . . . . . . . . . . 157
4.6.3 Tetrad induced local coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.6.4 The standard transformation between global and local coordinates . . . . 160
4.6.5 The description of tidal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.7 The gravitational N-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.7.1 Local evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.7.2 The translational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.8 Further developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
References 172
5 General Relativistic Theory of Light Propagation in Multipolar
Gravitational Fields 175
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.1.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.1.2 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.1.3 Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.2 The Metric Tensor, Gauges and Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 184
5.2.1 The canonical form of the metric tensor perturbation . . . . . . . . . . . . .184
5.2.2 The harmonic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.2.3 The ADM coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5.3 Equations of Propagation of Electromagnetic Signals . . . . . . . . . . . . . . . . . . . 189
5.3.1 Maxwell equations in curved spacetime . . . . . . . . . . . . . . . . . . . . . . . 189
5.3.2 Maxwell equations in the geometric optics approximation . . . . . . . . . . 190
5.3.3 Electromagnetic eikonal and light-ray geodesics . . . . . . . . . . . . . . . . . 191
5.3.4 Polarization of light and the Stokes parameters . . . . . . . . . . . . . . . . . . 200
5.4 Mathematical Technique for Analytic Integration of Light-Ray Equations . . . . 207
5.4.1 Monopole and dipole light-ray integrals . . . . . . . . . . . . . . . . . . . . . . . 208
5.4.2 Light-ray integrals from quadrupole and higher-order multipoles . . . . . 209
vi Contents
5.5 Gravitational Perturbations of the Light Ray . . . . . . . . . . . . . . . . . . . . . . . . .212

5.5.1 Relativistic perturbation of the electromagnetic eikonal . . . . . . . . . . . . 212
5.5.2 Relativistic perturbation of the coordinate velocity of light . . . . . . . . .215
5.5.3 Perturbation of the light-ray trajectory . . . . . . . . . . . . . . . . . . . . . . . . 216
5.6 Observable Relativistic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
5.6.1 Gravitational time delay of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
5.6.2 Gravitational deflection of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
5.6.3 Gravitational shift of frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5.6.4 Gravity-induced rotation of the plane of polarization of light . . . . . . . .230
5.7 Light Propagation through the Field of Gravitational Lens . . . . . . . . . . . . . . . 234
5.7.1 Small parameters and asymptotic expansions . . . . . . . . . . . . . . . . . . . 234
5.7.2 Asymptotic expressions for observable effects . . . . . . . . . . . . . . . . . . 237
5.8 Light Propagation through the Field of Plane Gravitational Waves . . . . . . . . . . 239
5.8.1 Plane-wave asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 239
5.8.2 Asymptotic expressions for observable effects . . . . . . . . . . . . . . . . . . 241
References 245
6 On the Backreaction Problem in Cosmology 253
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
6.2 Formulation and Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
6.3 Calculation in the Newtonian gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
6.4 Definition of the background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
References 261
7 Post-Newtonian Approximations in Cosmology 265
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
7.2 Derivatives on the geometric manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . .272
7.2.1 Variational derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
7.2.2 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275
7.3 Lagrangian and Field Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
7.3.1 Action Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
7.3.2 Lagrangian of the Ideal Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
7.3.3 Lagrangian of Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .279
7.3.4 Lagrangian of a Localized Astronomical System . . . . . . . . . . . . . . . . 280
7.4 Background manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
7.4.1 Hubble Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .281
7.4.2 Friedmann-Lemître-Robertson-Walker Metric . . . . . . . . . . . . . . . . . . 281
7.4.3 Christoffel Symbols and Covariant Derivatives . . . . . . . . . . . . . . . . . . 284
7.4.4 Riemann Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Contents vii
7.4.5 The Friedmann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

7.4.6 Hydrodynamic Equations of the Ideal Fluid . . . . . . . . . . . . . . . . . . . . 288
7.4.7 Scalar Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
7.4.8 Equations of Motion of Matter of the Localized Astronomical
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
7.5 Lagrangian Perturbations of FLRW Manifold . . . . . . . . . . . . . . . . . . . . . . . . 290
7.5.1 The Concept of Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
7.5.2 The Perturbative Expansion of the Lagrangian . . . . . . . . . . . . . . . . . . 292
7.5.3 The Background Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
7.5.4 The Lagrangian Equations for Gravitational Field Perturbations . . . . . 295
7.5.5 The Lagrangian Equations for Dark Matter Perturbations . . . . . . . . . . 298
7.5.6 The Lagrangian Equations for Dark Energy Perturbations . . . . . . . . . . 298
7.5.7 Linearised post-Newtonian Equations for Field Variables . . . . . . . . . .299
7.6 Gauge-invariant Scalars and Field Equations
in 1+3 Threading Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
7.6.1 Threading Decomposition of the Metric Perturbations. . . . . . . . . . . . .302
7.6.2 Gauge Transformation of the Field Variables . . . . . . . . . . . . . . . . . . . 304
7.6.3 Gauge-invariant Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
7.6.4 Field Equations for the Scalar Perturbations. . . . . . . . . . . . . . . . . . . . 307
7.6.5 Field Equations for Vector Perturbations . . . . . . . . . . . . . . . . . . . . . . 310
7.6.6 Field Equations for Tensor Perturbations . . . . . . . . . . . . . . . . . . . . . . 311
7.6.7 Residual Gauge Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
7.7 Post-Newtonian Field Equations in a Spatially-Flat Universe . . . . . . . . . . . . . 312
7.7.1 Cosmological Parameters and Scalar Field Potential . . . . . . . . . . . . . .312
7.7.2 Conformal Cosmological Perturbations . . . . . . . . . . . . . . . . . . . . . . . 314
7.7.3 Post-Newtonian Field Equations in Conformal Spacetime . . . . . . . . . .317
7.7.4 Residual Gauge Freedom in the Conformal Spacetime . . . . . . . . . . . .321
7.8 Decoupled System of the Post-Newtonian Field Equations . . . . . . . . . . . . . . . 322
7.8.1 The Universe Governed by Dark Matter and Cosmological Constant . . 322
7.8.2 The Universe Governed by Dark Energy . . . . . . . . . . . . . . . . . . . . . . 326
7.8.3 Post-Newtonian Potentials in the Linearized Hubble Approximation . . 327
7.8.4 Lorentz Invariance of Retarded Potentials . . . . . . . . . . . . . . . . . . . . . 332
7.8.5 Retarded Solution of the Sound-wave Equation . . . . . . . . . . . . . . . . . 335
References 338
List of Figures
1.1 Sketch of the correspondence between the quantized energy levels of the real and effec-
tive conservative dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 EOB waveform versus the most accurate numerical relativity waveform . . . . . . . . . 20
1.3 Close up around merger of the waveforms of Fig. 1.2 . . . . . . . . . . . . . . . . . . . 21
1.4 Comparison between various analytical estimates of the energy-angular momentum func-
tional relation and its numerical-relativity estimate . . . . . . . . . . . . . . . . . . . . 22
4.1 One global and N local coordinate systems are used for the description of the gravita-
tional N -body problem (from [44]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4.2 Three events of importance for the inversion of the coordinate transformation, eX , et
and eT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.1 The Penrose Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

5.2 Astronomical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
5.3 Gravitational Lens Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
5.4 Plane-Wave Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
Preface
The science of relativistic celestial mechanics is an essential branch of the modern gravitational physics,
a branch exploring the fundamental structure of spacetime by studying motion of massive bodies like
black holes, stars, planets as well as elementary particles, including photons, in gravitational field. It
establishes basic theoretical principles for calculation and interpretation of various relativistic effects
and phenomena observed in astrophysical stellar systems and in the solar system. Relativistic celestial
mechanics of massless particles like photons is more known among astronomers as relativistic astrom-
etry. An indefeasible branch of gravitational physics, it is required to map the coordinate description of
motion of celestial bodies into parameter space of observables. Theoretical progress in understanding
the orbital motion of celestial bodies would be inconceivable without a corresponding improvement in
mathematical description of motion of light rays in stationary and time-dependent gravitational field.
Relativistic celestial mechanics has received a special attention in the gravitational-wave astronomy.
Being on its way to direct detection of gravitational waves emitted by coalescing binary stars, the
gravitational-wave astronomy urgently needs highly-precise templates of gravitational waves emitted
by the stars at the very last stage of their orbital motion, just a few seconds before the stars collide and a
catastrophic supernova explosion takes place. Therefore, development of theoretical tools of relativistic
celestial mechanics has a fundamental significance for achieving further progress in gravitational-wave
astronomy which is expected to become a primary experimental tool bringing much deeper understand-
ing of the nature of gravitational field and the underlying geometric structure of the spacetime manifold.
Relativistic celestial mechanics was a subject of active research by many notable scientists, including
A. Einstein, H. Lorentz, V. A. Fock, T. Levi-Civita, L. Infeld, S. Chandrasekhar, J. Ehlers, G. C. McVit-
tie, and others who elaborated on various approaches to the equations of motion of celestial bodies and
the theory of astronomical observations in general relativity. More recently, a valuable contribution to
relativistic celestial mechanics was made by T. Damour, G. Schäfer, M. Soffel, C. M. Will, K. Nordtvedt,
T. Futamase, K. S. Thorne, W. G. Dixon, L. Blanchet, I. Rothstein. A key figure of relativistic celestial
mechanics of the second half of XX century has been Victor A. Brumberg, a scholar who presently lives
in Boston (USA) and who is still active in research. Victor A. Brumberg has made a significant con-
tribution to general relativity and the science of relativistic planetary ephemerides of the solar system.
He mentored and inspired many researchers around the globe (including the Editor of this book) to start
working in the field of relativistic celestial mechanics. The very term “relativistic celestial mechanics”
was introduced by Victor. A. Brumberg in his famous monograph “Relativistic Celestial Mechanics”
published in 1972 by Nauka (Science) – the main scientific publisher of the USSR – in Moscow. For
next two decades this monograph remained the most authoritative reference and the source of invalu-
able information for researchers working on relativistic equations of motion and experimental testing of
general relativity. Victor A. Brumberg received the 2008 Brower Award from the Division of Dynamic
Astronomy of the American Astronomical Society. The Brouwer Award was established to recognize
outstanding contributions to the field of dynamical astronomy, including celestial mechanics, astrometry,
geophysics, stellar systems, galactic and extragalactic dynamics.
The present book is a first volume of Festschrift aimed to honour the scientific influence and achieve-
vi
ments of V. A. Brumberg, and to celebrate his 80-th birthday which took place on February 12, 2013.
The book appears on the eve of another remarkable date – 100 years of Einstein’s general relativity
– the theory which dramatically changed the world of theoretical physics by opening new fascinating
opportunities in the scientific study of fundamental laws of Nature. The volume consists of seven chap-
ters discussing the recent theoretical advances in relativistic celestial mechanics and related areas of
theoretical physics and astronomy.
Chapter 1, written by T. Damour, introduces the amazingly rich mathematics of the relativistic two-
body problem. Solution of this problem within the Newtonian mechanics is cornerstone material that
can be found in any textbook on celestial mechanics. On the other hand, complete solution of this prob-
lem within general relativity has not been yet obtained, even though it has been subject of numerous
analytical investigations. The root of the difficulty is lying in the non-linear character of gravitational
interaction in Einstein’s theory of gravity, which prevents us from finding an exact solution to the prob-
lem. Hence, the analytic solution can be ascertained only by making use of successive approximations.
The method includes complicated, often diverging integrals which require development of regularisa-
tion technique based on the theory of distributions. Additional difficulties arise due to the emission
of gravitational waves by the two-body system, an effect generating a back reaction on the motion of
the bodies - the so-called radiation-reaction force. After reviewing some of the methods used to tackle
these problems, Chapter 1 focuses on a new, recently introduced approach to the motion and radiation of
(comparable-mass) binary systems: the effective-onebody (EOB) formalism. The basic elements of this
formalism are reviewed, and some of its recent developments are discussed. Several recent tests of EOB
predictions against numerical simulations have shown the aptitude of the EOB formalism to provide
accurate description of the dynamics and radiation of various binary systems (comprising black holes
or neutron stars) in regimes that are inaccessible to other analytical approaches such as the last orbits
and the merger of comparable mass black holes. Chapter 1 provides weighty arguments that, in synergy
with numerical simulations, the post-Newtonian theory and gravitational self-force (GSF) computations,
the EOB formalism is likely to provide an efficient way of accurately computing the numerous template
waveforms that are needed for the purposes of gravitational wave data analysis.
Chapter 2, written by G. Schäfer, continues theoretical analysis of the two-body problem in gen-
eral relativity, by making use of the advanced Hamiltonian technique introduced by Arnovitt, Deser
and Misner (ADM formalism). The Hamiltonian setting of general relativity allows a very elegant and
transparent treatment of the dynamics and motion of gravitating systems. Crucial in that context is the
computation of the reduced Hamiltonian which generates the dynamics of both the gravitating objects
and the gravitational field. Based on the framework of post-Newtonian approximation, the Chapter 2
covers the dynamics and motion of spinning compact binaries up to the fourth post-Newtonian approxi-
mation.
Chapter 3, written by M. Soffel, furnishes an account of the Damour-Soffel-Xu (DSX) formalism of
relativistic reference frames in N-body system. The DSX formalism is an extension of the formalism
advanced in 1988 by Brumberg and Kopeikin (the BK formalism) to build the post-Newtonian theory of
astronomical reference frames in the solar system. The BK-DSX theory is based on the complementary
use of N local coordinate charts attached to each body, which are used to describe rotation and local
dynamics of the body, and of a global coordinate chart, which is intended to describe the orbital motion
of the bodies. The advantage of the DSX formalism, compared to the BK formalism, is in the systematic
use of well-defined mass-type and spin-type multipole moments of the extended bodies. Chapter 3
explains the DSX formalism in a concise but mathematically rigorous form.
Chapter 4, written by Y. Xie and S. Kopeikin, presents a covariant theory of post-Newtonian equations
of translational motion of extended bodies in an N-body system. It significantly extends the results ob-
tained in 1970-80th by W. G. Dixon. The new theory is based on the combined BK-DSX theory extended
vii
to the realm of the scalar-tensor theory of gravity. It introduces one more type of multipole moments
to the formalism - the scalar-type moments. The chapter explains how to build the local and global
coordinates in a system of N extended bodies, and offers a procedure intended to derive the translational
equations of motion of the bodies, including all internal multipoles. It is proven that any integral moment,
which depends on the internal structure of the bodies in a way different from the "canonical" Blanchet-
Damour moments, vanishes from the translational equations of motion. Finally, a covariant form of the
post-Newtonian equations of motion of extended bodies, with all internal multipoles taken into account,
is derived by applying a technique proposed by Thorne and Hartle in 1985. The translational equations
of motion derived in this way represent a profound generalisation of the Mathisson-Papapetrou-Dixon
equations of motion.
Chapter 5, written by P. Korobkov and S. Kopeikin, delivers theoretical tools for solving the problem
of propagation of photons through multipolar gravitational field of an isolated astronomical system emit-
ting gravitational waves. The solution is written in the first post-Minkowskian approximation of general
relativity. The Chapter opens with an introduction to the linearised theory of retarded gravitational po-
tentials of the Lienard-Wiechert type. The Chapter then deals with derivation of differential equations
of light geodesics with retarded argument. Mathematical technique of integrating these equations is
proposed, and a solution is found in a closed form. It is demonstrated that the leading-order observable
relativistic effects depend on the value of the multipoles of the isolated system and their time derivatives
taken at the retarded instant of time. This retardation is caused by finite speed of propagation of gravity,
and for this reason the relativistic effects do not depend on the integrated values of the multipoles taken
along the past worldline of the isolated system. The integration technique reproduces the known results
of integration of equations of light rays in the stationary approximation of a gravitational lens and in the
approximation of a plane gravitational wave. Two limiting cases of small and large impact parameters
of a light ray with respect to the isolated system are worked out in more detail. It is shown that in case of
a small impact parameter the leading-order terms in the solution for light propagation depend neither on
radiative nor on intermediate zone components of the gravitational field, but the main effect comes from
the near-zone values of the multipole moments. This radiative-zone effacing property makes it much
more difficult (but not impossible!) to directly detect gravitational waves by astronomical instruments
than it was assumed by some researchers. Chapter 5 also presents analytical treatment of time-delay
and light-ray bending in the case of large impact parameter corresponding to the approximation of plane
gravitational wave. Explicit expressions for the time delay and the deflection angle of the light ray are
obtained in terms of the transverse-traceless (TT) multipole moments of the gravitating system. This
result can be directly applied to interpretation of observables in gravitational wave interferometers.
Development of the canonical theory of post-Newtonian approximations in relativistic celestial me-
chanics relies upon the key concept of an isolated astronomical system, under assumption that back-
ground spacetime is flat. The standard post-Newtonian theory of motion is instrumental in explanation
of the existing experimental data on binary pulsars, satellite and lunar laser ranging, and in building pre-
cise ephemerides of planets in the solar system. Recent cosmological studies indicate that the standard
post-Newtonian mechanics fails to describe more subtle dynamical effects in the small-scale structure
formation and in the motion of galaxy clusters comprising astronomical systems. In those settings, the
curvature of the expanding universe interacts with the local gravitational field of the astronomical system
and, as such, can not be ignored. Therefore, working out theoretical foundations of relativistic celestial
mechanics of isolated astronomical system residing on cosmological manifold is worthwhile. Additional
motivation for this comes from the gravitational wave astronomy which will study relativistic celestial
mechanics of binary systems in very distant galaxies residing at the edge of the visible universe. Dy-
namical evolution of the binaries on a cosmological background is primarily governed by multipolar
structure of its own gravitational field, but is also intrinsically connected with the cosmological parame-
viii
ters of the background manifold. These parameters are determined by the content of the substance filling
up the universe, whose most enigmatic components are the dark matter and dark energy. Tracking down
the orbital motion of binary systems in distant galaxies at gravitational wave observatories is promising
for doing precise cosmology. It is very likely that observation of binaries with gravitational wave de-
tectors will supersede the precision of measurement of cosmological parameters by radio astronomical
technique. These interesting questions are illuminated in Chapters 6 and 7 of the present book.
Chapter 6, authored by T. Futamase, outlines the results of his research on the emergence of the
cosmological metric in a lumpy universe, a line of study known as the averaging problem in cosmol-
ogy. The Chapter also discusses the gravitational backreaction by local nonlinear inhomogeneities on
the cosmic expansion, in the framework of general relativity. The problem became important after the
discovery of the cosmic acceleration associated with the presence of dark energy. After a brief re-
view of the subject, T. Futamase presents in detail his own approach to analytical calculation of the
backreaction, which allows him to overview the apparent discrepancies between previous works us-
ing different approaches and gauges. Chapter 6 partially resolves these discrepancies by defining the
spatially-averaged energy density of matter as a conserved quantity referred to a sufficiently large vol-
ume of comoving space. It is shown that the backreaction behaves like a positive-curvature term in the
averaged Friedmann-LemÃ®tre-Robertson-Walker (FLRW) universe. It neither accelerates nor decel-
erates the cosmic expansion in a matter-dominated universe, while the cosmological constant induces a
new type of backreaction with the equation-of-state parameter being - 4/3. However, the effective energy
density remains negative, and thus it decreases the acceleration.
Chapter 7, written by A. Petrov and S. Kopeikin, extends the post-Newtonian approximation of gen-
eral relativity to the realm of cosmology, by making use of a geometric theory of Lagrangian perturba-
tions of an FLRW cosmological manifold. The Lagrangian for a perturbed cosmological model includes
the dark matter, the dark energy, and the ordinary baryonic matter. The Lagrangian is decomposed in an
asymptotic Taylor series around a background FLRW manifold, with the small parameter being the mag-
nitude of the metric-tensor perturbation. Each term of the series decomposition is kept gauge-invariant.
The asymptotic nature of the Lagrangian decomposition does not require the post-Newtonian perturba-
tions to be small, though computationally it works most effectively when the perturbed metric is close to
the background one. The Lagrangian of dark matter is treated as an ideal fluid described by an auxiliary
scalar field called the Clebsch potential. The dark energy is associated with a single scalar field of an
unspecified potential energy. The scalar fields of dark matter and dark energy are taken as independent
dynamical variables which play the role of generalised coordinates in the Lagrangian formalism. This
allows the authors to implement the powerful methods of variational calculus, to derive gauge-invariant
field equations to be used in the post-Newtonian celestial mechanics in an expanding universe. The
equations generalise the field equations of the post-Newtonian theory in an asymptotically-flat space-
time, by taking into account the cosmological effects without assuming a rather artificial vacuole model
of an isolated system (like those proposed by Einstein and Strauss, McVittie, and Bonnor). A new cos-
mological gauge is proposed, which generalises the de Donder (harmonic) gauge of the post-Newtonian
theory in an asymptotically flat spacetime. The new gauge significantly simplifies the gravitational field
equations and reduces them to wave equations, the latter being differential equations of Bessel’s type.
The new gauge also allows the authors to find out the cosmological models wherein the field equations
are fully decoupled and can be solved analytically. The residual gauge freedom is explored and the resid-
ual gauge transformations are formulated in the form of wave equations for gauge functions. Chapter
7 demonstrates how cosmological effects interfere with the local distribution of matter of the isolated
system and its orbital dynamics. The Chapter also offers a precise mathematical definition of the New-
tonian limit for an isolated system residing on a cosmological manifold. The results of the Chapter can
be useful in the galactic astronomy, to study the dynamics of clusters of galaxies, and in the gravitational
ix
wave astronomy, for discussing the impact of cosmological effects on generation and propagation of
gravitational waves emitted by coalescing binaries.
Over the past 30 years, relativistic celestial mechanics has experienced radical progress both in theory
and in experimental testing of general relativity. The present volume cannot embrace it in its entirety. For
further reading on recent developments in relativistic celestial mechanics, we recommend the following
review articles and textbooks:
Asada, H., Futamase, T. and Hogan, P., “Equations of Motion in General Relativity”, Oxford University
Press: Oxford, 2011
Brumberg, V. A., “Celestial mechanics: past, present, future”, Solar System Research, Vol. 47, Issue 5,
pp. 347-358 (2013)
Brumberg, V. A., “Relativistic Celestial Mechanics on the verge of its 100 year anniversary” (Brouwer
Award lecture), Celestial Mechanics and Dynamical Astronomy, Vol. 106, Issue 3, pp. 209-234
(2010)
Brumberg, V. A., “Relativistic Celestial Mechanics”, Scholarpedia, Vol. 5, Issue 8, #10669. URL
(cited on Jan 12, 2014)
http://www.scholarpedia.org/article/Relativistic_Celestial_Mechanics
Brumberg, V. A., “Essential Relativistic Celestial Mechanics”, Adam Hilger: Bristol, 1991
Blanchet, L., “Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Bina-
ries”, Living Reviews in Relativity 5, 3 (2002). URL (cited on Jan. 12, 2014) http://www.
livingreviews.org/lrr-2002-3
Damour, T., “The problem of motion in Newtonian and Einsteinian gravity”, In: Three Hundred Years
of Gravitation, Eds. Hawking, S. W. and Israel, W., Cambridge University Press: Cambridge,
1987. pp. 128 - 198
Goldberger, W. D. and Rothstein, I. Z., “Effective field theory of gravity for extended objects”, Physical
Review D, Vol. 73, Issue 10, id. 104029 (2006)
Kopeikin, S., Efroimsky, M. and G. Kaplan, “Relativistic Celestial Mechanics of the Solar System”,
Wiley-VCH: Berlin, 2011
Soffel, M. and Langhans, R., “Space-Time Reference Systems”, Springer-Verlag: Berlin, 2013
Editor: Sergei Kopeikin University of Missouri, USA

February 12, 2014
1 The General Relativistic Two Body Problem
Thibault Damour
Institut des Hautes Études Scientifiques (IHÉS), 35 route de Chartres, F-91440,
Bures sur Yvette, France
1.1 Introduction
The general relativistic problem of motion, i.e. the problem of describing the dynamics of N gravita-
tionally interacting extended bodies, is one of the cardinal problems of Einstein’s theory of gravitation.
This problem has been investigated from the early days of development of general relativity, notably
through the pioneering works of Einstein, Droste and De Sitter. These authors introduced the post-
Newtonian (PN) approximation method, which combines three different expansions: (i) a weak-field
expansion (gµν − ηµν ≡ hµν 1); (ii) a slow-motion expansion (v/c 1); and a near-zone ex-
pansion 1c ∂t hµν ∂x hµν . PN theory could be easily worked out to derive the first post-Newtonian

(1PN) approximation, i.e. the leading-order general relativistic corrections to Newtonian gravity (in-
volving one power of 1/c2 ). However, the use of the PN approximation for describing the dynamics
of N extended bodies turned out to be fraught with difficulties. Most of the early derivations of the
1PN-accurate equations of motion of N bodies turned out to involve errors: this is, in particular, the
case of the investigations of Droste [1], De Sitter [2], Chazy [3] and Levi-Civita [4]. These errors were
linked to incorrect treatments of the internal structures of the bodies. Apart from the remarkable 1917
work of Lorentz and Droste [5] (which seems to have remained unnoticed during many years), the first
correct derivations of the 1PN-accurate equations of motion date from 1938, and were obtained by Ein-
stein, Infeld and Hoffmann [6], and Eddington and Clark [7]. After these pioneering works (and the
investigations they triggered, notably in Russia [8] and Poland), the general relativistic N -body problem
reached a first stage of maturity and became codified in various books, notably in the books of Fock [9],
Infeld and Plebanski [10], and in the second volume of the treatise of Landau and Lifshitz (starting, at
least, with the 1962 second English edition).
We have started by recalling the early history of the general relativistic problem of motion both
because Victor Brumberg has always shown a deep knowledge of this history, and because, as we shall
discuss below, some of his research work has contributed to clarifying several of the weak points of the
early PN investigations (notably those linked to the treatment of the internal structures of the N bodies).
For many years, the 1PN approximation turned out to be accurate enough for applying Einstein’s
theory to known N -body systems, such as the solar system, and various binary stars. It is still true today
that the 1PN approximation (especially when used in its multi-chart version, see below) is adequate for
describing general relativistic effects in the solar system. However, the discovery in the 1970’s of binary
systems comprising strongly self-gravitating bodies (black holes or neutron stars) has obliged theorists
to develop improved approaches to the N -body problem. These improved approaches are not limited (as
the traditional PN method) to the case of weakly self-gravitating bodies and can be viewed as modern
versions of the Einstein-Infeld-Hoffmann classic work [6].
In addition to the need of considering strongly self-gravitating bodies, the discovery of binary pulsars
in the mid 1970’s (starting with the Hulse-Taylor pulsar PSR 1913 + 16) obliged theorists to go beyond
2 Chapter 1 The General Relativistic Two Body Problem
the 1PN (O(v 2 /c2 )) relativistic effects in the equations of motion. More precisely, it was necessary to
go to the 2.5PN approximation level, i.e. to include terms O(v 5 /c5 ) beyond Newton in the equations of
motion. This was achieved in the 1980’s by several groups [11–15]. [Let us note that important progress
in obtaining the N -body metric and equations of motion at the 2PN level was achieved by the Japanese
school in the 1970’s [16–18].]
Motivation for pushing the accuracy of the equations of motion beyond the 2.5PN level came from the
prospect of detecting the gravitational wave signal emitted by inspiralling and coalescing binary systems,
notably binary neutron star (BNS) and binary black hole (BBH) systems. The 3PN-level equations of
motion (including terms O(v 6 /c6 ) beyond Newton) were derived in the late 1990’s and early 2000’s
[19–22, 80] (they have been recently rederived in [24]). Recently, the 4PN-level dynamics has been
tackled in [25–28].
Separately from these purely analytical approaches to the motion and radiation of binary systems,
which have been developed since the early days of Einstein’s theory, Numerical Relativity (NR) sim-
ulations of Einstein’s equations have relatively recently (2005) succeeded (after more than thirty years
of developmental progress) to stably evolve binary systems made of comparable mass black holes [29–
32]. This has led to an explosion of works exploring many different aspects of strong-field dynamics in
general relativity, such as spin effects, recoil, relaxation of the deformed horizon formed during the co-
alescence of two black holes to a stationary Kerr black hole, high-velocity encounters, etc.; see [33] for
a review and [34] for an impressive example of the present capability of NR codes. In addition, recently
developed codes now allow one to accurately study the orbital dynamics, and the coalescence of binary
neutron stars [35]. Much physics remains to be explored in these systems, especially during and after the
merger of the neutron stars (which involves a much more complex physics than the pure-gravity merger
of two black holes).
Recently, a new source of information on the general relativistic two-body problem has opened: grav-
itational self-force (GSF) theory. This approach goes one step beyond the test-particle approximation
(already used by Einstein in 1915) by taking into account self-field effects that modify the leading-order
geodetic motion of a small mass m1 moving in the background geometry generated by a large mass
m2 . After some ground work (notably by DeWitt and Brehme) in the 1960’s, GSF theory has recently
undergone rapid developments (mixing theoretical and numerical methods) and can now yield numeri-
cal results that yield access to new information on strong-field dynamics in the extreme mass-ratio limit
m1 m2 . See Ref. [36] for a review.
Each of the approaches to the two-body problem mentioned so far, PN theory, NR simulations and
GSF theory, have their advantages and their drawbacks. It has become recently clear that the best
way to meet the challenge of accurately computing the gravitational waveforms (depending on several
continuous parameters) that are needed for a successful detection and data analysis of GW signals in
the upcoming LIGO/Virgo/GEO/. . . network of GW detectors is to combine knowledge from all the
available approximation methods: PN, NR and GSF. Several ways of doing so are a priori possible.
For instance, one could try to directly combine PN-computed waveforms (approximately valid for large
enough separations, say r & 10 G(m1 +m2 )/c2 ) with NR waveforms (computed with initial separations
r0 > 10 G(m1 + m2 )/c2 and evolved up to merger and ringdown). However, this method still requires
too much computational time, and is likely to lead to waveforms of rather poor accuracy, see, e.g.,
[37, 38].
On the other hand, five years before NR succeeded in simulating the late inspiral and the coalescence
of binary black holes, a new approach to the two-body problem was proposed: the Effective One Body
(EOB) formalism [39–42]. The basic aim of the EOB formalism is to provide an analytical description
of both the motion and the radiation of coalescing binary systems over the entire merger process, from
Section 1.2 Multi-chart approach to the N -body problem 3
the early inspiral, right through the plunge, merger and final ringdown. As early as 2000 [40] this method
made several quantitative and qualitative predictions concerning the dynamics of the coalescence, and
the corresponding GW radiation, notably: (i) a blurred transition from inspiral to a ‘plunge’ that is just a
smooth continuation of the inspiral, (ii) a sharp transition, around the merger of the black holes, between
a continued inspiral and a ring-down signal, and (iii) estimates of the radiated energy and of the spin of
the final black hole. In addition, the effects of the individual spins of the black holes were investigated
within the EOB [42, 43] and were shown to lead to a larger energy release for spins parallel to the
orbital angular momentum, and to a dimensionless rotation parameter J/E 2 always smaller than unity
at the end of the inspiral (so that a Kerr black hole can form right after the inspiral phase). All those
predictions have been broadly confirmed by the results of the recent numerical simulations performed
by several independent groups (for a review of numerical relativity results and references see [33]).
Note that, in spite of the high computer power used in NR simulations, the calculation, checking and
processing of one sufficiently long waveform (corresponding to specific values of the many continuous
parameters describing the two arbitrary masses, the initial spin vectors, and other initial data) takes on
the order of one month. This is a very strong argument for developing analytical models of waveforms.
For a recent comprehensive comparison between analytical models and numerical waveforms see [44].
In the present work, we shall briefly review only a few facets of the general relativistic two body
problem. [See, e.g., [45] and [46] for recent reviews dealing with other facets of, or approaches to,
the general relativistic two-body problem.] First, we shall recall the essential ideas of the multi-chart
approach to the problem of motion, having especially in mind its application to the motion of compact
binaries, such as BNS or BBH systems. Then we shall focus on the Effective One Body (EOB) approach
to the motion and radiation of binary systems, from its conceptual framework to its comparison to NR
simulations.
1.2 Multi-chart approach to the N -body problem

The traditional (text book) approach to the problem of motion of N separate bodies in GR consists of
solving, by successive approximations, Einstein’s field equations (we use the signature − + ++)
1 8π G
Rµν − R gµν = 4 Tµν , (1.2.1)
2 c
together with their consequence
∇ν T µν = 0 . (1.2.2)
To do so, one assumes some specific matter model, say a perfect fluid,
T µν = (ε + p) uµ uν + p g µν . (1.2.3)
One expands (say in powers of Newton’s constant) the metric,

(1)
gµν (xλ ) = ηµν + hµν + h(2)
µν + . . . , (1.2.4)
and use the simplifications brought by the ‘Post-Newtonian’ approximation (∂0 hµν = c−1 ∂t hµν
∂i hµν ; v/c 1, p ε). Then one integrates the local material equation of motion (1.2.2) over
the volume of each separate body, labelled say by a = 1,2, . . . , N . In so doing, one must define some
‘center of mass’ zai of body a, as well as some (approximately conserved) ‘mass’ ma of body a, together
with some corresponding ‘spin vector’ Sai and, possibly, higher multipole moments.
An important feature of this traditional method is to use a unique coordinate chart xµ to describe the
full N -body system. For instance, the center of mass, shape and spin of each body a are all described
within this common coordinate system xµ . This use of a single chart has several inconvenient aspects,
even in the case of weakly self-gravitating bodies (as in the solar system case). Indeed, it means for
instance that a body which is, say, spherically symmetric in its own ‘rest frame’ X α will appear as
deformed into some kind of ellipsoid in the common coordinate chart xµ . Moreover, it is not clear how
to construct ‘good definitions’ of the center of mass, spin vector, and higher multipole moments of body
a, when described in the common coordinate chart xµ . In addition, as we are possibly interested in the
motion of strongly self-gravitating bodies, it is not a priori justified to use a simple expansion of the type
(1)
Gma /(c2 |x − za |) will not be uniformly small in the common coordinate
P
(1.2.4) because hµν ∼
a
system xµ . It will be small if one stays far away from each object a, but, it will become of order unity
on the surface of a compact body.
These two shortcomings of the traditional ‘one-chart’ approach to the relativistic problem of motion
can be cured by using a ‘multi-chart’ approach.The multi-chart approach describes the motion of N
(possibly, but not necessarily, compact) bodies by using N +1 separate coordinate systems: (i) one global
coordinate chart xµ (µ = 0,1,2,3) used to describe the spacetime outside N ‘tubes’, each containing
one body, and (ii) N local coordinate charts Xaα (α = 0,1,2,3; a = 1,2, . . . , N ) used to describe the
spacetime in and around each body a. The multi-chart approach was first used to discuss the motion
of black holes and other compact objects [47–54]. Then it was also found to be very convenient for
describing, with the high-accuracy required for dealing with modern technologies such as VLBI, systems
of N weakly self-gravitating bodies, such as the solar system [55, 56].
The essential idea of the multi-chart approach is to combine the information contained in several
expansions. One uses both a global expansion of the type (1.2.4) and several local expansions of the
type
(0) (1)
Gαβ (Xaγ ) = Gαβ (Xaγ ; ma ) + Hαβ (Xaγ ; ma , mb ) + · · · , (1.2.5)
(0)
where Gαβ (X; ma ) denotes the (possibly strong-field) metric generated by an isolated body of mass
ma (possibly with the additional effect of spin).
The separate expansions (1.2.4) and (1.2.5) are then ‘matched’ in some overlapping domain of com-
mon validity of the type Gma /c2 . Ra |x − za | d ∼ |xa − xb | (with b 6= a), where one can
relate the different coordinate systems by expansions of the form
1 µ
xµ = zaµ (Ta ) + eµ i
i (Ta ) Xa + f (Ta ) Xai Xaj + · · · (1.2.6)
2 ij
The multi-chart approach becomes simplified if one considers compact bodies (of radius Ra compa-
rable to 2 Gma /c2 ). In this case, it was shown [52], by considering how the ‘internal expansion’ (1.2.5)
propagates into the ‘external’ one (1.2.4) via the matching (1.2.6), that, in general relativity, the internal
structure of each compact body was effaced to a very high degree, when seen in the external expansion
(1.2.4). For instance, for non spinning bodies, the internal structure of each body (notably the way it
responds to an external tidal excitation) shows up in the external problem of motion only at the fifth
post-Newtonian (5PN) approximation, i.e. in terms of order (v/c)10 in the equations of motion.
This ‘effacement of internal structure’ indicates that it should be possible to simplify the rigorous
multi-chart approach by skeletonizing each compact body by means of some delta-function source.
Mathematically, the use of distributional sources is delicate in a nonlinear theory such as GR. However,
it was found that one can reproduce the results of the more rigorous matched-multi-chart approach by
treating the divergent integrals generated by the use of delta-function sources by means of (complex)
analytic continuation [52]. In particular, analytic continuation in the dimension of space d [57] is very
efficient (especially at high PN orders).
Section 1.3 EOB description of the conservative dynamics of two body systems 5
Finally, the most efficient way to derive the general relativistic equations of motion of N compact
bodies consists of solving the equations derived from the action (where g ≡ − det(gµν ))
dd+1 x √ c4
Z X Z q
S= g R(g) − ma c −gµν (zaλ ) dzaµ dzaν , (1.2.7)
c 16π G a
formally using the standard weak-field expansion (1.2.4), but considering the space dimension d as an
arbitrary complex number which is sent to its physical value d = 3 only at the end of the calculation.
This ‘skeletonized’ effective action approach to the motion of compact bodies has been extended to other
theories of gravity [50, 51]. Finite-size corrections can be taken into account by adding nonminimal
worldline couplings to the effective action (1.2.7) [58, 59].
As we shall further discuss below, in the case of coalescing BNS systems, finite-size corrections
(linked to tidal interactions) become relevant during late inspiral and must be included to accurately
describe the dynamics of coalescing neutron stars.
Here, we shall not try to describe the results of the application of the multi-chart method to N -body
(or 2-body) systems. For applications to the solar system see the book [60] of V. Brumberg; see also
several articles (notably by M. Soffel) in [61]. For applications of this method to binary pulsar systems
(and to their use as tests of gravity theories) see the articles by T. Damour and M. Kramer in [62].
1.3 EOB description of the conservative dynamics of two

body systems
Before reviewing some of the technical aspects of the EOB method, let us indicate the historical roots of
this method. First, we note that the EOB approach comprises three, rather separate, ingredients:
1. a description of the conservative (Hamiltonian) part of the dynamics of two bodies;
2. an expression for the radiation-reaction part of the dynamics;
3. a description of the GW waveform emitted by a coalescing binary system.
For each one of these ingredients, the essential inputs that are used in EOB works are high-order post-
Newtonian (PN) expanded results which have been obtained by many years of work, by many researchers
(see the review [46]). However, one of the key ideas in the EOB philosophy is to avoid using PN results
in their original “Taylor-expanded” form (i.e. c0 + c1 v/c + c2 v 2 /c2 + c3 v 3 /c3 + · · · + cn v n /cn ), but
to use them instead in some resummed form (i.e. some non-polynomial function of v/c, defined so as
to incorporate some of the expected non-perturbative features of the exact result). The basic ideas and
techniques for resumming each ingredient of the EOB are different and have different historical roots.
Concerning the first ingredient, i.e. the EOB Hamiltonian, it was inspired by an approach to electro-
magnetically interacting quantum two-body systems introduced by Brézin, Itzykson and Zinn-Justin [63].
The resummation of the second ingredient, i.e. the EOB radiation-reaction force F, was initially in-
spired by the Padé resummation of the flux function introduced by Damour, Iyer and Sathyaprakash [64].
More recently, a new and more sophisticated resummation technique for the (waveform and the) radia-
tion reaction force F has been introduced by Damour, Iyer and Nagar [65, 66]. It will be discussed in
detail below.
As for the third ingredient, i.e. the EOB description of the waveform emitted by a coalescing black
hole binary, it was mainly inspired by the work of Davis, Ruffini and Tiomno [67] which discovered the
transition between the plunge signal and a ringing tail when a particle falls into a black hole. Additional
motivation for the EOB treatment of the transition from plunge to ring-down came from work on the,
so-called, “close limit approximation” [68].
Within the usual PN formalism, the conservative dynamics of a two-body system is currently fully
known up to the 3PN level [19–24] (see below for the partial knowledge beyond the 3PN level). Going
to the center of mass of the system (p1 + p2 = 0), the 3PN-accurate Hamiltonian (in Arnowitt-Deser-
Misner-type coordinates) describing the relative motion, q = q1 − q2 , p = p1 = −p2 , has the
structure
relative 1 1 1
H3PN (q, p) = H0 (q, p) + H2 (q, p) + 4 H4 (q, p) + 6 H6 (q, p) , (1.3.1)
c2 c c
where
1 2 GM µ
H0 (q, p) = p − , (1.3.2)
2µ |q|
with
M ≡ m1 + m2 and µ ≡ m1 m2 /M , (1.3.3)
corresponds to the Newtonian approximation to the relative motion, while H2 describes 1PN corrections,
H4 2PN ones and H6 3PN ones. In terms of the rescaled variables q 0 ≡ q/GM , p0 ≡ p/µ, the explicit
form (after dropping the primes for readability) of the 3PN-accurate rescaled Hamiltonian Hb ≡ H/µ
reads [21, 70, 71]
2
Hb N (q, p) = p − 1 , (1.3.4)
2 q
b 1PN (q, p) = 1 (3ν − 1)(p2 )2 − 1 [(3 + ν)p2 + ν(n · p)2 ] 1 + 1 ,

H (1.3.5)
8 2 q 2q 2
b 2PN (q, p) = 1 (1 − 5ν + 5ν 2 )(p2 )3

H
16
1 1
+ [(5 − 20ν − 3ν 2 )(p2 )2 − 2ν 2 (n · p)2 p2 − 3ν 2 (n · p)4 ]
8 q
1 1 1 1
+ [(5 + 8ν)p2 + 3ν(n · p)2 ] 2 − (1 + 3ν) 3 , (1.3.6)
2 q 4 q
b 3PN (q, p) = 1 (−5 + 35ν − 70ν 2 + 35ν 3 )(p2 )4

H
128
1
+ [(−7 + 42ν − 53ν 2 − 5ν 3 )(p2 )3 + (2 − 3ν)ν 2 (n · p)2 (p2 )2
16
1
+ 3(1 − ν)ν 2 (n · p)4 p2 − 5ν 3 (n · p)6 ]
q

1 2 2 2 1
+ (−27 + 136ν + 109ν )(p ) + (17 + 30ν)ν(n · p)2 p2
16 16

1 1
+ (5 + 43ν)ν(n · p)4 2
12 q

25 1 2 335 23 2 2
+ − + π − ν− ν p
8 64 48 8

85 3 2 7 1
+ − − π − ν ν(n · p)2
16 64 4 q3

1 109 21 2 1
+ + − π ν 4. (1.3.7)
8 12 32 q
In these formulas ν denotes the symmetric mass ratio:
µ m1 m2
ν≡ ≡ . (1.3.8)
M (m1 + m2 )2
1
The dimensionless parameter ν varies between 0 (extreme mass ratio case) and 4
(equal mass case) and
plays the rôle of a deformation parameter away from the test-mass limit.
It is well known that, at the Newtonian approximation, H0 (q, p) can be thought of as describing a
‘test particle’ of mass µ orbiting around an ‘external mass’ GM . The EOB approach is a general rela-
tivistic generalization of this fact. It consists in looking for an ‘effective external spacetime geometry’
eff
gµν (xλ ; GM,ν) such that the geodesic dynamics of a ‘test particle’ of mass µ within gµνeff
(xλ , GM,ν) is
equivalent (when expanded in powers of 1/c2 ) to the original, relative PN-expanded dynamics (1.3.1).
Let us explain the idea, proposed in [39], for establishing a ‘dictionary’ between the real relative-
motion dynamics, (1.3.1), and the dynamics of an ‘effective’ particle of mass µ moving in gµν eff
(xλ , GM,ν).
1
The idea consists in ‘thinking quantum mechanically’ . Instead of thinking in terms of a classical Hamil-
relative
tonian, H(q, p) (such as H3PN , Eq. (1.3.1)), and of its classical bound orbits, we can think in terms of
the quantized energy levels E(n,`) of the quantum bound states of the Hamiltonian operator H(q̂,p̂).
These energy levels will depend on two (integer valued) quantum numbers n and `. Here (for a spheri-
cally symmetric interaction, as appropriate to H relative ), ` parametrizes the total orbital angular momen-
tum (L2 = `(` + 1) ~2 ), while n represents the ‘principal quantum number’ n = ` + nr + 1, where nr
(the ‘radial quantum number’) denotes the number of nodes in the radial wave function. The third ‘mag-
netic quantum number’ m (with −` ≤ m ≤ `) does not enter the energy levels because of the spherical
symmetry of the two-body interaction (in the center of of mass frame). For instance, the non-relativistic
Newton interaction Eq. (1.3.2) gives rise to the well-known result
2
1 GM µ
E0 (n,`) = − µ , (1.3.9)
2 n~
which depends only on n (this is the famous Coulomb degeneracy). When considering the PN correc-
tions to H0 , as in Eq. (1.3.1), one gets a more complicated expression of the form
1 α2 α2 c11

relative c20
E3PN (n,`) = − µ 2 1 + 2 + 2
2 n c n` n
4 6

α c13 c22 c31 c40 α c15 c60
+ 4 + + + + + . . . + , (1.3.10)
c n`3 n2 `2 n3 ` n4 c6 n`5 n6
where we have set α ≡ GM µ/~ = G m1 m2 /~, and where we consider, for simplicity, the (quasi-
classical) limit where n and ` are large numbers. The 2PN-accurate version of Eq. (1.3.10) had been
derived by Damour and Schäfer [69] as early as 1988 while its 3PN-accurate version was derived by
Damour, Jaranowski and Schäfer in 1999 [70]. The dimensionless coefficients cpq are functions of the
symmetric mass ratio ν ≡ µ/M , for instance c40 = 81 (145 − 15ν + ν 2 ). In classical mechanics (i.e.
for large n and `), it is called the ‘Delaunay Hamiltonian’, i.e. the Hamiltonian expressed in terms of the
action variables2 J = `~ = 2π 1 1
H H
pϕ dϕ, and N = n~ = Ir + J, with Ir = 2π pr dr.
1 This is related to an idea emphasized many times by John Archibald Wheeler: quantum mechanics can often help
us in going to the essence of classical mechanics.
2 We consider, for simplicity, ‘equatorial’ motions with m = `, i.e., classically, θ = π2 .
The energy levels (1.3.10) encode, in a gauge-invariant way, the 3PN-accurate relative dynamics of a
‘real’ binary. Let us now consider an auxiliary problem: the ‘effective’ dynamics of one body, of mass
µ, following (modulo the Q term discussed below) a geodesic in some ν-dependent ‘effective external’
(spherically symmetric) metric3
eff
gµν dxµ dxν = −A(R; ν) c2 dT 2 + B(R; ν) dR2 + R2 (dθ2 + sin2 θ dϕ2 ) . (1.3.11)
Here, the a priori unknown metric functions A(R; ν) and B(R; ν) will be constructed in the form of
expansions in GM/c2 R:
2
3 4
GM GM
GM GM
A(R; ν) = 1+e
a1 +e
a2 + +e
aa3
4 + ··· ;
c2 R c2 R
c2 R c2 R
e
2 3
GM GM GM
B(R; ν) = b1 2 + e
1 +e b2 + b3 + ··· , (1.3.12)
c R c2 R c2 R
where the dimensionless coefficients e an , e

bn depend on ν. From the Newtonian limit, it is clear that
we should set ea1 = −2. In addition, as ν can be viewed as a deformation parameter away from the
test-mass limit, we require that the effective metric (1.3.11) tend to the Schwarzschild metric (of mass
M ) as ν → 0, i.e. that
A(R; ν = 0) = 1 − 2GM/c2 R = B −1 (R; ν = 0) .
eff
Let us now require that the dynamics of the “one body” µ within the effective metric gµν be described
by an “effective” mass-shell condition of the form
µν eff eff
geff pµ pν + µ2 c2 + Q(peff
µ ) = 0,
where Q(p) is (at least) quartic in p. Then by solving (by separation of variables) the corresponding
‘effective’ Hamilton-Jacobi equation

µν ∂Seff ∂Seff 2 2 ∂Seff
geff + µ c + Q = 0,
∂xµ ∂xν ∂xµ
Seff = −Eeff t + Jeff ϕ + Seff (R) , (1.3.13)

one can straightforwardly compute (in the quasi-classical, large quantum numbers limit) the effective
Delaunay Hamiltonian Eeff (Neff , Jeff ), with Neff = neff ~, Jeff = èff ~ (where Neff = Jeff + IR
eff
, with
eff 1
H eff eff
IR = 2π pR dR, PR = ∂Seff (R)/dR). This yields a result of the form
1 α2 α2
eff
ceff

c11
Eeff (neff ,èff ) = µc2 − µ 2 1+ 2 + 20 2
2 neff c neff èff neff
α4
eff eff eff
ceff

c13 c22 c31 40
+ + + +
c4 neff `3eff n2eff `2eff n3eff èff n4eff
α6
eff eff

c15 c
+ + . . . + 60 , (1.3.14)
c6 neff `5eff n6eff
3 It is convenient to write the ‘effective metric’ in Schwarzschild-like coordinates. Note that the effective radial
coordinate R differs from the two-body ADM-coordinate relative distance RADM = |q|. The transformation
between the two coordinate systems has been determined in Refs. [39, 41].
Figure 1.1. Sketch of the correspondence between the quantized energy levels of the real and effective
conservative dynamics. n denotes the ‘principal quantum number’ (n = nr + ` + 1, with nr =
0,1, . . . denoting the number of nodes in the radial function), while ` denotes the (relative) orbital angular
momentum (L2 = `(` + 1) ~2 ). Though the EOB method is purely classical, it is conceptually useful
to think in terms of the underlying (Bohr-Sommerfeld) quantization conditions of the action variables
IR and J to motivate the identification between n and ` in the two dynamics.
where the dimensionless coefficients ceff an , e

pq are now functions of the unknown coefficients e bn entering
the looked for ‘external’ metric coefficients (1.3.12).
At this stage, one needs to define a ‘dictionary’ between the real (relative) two-body dynamics, sum-
marized in Eq. (1.3.10), and the effective one-body one, summarized in Eq. (1.3.14). As, on both sides,
quantum mechanics tells us that the action variables are quantized in integers (Nreal = n~, Neff = neff ~,
etc.) it is most natural to identify n = neff and ` = èff . One then still needs a rule for relating the two
relative
different energies Ereal and Eeff . Ref. [39] proposed to look for a general map between the real energy
levels and the effective ones (which, as seen when comparing (1.3.10) and (1.3.14), cannot be directly
identified because they do not include the same rest-mass contribution4 ), namely
relative 2
Eeff relative relative relative

Ereal Ereal Ereal Ereal
−1=f = 1 + α1 + α2
µc2 µc2 µc2 µc2 µc2
relative
3
Ereal
+ α3 + ... . (1.3.15)
µc2
The ‘correspondence’ between the real and effective energy levels is illustrated in Fig. 1.1.
Finally, identifying Eeff (n,`)/µc2 to 1 + f (Ereal

relative
(n,`)/µc2 ) yields a system of equations for deter-
mining the unknown EOB coefficients e an , bn , αn , as well as the three coefficients z1 , z2 , z3 parametriz-
e
ing a general 3PN-level quartic mass-shell deformation:
2
1 1 GM
z1 p4 + z2 p2 (n · p)2 + z3 (n · p)4 .

Q3PN (p) =
c6 µ2 R
4 total = M c2 + E relative = M c2 + Newtonian terms + 1PN/c2 + · · · , while E 2

Indeed Ereal real effective = µc + N +
1PN/c2 + · · · .
[The need for introducing a quartic mass-shell deformation Q only arises at the 3PN level.]
The above system of equations for e an , e
bn , αn (and zi at 3PN) was studied at the 2PN level in Ref. [39],
and at the 3PN level in Ref. [41]. At the 2PN level it was found that, if one further imposes the natural
condition eb1 = +2 (so that the linearized effective metric coincides with the linearized Schwarzschild
metric with mass M = m1 + m2 ), there exists a unique solution for the remaining five unknown
a2 , e
coefficients e a3 , e
b2 , α1 and α2 . This solution is very simple:
ν
a2 = 0 ,
e a3 = 2ν ,
e b2 = 4 − 6ν ,
e α1 = , α2 = 0 . (1.3.16)
2
At the 3PN level, it was found that the system of equations is consistent, and underdetermined in that
the general solution can be parametrized by the arbitrary values of z1 and z2 . It was then argued that it
is natural to impose the simplifying requirements z1 = 0 = z2 , so that Q is proportional to the fourth
power of the (effective) radial momentum pr . With these conditions, the solution is unique at the 3PN
level, and is still remarkably simple, namely
a4 = a4 ν , de3 = 2(3ν − 26)ν , α3 = 0 , z3 = 2(4 − 3ν)ν .

e
Here, a4 denotes the number

94 41 2
a4 = − π ' 18.6879027 , (1.3.17)
3 32
while de3 denotes the coefficient of (GM/c2 R)3 in the PN expansion of the combined metric coefficient
D(R) ≡ A(R) B(R) .
Replacing B(R) by D(R) is convenient because (as was mentioned above), in the test-mass limit ν → 0,
the effective metric must reduce to the Schwarzschild metric, namely

GM
A(R; ν = 0) = B −1 (R; ν = 0) = 1 − 2 ,
c2 R
so that
D(R; ν = 0) = 1 .
The final result is that the three EOB potentials A,D,Q describing the 3PN two-body dynamics are
given by the following very simple results. In terms of the EOB “gravitational potential”
GM
u≡ ,
c2 R
A3PN (R) = 1 − 2u + 2 ν u3 + a4 ν u4 , (1.3.18)

2 3
D3PN (R) ≡ (A(R)B(R))3PN = 1 − 6νu + 2(3ν − 26)νu , (1.3.19)
1 p4
Q3PN (q, p) = 2(4 − 3ν)ν u2 r2 . (1.3.20)
c2 µ
relative
In addition, the map between the (real) center-of-mass energy of the binary system Ereal = H relative =
Etot
relative − M c and the effective one Eeff is found to have the very simple (but non trivial) form
2
Eeff relative relative

s − m21 c4 − m22 c4

Ereal ν Ereal
2
= 1 + 2
1 + 2
= (1.3.21)
µc µc 2 µc 2 m1 m2 c4
2 2 relative 2
where s = (Etot
real ) ≡ (M c + Ereal ) is Mandelstam’s invariant s = −(p1 + p2 )2 .
It is truly remarkable that the EOB formalism succeeds in condensing the complicated, original 3PN
Hamiltonian, Eqs. (1.3.4)–(1.3.7), into the very simple potentials A, D and Q displayed above, together
with the simple energy map Eq. (1.3.21). For instance, at the 1PN level, the already somewhat involved
Lorentz-Droste-Einstein-Infeld-Hoffmann 1PN dynamics (Eqs. (1.3.4) and (1.3.5)) is simply described,
within the EOB formalism, as a test particle of mass µ moving in an external Schwarzschild background
of mass M = m1 + m2 , together with the (crucial but quite simple) energy transformation (1.3.21).
[Indeed, the ν-dependent corrections to A and D start only at the 2PN level.] At the 2PN level, the
seven rather complicated ν-dependent coefficients of H b 2PN (q, p), Eq. (1.3.6), get condensed into the
two very simple additional contributions + 2νu in A(u), and − 6νu2 in D(u). At the 3PN level, the
3
eleven quite complicated ν-dependent coefficients of H b 3PN , Eq. (1.3.7), get condensed into only three
simple contributions: + a4 νu4 in A(u), + 2(3ν − 26)νu3 in D(u), and Q3PN given by Eq. (1.3.20).
This simplicity of the EOB results is not only due to the reformulation of the PN-expanded Hamiltonian
into an effective dynamics. Notably, the A-potential is much simpler that it could a priori have been: (i)
as already noted it is not modified at the 1PN level, while one would a priori expect to have found a 1PN
potential A1PN (u) = 1 − 2u + νa2 u2 with some non zero a2 ; and (ii) there are striking cancellations
taking place in the calculation of the 2PN and 3PN coefficients e a2 (ν) and e a3 (ν), which were a priori
of the form e a2 (ν) = a2 ν + a02 ν 2 , and e a3 (ν) = a3 ν + a03 ν 2 + a003 ν 3 , but for which the ν-nonlinear
contributions a02 ν 2 , a03 ν 2 and a003 ν 3 precisely cancelled out. Similar cancellations take place at the 4PN
level (level at which it was recently possible to compute the A-potential, see below). Let us note for
completeness that, starting at the 4PN level, the Taylor expansions of the A and D potentials depend
on the logarithm of u. The corresponding logarithmic contributions have been computed at the 4PN
level [72, 73] and even the 5PN one [74, 75]. They have been incorporated in a recent, improved
implementation of the EOB formalism [76].
The fact that the 3PN coefficient a4 in the crucial ‘effective radial potential’ A3PN (R), Eq. (1.3.18),
is rather large and positive indicates that the ν-dependent nonlinear gravitational effects lead, for com-
parable masses (ν ∼ 41 ), to a last stable (circular) orbit (LSO) which has a higher frequency and a larger
binding energy than what a naive scaling from the test-particle limit (ν → 0) would suggest. Actu-
ally, the PN-expanded form (1.3.18) of A3PN (R) does not seem to be a good representation of the (un-
known) exact function AEOB (R) when the (Schwarzschild-like) relative coordinate R becomes smaller
than about 6GM/c2 (which is the radius of the LSO in the test-mass limit). In fact, by continuity with
the test-mass case, one a priori expects that A3PN (R) always exhibits a simple zero defining an EOB
“effective horizon” that is smoothly connected to the Schwarzschild event horizon at R = 2GM/c2
when ν → 0. However, the large value of the a4 coefficient does actually prevent A3PN to have this
property when ν is too large, and in particular when ν = 1/4. It was therefore suggested [41] to further
resum5 A3PN (R) by replacing it by a suitable Padé (P ) approximant. For instance, the replacement of
A3PN (R) by6
1 + n1 u
A13 (R) ≡ P31 [A3PN (R)] = (1.3.22)
1 + d 1 u + d 2 u2 + d 3 u3
1
ensures that the ν = 4
case is smoothly connected with the ν = 0 limit.
The same kind of ν-continuity argument, discussed so far for the A function, needs to be applied also
to the D3PN (R) function defined in Eq. (1.3.19). A straightforward way to ensure that the D function
5 The PN-expanded EOB building blocks A3PN (R), B3PN (R), . . . already represent a resummation of the PN dy-
namics in the sense that they have “condensed” the many terms of the original PN-expanded Hamiltonian within
a very concise format. But one should not refrain to further resum the EOB building blocks themselves, if this is
physically motivated.
6 We recall that the coefficients n1 and (d1 ,d2 ,d3 ) of the (1,3) Padé approximant P31 [A3PN (u)] are determined by
the condition that the first four terms of the Taylor expansion of A13 in powers of u = GM/(c2 R) coincide with
A3PN .
stays positive when R decreases (since it is D = 1 when ν → 0) is to replace D3PN (R) by D30 (R) ≡
P30 [D3PN (R)], where P30 indicates the (0,3) Padé approximant and explicitly reads
1
D30 (R) = . (1.3.23)
1 + 6νu2 − 2(3ν − 26)νu3
1.4 EOB description of radiation reaction and of the

emitted waveform during inspiral
In the previous Section we have described how the EOB method encodes the conservative part of the
relative orbital dynamics into the dynamics of an ’effective’ particle. Let us now briefly discuss how
to complete the EOB dynamics by defining some resummed expressions describing radiation reaction
effects, and the corresponding waveform emitted at infinity. One is interested in circularized binaries,
which have lost their initial eccentricity under the influence of radiation reaction. For such systems,
it is enough (in first approximation [40]; see, however, the recent results of Bini and Damour [77]) to
include a radiation reaction force in the pϕ equation of motion only. More precisely, we are using phase
space variables r, pr , ϕ, pϕ associated to polar coordinates (in the equatorial plane θ = π2 ). Actually
it is convenient to replace the radial momentum pr by the momentum conjugate to the ‘tortoise’ radial
coordinate R∗ = dR(B/A)1/2 , i.e. PR∗ = (A/B)1/2 PR . The real EOB Hamiltonian is obtained by
R
√
total
first solving Eq. (1.3.21) to get Hreal = s in terms of Eeff , and then by solving the effective Hamilton-
Jacobi equation to get Eeff in terms of the effective phase space coordinates qeff and peff . The result is
given by two nested square roots (we henceforth set c = 1):
real
HEOB 1
q
ĤEOB (r,pr∗ , ϕ) = = 1 + 2ν (Ĥeff − 1) , (1.4.1)
µ ν
where s
p2ϕ

p4r
Ĥeff = p2r∗ + A(r) 1 + 2 + z3 2∗ , (1.4.2)
r r
with z3 = 2ν (4 − 3ν). Here, we are using suitably rescaled dimensionless (effective) variables: r =
R/GM , pr∗ = PR∗ /µ, pϕ = Pϕ /µ GM , as well as a rescaled time t = T /GM . This leads to
equations of motion for (r,ϕ,pr∗ ,pϕ ) of the form
dϕ ∂ ĤEOB
= ≡ Ω, (1.4.3)
dt ∂ pϕ
1/2
dr A ∂ ĤEOB
= , (1.4.4)
dt B ∂ p r∗
dpϕ
= F̂ϕ , (1.4.5)
dt
1/2
dpr∗ A ∂ ĤEOB
= − , (1.4.6)
dt B ∂r
which explicitly read
dϕ Apϕ
= ≡Ω, (1.4.7)
dt νr2 Ĥ Ĥeff
1/2
dr A 1 2A
= pr∗ + z3 2 p3r∗ , (1.4.8)
dt B ν Ĥ Ĥeff r
Section 1.4 EOB description of radiation reaction and of the emitted waveform during inspiral 13
dpϕ
= F̂ϕ , (1.4.9)
dt
1/2
dpr∗ A 1
= −
dt B 2ν Ĥ Ĥeff
p2ϕ
0
2A A 2A
A0 + 2 A0 − + z3 − p 4
r∗
, (1.4.10)
r r r2 r3
where A0 = dA/dr. As explained above the EOB metric function A(r) is defined by Padé resumming
the Taylor-expanded result (1.3.12) obtained from the matching between the real and effective energy
levels (as we were mentioning, one uses a similar Padé resumming for D(r) ≡ A(r) B(r)). One
similarly needs to resum F̂ϕ , i.e., the ϕ component of the radiation reaction which has been introduced
on the r.h.s. of Eq. (1.4.5).
Several methods have been tried during the development of the EOB formalism to resum the radiation
reaction F
b ϕ (starting from the high-order PN-expanded results that have been obtained in the literature).
Here, we shall briefly explain the new, parameter-free resummation technique for the multipolar wave-
form (and thus for the energy flux) introduced in Ref. [78, 79] and perfected in [65]. To be precise, the
new results discussed in Ref. [65] are twofold: on the one hand, that work generalized the ` = m = 2
resummed factorized waveform of [78, 79] to higher multipoles by using the most accurate currently
known PN-expanded results [80–83] as well as the higher PN terms which are known in the test-mass
limit [84, 85]; on the other hand, it introduced a new resummation procedure which consists in consid-
ering a new theoretical quantity, denoted as ρ`m (x), which enters the (`,m) waveform (together with
other building blocks, see below) only through its `-th power: h`m ∝ (ρ`m (x))` . Here, and below, x
denotes the invariant PN-ordering parameter given during inspiral by x ≡ (GM Ω/c3 )2/3 .
The main novelty introduced by Ref. [65] is to write the (`,m) multipolar waveform emitted by a
circular nonspinning compact binary as the product of several factors, namely
() GM ν () (`+)/2 `−,−m π

()
h`m = n`m cλ+ (ν)x Y ,Φ Ŝeff T`m eiδ`m ρ``m . (1.4.11)
c2 R 2
Here denotes the parity of ` + m ( = π(` + m)), i.e. = 0 for “even-parity” (mass-generated)
()
multipoles (` + m even), and = 1 for “odd-parity” (current-generated) ones (` + m odd); n`m and
()
cλ+ (ν) are numerical coefficients; Ŝeff is a µ-normalized effective source (whose definition comes from
the EOB formalism); T`m is a resummed version [78, 79] of an infinite number of “leading logarithms”
entering the tail effects [86, 87]; δ`m is a supplementary phase (which corrects the phase effects not
included in the complex tail factor T`m ), and, finally, (ρ`m )` denotes the `-th power of the quantity ρ`m
which is the new building block introduced in [65]. Note that in previous papers [78, 79] the quantity
(ρ`m )` was denoted as f`m and we will often use this notation below. Before introducing explicitly the
various elements entering the waveform (1.4.11) it is convenient to decompose h`m as
() (N,) ()
h`m = h`m ĥ`m , (1.4.12)
(N,)
where h`m is the Newtonian contribution (i.e. the product of the first five factors in Eq. (1.4.11)) and
() ()
ĥ`m ≡ Ŝeff T`m eiδ`m f`m (1.4.13)
()
represents a resummed version of all the PN corrections. The PN correcting factor ĥ`m , as well as all its
()
building blocks, has the structure ĥ`m = 1 + O(x).
The reader will find in Ref. [65] the definitions of the quantities entering the “Newtonian” waveform
(N,) ()
h`m , as well as the precise definition of the effective source factor Sbeff , which constitutes the first
() ()
h`m . Let us only note here that the definition of Sbeff makes use of
factor in the PN-correcting factor b
(0)
EOB-defined quantities. For instance, for even-parity waves ( = 0) Seff is defined as the µ-scaled
b
()
effective energy Eeff /µc2 . [We use the “J-factorization” definition of Sbeff when = 1, i.e. for odd
parity waves.]
The second building block in the factorized decomposition is the “tail factor” T`m (introduced in
Refs. [78, 79]). As mentioned above, T`m is a resummed version of an infinite number of “leading
logarithms” entering the transfer function between the near-zone multipolar wave and the far-zone one,
real
due to tail effects linked to its propagation in a Schwarzschild background of mass MADM = HEOB . Its
explicit expression reads
ˆ
Γ(` + 1 − 2ik̂) ˆ ˆ
T`m = eπk̂ e2ik̂ log(2kr0 ) , (1.4.14)
Γ(` + 1)
√
where r0 = 2GM/ e and k̂ˆ ≡ GHEOB real
mΩ and k ≡ mΩ. Note that k̂ˆ differs from k by a rescaling
involving the real (rather than the effective) EOB Hamiltonian, computed at this stage along the sequence
of circular orbits.
The tail factor T`m is a complex number which already takes into account some of the dephasing
of the partial waves as they propagate out from the near zone to infinity. However, as the tail factor
only takes into account the leading logarithms, one needs to correct it by a complementary dephasing
term, eiδ`m , linked to subleading logarithms and other effects. This subleading phase correction can be
()
computed as being the phase δ`m of the complex ratio between the PN-expanded ĥ`m and the above
defined source and tail factors. In the comparable-mass case (ν 6= 0), the 3PN δ22 phase correction to
the leading quadrupolar wave was originally computed in Ref. [79] (see also Ref. [78] for the ν = 0
limit). Full results for the subleading partial waves to the highest possible PN-accuracy by starting from
the currently known 3PN-accurate ν-dependent waveform [83] have been obtained in [65]. For higher-
order test-mass (ν → 0) contributions, see [88, 89]. For extensions of the (non spinning) factorized
waveform of [65] see [90–92].
The last factor in the multiplicative decomposition of the multipolar waveform can be computed as
()
being the modulus f`m of the complex ratio between the PN-expanded ĥ`m and the above defined
source and tail factors. In the comparable mass case (ν 6= 0), the f22 modulus correction to the leading
quadrupolar wave was computed in Ref. [79] (see also Ref. [78] for the ν = 0 limit). For the subleading
partial waves, Ref. [65] explicitly computed the other f`m ’s to the highest possible PN-accuracy by
starting from the currently known 3PN-accurate ν-dependent waveform [83]. In addition, as originally
proposed in Ref. [79], to reach greater accuracy the f`m (x; ν)’s extracted from the 3PN-accurate ν 6= 0
results are completed by adding higher order contributions coming from the ν = 0 results [84, 85]. In
the particular f22 case discussed in [79], this amounted to adding 4PN and 5PN ν = 0 terms. This
“hybridization” procedure was then systematically pursued for all the other multipoles, using the 5.5PN
accurate calculation of the multipolar decomposition of the gravitational wave energy flux of Refs. [84,
85].
()
The decomposition of the total PN-correction factor ĥ`m into several factors is in itself a resummation
procedure which already improves the convergence of the PN series one has to deal with: indeed, one
can see that the coefficients entering increasing powers of x in the PN expansion of the f`m ’s tend
()
to be systematically smaller than the coefficients appearing in the usual PN expansion of ĥ`m . The
reason for this is essentially twofold: (i) the factorization of T`m has absorbed powers of mπ which
()
contributed to make large coefficients in ĥ`m , and (ii) the factorization of either Ĥeff or ĵ has (in the
ν = 0 case) removed the presence of an inverse square-root singularity located at x = 1/3 which
caused the coefficient of xn in any PN-expanded quantity to grow as 3n as n → ∞.
Section 1.4 EOB description of radiation reaction and of the emitted waveform during inspiral 15
To further improve the convergence of the waveform several resummations of the factor f`m (x) =
1 + c`m `m 2
1 x + c2 x + . . . have been suggested. First, Refs. [78, 79] proposed to further resum the
f22 (x) function via a Padé (3,2) approximant, P23 {f22 (x; ν)}, so as to improve its behavior in the
strong-field-fast-motion regime. Such a resummation gave an excellent agreement with numerically
computed waveforms, near the end of the inspiral and during the beginning of the plunge, for different
mass ratios [78, 93, 94]. As we were mentioning above, a new route for resumming f`m was explored
in Ref. [65]. It is based on replacing f`m by its `-th root, say
ρ`m (x; ν) = [f`m (x; ν)]1/` . (1.4.15)
The basic motivation for replacing f`m by ρ`m is the following: the leading “Newtonian-level” contribu-
()
tion to the waveform h`m contains a factor ω ` rharm
`
v where rharm is the harmonic radial coordinate used
in the MPM formalism [95, 96]. When computing the PN expansion of this factor one has to insert the
PN expansion of the (dimensionless) harmonic radial coordinate rharm , rharm = x−1 (1 + c1 x + O(x2 )),
as a function of the gauge-independent frequency parameter x. The PN re-expansion of [rharm (x)]` then
generates terms of the type x−` (1 + `c1 x + ....). This is one (though not the only one) of the origins of
1PN corrections in h`m and f`m whose coefficients grow linearly with `. The study of [65] has pointed
out that these `-growing terms are problematic for the accuracy of the PN-expansions. The replacement
of f`m by ρ`m is a cure for this problem.
Several studies, both in the test-mass limit, ν → 0 (see Fig. 1 in [65]) and in the comparable-mass
case (see notably Fig. 4 in [66]), have shown that the resummed factorized (inspiral) EOB waveforms
defined above provided remarkably accurate analytical approximations to the “exact” inspiral waveforms
computed by numerical simulations. These resummed multipolar EOB waveforms are much closer
(especially during late inspiral) to the exact ones than the standard PN-expanded waveforms given by
Eq. (1.4.12) with a PN-correction factor of the usual “Taylor-expanded” form
()PN
h`m = 1 + c`m
b `m 3/2
1 x + c3/2 x + c`m 2
2 x + ...
See Fig. 1 in [65].
Finally, one uses the newly resummed multipolar waveforms (1.4.11) to define a resummation of the
radiation reaction force Fϕ defined as
1 (`max )
Fϕ = − F , (1.4.16)
Ω
where the (instantaneous, circular) GW flux F (`max ) is defined as

`max `
2 XX
F (`max ) = (mΩ)2 |Rh`m |2 . (1.4.17)
16πG m=1
`=2
Summarizing: Eqs. (1.4.11) and (1.4.16), (1.4.17) define resummed EOB versions of the waveform
h`m , and of the radiation reaction Fb ϕ , during inspiral. A crucial point is that these resummed expres-
sions are parameter-free. Given some current approximation to the conservative EOB dynamics (i.e.
some expressions for the A,D,Q potentials) they complete the EOB formalism by giving explicit pre-
dictions for the radiation reaction (thereby completing the dynamics, see Eqs. (1.4.3)–(1.4.6)), and for
the emitted inspiral waveform.
1.5 EOB description of the merger of binary black holes

and of the ringdown of the final black hole
Up to now we have reviewed how the EOB formalism, starting only from analytical information ob-
tained from PN theory, and adding extra resummation requirements (both for the EOB conservative
potentials A, Eq. (1.3.22), and D, Eq. (1.3.23), and for the waveform, Eq. (1.4.11), and its associated
radiation reaction force, Eqs. (1.4.16), (1.4.17)) makes specific predictions, both for the motion and the
radiation of binary black holes. The analytical calculations underlying such an EOB description are
essentially based on skeletonizing the two black holes as two, sufficiently separated point masses, and
therefore seem unable to describe the merger of the two black holes, and the subsequent ringdown of the
final, single black hole formed during the merger. However, as early as 2000 [40], the EOB formalism
went one step further and proposed a specific strategy for describing the complete waveform emitted
during the entire coalescence process, covering inspiral, merger and ringdown. This EOB proposal is
somewhat crude. However, the predictions it has made (years before NR simulations could accurately
describe the late inspiral and merger of binary black holes) have been broadly confirmed by subsequent
NR simulations. [See the Introduction for a list of EOB predictions.] Essentially, the EOB proposal
eff
(which was motivated partly by the closeness between the 2PN-accurate effective metric gµν [39] and
the Schwarzschild metric, and by the results of Refs. [67] and [68]) consists of:
(i) defining, within EOB theory, the instant of (effective) “merger” of the two black holes as the
(dynamical) EOB time tm where the orbital frequency Ω(t) reaches its maximum;
(ii) describing (for t ≤ tm ) the inspiral-plus-plunge (or simply insplunge) waveform, hinsplunge (t), by
using the inspiral EOB dynamics and waveform reviewed in the previous Section; and
(iii) describing (for t ≥ tm ) the merger-plus-ringdown waveform as a superposition of several quasi-
normal-mode (QNM) complex frequencies of a final Kerr black hole (of mass Mf and spin parameter
af , self-consistency estimated within the EOB formalism), say
2
Rc X + −σ+ (t−t )
hringdown
`m (t) = CN e N m
, (1.5.1)
GM N
+
with σN = αN + i ωN , and where the label N refers to indices (`, `0 , m, n), with (`, m) being the
Schwarzschild-background multipolarity of the considered (metric) waveform h`m , with n = 0,1,2 . . .
being the ‘overtone number’ of the considered Kerr-background Quasi-Normal-Mode, and `0 the degree
of its associated spheroidal harmonics S`0 m (aσ, θ);
+
(iv) determining the excitation coefficients CN of the QNM’s in Eq. (1.5.1) by using a simplified rep-
resentation of the transition between plunge and ring-down obtained by smoothly matching (following
Ref. [78]), on a (2p + 1)-toothed “comb” (tm − pδ, . . . , tm − δ, tm , tm + δ, . . . , tm + pδ) centered
around the merger (and matching) time tm , the inspiral-plus-plunge waveform to the above ring-down
waveform.
Finally, one defines a complete, quasi-analytical EOB waveform (covering the full process from in-
spiral to ring-down) as:
insplunge
hEOB
`m (t) = θ(tm − t) h`m (t) + θ(t − tm ) hringdown
`m (t) , (1.5.2)
where θ(t) denotes Heaviside’s step function. The final result is a waveform that essentially depends only
on the choice of a resummed EOB A(u) potential, and, less importantly, on the choice of resummation
of the main waveform amplitude factor f22 = (ρ22 )2 .
Section 1.5 EOB description of the merger of binary black holes and of the ringdown of the final black
hole 17
We have emphasized here that the EOB formalism is able, in principle, starting only from the best
currently known analytical information, to predict the full waveform emitted by coalescing binary black
holes. The early comparisons between 3PN-accurate EOB predicted waveforms7 and NR-computed
waveforms showed a satisfactory agreement between the two, within the (then relatively large) NR
uncertainties [97, 98]. Moreover, as we shall discuss below, it has been recently shown that the currently
known Padé-resummed 3PN-accurate A(u) potential is able, as is, to describe with remarkable accuracy
several aspects of the dynamics of coalescing binary black holes, [99, 100].
On the other hand, when NR started delivering high-accuracy waveforms, it became clear that the
3PN-level analytical knowledge incorporated in EOB theory was not accurate enough for providing
waveforms agreeing with NR ones within the high-accuracy needed for detection, and data analysis of
upcoming GW signals. [See, e.g., the discussion in Section II of Ref. [91].] At that point, one made
use of the natural flexibility of the EOB formalism. Indeed, as already emphasized in early EOB work
[42, 101], we know from the analytical point of view that there are (yet uncalculated) further terms in
the u-expansions of the EOB potentials A(u), D(u), . . . (and in the x-expansion of the waveform), so
that these terms can be introduced either as “free parameter(s) in constructing a bank of templates, and
[one should] wait until” GW observations determine their value(s) [42], or as “fitting parameters and
adjusted so as to reproduce other information one has about the exact results” (to quote Ref. [101]). For
instance, modulo logarithmic corrections that will be further discussed below, the Taylor expansion in
powers of u of the main EOB potential A(u) reads
a3 (ν)u3 + e
ATaylor (u; ν) = 1 − 2u + e a4 (ν)u4 + e
a5 (ν)u5 + e
a6 (ν)u6 + . . .
where the 2PN and 3PN coefficients e a3 (ν) = 2ν and e a4 (ν) = a4 ν have been known since 2001, but
where the 4PN, 5PN,. . . coefficients, e a5 (ν), e
a6 (ν), . . . were not known at the time (see below for the
recent determination of ea5 (ν)). A first attempt was made in [101] to use numerical data (on circular
orbits of corotating black holes) to fit for the value of a (single, effective) 4PN parameter of the simple
form ea5 (ν) = a5 ν entering a Padé-resummed 4PN-level A potential, i.e.
A14 (u; a5 ,ν) = P41 A3PN (u) + νa5 u5 .

(1.5.3)
This strategy was pursued in Ref. [79, 102] and many subsequent works. It was pointed out in Ref. [66]
that the introduction of a further 5PN coefficient ea6 (ν) = a6 ν, entering a Padé-resummed 5PN-level A
potential, i.e.
A15 (u; a5 ,a6 ,ν) = P51 A3PN (u) + νa5 u5 + νa6 u6 ,

(1.5.4)
helped in having a closer agreement with accurate NR waveforms.
In addition, Refs. [78, 79] introduced another type of flexibility parameters of the EOB formalism:
the non quasi-circular (NQC) parameters accounting for uncalculated modifications of the quasi-circular
inspiral waveform presented above, linked to deviations from an adiabatic quasi-circular motion. These
NQC parameters are of various types, and subsequent works [66, 91, 93, 94, 103, 104] have explored
several ways of introducing them. They enter the EOB waveform in two separate ways. First, through
an explicit, additional complex factor multiplying h`m , e.g.
NQC
f`m = (1 + a`m `m `m `m
1 n1 + a2 n2 ) exp[i(a3 n3 + a4 n4 )]
where the ni ’s are dynamical functions that vanish in the quasi-circular limit (with n1 , n2 being time-
7 The new, resummed EOB waveform discussed above was not available at the time, so that these comparisons
(N,)
employed the coarser “Newtonian-level” EOB waveform h22 (x).
even, and n3 , n4 time-odd). For instance, one usually takes n1 = (pr∗ /rΩ)2 . Second, through the
(discrete) choice of the argument used during the plunge to replace the variable x of the quasi-circular
inspiral argument: e.g. either xΩ ≡ (GM Ω)2/3 , or (following [106]) xϕ ≡ vϕ 2
= (rω Ω)2 where
1/3
vϕ ≡ Ω rω , and rω ≡ r[ψ(r,pϕ )] is a modified EOB radius, with ψ being defined as
−1 " s !#
p2ϕ

2 dA(r)
ψ(r,pϕ ) = 2 1 + 2ν A(r) 1 + 2 − 1 . (1.5.5)
r dr r
For a given value of the symmetric mass ratio, and given values of the A-flexibility parameters ea5 (ν), e
a6 (ν)
one can determine the values of the NQC parameters a`m i ’s from accurate NR simulations of binary black
hole coalescence (with mass ratio ν) by imposing, say, that the complex EOB waveform hEOB `m (t
EOB
a6 ; a`m
a5 , e
;e i )
NR NR EOB
osculates the corresponding NR one h`m (t ) at their respective instants of “merger”, where tmerger ≡
tEOB
m was defined above (maximum of ΩEOB (t)), while tNR merger is defined as the (retarded) NR time
where the modulus |hNR 22 (t)| of the quadrupolar waveform reaches its maximum. The order of oscu-
lation that one requires between hEOB NR
`m (t) and h`m (t) (or, separately, between their moduli and their
phases or frequencies) depends on the number of NQC parameters a`m i . For instance, a1
`m
and a`m2
EOB EOB
affect only the modulus of h`m and allow one to match both |h`m | and its first time derivative, at
merger, to their NR counterparts, while a`m `m
3 , a4 affect only the phase of the EOB waveform, and allow
EOB
one to match the GW frequency ω`m (t) and its first time derivative, at merger, to their NR counter-
parts. The above EOB/NR matching scheme has been developed and declined in various versions in
Refs. [66, 76, 91, 93, 94, 103–105]. One has also extracted the needed matching data from accurate NR
simulations, and provided explicit, analytical ν-dependent fitting formulas for them [66, 76, 91].
Having so “calibrated” the values of the NQC parameters by extracting non-perturbative information
from a sample of NR simulations, one can then, for any choice of the A-flexibility parameters, compute a
full EOB waveform (from early inspiral to late ringdown). The comparison of the latter EOB waveform
to the results of NR simulations is discussed in the next Section.
1.6 EOB vs NR
There have been several different types of comparison between EOB and NR. For instance, the early
work [97] pioneered the comparison between a purely analytical EOB waveform (uncalibrated to any NR
information) and a NR wavform, while the early work [107] compared the predictions for the final spin
of a coalescing black hole binary made by EOB, completed by the knowledge of the energy and angular
momentum lost during ringdown by an extreme mass ratio binary (computed by the test-mass NR code of
[108]), to comparable-mass NR simulations [109]. Since then, many other EOB/NR comparisons have
been performed, both in the comparable-mass case [66, 79, 93, 94, 98, 102, 103], and in the small-mass-
ratio case [78, 104, 110, 111]. Note in this respect that the numerical simulations of the GW emission
by extreme mass-ratio binaries have provided (and still provide) a very useful “laboratory” for learning
about the motion and radiation of binary systems, and their description within the EOB formalism.
Here we shall discuss only two recent examples of EOB/NR comparisons, which illustrate different
facets of this comparison.
1.6.1 EOB[NR] waveforms vs NR ones

We explained above how one could complete the EOB formalism by calibrating some of the natural
EOB flexibility parameters against NR data. First, for any given mass ratio ν and any given values of the
Section 1.6 EOB vs NR 19
A-flexibility parameters ea5 (ν), e

a6 (ν), one can use NR data to uniquely determine the NQC flexibility
parameters ai ’s. In other words, we have (for a given ν)
ai = ai [NR data; a5 , a6 ] ,
where we defined a5 and a6 so that e a5 (ν) = a5 ν, e

a6 (ν) = a6 ν. [We allow for some residual ν-
dependence in a5 and a6 .] Inserting these values in the (analytical) EOB waveform then defines an
NR-completed EOB waveform which still depends on the two unknown flexibility parameters a5 and
a6 .
In Ref. [66] the (a5 ,a6 )-dependent predictions made by such a NR-completed EOB formalism were
compared to the high-accuracy waveform from an equal-mass binary black hole (ν = 1/4) computed
by the Caltech-Cornell-CITA group [112], (and then made available on the web). It was found that there
is a strong degeneracy between a5 and a6 in the sense that there is an excellent EOB-NR agreement for
an extended region in the (a5 ,a6 )-plane. More precisely, the phase difference between the EOB (metric)
waveform and the Caltech-Cornell-CITA one, considered between GW frequencies M ωL = 0.047 and
M ωR = 0.31 (i.e., the last 16 GW cycles before merger), stays smaller than 0.02 radians within a long
and thin banana-like region in the (a5 ,a6 )-plane. This “good region” approximately extends between
the points (a5 ,a6 ) = (0, − 20) and (a5 ,a6 ) = (−36, + 520). As an example (which actually lies on
the boundary of the “good region”), we shall consider here (following Ref. [113]) the specific values
a5 = 0, a6 = −20 (to which correspond, when ν = 1/4, a1 = −0.036347, a2 = 1.2468). [Ref. [66]
did not make use of the NQC phase flexibility; i.e. it took a3 = a4 = 0. In addition, it introduced a
NQC
(real) modulus NQC factor f`m only for the dominant quadrupolar wave ` = 2 = m.] We henceforth
use M as time unit. This result relies on the proper comparison between NR and EOB time series, which
is a delicate subject. In fact, to compare the NR and EOB phase time-series φNR EOB
22 (tNR ) and φ22 (tEOB )
one needs to shift, by additive constants, both one of the time variables, and one of the phases. In other
words, we need to determine τ and α such that the “shifted” EOB quantities
0
t0EOB = tEOB + τ , EOB
φ22 = φEOB
22 + α (1.6.1)
“best fit” the NR ones. One convenient way to do so is first to “pinch” (i.e. constrain to vanish) the
EOB/NR phase difference at two different instants (corresponding to two different frequencies ω1 and
ω2 ). Having so related the EOB time and phase variables to the NR ones we can straigthforwardly
compare the EOB time series to its NR correspondant.
0 In particular,
EOB 0EOB we can compute the (shifted)
∆ω1 ,ω2 φEOBNR
22 (tNR ) ≡ φ22 (t ) − φNR NR
22 (t ). (1.6.2)
EOB–NR phase difference
8 EOB
Figure 1.2 compares (the real part of) the analytical EOB metric quadrupolar waveform Ψ22 /ν to the
corresponding (Caltech-Cornell-CITA) NR metric waveform ΨNR 22√/ν. [Here, Ψ22 denotes the Zerilli-
normalized asymptotic quadrupolar waveform, i.e. Ψ22 ≡ Rh b 22 / 24 with R b = Rc2 /GM .] This NR
metric waveform has been obtained by a double time-integration (following the procedure of Ref. [94])
from the original, publicly available, curvature waveform ψ422 [112]. Such a curvature waveform has
been extrapolated both in resolution and in extraction radius. The agreement between the analytical
prediction and the NR result is striking, even around the merger. See Fig. 1.3 which closes up on
the merger. The vertical line indicates the location of the EOB-merger time, i.e., the location of the
maximum of the orbital frequency.
The phasing agreement between the waveforms is excellent over the full time span of the simulation
(which covers 32 cycles of inspiral and about 6 cycles of ringdown), while the modulus agreement is
excellent over the full span, apart from two cycles after merger where one can notice a difference. More
8 The two “pinching” frequencies used for this comparison are M ω1 = 0.047 and M ω2 = 0.31.
0.3
Numerical Relativity (Caltech-Cornell)
EOB (a 5 = 0; a 6 =-20)
0.2
0.1
ℜ[Ψ22]/ν
−0.1
−0.2
1:1 mass ratio
−0.3
500 1000 1500 2000 2500 3000 3500 4000

t
Figure 1.2. This figure illustrates the comparison (made in Refs. [66, 113]) between the (NR-completed)
EOB waveform (Zerilli-normalized quadrupolar (` = m = 2) metric waveform (1.5.2) with parameter-
free radiation reaction (1.4.16) and with a5 = 0,a6 = −20) and one of the most accurate numerical
relativity waveform (equal-mass case) nowadays available [112]. The phase difference between the two
is ∆φ ≤ ±0.01 radians during the entire inspiral and plunge, which is at the level of the numerical error.
precisely, the phase difference, ∆φ = φEOB NR

metric − φmetric , remains remarkably small (∼ ±0.02 radians)
during the entire inspiral and plunge (ω2 = 0.31 being quite near the merger). By comparison, the root-
sum of the various numerical errors on the phase (numerical truncation, outer boundary, extrapolation
to infinity) is about 0.023 radians during the inspiral [112]. At the merger, and during the ringdown,
∆φ takes somewhat larger values (∼ ±0.1 radians), but it oscillates around zero, so that, on average, it
stays very well in phase with the NR waveform whose error rises to ±0.05 radians during ringdown. In
addition, Ref. [66] compared the EOB waveform to accurate numerical relativity data (obtained by the
Jena group [94]) on the coalescence of unequal mass-ratio black-hole binaries. Again, the agreement
was good, and within the numerical error bars.
This type of high-accuracy comparison between NR waveforms and EOB[NR] ones (where EOB[NR]
denotes a EOB formalism which has been completed by fitting some EOB-flexibility parameters to
NR data) has been pursued and extended in Ref. [91]. The latter reference used the “improved” EOB
formalism of Ref. [66] with some variations (e.g. a third modulus NQC coefficient ai , two phase NQC
coefficients, the argument xΩ in (ρTaylor `
`m (x)) , eight QNM modes) and calibrated it to NR simulations of
mass ratios q = m2 /m1 = 1,2,3,4 and 6. They considered not only the leading (`, m) = (2,2) GW
mode, but the subleading ones (2,1), (3,3), (4,4) and (5,5). They found that, for this large range of mass
ratios, EOB[NR] (with suitably fitted, ν-dependent values of a5 and a6 ) was able to describe the NR
waveforms essentially within the NR errors. See also the recent Ref. [76] which incorporated several
analytical advances in the two-body problem. This confirms the usefulness of the EOB formalism in
helping the detection and analysis of upcoming GW signals.
Here, having in view GW observations from ground-based interferometric detectors we focussed on

comparable-mass systems. The EOB formalism has also been compared to NR results in the extreme
mass-ratio limit ν 1. In particular, Ref. [104] found an excellent agreement between the analytical
and numerical results.
Section 1.6 EOB vs NR 21
0.3
0.2
0.1
ℜ[Ψ22]/ν
−0.1
−0.2
−0.3 Merger time
3800 3820 3840 3860 3880 3900 3920 3940 3960 3980 4000
t
Figure 1.3. Close up around merger of the waveforms of Fig. 1.2. Note the excellent agreement between
both modulus and phasing also during the ringdown phase.
1.6.2 EOB[3PN] dynamics vs NR one

Let us also mention other types of EOB/NR comparisons. Several examples of EOB/NR comparisons
have been performed directly at the level of the dynamics of a binary black hole, rather than at the level
of the waveform. Moreover, contrary to the waveform comparisons of the previous subsection which
involved an NR-completed EOB formalism (“EOB[NR]”), several of the dynamical comparisons we are
going to discuss involve the purely analytical 3PN-accurate EOB formalism (“EOB[3PN]”), without any
NR-based improvement.
First, Le Tiec et al. [99] have extracted from accurate NR simulations of slightly eccentric binary
black-hole systems (for several mass ratios q = m1 /m2 between 1/8 and 1) the function relating the
periastron-advance parameter
∆Φ
K =1+ ,
2π
(where ∆Φ is the periastron advance per radial period) to the dimensionless averaged angular frequency
M Ωϕ (with M = m1 +m2 as above). Then they compared the NR-estimate of the mass-ratio dependent
functional relation
K = K(M Ωϕ ; ν) ,
where ν = q/(1 + q)2 , to the predictions of various analytic approximation schemes: PN theory, EOB
theory and two different ways of using GSF theory. Let us only mention here that the prediction from
the purely analytical EOB[3PN] formalism for K(M Ωϕ ; ν) [72] agreed remarkably well (essentially
within numerical errors) with its NR estimate for all mass ratios, while, by contrast, the PN-expanded
prediction for K(M Ωϕ ; ν) [70] showed a much poorer agreement, especially as q moved away from 1.
Second, Damour, Nagar, Pollney and Reisswig [100] have extracted from accurate NR simulations of
black-hole binaries (with mass ratios q = m2 /m1 = 1,2 and 3) the gauge-invariant relation between
−0.03
NR
3PN-Taylor
−0.04 EOB3PN
EOBadiabatic
3PN
−0.05
−0.06
−0.0375
−0.07
−0.038
E −0.08
−0.09 −0.0385
−0.1 −0.039
−0.11 −0.0395
−0.12
−0.04
3.92 3.93 3.94 3.95 3.96 3.97 3.98
−0.13
2.8 3 3.2 3.4 3.6 3.8 4
j
Figure 1.4. Comparison (made in [100]) between various analytical estimates of the energy-angular
momentum functional relation and its numerical-relativity estimate (equal-mass case). The standard
“Taylor-expanded” 3PN E(j) curve shows the largest deviation from NR results, especially at low j’s,
while the two (adiabatic and nonadiabatic) 3PN-accurate, non-NR-calibrated EOB E(j) curves agree
remarkably well with the NR one.
the (reduced) binding energy E = (Etot − M )/µ and the (reduced) angular momentum j = J/(GµM )
of the system. Then they compared the NR-estimate of the mass-ratio dependent functional relation
E = E(j; ν)
to the predictions of various analytic approximation schemes: PN theory and various versions of EOB
theory (some of these versions were NR-completed). Let us only mention here that the prediction from
the purely analytical, 3PN-accurate EOB[3PN] for E(j; ν) agreed remarkably well with its NR estimate
(for all mass ratios) essentially down to the merger. This is illustrated in Fig. 4 for the q = 1 case.
By contrast, the 3PN expansion in (powers of 1/c2 ) of the function E(j; ν) showed a much poorer
agreement (for all mass ratios).
Recently, several other works have (successfully) compared EOB dynamical predictions to NR re-
sults. Ref. [114] compared the EOB[NR] predictions for the dynamical state of a non-spinning, coalesc-
ing BBH at merger to NR results and found agreement at the per mil level. Ref. [115] compared the
predictions of an analytical (3.5PN-accurate) spinning EOB model to NR simulations and found a very
good agreement.
Section 1.7 Other developments 23
1.7 Other developments

1.7.1 EOB with spinning bodies
We do not wish to enter into a detailed discussion of the extension of the EOB formalism to binary sys-
tems made of spinning bodies. Let us only mention that the spin-extension of the EOB formalism was
initiated in Ref. [42], that the first EOB-based analytical calculation of a complete waveform from a spin-
ning binary was performed in Ref. [43], and that the first attempt at calibrating a spinning EOB model
to accurate NR simulations of spinning (non precessing) black-hole binaries was presented in [116]. In
addition, several formal aspects related to the inclusion of spins in the EOB formalism have been dis-
cussed in Refs. [117–121] (see references within these papers for PN works dealing with spin effects)
and a generalization of the factorized multipolar waveform of Ref. [65] to spinning, non-precessing
binaries has been constructed in Refs. [90, 92]. Comparisons between spinning-EOB models and NR
simulations have been obtained in [122, 123] and, recently, in the spinning, precessing case, in [124].
1.7.2 EOB with tidally deformed bodies
In binary systems comprising neutron stars, rather than black holes, the tidal deformation of the neu-
tron star(s) will significantly modify the phasing of the emitted gravitational waveform during the late
inspiral, thereby offering the possibility to measure the tidal polarizability of neutron stars [125–128].
As GW’s from binary neutron stars are expected sources for upcoming ground-based GW detectors, it
is important to extend the EOB formalism by including tidal effects. This extension has been defined in
Refs. [133, 134]. The comparison between this tidal-extended EOB and state-of-the-art NR simulations
of neutron-star binaries has been discussed in Refs. [129–132]. It appears from these comparisons that
the tidal-extended EOB formalism is able to describe the motion and radiation of neutron-star binaries
within NR errors. More accurate simulations will be needed to ascertain whether one needs to calibrate
some higher-order flexibility parameters of the tidal-EOB formalism, or whether the currently known
analytic accuracy is sufficient [35, 131].
1.7.3 EOB and GSF
We mentioned in the Introduction that GSF theory has recently opened a new source of information on
the general relativistic two-body problem. Let us briefly mention here that there has been a quite useful
transfer of information from GSF theory to EOB theory. The program of using GSF-theory to improve
EOB-theory was first highlighted in Ref. [72]. That work pointed to several concrete gauge-invariant
calculations (within GSF theory) that would provide accurate information about the O(ν) contributions
to several EOB potentials. More precisely, let us define the functions a(u) and d(u) ¯ as the ν-linear
−1
contributions to the EOB potentials A(u; ν) and D(u; ν) ≡ D (u; ν):
A(u; ν) = 1 − 2u + ν a(u) + O(ν 2 ) ,
D(u; ν) = (AB)−1 = 1 + ν d(u)

¯ + O(ν 2 ) .
Ref. [72] has shown that a computation of the GSF-induced correction to the periastron advance of
slightly eccentric orbits would allow one to compute the following combination of EOB functions
1
ρ̄(u) = a(u) + u a0 (u) + u(1 − 2u) a00 (u) + (1 − 6u) d(u)
¯ .
2
The GSF-calculation of the EOB function ρ̄(u) was then performed in Ref. [135] (in the range 0 ≤ u ≤
1
6
).
Later, a series of works by Le Tiec and collaborators [75, 136, 137] have (through an indirect route)
shown how GSF calculations could be used to compute the EOB ν-linear a(u) function separately from
¯
the d(u) one. Ref. [75] then gave a fitting formula for a(u) over the interval 0 ≤ u ≤ 51 as well as
accurate estimates of the coefficients of the Taylor expansion of a(u) around u = 0 (corresponding
to the knowledge of the PN expansion of a(u) to a very high PN order). More recently, Ackay et al.
[138] succeeded in accurately computing (through GSF theory) the EOB a(u) function over the larger
interval 0 ≤ u ≤ 31 . It was (surprisingly) found that a(u) diverges like a(u) ≈ 0.25(1 − 3u)−1/2 at
−
the light-ring limit u → 13 . The meaning for EOB theory of this singular behavior of a(u) at the
light-ring is discussed in detail in Ref. [138].
Let us finally mention that Ref. [28] has recently showed how to combine analytical GSF theory with
the partial 4PN-level results of Ref. [27] so as to obtain the complete analytical expression of the 4PN-
a5 (ν; ln a) of u5
level contribution to the A potential. Specifically, Ref. [28] found that the coefficient e
in the PN expansion, of A(u; ν),
a3 (ν)u3 + e
ATaylor (u; ν) = 1 − 2u + e a4 (ν)u4 + e
a5 (ν; ln u)u5 + e
a6 (ν; ln u)u6 + . . .
was equal to
64
a5 (ν; ln u) = (a5 +
e ln u)ν + a05 ν 2 ,
5
with
4237 2275 2 256 128
a5 = − + π + ln 2 + γ,
60 512 5 5
221 41 2
a05 = − + π .
6 32
a5 (ν) is no more than quadratic in ν, i.e. without contributions of degree ν 3 and ν 4 . [Contri-
Note that e
butions of degree ν 3 and ν 4 would a priori be expected in a 4PN level quantity; see, e.g., e4PN (ν; ln x)
below.] We recall that similar cancellations of higher ν n terms were found at lower PN orders in the
EOB A(u; ν) function. Namely, they were found to contain only terms linear in ν, while e a3 (ν) could
a priori have been quadratic in ν, and ea4 (ν) could a priori have been cubic in ν. The fact that similar
remarkable cancellations still hold, at the 4PN level, is a clear indication that the EOB packaging of
information of the dynamics in the A(u; ν) potential is quite compact. By contrast, the PN expansions
of other dynamical functions do not exhibit such cancellations. For instance, the coefficients entering
the PN expansion of the (gauge-invariant) function E(x; ν) relating the total energy to the frequency
parameter x ≡ (M Ωϕ )2/3 , namely
1
E(x; ν) = − µc2 x(1 + e1P N (ν)x + e2P N (ν)x2 + e3P N (ν)x3
2
+e4P N (ν; ln x)x4 + O(x5 ln x)),
contain all the a priori possible powers of ν. In particular, at the 4PN level e4PN (ν; ln x) is a polynomial
of fourth degree in ν.
Section 1.8 Conclusions 25
1.8 Conclusions
Though the present work did not attempt to expound the many different approaches to the general rela-
tivistic two-body problem but focussed only on a few approaches, we hope to have made it clear that there
is a complementarity between the various current ways of tackling this problem: post-Newtonian9 , effec-
tive one body, gravitational self-force, and numerical relativity simulations. Among these approaches,
the effective one body formalism plays a special role in that it allows one to combine, in a synergetic
manner, information coming from the other approaches. As we are approaching the 100th anniversary
of the discovery of general relativity, it is striking to see how this theory has not only passed with flying
colors many stringent tests, but has established itself as an essential tool for describing many aspects of
the Universe from, say, the Big Bang to an accurate description of planets and satellites. Though the
two-body (and, more generally, the N -body) problem is one of the oldest problems in general relativ-
ity, it is more lively than ever. Indeed, several domains of (astro-)physics and astronomy are providing
new incentives for improving the various ways of describing general relativistic N -body systems: the
developement of (ground-based and space-based) detectors of gravitational waves, the development of
improved techniques for observing binary pulsars, the prospect of observing soon (with Gaia) a billion
stars with ∼ 10−5 arcsec accuracy, . . . Together with our esteemed friend and colleague Victor Brum-
berg, who pioneered important developments in Relativistic Celestial Mechanics, we are all looking
forward to witnessing new applications of Einstein’s vision of gravity to the description and understand-
ing of physical reality.
9 including the effective-field-theory reformulation of the computation of the PN-expanded Fokker-action [45, 139].
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2 Hamiltonian Dynamics of Spinning Compact
Binaries Through High Post-Newtonian Ap-
proximations
Gerhard Schäfer
Friedrich-Schiller-Universität Jena, Theoretisch-Physikalisches Institut, Max-Wien-Platz 1,
D-07743 Jena, Germany
2.1 Introduction
The book by Professor Brumberg entitled “Essential Relativistic Celestial Mechanics”, published in
1991 [1], was the long-awaited successor of his much earlier book “Relativistic Celestial Mechanics”
of 1972 [2], published in russian only. In these accounts the celestial mechanics has been presented
with unprecedented details based on relativistic physics. Fully in their tradition, the recent book by
Kopeikin, Efroimsky, and Kaplan entitled “Relativistic Celestial Mechanics of the Solar System” [3] is
the most advanced account on the dynamics of the solar system. The mentioned books mainly rely on
the first post-Newtonian (1pN) approximation of Einstein’s general relativity theory but in part and to
some extent also results are presented from the 2pN and 2.5pN approximations and the leading order
gravitational spin coupling. Focussing on compact binary systems, black holes and neutron stars, the
present article covers the dynamics and motion up to the 4pN approximation including proper rotation
(spin) of the bodies.
The Einstein theory of gravitation delivers an excellent and well tested description of the dynamics of
gravitating systems despite its very intricate and complex structure from both mathematical and concep-
tual points of view [4]. The very reason for this complication is the fact that the gravitational potential
functions, which are identical with the metric coefficients of spacetime, together with their first deriva-
tives, the so-called Christoffel symbols, are no observables but depend on the chosen coordinate system.
Selecting a specific coordinate system thus means choosing a specific gauge. The freedom of choice of
the coordinate system renders the highly non-linear Einstein field equations degenerate, e.g. [5]. A most
elegant way to cope with these difficulties is the Hamiltonian approach where finally one functional, the
non-degenerate physical Hamiltonian, determines the dynamics of the whole gravitating system. It is a
Hamiltonian setting of general relativity which is used in the present contribution. If at various places in
the following the Newton’s gravitational constant, G, and the speed of light, c, do not show up explicitly,
the units 16πG = 1 and c = 1 are employed. Notice should also be made to the fact that the article
is devoted to the Hamiltonian treatment of spinning and non-spinning binaries; for related results from
other formalisms, the reader is adviced to consult [6–8].
2.2 Hamiltonian formulation of general relativity

The most often used Hamiltonian formulation of general relativity is the one developed by Arnowitt,
Deser, and Misner (ADM) around 1960, [9], which is nearly half a century after Einstein’s publication
38 Chapter 2 Hamiltonian Dynamics of Spinning Compact Binaries
of his theory in 1915. Their primary interest was the quantization of the gravitational field which is
still an unsolved problem. On the other side, the application of the ADM approach in classical general
relativity has turned out very efficient as can be clearly seen in the following.
In the ADM formalism, the full Einstein field equations
1 µν
Gµν = T , (2.2.1)
2
where Gµν denotes the Einstein tensor and T µν the energy-momentum tensor (µ,ν = 0,1,2,3; i,j =
1,2,3), are obtained by applying functional derivatives to the formal Hamiltonian (symbolic writing in
part) Z
H= d3 x[N H(γ, ∂γ, ∂∂γ, π, M V ) − N i Hi (γ, ∂γ, π, ∂π, M V )] (2.2.2)
with compact-support perturbations of the field functions N and N i , coined lapse and shift functions by
John Wheeler [10], respectively, as well as of the three-dimensional metric coefficients γij ≡ gij and
their canonical conjugates π ij . The field functions γij and π ij are dynamical variables whereas the field
functions N and N i are Lagrange multipliers which do not show up in the functions H, called super
Hamiltonian, and Hi , called super momentum (the minus sign in Eq. (2) results from the definition of
Hi to contain the momentum density of the matter and not the negative of it). The canonical conjugates
of N and N i do not appear in the Hamiltonian. Variation of the formal Hamiltonian with respect to N
and N i results in the constraint equations which are of fundamental importance in the ADM formalism.
M V is an abbreviation for the canonical matter variables. In terms of the metric coefficients gµν , or
their inverse g µν , the lapse and shift functions can easily be deduced from the following equations, see
e.g. [11],
ds2 ≡ gµν dxµ dxν = −N 2 dx0 dx0 + γij (dxi + N i dx0 )(dxj + N j dx0 ) , (2.2.3)
2
∂S ∂S 1 ∂S ∂S ∂S ∂S
g µν =− 2 − Ni i + γ ij , (2.2.4)
∂xµ ∂xν N ∂x0 ∂x ∂xi ∂xj
where γ ij denotes the inverse three-dimensional metric, γ ik γkj = δij . ds2 is the line element and S is a
scalar function; if applied to a particle, their meanings are particle’s proper time τ (ds2 = −c2 dτ 2 ) and
action, respectively. The motion orthogonal to x0 = const reads dxi + N i dx0 = 0 and the derivative
orthogonal to x0 = const takes the form ∂0 − N i ∂i , where ∂µ = ∂x∂µ .
The following identities hold,
√ √
H≡ −gN (T 00 − 2G00 ), Hi ≡ −g(Ti0 − 2G0i ) (2.2.5)
and
√
π ij ≡ − γ(γ ik γ jl − γ ij γ kl )Kkl , (2.2.6)
k l 0
where Kkl dx dx is the second fundamental form or extrinsic curvature of a hypersurface x = const
whereas the first fundamental form or intrinsic metric is given by γkl dxk dxl . The determinants of
gµν and γij are denoted by g and γ, respectively. In terms of Christoffel symbols, Kij = −N Γ0ij
holds showing that the extrinsic curvature depends on the first partial spacetime derivatives of the metric
coefficients. In contrast herewith, the intrinsic curvature form Rkl dxk dxl of a hypersurface x0 = const
depends on partial space derivatives of the metric coefficients through second order.
If the field functions are assumed to decay at spacelike flat infinity (denoted by i0 ) in the way physical
gravitational fields have to decay, as being inferred from physical solutions of the Einstein field equations
Section 2.2 Hamiltonian formulation of general relativity 39
for isolated systems, namely
N = 1 + O(1/r), N i = O(1/r), γij = δij + O(1/r), π ij = O(1/r2 ) , (2.2.7)
a surface term has to be added to the Hamiltonian for not spoiling the Hamilton principle, reading, see
[9, 12, 13], Z I
H= d3 x(N H − N i Hi ) + d2 si ∂j (γij − δij γkk ) , (2.2.8)
i0
where d2 si denotes the two-dimensional surface area element with outward-pointing unit normal vector.
Splitting naturally H = Hmatter + Hfield and Hi = Himatter + Hifield , it is interesting to point out that
the Hamiltonian for test matter (TM) in an external gravitational field (passive interaction) reads
Z
HTM = d3 x(N Hmatter − N i Himatter ) . (2.2.9)
For the treatment of gravitationally passively and actively interacting matter the free-field (FF) Hamilto-
nian Z
HFF = d3 x(N Hfield − N i Hifield ) (2.2.10)
has to be added as well as the above surface term because gravitational interaction implies HTM + HFF =
0 so otherwise the Hamiltonian of the total system would vanish, e.g. [12]. Indeed, on the submanifold
where the constraint equations hold,
H = 0, Hi = 0 , (2.2.11)
the total Hamiltonian reduces to a surface term only,

I
H= d2 si ∂j (γij − δij γkk ) . (2.2.12)
i0
The present considerations clearly show that gravitational energy is not localizable. The rewriting of
the energy surface integral as volume integral, using the Gaussian theorem, i.e.
Z
H = d3 x ∂i ∂j (γij − δij γkk ) , (2.2.13)
does neither give a unique integrand nor a one which transforms as some tensor-density component
under general coordinate transformations. The Hamiltonian within the Dirac approach reads
Z
HD = − d3 x ∂i (γ −1/2 ∂j (γγ ij )) (2.2.14)
and the one in the Schwinger formalism takes the form

Z
HS = − d3 x ∂i ∂j (γγ ij ) . (2.2.15)
The three Hamiltonians evidently fulfil H = HD = HS , taking into account the asymptotic behavior of
the metric coefficients.
The surface-integral Hamiltonian is still not able to generate the remaining Einstein field equations
and, thus, is not a Hamiltonian at all; it only delivers the total energy of the system and coincides with,
e. g., the energy expression from the Landau-Lifshitz pseudo-tensor complex for the gravitational field,
[14]. However, by the additional choice of four appropriate coordinate conditions, e.g.
π ii = 0 and 3∂j γij − ∂i γjj = 0 , (2.2.16)
resulting in
π ij = π̃ ij + π ijTT and γij = ψδij + hTT
ij (2.2.17)
with
2
π̃ ij = ∂j π i + ∂i π j − δij ∂k π k , (2.2.18)
3
the energy expression in question turns into a Hamitonian for the matter variables and the “true” degrees
ijTT
of freedom of the gravitational field, hTT
ij and their canonical conjugate π which both are tranverse
iiTT ijTT
and traceless, i.e. hTT
ii = π = ∂ h
j ij
TT
= ∂j π = 0. The outcome is the ADM-Hamiltonian.
Importantly, the derivation of the ADM-Hamiltonian needs the constraint equations and the coordinate
conditions only (the functions ψ and π i are determined by the constraint equations). The lapse and shift
functions do not enter; only the asymptotic value 1 of the lapse function entered into the obtention of
the ADM-Hamiltonian. For the calculation of the functions N and N i , the coordinate conditions are
needed as well as the field equations for gij and π ij . Importantly too, the resulting equations for N
and N i are of instantaneous type without any time derivatives. Only the variables hTT ij and π
ijTT
fulfil
evolution equations and result from functional derivatives of the ADM-Hamiltonian which can be put
into the form I Z
HADM = −2 d2 si ∂i ψ[hTT
kl ,π klTT
, M V ] = −2 d3 x∂i ∂i ψ . (2.2.19)
i0
The Poisson-bracket commutation relations read
{hTT
ij (x,t), π
klTT
(x0 ,t)} = TTkl
δij δ(x − x0 ) , (2.2.20)
TTkl
and zero otherwise, where δij , the transverse-traceless projection operator, is defined by
1
TTkl
δij = (Pil Pjk + Pik Pjl − Pkl Pij ), Pij = δij − ∆−1 ∂i ∂j , (2.2.21)
2
where ∆−1 denotes the usual inverse Laplacian in flat three-dimensional space.
2.2.1 Point particles
In regard to the matter variables M V , the canonical structure is simple for point masses and for perfect
fluids, e.g. [9, 15]. Also the inclusion of the electromagnetic field is straightforward, [9]. Quite involved,
however, is the canonical setting of spinning particles. For point particles, the total super Hamiltonian
and super momentum read,
X 1/2 1 1
H= m2a c2 + γ ij pai paj δa + (πji πij − πii πjj ) − γ 1/2 R , (2.2.22)
a
γ 1/2 2
X
Hi = pai δa − π kl γkl,i + 2πi,j
j
, (2.2.23)
a
where πji = γjk π ik , R = γ ij Rij , A,i = ∂i A, and δa = δ(xi − xia ) with d3 xδa = 1. The canonical
R
particle variables, positions and momenta, are simply xia and pai , with the particle label a = 1, 2, ...,
Section 2.2 Hamiltonian formulation of general relativity 41
fulfilling the standard Poisson-bracket relations,
{xia , paj } = δij , (2.2.24)
and zero otherwise.

The reader my wonder how the use of Dirac delta functions can make sense in a non-linear theory
such as Einstein’s general relativity theory. Here it is to be said that with dimensional regularization, e.g.
[16, 17], i.e. performimg calculations in d space-dimensions and making the limit d → 3 at the very end
of the calculations, all local singularities could be uniquely controlled through 4pN order including spin,
[16, 18, 19]. The problem with the badly defined outer near zone, starting at the 4pN approximation,
is not connected with the application of Dirac delta functions, [19]. Interestingly, Brill-Lindquist black
holes can be represented by Dirac delta functions with support in conformally related three-dimensional
space, i.e. fictitious point masses in fictitious space generate black holes (extended objects!) in physical
space, [20], similarly to the image charges in the electrodynamics which are also not located in the region
where the physical fields operate.
2.2.2 Spinning particles
In case of spinning particles, the matter super Hamiltonian and super momentum are given by, [21, 22],
X
1 Sali paj Sali paj pan pal

Hmatter = − npa − + γ mn γ kl γij,k
a
2 npa (npa )2 (ma − npa )
paj γ ji kl

pal
+ πa γkl,i δa − γ ij γ kl Sajk δa , (2.2.25)
npa ma − npa ,i
X 1

Himatter = (pai − πakl γkl,i )δa + (sij
a δa ),j , (2.2.26)
a
2
where
ma pa(k nSal) 1
πaij = γ ik γ jl + B ij γ km γ ln Samn , (2.2.27)
2npa (ma − npa ) 2 kl
1/2 pak γ kj Saji

npa = − m2a c2 + γ ij pai paj , nSai = − , (2.2.28)
m
2pal pa(i Sak)p

sij jk jk lp
a = γ Saik + γ γ . (2.2.29)
npa (ma − npa )
ij
If the symmetric root of the symmetric matrix γij is denoted by eij , i.e. eik ekj = γij , Bkl reads
ij ∂eml ∂emk
2Bkl = ekm − elm . (2.2.30)
∂γij ∂γij
The canonical field momentum is now given by

ij
X
πcan = π ij + πaij δa . (2.2.31)
a
Imposing coordinate conditions of the form

ii
πcan = 0 and 3∂j γij − ∂i γjj = 0 , (2.2.32)
a new decomposition
ij ij ijTT
πcan = π̃can + πcan (2.2.33)
arises. Also applying the constraint equations H = 0 and Hi = 0, wherein π ij has to be substituted
ij
through πcan , finally, the reduced total action W for spinning particles turns out to be
Z Z
X 1
W = dt (pai ẋia + Sa(i)(j) θ̇a(i)(j) ) + d4 x πcan
ijTT TT
hij,0
a
2
Z h i
− dt HADM xia , pai ,Sa(i)(j) , hTT ijTT
ij ,πcan , (2.2.34)
where Saij = eik ejl Sa(k)(l) holds. The Poisson-bracket relations read,
{xia , paj } = δij , {Sa(i) , Sa(j) } = ijk Sa(k) , (2.2.35)
0
{hTT klTT
ij (x,t), πcan (x ,t)} = TTkl
δij δ(x − x0 ) , (2.2.36)
and zero otherwise, with Sa(k) = 12 kij Sa(i)(j) . The evolution equations for the canonical variables have
the forms,
d i
(xa , pai , Sa(i) ) = {(xia , pai , Sa(i) ), HADM } , (2.2.37)
dt
∂ TT ijTT ijTT
(hij ,πcan ) = {(hTT
ij ,πcan ), HADM } . (2.2.38)
∂t
2.2.3 Introducing the Routhian
A priori for self-gravitating objects, the total Hamiltonian results in the form H = H[p,q,S,hTT ,π TT ].
The transition to a Routhian description of the type R = R[p,q,S,hTT ,ḣTT ] allows the derivation of an
autonomous matter Hamiltonian for the conservative dynamics,
Hc ≡ R[p,q,S,hTT (x; p,q,S),ḣTT (x; p,q,S)] = Hc (p,q,S) (2.2.39)
as well as a non-autonomous or non-conservative one resulting in
Hnc (p,q,S; t) = R[p,q,S,hTT (x; p0 ,q 0 ,S 0 ),ḣTT (x; p0 ,q 0 ,S 0 )]
= Hnc (p,q,S; p0 ,q 0 ,S 0 ) . (2.2.40)
In more detail, the definition of the Routhian in question is given by, in the pN context first applied in
[23],
h i Z
R xia , pai ,Sa(i) , hTT TT
ij , ∂t hij = H − d3 x πcan
ijTT
∂t hTT
ij . (2.2.41)
Section 2.3 The Poincaré algebra 43
The equations of motion for the matter read,
∂R ∂R
ṗai = − , ẋia = , Ṡa(i) = {Sa(i) , R} , (2.2.42)
∂xia ∂pai
and the field equations take the form
δ R(t0 )dt0
R
= 0. (2.2.43)
δhTT
ij (x ,t)
k
The insertion of the solution of the field equations into the Routhian results in the mentioned autonomous
and non-autonomous Hamiltonians for the matter degrees of freedom. The use of equations of motion is
allowed on the action or Hamiltonian level. However, one should keep in mind that a change of variables
is getting performed implicitly, [24, 25]. The final equations of motion are of the form,
∂Hc;nc ∂Hc;nc
ṗai = − , ẋia = , (2.2.44)
∂xia ∂pai
∂Hc,nc
Ṡa(i) = −ijk Sa(j) Ωa(k) with Ωa(i) = . (2.2.45)
∂Sa(i)
2.3 The Poincaré algebra

In asymptotically flat spacetimes the Poincaré or inhomogeneous Lorentz group is a global symmetry
group. Its generators P µ and J µν are conserved in time and fulfil the Poincaré algebra, see e.g. [13],
{P µ , P ν } = 0, (2.3.1)
{P µ , J ρσ } = −η µρ P σ + η µσ P ρ , (2.3.2)
{J µν , P ρσ } = −η νρ J µσ + η µρ J νσ + η σµ J ρν − η σµ J ρµ . (2.3.3)
The meaning of the components are energy P 0 = H/c, linear momentum P i = Pi , angular momentum
J ij = Jij , and Lorentz boost J i0 /c ≡ K i = Gi − t P i . A center-of-energy vector is defined by
X i = c2 Gi /H. This vector, however, is not a canonical position vector, see e.g. [26]. In terms of
three-dimensional quantities, the Poincaré algebra reads, see e.g. [27], with Jij = ijk Jk ,
{Pi , H} = {Ji , H} = 0 , (2.3.4)
{Ji , Pj } = εijk Pk , {Ji , Jj } = εijk Jk , (2.3.5)
{Ji , Gj } = εijk Gk , (2.3.6)
{Gi , H} = Pi , (2.3.7)
1
{Gi , Pj } = H δij , (2.3.8)
c2
1
{Gi , Gj } = −εijk Jk . (2.3.9)
c2
In terms of the field function φ, with ψ = (1 + φ/8)4 , the energy or Hamiltonian H and the center-
of-energy vector Gi = Gi have the representations
Z I
H = − d3 x ∆φ = − r2 dΩni ∂i φ, (2.3.10)
i0
Z I
Gi = − d3 x xi ∆φ = − r2 dΩnj (xi ∂j − δij )φ , (2.3.11)
i0
whereas the total linear and angular momentum are given by

Z I
Pi = −2 d3 x∂j π̃ ij = −2 r2 dΩnj π̃ ij , (2.3.12)
i0
Z I
Ji = −2 d3 xijk xj ∂l π̃ kl = −2 r2 dΩnl ijk xj π̃ kl , (2.3.13)
i0
where r2 dΩni is the two-dimensional surface-area element and ni the outward-pointing radial unit
vector.
The Poincaré algebra has been extensively used in the calculations and checkings of higher order
pN Hamiltonians, [18, 19, 27–33]. Hereby the most important equation was (2.3.7) which tells that the
total linear momentum is a total time derivative. Once in previous calculations, this equation has finally
fixed the kinetic ambiguity in the non-dimensional regularization calculations in [27]. The reader might
perhaps be also interested in the construction of canonical center-of-energy and relative coordinates in
asymptotically flat spacetimes, [34].
2.4 Post-Newtonian binary Hamiltonians

2.4.1 Spinless binaries
For non-spinning compact objects, the binary dynamics is fully known up to the 3.5pN order and in part
even at 4pN order. On reasons of readability, the Hamiltonians for spinless particles are given in the
center-of-energy frame only and the 3.5pN part is not shown. The known but not shown Hamiltonians
can easily be found in the quoted literature below.
Introducing the reduced quantities, Ĥ = (H − M )/µ, µ = m1 m2 /M , M = m1 + m2 , ν = µ/M
with 0 ≤ ν ≤ 1/4 (test particle case ν = 0, equal mass case ν = 1/4), p = p1 /µ, r = r12 = |x1 −x2 |,
pr = (n · p), q = (x1 − x2 )/GM , and n = n12 = q/|q|, in the center-of-energy frame, p1 + p2 = 0,
the Hamiltonian
Ĥ(t) = ĤN + Ĥ1pN + Ĥ2pN + Ĥ3pN + Ĥ4pN + ...
+ Ĥ2.5pN (t) + Ĥ3.5pN (t) + ... , (2.4.1)
where the 2.5pN and 3.5pN Hamiltonians are non-conservative (dissipative) ones, [35–38], contains the
following conservative terms, [16, 19, 39, 40],
p2 1
ĤN = − , (2.4.2)
2 q
1 1 1 1
Ĥ1pN = (3ν − 1)p4 − [(3 + ν)p2 + νp2r ] + 2 , (2.4.3)
8 2 q 2q
Section 2.4 Post-Newtonian binary Hamiltonians 45
1
Ĥ2pN = (1 − 5ν + 5ν 2 )p6
16
1 1
+ [(5 − 20ν − 3ν 2 )p4 − 2ν 2 p2r p2 − 3ν 2 p4r ]
8 q
1 1 1 1
+ [(5 + 8ν)p2 + 3νp2r ] 2 − (1 + 3ν) 3 , (2.4.4)
2 q 4 q
1
Ĥ3pN = (−5 + 35ν − 70ν 2 + 35ν 3 )p8
128
1
+ [(−7 + 42ν − 53ν 2 − 5ν 3 )p6 + (2 − 3ν)ν 2 p2r p4
16
1
+ 3(1 − ν)ν 2 p4r p2 − 5ν 3 p6r ]
q
1 1
+ [ (−27 + 136ν + 109ν 2 )p4 + (17 + 30ν)νp2r p2
16 16
1 1
+ (5 + 43ν)νp4r ] 2
12 q

25 1 2 335 23 2 2
+ − + π − ν− ν p
8 64 48 8

85 3 2 7 1
+ − − π − ν νp2r 3
16 64 4 q

1 109 21 2 1
+ + − π ν 4, (2.4.5)
8 12 32 q

7 63 189 2 105 3 63 4 10
Ĥ4pN = − ν+ ν − ν + ν p
256 256 256 128 256
(
45 8 45 8 423 8 3 2 6 9 4 4
+ p − p ν+ p − pr p − pr p ν 2
128 16 64 32 64

1013 8 23 2 6 69 4 4 5 6 2 35 8
+ − p + pr p + pr p − pr p + pr ν 3
256 64 128 64 256
)
35 8 5 2 6 9 4 4 5 6 2 35 8 1
+ − p − pr p − pr p − pr p − pr ν 4
128 32 64 32 128 q
(
13 6 791 6 49 2 4 889 4 2 369 6
+ p + − p + pr p − pr p + pr ν
8 64 16 192 160

4857 6 545 2 4 9475 4 2 1151 6
+ p − pr p + pr p − pr ν 2
256 64 768 128
)
2335 6 1135 2 4 1649 4 2 10353 6 1
+ p + pr p − pr p + pr ν 3
256 256 768 1280 q2
(
105 4 237 4 1293 2 2 97 4 q
+ p + C41 (n,p) + p − pr p + pr ln ν
32 40 40 4 ŝ
)
2 553 4 225 2 2 381 4 1
+ C42 (n,p) ν + − p − pr p − pr ν 3
128 64 128 q3
( )
105 2 233 2 29 2 q 1
+ p + C21 (n,p) + p − pr ln ν + C22 (n,p) ν 2
32 40 6 ŝ q4
( )
1 21 q 2 1
+ − + c01 + ln ν + c02 ν , (2.4.6)
16 20 ŝ q5

1189789 18491 2 4 127 4035 2 2 2
C42 (n,p) = − + π p + − − π pr p
28800 16384 3 2048

57563 38655 2 4
+ − π pr , (2.4.7)
1920 16384

672811 158177 2 2
C22 (n,p) = − π p
19200 49152

21827 110099 2 2
+ − + π pr , (2.4.8)
3840 49152
1256 7403 2
C02 = − + π , (2.4.9)
45 3072
and the terms C41 (n,p) and C21 (n,p) have the structure
C41 (n,p) = c411 p4 + c412 p2r p2 + c413 p4r , (2.4.10)
C21 (n,p) = c211 p2 + c212 p2r . (2.4.11)
The 6 constants c411 , c412 , c413 , c211 , c212 , c01 are still unknown. Their calculations will need an in-
tricate treatment of spacelike infinity in a post-Newtonian setting. The both articles [41, 42] clearly
indicate this intricate problem.
To complete, the center-of-energy vectors through 3pN order can be found in [27]: they, of course, do
fulfil the Poincaré algebra through this order.
Finally, also the n-body Hamiltonian through linear order in Newton’s gravitational constant G,
coined first post-Minkowskian approximation, is known, [46].
Section 2.4 Post-Newtonian binary Hamiltonians 47
Radiation reaction
The dissipative 2.5pN Hamiltonian from the gravitational radiation reaction is given by, [35, 36],
ni nj d3 Q̂ij (t̂)

2
Ĥ2.5pN (t̂) = pi pj − , (2.4.12)
5 q dt̂3
where Q̂ij (t̂) = ν(q 0i q 0j − δij q 02 /3) holds. The time derivatives (t̂ = t/GM ) in the Hamiltonian are
allowed to be eliminated using the equations of motion. Only after the performance of the phase-space
derivatives, the primed variables are allowed to be identified with the unprimed ones. The resulting
reaction force can be found in [43, 44] (in [43], there is a factor 1/2 misprinted in the definition of the
center-of-energy reaction force and in the 2pN expressions the factor 7m1 m2 has to be replaced through
5m1 m2 , [39, 45]). In the section “Radiation damping” below the orbital period change resulting from
the 2.5 Hamiltonian dynamics is presented.
2.4.2 Spinning binaries

Often in the literature, the spin is counted of 0.5pN order using typical black hole dimensions, S ∼
m v r ∼ m c Gm/c2 ∼ m2 G/c. This ordering will also be applied in the present contribution.
The leading order spin-orbit Hamiltonian reads, e.g. [47, 48],

1.5pN G XX 1 3mb
HSO = 2 2
(S a × n ab ) · pa − 2p b . (2.4.13)
c a rab 2ma
b6=a
The leading order spin(1)-spin(2) Hamiltonian takes the form, e.g. [47, 48],
XX 1
HS2pN
1 S2
=G 3
[3(Sa · nab )(Sb · nab ) − (Sa · Sb )] (2.4.14)
2rab
a b6=a
and the leading order spin(1)-spin(1) dynamics is given by, going beyond linear order in spin, e.g. [47,
48], r = r12 , n = n12 ,
1
HS2pN
1 S1
=G [3(S1 · n)(S1 · n) − (S1 · S1 )] . (2.4.15)
2r3
The next-to-leading order spin-orbit Hamiltonian reads, [29],
" "
2.5pN G 5m2 p21 3(p1 · p2 )
HSO = − ((p1 × S1 ) · n) +
r2 8m31 4m21
#
3p22 3(p1 · n)(p2 · n) 3(p2 · n)2
− + +
4m1 m2 4m21 2m1 m2

(p1 · p2 ) 3(p1 · n)(p2 · n)
+ ((p2 × S1 ) · n) +
m1 m2 m1 m2
#
2(p2 · n) 3(p1 · n)
+ ((p1 × S1 ) · p2 ) −
m1 m2 4m21
"
G2 5m22

11m2
+ − ((p1 × S1 ) · n) +
c4 r 3 2 m1
#
15m2
+ ((p2 × S1 ) · n) 6m1 + + (1 ↔ 2) (2.4.16)
2
and the next-to-leading order spin(1)-spin(2) Hamiltonian is given by, [30],
G
HS3pN
1 S2
= [3((p1 × S1 ) · n)((p2 × S2 ) · n)/2
2m1 m2 r3
+ 6((p2 × S1 ) · n)((p1 × S2 ) · n)
− 15(S1 · n)(S2 · n)(p1 · n)(p2 · n)
− 3(S1 · n)(S2 · n)(p1 · p2 ) + 3(S1 · p2 )(S2 · n)(p1 · n)
+ 3(S2 · p1 )(S1 · n)(p2 · n) + 3(S1 · p1 )(S2 · n)(p2 · n)
+ 3(S2 · p2 )(S1 · n)(p1 · n) − 3(S1 · S2 )(p1 · n)(p2 · n)
+ (S1 · p1 )(S2 · p2 ) − (S1 · p2 )(S2 · p1 )/2 + (S1 · S2 )(p1 · p2 )/2]
3
+ [−((p1 × S1 ) · n)((p1 × S2 ) · n)
2m21 r3
+ (S1 · S2 )(p1 · n)2 − (S1 · n)(S2 · p1 )(p1 · n)]
3
+ [−((p2 × S2 ) · n)((p2 × S1 ) · n)
2m22 r3
+ (S1 · S2 )(p2 · n)2 − (S2 · n)(S1 · p2 )(p2 · n)]
6G2 (m1 + m2 )
+ [(S1 · S2 ) − 2(S1 · n)(S2 · n)] . (2.4.17)
c4 r4
Finally, the next-to-leading order spin(1)-spin(1) dynamics reads, [31],

G 5m2 m2 21m2
HS3pN
1 S1
= − (p1 · S1 )2 + 3 p21 S21 − (p1 · n)2 S21
r3 4m31 m1 8m31
3m2 2 15m2 3
− p1 (S1 · n)2 + (p1 · n) (S1 · n) (p1 · S1 ) − p22 S21
8m31 4m31 4m1 m2
9 1 9
+ p22 (S1 · n)2 − (p1 · p2 ) S21 − (p1 · p2 ) (S1 · n)2
4m1 m2 4m21 4m21
3 3
+ (p1 · S1 ) (p2 · S1 ) − (p1 · n) (p2 · S1 ) (S1 · n)
2m21 2m21
3 15
− (p2 · n) (p1 · S1 ) (S1 · n) + (p1 · n) (p2 · n) S21
2m21 4m21

15 2
− (p 1 · n) (p 2 · n) (S 1 · n)
4m21
G2 m2

m2 2 2 2m2 2
− 5 1 + (S 1 · n) − S 1 + 4 1 + (S1 · n) . (2.4.18)
2r4 m1 m1
Also this Hamiltonian goes beyond linear order in spin. For its derivation the Tulczyjew energy-
Section 2.5 Binary motion 49
momentum tensor was needed one step beyond pole-dipole-particle approximation. All the previous
Hamiltonians, spinless and spinning, were valid for both neutron stars and black holes with the excep-
tion of the next-to-leading order spin(1)-spin(1) one which is valid for black holes only. Its generalization
to neutron stars were achieved in [49, 50]. The next-to-leading order Hamiltonians found confimations
through [49, 51–54], see [55].
As examples, the up to the next-to-leading order spin-orbit center-of-energy vector is given by, [29,
56, 57],
p2
G1.5pN + G2.5pN 1
P P
SO SO = a 2ma (pa × Sa ) − (pa
× Sa )
a
a 8m3a

mb +xb
((pa × Sa ) · nab ) 5xraab
P P
+ a b6=a 4ma rab
− 5(pa × Sa )

1 3
× Sa ) − 12 (nab × Sa )(pb · nab )
P P
+ a b6=a rab 2
(pb

+xb
−((pa × Sa ) · nab ) xarab , (2.4.19)
and the spin(1)-spin(2) one reads, [30, 56, 57],

1 XX xa Sa
G3pN
S1 S2 = [3(Sa · nab )(Sb · nab ) − (Sa · Sb )] 3 + (Sb · nab ) 2 . (2.4.20)
2 a rab rab
b6=a
The leading order spin(1)-spin(2) expression of the spin-orbit center-of-energy vector renders zero. The
contribution to the center-of-energy vector from the spin(a)-spin(a) interaction can be found in [57].
To summarize, the following pN-Hamiltonians with spin are known fully explicitly,
1.5pN 2.5pN 3.5pN 4pN
H(t) = HSO + HSO + HSO + HSO (t)
+ HS2pN
1 S2
+ HS3pN
1 S2
+ HS4pN
1 S2
+ HS4.5PN
1 S2
(t)
+ HS2pN
1 S1
+ HS2pN
2 S2
+ HS3pN
1 S1
+ HS3pN
2 S2
. (2.4.21)
The conservative Hamiltonians not given above can be found in [18, 32, 33], the non-conservative ones in
[58, 59]. Related results for the conservative parts were achieved in [60, 61] and for the non-conservative
ones in [62, 63]. Also the test-spin Hamiltonian in the Kerr metric is known, [64].
2.5 Binary motion

It is a remarkable fact that the high pN motion of spinless binaries can be given in full-analytic form if
restricted to the conservative part of the motion. Taking the radiation reaction into account, the secular
aspects of the motion can be analytically treated too, apart from trivial numerical integrations. Of par-
ticular interest is the circular motion and its innermost stable circular orbit (ISCO) where plunge motion
happens beyond. During the inspiral phase an initial eccentricity is damped out so that the orbits are
getting more and more circular. Only through external actions, e.g. third-body collision, tight orbits can
have high eccentricities.
2.5.1 Spinless two-body systems

Circular motion
For circular motion, the energy of a binary depends on the orbital frequency ω only. Introducing the
dimensionless variable x = (GM ω/c3 )2/3 and identifying ŝ = cP/GM , where P = 2π/ω is the
orbital period, i.e. ŝ = 2πc−2 x−3/2 , the 4pN-accurate conservative binding energy, Eq. (2.4.1), can be
put into the form
c2 x

Ê4PN (x; ν) = − 1 + e1 (ν) x + e2 (ν) x2
2

448
+ e3 (ν) x3 + e4 (ν) + ν ln x x4 + O x5 ,

(2.5.1)
15
where the fractional corrections to the Newtonian energy at various pN orders read
3 1
e1 (ν) = − − ν, (2.5.2)
4 12
27 19 1 2
e2 (ν) = − + ν− ν , (2.5.3)
8 8 24

675 34445 205 2 155 2 35 3
e3 (ν) = − + − π ν− ν − ν , (2.5.4)
64 576 96 96 5184

3969 498449 3157 2
e4 (ν) = − + c1 ν + − + π ν2
128 3456 576
301 3 77 4
+ ν + ν (2.5.5)
1728 31104
with
123671 9037 2 1792 896
c1 = − + π + ln 2 + γ, (2.5.6)
5760 1536 15 15
where γ is the Euler number. The lnx-term has been calculated for the first time in [65, 66] and later
verified in [28, 67]. The terms in e4 proportional to ν 4 and ν 3 have been obtained for the first time
in [28] and were later confirmed by [68], and the term proportional to ν 3 has been obtained only quite
recently, [19]. The same holds with c1 which has been derived in [42]. A numerical value of c1 was
known before, [41], coinciding, however, only in the first five digits, 153.88, with the analytic result.
In the test-mass limit ν = 0, the exact ISCO occurs for xISCO = 1/6 corresponding to the minimum
of the function
Ê(x) 1−x
= − 1. (2.5.7)
c2 (1 − 3x)1/2
The 4pN prediction for the location of the ISCO in the test-mass limit is 0.179... which deviates by
∼7.7% from the exact value. In the equal-mass case (ν = 1/4), the corresponding values are 0.648...
(1pN), 0.265... (2pN), 0.254... (3pN), 0.236... (4pN),[19].
Eccentric orbital motion

Dynamical invariants related to our previous dynamics are easily calculated within a Hamiltonian frame-
work, [69]. Let us denote the radial action by ir (Ê,j) with Ê = Ĥ and p2 = p2r + j 2 /r2 (p =
pr er + pϕ eϕ with orthonormal basis er , eϕ in the orbital plane). Then it holds

I
1
ir (Ê,j) = dr pr , (2.5.8)
2π
where the integration is defined from minimum to minimum of the radial distance. Thus all expressions
derived hereof relate to orbits completed in this sense. From analytical mechanics it is known that the
phase of the completed orbit revolution Φ is given by
Φ ∂
= 1 + k = − ir (Ê,j) (2.5.9)
2π ∂j
and the orbital period P reads
P ∂
= ir (Ê,j) . (2.5.10)
2πGm ∂ Ê
Explicitly, up to 3pN order, we get for the periastron advance parameter k, [16, 69],

1 3 1 5 1 1
k = 1 + (7 − 2ν) + (5 − 2ν) Ê
c2 j 2 c2 4 j2 2

1 1 Ê 2
+ a1 (ν) + a2 (ν) + a3 (ν) Ê , (2.5.11)
c4 j4 j2
and for the orbital period, [16, 69],

P 1 1 1
= 1− (15 − ν)Ê
2πGm (−2Ê)3/2 c2 4
(−2Ê)3/2

1 3 3
+ (5 − 2ν) − (35 + 30ν + 3ν 2 ) Ê 2
c4 2 j 32
(−2Ê)3/2 (−2Ê)5/2

1 3
+ a2 (ν) − 3a3 (ν) + a4 (ν) Ê , (2.5.12)
c6 j3 j
where

5 77 41 2 125 7 2
a1 (ν) = + π − ν+ ν , (2.5.13)
2 2 64 3 4

105 41 2 218 45 2
a2 (ν) = + π − ν+ ν , (2.5.14)
2 64 3 6
1
a3 (ν) = (5 − 5ν + 4ν 2 ), (2.5.15)
4
5
a4 (ν) = (21 − 105ν + 15ν 2 + 5ν 3 ) . (2.5.16)
128
Through 2pN order, applications to binary pulsar systems have been worked out, particularly for the
binary pulsar PSR 1913+16, [39].
Explicit analytical orbit solutions of the conservative dynamics through 3pN order are given in [70].
Up to 2pN order, these expressions read, [39, 71],
r = ar (1 − er cosu), (2.5.17)
2π
(t − t0 ) = u − et sinu + (v − u)F + Fv sinv , (2.5.18)
P
2π
(φ − φ0 ) = v + G2v sin(2v) + G3v sin(3v) , (2.5.19)
Φ
"s #
1 + eφ u
v = 2arctan tan , (2.5.20)
1 − eφ 2
where the coefficients on the right sides depend on energy and angular momentum in the form
1 (−2Ê) (−2Ê)2
ar = 1 + (−7 + ν) + (1 + 10ν + ν 2 )
(−2Ê) 4c2 16c4
!
1
+ (−68 + 44ν) , (2.5.21)
(−2Êj 2 )
(−2Ê)
e2r = 1 + 2Êj 2 + 24 − 4ν + 5(−3 + ν)(−2 Êj 2
)
4c2
(−2Ê)2
+ 52 + 2ν + 2ν 2 − (80 − 55ν + 4ν 2 )(−2Êj 2 )
8c4
8
− (−17 + 11ν) , (2.5.22)
(−2Êj 2 )
(−2Ê)
e2t = 1 + 2Êj 2 + 2
− 8 + 8ν − (−17 + 7ν)(−2Êj 2 )
4c
(−2Ê)2
+ 8 + 4ν + 20ν 2 − (112 − 47ν + 16ν 2 )(−2Êj 2 )
8c4
4
q
− 24 (−2Êj 2 )(−5 + 2ν) − (−17 + 11ν)
(−2Êj 2 )
24
− q (5 − 2ν) , (2.5.23)
(−2Êj 2 )
(−2Ê)
e2φ = 1 + 2Êj 2 + 24 + (−15 + ν)(−2Êj 2 )
4c2
(−2Ê)2
+ − 32 + 176ν + 18ν 2 − (160 − 30ν + 3ν 2 )(−2Êj 2 )
16c4
1
+ (408 − 232ν − 15ν 2 ) , (2.5.24)
(−2Êj 2 )
3(−2Ê)2 5 − 2ν
F = , (2.5.25)
2c4
q
(−2Êj 2 )
1 (−2Ê)2
q
Fv = − (4 + ν)ν 1 + 2Êj 2 , (2.5.26)
8c4
q
(−2Êj 2 )
(−2Ê)2 1 + 2Êj 2
G2v = ν(1 − 3ν) , (2.5.27)
8c4 (−2Êj 2 )2
3(−2Ê)2 (1 + 2Êj 2 )3/2 2

G3v = − ν . (2.5.28)
32c4 (−2Êj 2 )2
The given parametrization of the orbits is based on the 1pN parametrization by [72]. The latter parametriza-
tion is different from the one in [1] but identical to the one in [73] in the test-body limit. Through 1pN
order the parametrization given above plays an important role in current observations of binary pulsar
systems, e.g. [74].
Radiation damping
The 2.5pN gravitational radiation damping of the orbital period reads, e.g. [75, 76], where also the
3.5pN damping is treated via balance equations,
96G3 µM 2

Ṗ 73 2 37 4
=− 1 + e + e , (2.5.29)
P 5c5 a4 (1 − e2 )7/2 24 96
where for the orbital elements the Newtonian approximation is sufficient, a = ar and e = er = et = eφ .
Current confirmations of this damping formula at 2.5pN order were obtained from the binary pulsar PSR
B1913+16 with uncertainty 2 × 10−3 , [74], and from the double pulsar PSR J0737-3039A/B within an
error bar of 3 × 10−4 , M. Kramer and N. Wex (personal communication). The 3.5pN Hamiltonian
dynamics has been worked out in [37, 38] but it is far too small for current observations.
2.5.2 Particle motion in Kerr geometry
The binding energy of a test-particle in the Kerr geometry can be put into the form, taking into account
linear and quadratic spin terms, each up to some pN order, of the central black hole, [77],
"
p2r C2 C2 M 3M p2r

GM m 1
E = + 2 − + 2 G − 3 −
2m 2r m r c 2r m 2rm
#
p4r C4 C2 p2r G2 M 2 m 1 2aG2 Lz M 2
− − 4 3 − 2 3 − + 3
8m3 8r m 4r m 2r2 c r3
C M
4
3C2 M p2r 5M p4r G3 M 3 m

1
+ G + + −
c4 8r5 m3 4r3 m3 8rm3 2r3
C2 M 2 3M 2 p2r C6 3C4 p2r p6r

+ G2 − 4 + 2
+ 6 5
+ 4 5
+
4r m 4r m 16r m 16r m 16m5
3C2 p4r L2z M 2 M 2 p2r

2 2 1
+ + a G − + + O , (2.5.30)
16r2 m5 2r4 m 2r2 m c6
with the Carter-like constant, C,
C2 = L2 + α2 a2 cos2 θ with
2
2E GM m 1
α2 = − +O . (2.5.31)
mc2 c c6
where the dimensionless spin-parameter a = Sc/GM 2 has the range a ∈ [−1,1]. The canonical
variables are (pr , L, Lz ; r, v, Ω), where r and pr denote the radial coordinate and its canonical conjugate
momentum, respectively, v being the true anomaly, i.e., the angle between the position vector r = rn
and the direction of the ascending node N = (ez × L) /|ez × L|: cos v = n · N and sin v = n · W with
W = L × N/L. Ω is the angle of the ascending node as measured from the x-axis of the non-rotating
orthonormal basis (ex ,ey ,ez ): cos Ω = N · ex and sin Ω = N · ey , measuring the precession of the
orbital plane about Lz which is tilted from the equatorial plane with the inclination angle i, the angle
between the background z-axis and the orbital angular momentum vector L, so Lz = L cos i.
The canonical Poisson-bracket relations read
{r,pr } = {v,L} = {Ω,Lz } = 1 (2.5.32)
with all other brackets being zero.

The derivation of observables is straighforward with the knowledge of the action of a completed
revolution, from periastron to periastron,
S = S(E,Lz ,C, P,Φ, U )

I I
= −Eb P + Lz Φ + pr dr + Ldv . (2.5.33)
The result depends on the three constants of motion, E,Lz ,C, and three orbit-completed variables
I I I
P = dt , Φ = dΩ , U = dv , (2.5.34)
P being the orbital period, Φ the precession of the orbital plane per revolution, and U the intrinsic
periastron advance. The relation between the variable v and the space-fixed Boyer-Lindquist coordinate
θ reads,
sin v sin i = cos θ . (2.5.35)
Solving Eq. (2.5.30) for pr and Eq. (2.5.31) for L allows the calculation of the action, Eq. (2.5.33).
Within leading order calculations, the variable L in the expression for cos i can be replaced by C. The
next step is to calculate the remaining integrals in the action (2.5.33), resulting in
1 3G2 M 2 m2
I
1 GM m
pr dr = −C + q + 2
2π − 2E c C
m
r
1 2aG3 Lz M 3 m3

15GM m −E
+ −
4 2m c3 C3
"
1 15EG2 M 2 m4 35G4 M 4 m4
+ +
c4 2C 4C3
r
M 2m L2 M 2 m

35EGM Eb
+ − + a2 EG2 − + z 3
32 2m 2C 2C
#
G4 M 4 m4 3G4 L2z M 4 m4
− +
4C3 4C5
12EG3 M 3 Lz m2 21G5 M 5 Lz m5

a
− + (2.5.36)
c5 C 3 C5
and
I I p
Ldv = C2 − α2 a2 sin2 i sin2 v dv
L2

π 2 2
= CU − a α 1 − z2 + O(a2 /c6 ) , (2.5.37)
2C C
where integration over a closed Newtonian orbit has been performed for the α2 -term.
The action principle tells that
∂S ∂S ∂S
= = = 0. (2.5.38)
∂C ∂Lz ∂E
Hereof the two periastron shifts, ∆Ū and ∆Φ̄, the intrinsic and the one related with the precession of the
orbital plane, respectively, result in the form
1 3G2 M 2 m2 6aG3 M 3 m3
∆Ū = (U − 2π) = − cos i
2π c C
2 2 c3 C3
"
1 15EG2 M 2 m 105G4 M 4 m4
+ 4 2
+
c 2C 4C4
#
3G4 M 4 m4
a2 2
+ 5 cos i − 1
4C4
aG3 M 3 m2 G2 M 2 m3

− 36E + 105 cos i (2.5.39)
c5 C3 C2
and
2aG3 M 3 m3 a2 3G4 M 4 m4

1
∆Φ̄ = Φ= + cos i −
2π c3 C3 c4 2C4
12EG3 M 3 m2 21G5 M 5 m5

a
+ + . (2.5.40)
c5 C3 C5
These periastron advances agree with corresponding results in [39, 47] in the test-mass limit, apart from
the higher pN-order term linear in a which does not appear therein. For the orbital period we get

2πGM 1 15 E
P = 3/2
1− 2
(−2E/m) c 4 m
3/2 !
105 E 2

1 15 GM m −2E
+ −
c4 2 C m 32 m2
3/2
12aG2 M 2 m2

−2E
− cos i . (2.5.41)
c5 C2 m
In the case of equatorial motion the resulting shift, say ∆Φ̃, is given by
I
1 ∂ ∂
∆Φ̃ = ∆Ū + ∆Φ̄ = −1 − + pr dr
2π ∂C ∂Lz
3G2 M 2 m2 4aG3 M 3 m3
= −
c C
2 2 c3 C3
1 15EG2 M 2 m 105G4 M 4 m4 3G4 M 4 m4

+ 4 2
+ 4
+ a2 4
c 2C 4C 2C
24EG3 M 3 m2 84G5 M 5 m5

a
+ − − . (2.5.42)
c5 C3 C5
This quantity can also be deduced from [78] apart from the higher pN-order term linear in a which has
not been treated therein. As C deviates from L only at the order a2 /c4 , in the obtained results, C may
always be replaced with L.
2.5.3 Two-body systems with spinning components

The motion of spins and precession of orbits of spinning binaries are important indicators for the grav-
itational interaction. From an observational point of view, only the leading pN-order interactions are
important, so in the following we will restrict ourselves to those interactions, particularly to the leading
order spin-orbit and spin(1)-spin(2) interactions. Also, we shall restrict ourselves to the center-of-energy
systems.
Introducing the Runge-Lenz-Laplace-Lagrange vector (often only called Laplace-Runge-Lenz vector
or even only Runge-Lenz vector as coined by Pauli in the quantum mechanics; but Lagrange introduced
it first, cf. [39, 79]),
GM µ2
A≡p×L− r, (2.5.43)
r
the time-averaged or secular precession of the binary orbit takes the form
dL SO
< >t = ΩSO × L , (2.5.44)
dt
dA SO
< >t = ΩSO × A , (2.5.45)
dt
where L = r × p denotes the orbital angular momentum and < ... >t orbital averaging over time. As
above, M = M1 + M2 is the total mass and µ the reduced one; only the single masses are now put with
capital letters. The precessional frequency vector reads
2G 1 (L · Seff )L
ΩSO = 2
S eff − 3 , (2.5.46)
c a3 (1 − e2 )3/2 L2
3 M2 M1
Seff = S+ S1 + S2 , (2.5.47)
4 M1 M2
where for the orbital elements, in Newtonian approximation, a = ar and e = er = et = eφ hold. This
is the famous Lense-Thirring effect. With the satellites LAGEOS(1&2) the predicted effect of 31 mas/yr
has been measured with an accuracy of 1 × 10−1 , [80].
The Schiff effect, also called Lense-Thirring effect for spin or frame-dragging effect, is given by
dS S1 S2
1
= ΩS1 S2 × S1 , (2.5.48)
dt
where
G 3(r · S2 )r
ΩS1 S2 = − S2 . (2.5.49)
c2 r3 r2
The space mission GP-B succeeded in measuring the predicted effect of 39.2 mas/yr with an accuracy
of 2 × 10−1 , [81].
A further effect is the de Sitter effect, also called Fokker effect or geodetic precession, which results
from the spin(1)-spin(2) coupling in the form
dS SO
1
= ΩsSO × S1 , (2.5.50)
dt
2G 3M2
ΩsSO = 2 3
1+ L. (2.5.51)
c r 4M1
The precession of the “spin” (orbital angular momentum) of the Earth-Moon system in the gravitational
field of the Sun has the value 19 mas/yr and could be verified with precision of 2 × 10−2 , [82]. The
precession rate in the GP-B mission of 6,606 mas/yr could be seen with a precision of 3 × 10−3 by [81].
Also in case of binary pulsar systems, the spin precession has been observed, e. g. [83].
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3 Covariant Theory of the Post-Newtonian Equa-
tions of Motion of Extended Bodies
Yi Xie1,2 and Sergei Kopeikin3
1
Department of Astronomy, Nanjing University, 22 Hankou Road,
Nanjing Jiangsu 210093, China
2
Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University,
Nanjing Jiangsu 210093, China
3
Department of Physics & Astronomy, University of Missouri, 322 Physics Bldg., Columbia,
Missouri 65211, USA
3.1 Introduction
Theoretical formulation of relativistic equations of motion of massive particles and extended bodies have
been an important research problem since the discovery of general theory of relativity by Albert Einstein.
It continues to be of paramount importance for further theoretical development of general theory of
relativity and its experimental testing in the solar system [93, 154] and in the binary pulsars [37, 40, 106,
151]. Rapidly growing new branch of relativistic astrophysics – gravitational wave astronomy – urgently
demands significantly better understanding of the theoretical principles underlying the derivation of
relativistic equations of motion of compact and extended bodies comprising the astrophysical N -body
system [4, 10, 36, 134].
During the last three decades the most theoretical efforts in solving the problem of motion in general
relativity have been focused on the astronomical systems consisting of point-like particles with the goal
to advance the analytic derivation of the higher-order post-Newtonian corrections beyond the famous
quadrupole formula of Landau and Lifshitz [107] describing the loss of energy and angular momentum
from the system due to the emission of gravitational waves. Theoretical difficulties in solving this
problem are mainly caused by the non-linear nature of the gravitational field leading to its self-interaction
and appearance of divergent integrals which have to be regularized [13, 15, 16]. Additional problem is
to take into account the internal structure of the bodies comprising the astronomical N -body system
because the point-particle limit is not sufficient to correctly predict the templates of gravitational waves
emitted by coalescing binaries consisting of neutron stars or black holes – the multipole moments of
the moving bodies – spin, quadrupole, etc. – are a matter of importance and are to be taken into account
especially at the very last stage of the coalescing process [14, 145, 159].
Calculations of equations of motion of binary systems for gravitational wave astronomy are primarily
pursued either in the Arnovitt-Deser-Misner (ADM) or in harmonic coordinates [84, 114]. Hence, the
resulting equations of motion are not covariant that requires careful consideration not only the process of
generation of gravitational waves but also their propagation and eventual detection by gravitational wave
observatories. Under assumption that background spacetime of a binary system emitting gravitational
waves is asymptotically flat the gauge invariant quantities are orbital period and periastron advance of
the binary system that can be expressed in terms of the energy and angular momentum of the system
representing the integrals of motion in non-radiative post-Newtonian approximations [41, 45, 76]. These
quantities are adiabatic invariants under a secular change of the orbital parameters of the binary system
64 Chapter 3 Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies
caused by the emission of gravitational waves.

In this chapter we do not discuss the radiative celestial mechanics of astronomical N -body systems
and restrict ourselves with a conservative, post-Newtonian approximation of the first order but with a
full account of internal structure of the bodies. The goal is to build the covariant post-Newtonian theory
of motion of extended bodies and to find out relativistic corrections to the point-like particle limit which
accounts for all multipole moments characterizing the interior structure of the extended bodies. This
programmatic task was put forward by Mathisson [111, 112, 133] and originally explored by Fock [73],
Papapetrou [123–125], Tulczyjew [149]. Significant progress in advancing solution of this problem was
achieved by Dixon [57, 58, 60, 62] who proposed to derive an exact covariant equations of motion of
extended bodies from microscopic conservation law 1
∇α T αβ = 0 , (3.1.1)
where ∇α denotes a covariant derivative and T aβ is the stress-energy-momentum tensor of matter, by

making use of a novel method of integration of the linear connection in general relativity (Christoffel
symbols). The generic mathematical techniques used by Dixon to achieve this goal were the formal-
ism of two-point world function σ of Synge[144] and its derivatives (called sometimes bi-tensors), the
horizontal and vertical covariant derivatives defined on the tangent bundle of spacetime manifold (also
known as Ehresmann’s connection [91]) ) and the distributional theory of multipole moments.
An extended body in Dixon’s approach is idealized as a time-like world tube filled with matter which
stress-energy-momentum tensor T αβ vanishes outside the tube. By making use of the bi-tensor prop-
agators, K α µ0 and H α µ0 , composed out of the second-order derivatives of the world function, Dixon
defined the total linear momentum pα and spin S αβ of the extended body by integrals over a spacelike
hypersurface Σ, [62, Eqs. 66, 67]
Z
0 0p
pα (z,Σ) ≡ K α µ0 T µ ν −g(x0 )dΣν 0 , (3.1.2)
Σ
Z
0 0
X [α H β] µ0 T µ
αβ ν
p
S (z,Σ) ≡ −2 −g(x0 )dΣν 0 , (3.1.3)
Σ
where the primed indices belong to the point of integration x0 on Σ, z = z α (s) is a reference world line
Z of a representative point that is associated with the center of mass of the body, s is a parameter on
this world line, X α is a partial derivative of the world function that can be viewed as a vector in tangent
space attached to each point z of the world line Z. The cross section Σ = Σ(s) of the world tube of the
body consists of all spacelike geodesics passing through z(s) orthogonal to the dynamical four-velocity,
nα of z(s).
Definition of mass M = M (z,Σ), and the mass dipole moment mα = mα (z,Σ), of the body are
given by
pα ≡ M nα , (3.1.4)
α αβ
m ≡ S nβ , (3.1.5)
where nα is the dynamical velocity of the body moving along the world line Z related to its kinematic
velocity uα = dz α /ds by nα uα = −1, and the normalization condition is uα uα = −1. The world
line z = z(s) of the center of mass of the body is, then, defined by chosing mα = 0. Due to (3.1.4) it is
1 Greek indices take values from 0 to 3 and numerate spacetime coordinates. Roman indices take values 1,2,3 and
numerate spatial coordinates only. Repeated indices indicate application of Einstein’s summation rule.
Section 3.1 Introduction 65
equivalent to
S αβ pβ = 0 , (3.1.6)
which is known as Dixon’s supplementary condition [62, Eq. 81].
High-order multipoles of the body are defined by means of 2l -pole moment tensor [62, Eqs. 140–
145], Z
I α1 ...αl µν (z) = X α1 ...X αl T̂ µν (z,X)DX , (3.1.7)
where X α is the same vector

p as in (3.1.3), T̂ µν (z,X) is a skeleton of the stress-energy-momentum tensor
µν 4
T of the body, DX ≡ −g(z)d X is the invariant volume of integration on the tangent spacetime
at point z on Z, and the integration is performed over the tangent spacetime. According to Dixon, the
skeleton T̂ µν (z,X) is a distribution (called a generalized function in Russian [136]) defined on tangent
spacetime in such a way that it contains only information about the body but is entirely independent of
the geometry of the surrounding spacetime which contains the bodies. Definition (3.1.7) assumes that
I α1 ...αl µν = I (α1 ...αl )(µν) , (3.1.8)
where the round parentheses around the tensor indices denote a full symmetrization. Furthermore, the
stress-energy conservation law imposes one more constrain [62]
I (α1 ...αl µ)ν = 0 . (3.1.9)
The 2l -pole moments are coupled to the Riemann tensor Rα µβν characterizing the curvature of space-
time. Therefore, they can be replaced with a more suitable set of reduced moments J α1 ...αl λµνρ which
are defined by the following formulas
J α1 ...αl λµνρ = I (α1 ...αl )[λ[νµ]ρ] , (l ≥ 0) (3.1.10a)

α1 ...αl λ[µνρ]
J = 0, (l ≥ 0) (3.1.10b)
J α1 ...[αl λµ]νρ = 0, (l ≥ 1) (3.1.10c)
where the square parentheses around the tensor indices denote a full anti-symmetrization, and the nested
square brackets in (3.1.10a) denote anti-simmetrization on pairs of indices λ, µ and ν, ρ independently.
Inverting (3.1.10) yields
4(l − 1) (α1 ...αl−1 /µ/αl )ν

I α1 ...αl µν = J , (3.1.11)
l+1
where the forward slashes embracing the index µ mean that it is excluded from the symmetrization
denoted with the round parentheses. The moments satisfy the orthogonality condition
J α1 ...αl λµνρ pα1 = 0 , (3.1.12)
being valid for any index from the set α1 ,α2 ,...αl due to the property of symmetry (3.1.7).
Dixon presented a number of arguments suggesting that the covariant equations of motion of the
extended body must have the following exact form [59, Eqs. 4.9, 4.10]
∞
dpα 1 β µν 1X1
= v S R̄αβµν − ∇(α Aβ1 ...βl )µν I β1 ...βl µν (3.1.13)
ds 2 2 l!
l=2
∞
dS αβ X 1
= 2p[α v β] − Bγ ...γ µν [α I β]γ1 ...γl µν , (3.1.14)
ds l! 1 l
l=1
where d/ds ≡ v α ∇α is the (horizontal) covariant derivative along the reference line z = z(s),
Aβ1 ...βl µν and Bβ1 ...βl µνσ are the l-fold symmetrized tensors composed from the covariant derivatives
of the metric tensor and the world function at point x taken, then, in the limit x → z and the bar above
the Riemann tensor indicates that it is calculated on the background spacetime manifold from which
the body under consideration is excluded. These tensors are expressed in terms of the Riemann tensor,
R̄α µβν , of the background spacetime and its covariant derivatives. We follow [147] and call tensors,
Aβ1 ...βl µν and Bβ1 ...βl µνσ , the external multipole moments of the background spacetime. The multi-
pole moments characterizing the interior structure and gravitational field of the body itself are called the
internal multipole moments. Analytic calculations of tensors Aβ1 ...βl µν and Bβ1 ...βl µνσ in the frame-
work of Dixon’s theory are lengthy and were never performed by Dixon or anybody else except the
quadrupole approximation in which equations of motion (3.1.13), (3.1.14) read [62, Eqs. 171, 172], [59,
Eqs. 4.11, 4.12]
dpα 1 β µν α 1
= v S R̄ βµν + ∇α R̄σρµν J σρµν + ... , (3.1.15)
ds 2 6
dS αβ 4
= 2p[α v β] + J µνσ[α R̄µνσ β] + ... , (3.1.16)
ds 3
where the ellipsis indicate the presence of higher-order multipole terms.
The mathematical elegance and the apparently covariant nature of the Dixon theory of equations
of motion is striking and attracted a peer attention of a number of experts in general relativity after
it was proposed [6, 7, 66, 67, 135]. However, it was soon realized [147] that there are several issues
which make the theory look rather abstract and not directly suitable for astrophysical applications or in
relativistic celestial mechanics of the solar system. It reduced the number of publications pursuing a
la Dixon’s study of equations of motion. The main problem is that Dixon’s theory does not deal with
the gravitational field equations. It tacitly assumes that they are solved and the metric tensor is known.
However, the Einstein equations and the equations of motion of the bodies creating the field are bounded
in a closed tie – matter generates gravity while gravity governs the motion of matter. Dixon’s definitions
of the multipole moments of the bodies and other auxiliary geometric structures depend on the metric
tensor which, in its own turn and in accordance with Einstein’s equations, depends on the multipole
moments and those structures. Their general-relativistic interaction is non-linear which assumes that
the field equations cannot be mapped to the linear multipolar structure of equations of motion (3.1.13),
(3.1.14) that seems are missing the non-linear contributions of the body’s moments. This is indeed
the case as we shall see later on in section 3.8. Moreover, the Dixon’s equations are incomplete even
at a linear level as they include neither the spin-type (internal) multipole moments of extended bodies
besides their spins nor the time derivatives of the internal multipoles [see equations (3.8.47)–(3.8.59)
in this chapter]. Because of these drawbacks the Dixon theory is imperfect and can hardly be called
“standard” as some authors have been recently claiming [9].
Dixon’s approach is missing several critical ingredients which has to be added in order to make the
theory complete and capable to determine the motion of extended bodies in practical situations. More
precisely, the following additional ingredients of the theory are required:
1. the procedure of finding the metric tensor from the field equations;
2. the procedure of selection of the center-of-mass world line Z within each body;
3. the momentum-velocity relation;
4. the procedure of unambiguous characterization and determination of the gravitational self-field and
self-force inside the body;
5. the precise algorithm for calculation the body’s internal multipoles (3.1.7) as the stress-energy-
momentum tensor’s skeleton, T̂ αβ , introduced by Dixon, is lacking this algorithm;
6. the precise algorithm for calculation the external multipoles Aβ1 ...βl µν and Bβ1 ...βl µνσ of the
background spacetime manifold in the right side of (3.1.13) and (3.1.14) in terms of the internal
multipoles of the bodies.
The goal of this chapter is to discuss the missing elements of Dixon’s approach and to present a com-
prehensive covariant theory of post-Newtonian equations of motion. Herein, we focus on translational
equations of the bodies. Covariant rotational equations of motion will be considered somewhere else.
We shall step beyond general theory of relativity and work with the field equations of a scalar-tensor
theory of gravity. Therefore, in addition to the metric tensor, a scalar field will be also a carrier of
a long-range gravitational interaction. This introduces some additional complications compared with
Dixon’s general-relativistic theory of equations of motion. In particular, instead of two sets of general-
relativistic multipole moments we shall have to define additional set associated with the scalar field.
The introduction of the scalar field has certain advantages for experimental tests of general relativity
as it allows us to parametrize the field equations and equations of motion with two parameters, β and
γ, of the parametrized post-Newtonain (PPN) theory which is a covariant generalization of a purely
phenomenological parametrization of general-relativistic metric of the “canonical” Will-Nordtvedt PPN
formalism [153]. We have started development of this theory in [93, 94, 158]. Here, we present new
results completing Dixon’s program of derivation of translational equations of motion with accounting
for all internal and external multipoles of extended bodies in N -body system. We also give a covariant
form of the equations of motion which generalize Dixon’s equations (3.1.13), (3.1.14).
The brief content of our study is as follows. In next section 3.2 we discuss a scalar-tensor theory of
gravity for post-Newtonian celestial mechanics of N -body system. Parametrization of the field equa-
tions, small parameters, and post-Newtonian approximations and gauges are introduced in section 3.3.
Parametrized post-Newtonian coordinate charts covering the spacetime manifold globally and in a local
neighbourhood of each body are set up in section 3.4. They make up an atlas of spacetime manifold.
The coordinates in relativity are characterized in terms of the metric tensor and its parameters - multi-
pole moments which are also explained in section 3.4. The differential structure of spacetime manifold
presupposes that the parametric descriptions of the metric tensor given in different coordinates must
smoothly match in the regions where the coordinate charts overlap. The matching procedure is de-
scribed in section 3.5 and gives relations between the multipole moments and the metric functionals
from the distribution of matter density, its current, pressure, etc. It also establishes parametrization of
the world line W of the origin of each local coordinates with respect to the global coordinate chart and
defines the effective background metric, ḡαβ , for each extended body that is used later on for derivation
of covariant equations of motion. The local coordinate chart introduced around each extended body
is used for detailed description of the body’s gravitational field and for definition of the body’s mass
and the center of mass in section 3.6. It is also used to derive the local equations of motion of the
body’s center of mass with respect to the origin of the local coordinates. Because the world line W of
the origin of the local coordinates has been derived by the method of asymptotic matching in section
3.5, the equations of motion of the body follow immediately after substitution of the local equations
of motion to the parametric description of the world line W. This calculation is performed in section
3.7 which derives equations of translational motion (3.7.20) of each body from N -body astronomical
system in terms of their internal multipoles as well as global coordinates and velocities of their centers
of mass. Finally, section 3.8 establishes covariant form (3.8.47) of the translational equations of motion
of extended bodies which completes the Dixon’s program.
3.2 A Theory of Gravity for Post-Newtonian Celestial

Mechanics
We consider an isolated N -body system comprised of N extended bodies with non-singular interior
described by the stress-energy-momentum tensor T αβ . The bodies have a localized matter support and
are supposed to be well-isolated one from another in space. Accretion and other fluxes of matter outside
of the bodies are absent or neglected.
Post-Newtonian celestial mechanics describes orbital and rotational motions of the bodies on a curved
spacetime manifold described by the metric tensor, gαβ obtained as a solution of the field equations of
a metric-based theory of gravitation in the slow-motion and weak-gravitational field approximation.
Class of the viable metric theories of gravity, which can be employed for developing relativistic celestial
mechanics, ranges from general theory of relativity [21, 107] to a scalar-vector-tensor theory of gravity
recently proposed by [8] for description the motion of galaxies at cosmological scale. It is inconceivable
to review all these theories in the present chapter and we refer the reader to [155] for further details
on those alternative theories. As for our goal, we shall build a parametrized post-Newtonian theory of
celestial mechanics in the framework of a scalar-tensor theory of gravity introduced by Jordan and Fierz
[71, 86, 87] and re-discovered later by Brans and Dicke [17, 52, 53]. The Jordan-Fierz-Brans-Dicke
(JFBD) theory extends the Lagrangian of general relativity by introducing a long range, non-linear scalar
field (or fields [38]) minimally coupled to gravity. The presence of the scalar field causes deviation of
the metric-based gravity theory from pure geometry. The scalar field highlights the geometric role of the
metric tensor and makes the physical content of the gravitational theory richer. Recent discovery of the
scalar Higgs boson at LHC [55] and its possible connection to the JFBD scalar field in gravitation and
cosmology [50] reinforces the theoretical basis and value of the scalar-tensor theory in astrophysics and
relativistic celestial mechanics.
3.2.1 The Field Equations

Gravitational field in the scalar-tensor theory of gravity is described by the metric tensor gαβ and a long-
range scalar field φ with a self-interaction described by means of a coupling function θ(φ). The field
equations in the scalar-tensor theory are derived from the action [153]
c3 φ,α φ,α √
Z
16π
S= φR − θ(φ) − 4 L(gµν ,Ψ) −g d4 x, (3.2.1)
16π φ c
where the first, second and third terms in the integrand of (3.2.1) are the Lagrangian densities of the
gravitational field, scalar field and matter of the N -body system respectively, g = det[gαβ ] < 0 is the
determinant of the metric tensor gαβ , R = g αβ Rαβ is the Ricci scalar, Rαβ is the Ricci tensor, Ψ
indicates the set of matter variables, and θ(φ) is the coupling function of the scalar field which is kept
unspecified for the purpose of further parametrization of the possible deviation from general relativity.
The action (3.2.1) is written in the Jordan-Fierz frame in which the metric tensor gαβ has a standard
physical meaning of observable quantity in definition of the proper time and proper length [153].
For the sake of simplicity we postulate that the potential V (φ) of the scalar field φ is nil so that
the scalar field propagates freely. Discarding the potential V (φ) is justified in the first post-Newtonian
approximation as it does not yield any measurable relativistic effects within the boundaries of the solar
system [155]. However, if the potential is non-linear it can be important in strong gravitational fields of
neutron stars and black holes, and its inclusion to the theory leads to interesting physical consequences
[38, 39]. We shall not analyse in this chapter the non-linear effects of the scalar field φ.
Field equations for the metric tensor are obtained by variation of action (3.2.1) with respect to gαβ . It
Section 3.2 A Theory of Gravity for Post-Newtonian Celestial Mechanics 69
yields [153]

8π 1 φ,µ φ,ν 1 1
Rµν = Tµν − gµν T + θ(φ) + φ ;µν + g µν g φ , (3.2.2)
φc2 2 φ2 φ 2
where
∂2 ∂
g ≡ g µν − g µν Γα
µν (3.2.3)
∂xµ ∂xν ∂xα
is the differential Laplace-Beltrami operator [69, 115], and Tµν is the stress-energy-momentum tensor
of matter comprising the N -body system defined by the variational derivative from the Lagrangian L of
matter [107] √ √
c2 √ ∂( −gL) ∂ ∂( −gL)
−g Tµν ≡ − . (3.2.4)
2 ∂g µν ∂xα ∂g µν,α
Equation for the scalar field φ is obtained by variation of action (3.2.1) with respect to φ. After making
use of the contracted form of (3.2.2) it yields [153]

1 8π dθ
g φ = T − φ,α φ,α , (3.2.5)
3 + 2θ(φ) c 2 dφ
where T = g αβ Tαβ is the trace of the stress-energy-momentum tensor of matter which serves as a
source of the scalar field. The last term in (3.2.5) shows that θ(φ) is responsible for self-interaction of
the field φ.
3.2.2 The Energy-Momentum Tensor
Gravitational field and matter are tightly connected via the Bianchi identity of the field equations for the
metric tensor [107, 115]. The Bianchi identity makes four out of ten components of the metric tensor
fully independent so that they can be chosen arbitrary. This freedom is usually fixed by picking up a
specific gauge condition, which imposes four restrictions on four components of the metric tensor and/or
its first derivatives but no restriction on the scalar field. On the other hand, the Bianchi identity imposes
four differential constraints (3.1.1) on the stress-energy-momentum tensor of matter which constitutes
microscopic equations of motion of matter [107].
We have assumed that the astronomical N -body system is isolated, which means that we neglect any
influence from the gravitational environment outside the system and ignore cosmological effects. This
makes the spacetime asymptotically-flat so that the center of mass of the system can be set at rest with
respect to an inertial coordinate chart at infinity. We postulate that the matter of the system is described
by the stress energy tensor
T αβ = ρ 1 + c−2 Π uα uβ + c−2 π αβ ,

(3.2.6)
where ρ and Π are the density and the specific internal energy of matter, uα = dxα /cdτ is 4-velocity
of the matter with τ being the proper time along the world line of matter’s volume element, and π αβ is
a symmetric tensor of spatial stresses being orthogonal to the 4-velocity of matter
uα παβ = 0. (3.2.7)
Equation (3.2.7) means that the stress tensor has only spatial components in the frame co-moving with
matter.
Due to the Bianchi identity the energy-momentum tensor is conserved, that is satisfies the microscopic
equation of motion (3.1.1) or

T αβ ;β = 0 , (3.2.8)
where here and everywhere else the semicolon denotes a covariant derivative on the spacetime manifold
with respect to the Christoffel symbols calculated from the metric gαβ . The conservation of the energy-
momentum tensor leads to the equation of continuity
1 √
(ρuα );α = √ (ρ −guα ),α = 0 , (3.2.9)
−g
and to the second law of thermodynamics that is expressed as a differential relation between the specific
internal energy and the tensor of stresses
ρuα Π,α + π αβ uα;β = 0 . (3.2.10)
These equations will be employed later for solving the field equations and for derivation of the equations
of motion of the bodies.
3.3 Parametrized Post-Newtonian Celestial Mechanics

Post-newtonian celestial mechanics is an extension of Newtonian mechanics on a curved spacetime
manifold [19, 21]. Mathematical description of the manifold requires a particular theory of gravity
going beyond the Newtonian theory. General-relativistic celestial mechanics operates with Einstein’s
general relativity having a minimal number of free fundamental parameters characterizing geometry of
spacetime. This chapter deals with JFBD scalar-tensor theory of gravity which can be considered as a
covariant extension of general relativity parametrizing possible deviations from a pure geometric the-
ory of spacetime. The basic principles of the parametrized post-Newtonian (PPN) celestial mechanics
remains the same as in general relativity and in the Newtonian gravitational theory though the mathe-
matical derivation of equations of motion of extended bodies become more complicated. PPN formalism
described in [153] contains more parameters than the scalar-tensor theory. However, it is not covariant
(though Lorentz-invariant) and is limited to consideration of gravitational field of massive point-like
particles only. We do not apply it anywhere in this chapter to avoid non-covariant conclusions which
can easily become misleading and/or controversial. We explain the basic principle of PPN celestial
mechanics in this section. More details are given in textbook [93].
3.3.1 External and Internal Problems of Motion

Celestial mechanics subdivides equations of motion of N -body system in three classes corresponding
to the number of degrees of freedom of matter of the bodies [73]. The main degrees of freedom are
characterized by motion of linear and angular momenta of the bodies. Corresponding equations are:
1. translational equations of motion consisting of
• the equation of motion of the total linear momentum of the entire system,
• the equations of motion of the linear momentum of each body,
2. rotational equations of motion consisting of
• the equation of motion of the total angular momentum of the entire system;
• the equations of motion of the angular momentum (spin) of each body.
Section 3.3 Parametrized Post-Newtonian Celestial Mechanics 71
3. equations of temporal evolution of multipole moments of the system as a whole and each body
separately.
The translational and rotational equations are typically refer to the translational motion of rigidly rotating
bodies. If the bodies have time-dependent internal structure, the equations of motion of higher-order
internal multipoles must be included.
Derivation of the translational and rotational equations of motion naturally suggest separation of the
problem of motion in two parts: external and internal. The external problem refers to the derivation of
the translational equations, and the internal problem deals with the derivation of the rotational equations
of motion as well as equations of motion of the body’s internal multipoles. Though, the final form of
the equations of motion must be covariant and is not expected to depend on the choice of a particular
(privileged) coordinate chart on spacetime manifold, the coordinate approach is the most effective for
unambiguous separation of the internal and external degrees of motion and for building the background
effective manifold with metric, ḡαβ , admitting the covariant formulation of the equations of motion for
the linear momentum, spin and other internal multipoles.
The absence of a privileged coordinate chart is ensured by the covariant nature of the field equations
of the scalar-tensor theory of gravity used in this chapter. In particular, the theory is Lorentz-invariant
which means that in case of an isolated astronomical system embedded to asymptotically-flat spacetime
we can always introduce a global coordinate chart with the origin located at the center of mass of the
system. In what follows, we shall systematically neglect the cosmological effects in the motion of the
bodies and assume that (1) the metric tensor, gαβ , in the global coordinates approaches the Minkowski
metric, ηαβ , at infinity, and (2) the global coordinates smoothly match the inertial coordinates of the
Minkowski spacetime at infinity. However, the global coordinate chart is not sufficient for solving the
problem of motion of extended bodies as it is not well suited for description of the internal structure of
each body from the astronomical N -body system.
Indeed, the motion of matter is naturally split in two components – the orbital motion of the center
of mass of each body and the internal motion of matter with respect to the body’s center of mass. The
global chart is adequate for describing the orbital dynamics. On the other hand, description of the inter-
nal motion of matter demands introduction of a local coordinate chart attached to each gravitating body
as it excludes spurious effects (like Lorentz contraction, etc.) which have no relation to internal struc-
ture of the body [97]. Construction of the local coordinates should be reconciled with the principle of
equivalence as long as it is allowed by the scalar-tensor theory of gravity. The body-related coordinates
replicate the inertial coordinates only locally and cover a limited domain of spacetime manifold around
the body under consideration. Thus, a full solution of the external and internal problems of celestial
mechanics is based on consideration of N + 1 coordinate charts – one global and N local ones. It agrees
with the topological structure of manifold defined by a set of overlapping coordinate charts making the
atlas of the manifold [3, 63]. The equations of motion in the post-Newtonian celestial mechanics are
intimately connected to the differential structure of the spacetime manifold characterized by the metric
tensor and affine connection. It means that the mathematical presentations of the metric tensor in the lo-
cal and global coordinates must be diffeomorphically equivalent that is the transition functions defining
the transformation from local to global coordinates must map the components of the metric tensor of the
internal problem of motion to the external one and vice versa.
Newtonian mechanics of N -body system describes translational motion of the bodies in global coor-
dinates which origin is placed at the Newtonian center of mass of the system. Local coordinates for each
body are constructed by a spatial translation of the origin of the global coordinates to the Newtonian cen-
ter of mass of the body. Time in the Newtonian theory is absolute, and, hence, does not change when one
transforms it from global to local coordinates. Newtonian space is also absolute, which makes the differ-
ence between the global and local coordinates physically insignificant. The theory changes dramatically
as one switches from the Newtonian concepts to a consistent relativistic theory of gravity. There is no
longer the absolute time nor the absolute space which are replaced with a pseudo-Riemannian spacetime
manifold endowed with a rather complicated set of differential equations for gravitational variables and
matter fields accounting for various relativistic effects. Construction of the post-Newtonian global and
local coordinates depends on the boundary conditions imposed on the field equations [93]. The principle
of relativity should be satisfied when the law of transformation from the global to local coordinates is
derived. Not only should it be consistent with the Lorentz transformation of special relativity but must
account for the full gauge freedom of the relativistic theory of gravity as well [96]. Time and spatial
coordinates are transformed simultaneously making up a class of non-linear coordinate transformations
establishing mutual functional relations between various geometric objects and world lines of the bodies
[21, 101, 137].
At the first glance the coordinate-related approach to the problem of motion looks quite disentangled
from the covariant approach advocated by Dixon. In fact, the two approaches complement each other in
a constructive way. In order to prove its consistency the Dixon’s theory must be linked to the differential
structure of spacetime manifold which is given in terms of the transition functions between coordinate
charts. Moreover, it must be connected to the solutions of the field equations in order to identify the
background spacetime manifold used to build skeleton of the stress-energy-momentum tensor, and to
link it to observational values of multipole moments in astrophysics. It is impossible to identify the
center of mass of an extended body and derive the equation of its world line without solving the internal
problem of motion in the local coordinates with its subsequent matching to the external solution. Dixon’s
covariant formulation merely states that the center of mass inside of each body in N -body system exists
but it does not provide the criterion for making the unique choice of the center of mass. This problem
cannot be solved without resorting to the method of asymptotic matching of the solutions of the field
equations. For these reasons we tackle the problem of covariant formulation of equations of motion, first,
from the coordinate-related approach which allows us to find out the unique definition of the center of
mass of the body at least in the post-Newtonian approximation. Covariant description of the equations of
motion is achieved in our theoretical scheme later on, at last stage of calculations, by mapping back the
locally defined quantities and equations to arbitrary coordinates on the effective background manifold
with metric ḡαβ (see section 3.8 for more detail). This procedure has been proposed by Landau and
Lifshitz [107] and applied to the problem of motion by Thorne and Hartle [147]. It works perfect
on torsionless manifolds with the affine connection being fully determined by the metric tensor. Its
extension to the manifolds with torsion and/or non-minimal coupling of matter with gravity requires
further theoretical study. Some steps forward in this direction have been made, for example, in [72, 83,
109, 110, 160] and [128, 129]. We do not discuss these extensions over here.
3.3.2 Solving the Field Equations by Post-Newtonian Approximations

Small Parameters
Field equations (3.2.2) and (3.2.5) of the scalar-tensor theory of gravity represent a system of eleventh
non-linear differential equations in partial derivatives. It is challenging to find their solution in the case
of N -body system made of self-gravitating extended bodies which back reaction on the geometry of
spacetime manifold cannot be neglected. Like in general relativity, an exact solution of this problem is
not known and may not be available in analytic form. Hence, one has to resort to approximations to apply
the analytic methods. Two basic methods are known in asymptotically-flat spacetime, namely, the post-
Minkowskian (PMA) and the post-Newtonian (PNA) approximations [35]. Post-Newtonian approxima-
tions are applicable in case when matter moves slowly and gravitational field is weak everywhere – the
conditions, which are satisfied, e.g., within the solar system. Post-Minkowskian approximations relax
the requirement of the slow motion but the weak-field limitation remains. We use the post-Newtonian
approximations in this chapter.
Post-Newtonian approximations assume that the metric tensor can be expanded in the near zone of
N -body system with respect to the ultimate speed of gravity c called the speed of light for historical
reasons [93]. This expansion may be not analytic in higher post-Newtonian approximations in a certain
class of coordinates including the harmonic coordinates [11, 88]. Exact formulation of basic axioms
underlying the post-Newtonian expansion was given by Rendall [132]. Practically, it requires to have
several small parameters characterizing N -body system and the interior structure of the bodies. They
are: i ∼ vi /c, e ∼ ve /c, and ηi ∼ Ui /c2 , ηe ∼ Ue /c2 , where vi is a characteristic internal velocity
of motion of matter inside an extended body, ve is a characteristic velocity of the relative motion of the
bodies with respect to each other, Ui is the internal gravitational potential inside each body, and Ue is the
external gravitational potential in the space between the bodies. If we denote a characteristic radius of
body as L and a characteristic distance between the bodies as R, the internal and external gravitational
potentials will be Ui ' GM/L and Ue ' GM/R, where M is a characteristic mass of the body.
Due to the virial theorem of the Newtonian gravity [107] the small parameters are not fully indepen-
dent. Specifically, one has 2i ∼ ηi and 2e ∼ ηe . Hence, two parameters i and e are sufficient in doing
the post-Newtonian expansion. In what follows, we shall use notation ≡ 1/c to mark the presence of
the post-Newtonian parameter and the fundamental speed c in the post-Newtonian series.
Besides the small relativistic parameters and η, the post-Newtonian approximations utilize one
more small parameter which, in fact, is not relativistic parameter as it presents already in the Newtonian
gravity. This parameter is denoted δ ∼ L/R, and it characterizes the influence of the finite size of
the body on the gravitational field outside of it. It is well-known that in the Newtonian mechanics
gravitational field of a spherically-symmetric body is the same as the field of a point-like particle having
the same mass [28]. It suggests that for spherically-symmetric bodies parameter δ = L/R does not
play any role in the Newtonian approximation. However, it may appear directly in PNA expansion even
if the body is spherically-symmetric. Such appearance of the parameter δ is not supported in general
relativity which satisfy the, so-called, effacing principle [34, 35, 97], but is allowed in scalar-tensor
theory of gravity where terms of the order of (β − 1)2 δ 2 appear in the translational equations of motion
of spherically-symmetric bodies [95].
If the bodies are not spherically-symmetric, parameter δ appears in both Newtonian and post-Newtonian
approximations as a result of the multipolar expansion of gravitational field with respect to body’s in-
ternal multipoles. A multipole of order l depends on the parameter of ellipticity of the body, Jl that is
related to the elastic properties of matter which are characterized by Love’s numbers κnl (n = 1,2,3).
Generally, they are different for each multipole [29, 78, 162]. The present chapter will account for all
gravitational multipoles of the massive bodies without truncation of the multipolar series at some finite
order l.
The Post-Newtonian Series

The post-Newtonian series are expansions of the geometric, scalar field and matter variables around their
background values with respect to the small parameters introduced above. We denote φ0 the background
value of the scalar field φ and assume that the dimensionless perturbation of the field, ζ, is small. The,
we can write an exact decomposition
φ = φ0 (1 + ζ) . (3.3.1)
In principle, the background value φ0 of the scalar field depends on time due to the Hubble expansion
of the universe. Because φ0 defines the current value of the universal gravitational constant, G, the time
dependence of φ0 causes a secular drift of G = G0 + Ġ(t − t0 ) [see (3.3.29)]. However, in our model
of N -body system, the spacetime is assumed to be asymptotically flat which excludes cosmological
scenario and makes φ0 = const.
According to theoretical expectations [43] and experimental limitation on PPN parameters [155], the
post-Newtonian perturbation ζ of the scalar field have a very small magnitude, so that we can expand all
quantities depending on the scalar field in the Taylor series with respect to ζ using it as a small parameter
in the expansion. In particular, the post-Newtonian decomposition of the coupling function θ(φ) can be
written as
θ(φ) = ω + ω 0 ζ + O(ζ 2 ), (3.3.2)
where ω ≡ θ(φ0 ), ω 0 ≡ (dθ/dζ)φ=φ0 , and we impose the boundary condition on the scalar field
such that ζ approaches zero as the distance from the N -body system approaches spatial infinity. The
post-Newtonian expansion of the perturbation ζ is given in the form
(1) (2)
ζ = ζ + 2 ζ + O(3 ) , (3.3.3)
(1) (2)
where the post-Newtonian corrections ζ , ζ , etc. will be defined below.
The background value of the metric tensor gαβ in our model of asymptotically-flat spacetime is the
Minkowski metric ηαβ . Cosmological post-Newtonian approximations with the background Friedmann-
Lemaître-Roberston-Walker metric is considered in chapter 7 of this book. The metric tensor is expanded
in the post-Newtonian series with respect to parameter ≡ 1/c as follows
(1) (2) (3) (4)
gαβ = ηαβ + h αβ + 2 h αβ + 3 h αβ + 4 h αβ + O(5 ). (3.3.4)
The generic post-Newtonian expansion of the metric tensor is not analytic [11, 35, 88]. However, the
non-analytic terms emerge only in higher post-Newtonian approximations and do not affect results of
the present chapter since we restrict ourselves with the first post-Newtonian approximation. Notice
also that the linear, with respect to , terms in the metric tensor expansion (3.3.4) can be eliminated by
making coordinate adjustments [147]. These terms do not originate from the field equations and are
pure coordinate-dependent effect. If we kept them, they would make the coordinate grid non-orthogonal
and rotating at classic level. Reference frames with such properties are rarely used in astronomy and
astrophysics. Therefore, we assume that the linear term in expansion (3.3.4) is absent.
After eliminating the linear terms in the post-Newtonian expansion of the metric tensor and substitut-
ing the expansion to the field equations (3.2.2) we can check by inspection that various components of
the metric tensor and the scalar field have in the first post-Newtonian approximation the following form
(2) (4)
g00 = −1 + 2 h 00 + 4 h 00 + O(5 ), (3.3.5)
(3)
g0i = 3 h 0i + O(5 ), (3.3.6)
(2)
gij = δij + 2 h ij + O(4 ), (3.3.7)
where each term of the expansions will be explained below. Scalar field equation (3.2.5) shows that there
is no linear term in (3.3.3) which reduces it in the first post-Newtonian approximation to
(2)
ζ = 2 ζ + O(4 ) . (3.3.8)
In order to simplify notations, we shall use the following abbreviations:

(2) (4) (3) (2) (2)
h00 ≡ h 00 , l00 ≡ h 00 , h0i ≡ h 0i , hij ≡ h ij , h ≡ h kk , (3.3.9)
and
(2)
ϕ ≡ (ω + 2) ζ . (3.3.10)
Post-Newtonian expansion of the metric tensor and the scalar field introduces a corresponding expan-
sion of the stress-energy-momentum tensor of matter
(0) (2)
T00 = T 00 + 2 T 00 + O(4 ), (3.3.11)
(1)
T0i = T 0i + O(3 ), (3.3.12)
(2)
Tij = 2 T ij + O(4 ), (3.3.13)
(n)
where T αβ (n = 0,1,2,3...) denote terms of the order n . In the first post-Newtonian approximation
the components of the energy-momentum tensor were derived by Fock [73]
(0)
T 00 = ρ∗ , (3.3.14)
(1)
T 0i = −ρ∗ v i , (3.3.15)
(2)
∗ i j ij
T ij = ρ v v +π , (3.3.16)
(2)
2
v h
T 00 = ρ∗ + Π − h00 − , (3.3.17)
2 2
where v i = cui /u0 is the 3-dimensional velocity of matter.

Fock also introduced the invariant density of matter [73]
√ 2
ρ∗ ≡ −gu0 ρ = ρ + ρ(v 2 + h) + O(4 ), (3.3.18)
2
which is a useful mathematical tool in relativistic hydrodynamics [115, 153]. The reason is that the
invariant density, unlike density ρ, obeys an exact equation of continuity (3.2.9) that can be recast to a
Newtonian-like form [73]
cρ∗,0 + (ρ∗ v i ),i = 0, (3.3.19)
where f,0 ≡ ∂f /∂t, and comma with an index behind it denotes a partial derivative with respect to a
corresponding coordinate. Equation (3.3.19) is valid in any post-Newtonian approximation and it makes
calculation of time derivative of a volume integral of any function f (t,x) simple
Z Z
d df (t,x) 3
ρ∗ (t,x)f (t,x)d3 x = ρ∗ (t,x) d x, (3.3.20)
dt dt
VA VA
where VA is a volume of body A, the total time derivative
d ∂ ∂
= + vi i , (3.3.21)
dt ∂t ∂x
and we have taken into account in derivation of (3.3.20) that body A moves, and its shape and internal
structure depend on time [93]. Equation (3.3.20) is exact.
3.3.3 The Post-Newtonian Field Equations

The post-Newtonian field equations for the post-Newtonian components of the metric tensor and scalar
field variables can be derived after substituting the post-Newtonian series of the previous section to the
covariant equations (3.2.2) and (3.2.5), and arranging the terms in the expansion in the order of smallness
with respect to parameter ≡ 1/c. The post-Newtonian equations are covariant like the original field
equations that is they are valid in arbitrary coordinates. Hence, their solution depends on four arbitrary
functions reflecting the gauge freedom. It is a common practice to limit this arbitrariness by imposing a
gauge condition which is equivalent to a choice of a specific set of coordinates on spacetime manifold.
The gauge condition does not fix the freedom in choosing coordinates completely – a restricted class of
coordinate transformations within the imposed gauge is still allowed. This class of transformations is
called the residual gauge freedom which plays an essential role in theoretical formulation of relativistic
celestial mechanics of a N -body system.
The most convenient gauge condition in the scalar-tensor theory of gravity were proposed by Nutku
[121, 122] as a generalization of the harmonic gauge
√

φ −g g µν =0, (3.3.22)
,ν
where the comma denotes a partial derivative with respect to a corresponding coordinate, f,ν = ∂f /∂xν .
By making use of the conformal metric tensor, g̃αβ ≡ φgαβ , equation (3.3.22) can be recast to the
harmonic gauge condition for the conformal metric tensor [73, 123]
( −g̃ g̃ µν ),ν = 0 .
p
(3.3.23)
Post-Newtonian expansion of gauge condition (3.3.22) yields

2φ
+ h00 + h = 2h0k,k , (3.3.24)
ω+2 ,0

2φ
− h00 + h = 2hik,k . (3.3.25)
ω+2 ,i
It is worth noting that in the first post-Newtonian approximation, equations (3.3.24), (3.3.25) do not
(4)
involved the metric tensor component h 00 ≡ l00 , which is directly obtained from the field equation and
is fixed.
The post-Newtonian field equations for the scalar field and the metric tensor are obtained from equa-
tions (3.2.5) and (3.2.2) after making use of the post-Newtonian expansions, given by equations(3.3.5)–
(3.3.13), and the gauge conditions (3.3.24), (3.3.25). The scalar-tensor theory of gravity with variable
coupling function θ(φ) has two constant parameters, ω and ω 0 , characterizing deviation from general
relativity. They are related to the standard PPN parameters γ and β as follows [153]
ω+1
γ = γ(ω) = , (3.3.26)
ω+2
0
ω
β = β(ω) = 1+ . (3.3.27)
(2ω + 3)(2ω + 4)2
General relativity is obtained as a limiting case of the scalar-tensor theory when parameters γ = β = 1
or ω = ∞. Notice that in order to get this limit convergent, parameter ω 0 must grow slower than ω 3
as ω approaches infinity. Currently, there are no experimental data restricting the functional behavior of
ω 0 ∼ ω 3 β(ω) which could help us to understand better the nature of the coupling function θ(φ). This
makes parameter β a primary target for experimental study in the near-future gravitational experiments
[1, 31, 56] including the advanced lunar laser ranging (LLR) [100].
The scalar field perturbation (3.3.10) is expressed in terms of γ as
(2)
ζ = (1 − γ)ϕ. (3.3.28)
The background scalar field φ0 and the parameter of coupling ω determine the observed numerical value
of the universal gravitational constant
2ω + 4 1
G= . (3.3.29)
2ω + 3 φ0
Had the background value φ0 of the scalar field been driven by cosmological evolution, the measured
value of the universal gravitational constant G would depend on time, G = G0 + Ġ(t − t0 ), and might
be detected experimentally. Currently, the best upper limit on time variability of G is imposed by LLR
as |Ġ/G| < (4 ± 9) × 10−13 yr−1 [156].
After making use of definition of the tensor of energy-momentum (3.3.14)–(3.3.17), and PPN param-
eters (3.3.26)–(3.3.29), one obtains the final form of the post-Newtonian field equations:
ϕ = −4πGρ∗ , (3.3.30)
h200

2 2
h00 + l00 + + 2(β − 1)ϕ = (3.3.31)
2
π kk

1 h
−8πGρ∗ 1 + 2 (γ + ) v 2 + Π + γ ∗ − − (2β − γ − 1)ϕ
2 ρ 6
+2 h<ij> h00,ij ,
h0i = 8πG(1 + γ)ρ∗ v i , (3.3.32)

hij = −8πGγρ∗ δij , (3.3.33)
where ≡ η µν ∂µ ∂ν is the D’Alembert (wave) operator of the Minkowski spacetime, h ≡ hii =

δ ij hij , and the angular brackets around tensor indices in h<ij> ≡ hij − δij h/3, denote the symmetric
trace-free (STF) part of the spatial components of the metric tensor [the STF tensors which are explained
in Chapter 5 of this book. They are also thoroughly discussed by Thorne [146], Blanchet and Damour
[11] and in a review article by Poisson et al. [127]. Equations (3.3.30)–(3.3.33) are valid in the class of
coordinates defined by the gauge condition (3.3.22).
3.3.4 Conformal Harmonic Gauge
Let us rewrite the Nutku gauge condition (3.3.22) as follows

,α
φ
g µν Γα
µν = ln . (3.3.34)
φ0
It reveals that the Laplace-Beltrami operator (3.2.3) depends on the scalar field,
∂2

1 ∂φ ∂
g ≡ g µν µ ν
− µ ν
. (3.3.35)
∂x ∂x φ ∂x ∂x
Any function F (xα ) satisfying the homogeneous Laplace-Beltrami equation, g F (xα ) = 0, is called
harmonic. The gauge condition (3.3.22) assumes that g xα = −(ln φ),α 6= 0, that is the coordinate xα
is not a harmonic function on spacetime manifold in the Jordan-Fierz frame metric. Nonetheless, such
non-harmonic coordinates are more convenient in the scalar-tensor theory of gravity because they allow
us to eliminate more spurious terms from the field equations than the harmonic gauge condition does.
We call the class of coordinates singled out by the Nutku gauge conditions (3.3.22), conformal harmonic
coordinates [94]. This is because these coordinates are harmonic functions in the conformal Einstein
frame metric g̃αβ .
The conformal harmonic coordinates have many properties similar to the harmonic coordinates in
general relativity. The choice of the conformal harmonic coordinates for constructing the theory of mo-
tion of extended bodies is justified by the following three factors: (1) the conformal harmonic coordinates
approach harmonic coordinates in general relativity when the scalar field φ → φ0 so that β = γ = 1,
(2) the conformal harmonic coordinates are natural for scalar-tensor parametrization of equations used
in resolutions of the IAU 2000 General Assembly [137] on relativistic reference frames, (3) the gauge
condition (3.3.22) significantly simplifies the post-Newtonian field equations, thus, facilitating their so-
lution. Harmonic coordinates were used by [90] for construction of post-Newtonian reference frames in
PPN formalism. Pitfalls associated with this choice have been analysed in [93, 94].
Gauge condition (3.3.34) does not fix the conformal harmonic coordinates uniquely. Let us change
the coordinates
xα 7→ wα = wα (xα ) , (3.3.36)
but keep the Nutku gauge condition (3.3.34) the same. Simple calculation shows that under such condi-
tion the new coordinates wα must satisfy a homogeneous wave equation
∂ 2 wα
g µν (xβ ) =0, (3.3.37)
∂xµ ∂xν
which means that the new coordinates, wα , are harmonic functions. Equation (3.3.37) describes the
residual gauge freedom and has an infinite number of non-trivial solutions defining the entire set of the
local coordinates wα on spacetime manifold of N -body system. This residual gauge freedom in the
scalar-tensor theory of gravity is the same as in the harmonic gauge in general relativity. We specify the
set of the conformal harmonic coordinates for N -body system in the next section.
3.4 Parametrized Post-Newtonian Coordinates

Our task in this chapter is to derive fully covariant post-Newtonian equations of translational motion
of extended bodies with accounting for all their multipole moments. Standard textbooks on the post-
Newtonian celestial mechanics [19, 21, 32, 73, 138] derives the post-Newtonian equations of motion in
a particular, usually harmonic, gauge which freezes the gauge-dependent modes and brings the equations
to a form which is suitable for practical applications. As we explained above, the post-Newtonian equa-
tions admit a large freedom in making the gauge (coordinate) transformations on spacetime manifold
[21, 61, 96, 124] as well as in the configuration space of orbital and other parameters characterizing the
motion of bodies [64, 65]. Therefore, single terms taken in such post-Newtonian equations separately
from the others, make no physical sense – they can be always changed (or even eliminated) by making
coordinate transformations. Only after the equations are solved and their solutions are substituted to
observables we can unambiguously discuss gravitational physics because the observables are invariantly
defined [20, 22, 93]. This subtle difference between coordinate and observable effects is not so explicit
in the Newtonian theory but should be carefully taken into account in relativistic celestial mechanics to
Section 3.4 Parametrized Post-Newtonian Coordinates 79
avoid misinterpretation of gravitational physics on spacetime manifold [92, 99]. Coordinate-independent

form of equations of motion is important to facilitate this task.
Unfortunately, the road to success is paved with failure, and Dixon’s straightforward approach to
build covariant theory of motion of celestial bodies faces with obstacles and, in fact, is inapplicable for
derivation of equations of motion of compact astrophysical objects like black holes where the only way
to proceed is the method of asymptotic matching of vacuum solutions of the field equations in global and
local coordinates [48, 49, 75, 79, 80]. The case of Dixon’s extended bodies is manageable but the price
to pay is to replace the interior structure of the bodies with a skeleton of the stress-energy-momentum
tensor of matter which definition admits a large freedom in interpretation and is not unique. The only
definite statement which could be made so far about the stress-energy-momentum tensor skeleton, is that
it may exist [see [62, Appendix]]. It is clear that this situation is unsatisfactory and the only remedy is
to make use of the local coordinates to identify Dixon’s multipole moments with the multipole moments
of gravitational field which emerge unambiguously in the process of solution of the field equations.
One more problem which was not solved satisfactory in Dixon’s covariant approach is the identifica-
tion of the center of mass of the extended body and its world line. Its solution is easily achieved in the
coordinate-based approached amended with the method of asymptotic matching as will be demonstrated
below. Nonetheless, the reader should understand that introduction of coordinates is just an auxiliary
intermediate step in building the covariant theory of motion of celestial bodies as they are required for
unique solution of the internal and external problems of celestial mechanics. The coordinates play a role
of scaffolding which is necessary to single out the world line of the center of mass of each body, say B,
and to separate a self-force from the external gravitational force exerted on the body B by other N − 1
bodies of the N -body system. The coordinate scaffolding is removed as soon as the theory is completed
in section 3.8.
As we have learned above, N -body problem requires introduction of one global and N local coordi-
nate charts – one for each body. Geometric attributes of the coordinate charts as well as their kinematic
and dynamic properties are distinguished in scalar-tensor theory of gravity by specification of the metric
tensor and the scalar field.
3.4.1 The Global Post-Newtonian Coordinates

Boundary Conditions and Kinematic Properties
We have assumed that N -body system is isolated and there is no matter outside of it. We have to decide
which bodies in the system should be considered as the sources of the gravitational field. It is clear
that the number N of such bodies depends on the accuracy of astronomical observations and on the
precision of calculation of their celestial ephemeris (position and velocities). Since there are no external
astronomical bodies outside the system, the spacetime can be considered as flat at infinity with the metric
tensor, gαβ , asymptotically approaching the Minkowski metric ηαβ = diag(−1, + 1, + 1, + 1). We
further assume that there are no singularities on the manifold like black holes, wormholes, etc., the
bodies move slowly and gravitational field is weak everywhere.
These limitations allow us to cover the whole spacetime manifold with a single coordinate chart
denoted as xα = (x0 ,xi ), where x0 = ct is a coordinate time and xi ≡ x are spatial coordinates.
The global coordinates are used for the description of orbital dynamics of the bodies with respect to the
center of mass of the N -body system. The coordinate time and spatial coordinates have no immediate
physical meaning in those domains of spacetime where gravitational field is not negligible. However,
when one approaches to infinity the global coordinates approximates the inertial coordinates of observer
in the Minkowski space. For this reason, one can think about the coordinate time t and the spatial
coordinates xi as the proper time and the proper distance measured by fictitious observers located at rest
at infinity with respect to the center of mass of the system [73]. These coordinates have been used some
researchers for invariant description of observable effects in astronomy [19, 85]. They are also useful
for formulation of conformal infinity in isolated astronomical systems that provides a very powerful
method within numerical relativity to study global problems such as gravitational wave propagation and
detection [74].
Precise mathematical definition of the global coordinates can be given in terms of the metric tensor,
which is a solution of the field equations (3.3.31)–(3.3.33) with boundary conditions imposed at infinity.
To formulate the boundary conditions, we introduce the metric perturbation
hαβ (t,x) ≡ gαβ (t,x) − ηαβ , (3.4.1)
where hαβ is the full post-Newtonian series defined in (3.3.4). The very existence of the global coordi-
nates matching asymptotically with the inertial coordinates of the Minkowski spacetime, presumes that
the products, rhαβ and r2 hαβ,γ , where r = |x|, are bounded at spatial infinity while at null past infinity
r→∞
lim hαβ (t,x) = 0 . (3.4.2)
t+r/c=const.
Additional boundary condition must be imposed on the first derivatives of the metric tensor to exclude
non-physical radiative solutions associated with gravitational waves incoming to N -body system [73].
This condition is imposed because there are no sources of gravitational field outside of the N -body
system. It is formulated as follows [34, 73]
lim
r→∞
[(rhαβ ),r + (rhαβ ),0 ] = 0 , (3.4.3)
t+r/c=const.
where the comma denotes a partial derivative with respect to the radial, r, and time, x0 = ct, coordinates.
Though, the first post-Newtonian approximation does not include gravitational waves, the boundary
condition (3.4.3) tells us to choose the retarded solution of the field equation (3.3.31)-(3.3.33).
Similar "no-incoming-radiation" conditions are imposed on the perturbation, ϕ, of the scalar field
defined in (3.3.28),
lim
r→∞
ϕ(t,x) = 0 , (3.4.4)
t+r/c=const.
lim
r→∞
[(rϕ),r + (rϕ),0 ] = 0 . (3.4.5)
t+r/c=const.
The origin of the global coordinates coincides with the center of mass of N -body system at any instant
of time. This condition can be satisfied after choosing a suitable definition of the post-Newtonian dipole
moment, Di , of N -body system and equating its numerical value to zero along with its first and second
time derivatives. This requirement can be fulfilled, at least in the first post-Newtonian approximation,
because of the law of conservation of the linear momentum, Pi , of the system [19, 21, 93]. The law of
conservation of the angular momentum of N -body system allows us to make the spatial axes of the global
coordinates non-rotating in space either in kinematic or dynamic sense [24, 25]. Coordinates are called
kinematically non-rotating if their spatial orientation does not change with respect to the Minkowskian
coordinates at infinity as time goes on [104, 105]. Dynamically non-rotating coordinates are defined by
the condition that equations of motion of test particles moving with respect to these coordinates do not
have terms that can be interpreted as the Coriolis or centripetal forces [104]. This definition operates
only with the local properties of the spacetime manifold. Because of the assumption that N -body system
is isolated the global coordinate grid do not rotate in any sense about the coordinate origin.
Metric Tensor and Scalar Field
The metric tensor gαβ (t,x) and the scalar field ϕ(t,x) are obtained in the global coordinates by solving
the field equations (3.3.30)–(3.3.33)with the boundary conditions (3.4.2)–(3.4.4). It yields [93, 94]
ϕ(t,x) = U (t,x) , (3.4.6)

h00 (t,x) = 2 U (t,x) , (3.4.7)
l00 (t,x) = 2Ψ(t,x) − 2(β − 1)ϕ2 (t,x) − 2U 2 (t,x) − χ,tt (t,x) , (3.4.8)
h0i (t,x) = −2(1 + γ) Ui (t,x) , (3.4.9)
hij (t,x) = 2γδij U (t,x) , (3.4.10)
where χ,tt ≡ ∂χ/∂t2 , the post-Newtonian potential

1 1
Ψ(t,x) ≡ γ + Ψ1 (t,x) − Ψ2 (t,x) + (1 + γ − 2β)Ψ3 (t,x) + Ψ4 (t,x) + γΨ5 (t,x) , (3.4.11)
2 6
and parameters γ and β have been defined in (3.3.26) and (3.3.27) respectively.
Gravitational potentials U, U i , χ, Ψ = 5k=1 Ψk are linear combinations of the respective gravita-
P
tional potentials of the bodies of N -body system,

X X i X X
U= UA , Ui = UA , Ψk = ΨAk , χ= χA , (3.4.12)
A A A A
where the summation index A = 1,2,...,N numerates the bodies of the astronomical system.
Each gravitational potential of body A is defined as an integral over the spatial volume VA occupied
by matter of this body, Z ∗
ρ (t,x0 ) 3 0
UA (t,x) = G d x, (3.4.13)
|x − x0 |
VA
ρ∗ (t,x0 )v i (t,x0 ) 3 0
Z
i
UA (t,x) = G d x, (3.4.14)
|x − x0 |
VA
Z
χA (t,x) = −G ρ∗ (t,x0 )|x − x0 |d3 x0 , (3.4.15)
VA
ρ∗ (t,x0 )v 2 (t,x0 ) 3 0
Z
ΨA1 (t,x) = G d x, (3.4.16)
|x − x0 |
VA
ρ∗ (t,x0 )h(t,x0 ) 3 0
Z
ΨA2 (t,x) = G d x, (3.4.17)
|x − x0 |
VA
ρ∗ (t,x0 )ϕ(t,x0 ) 3 0
Z
ΨA3 (t,x) = G d x, (3.4.18)
|x − x0 |
VA
ρ∗ (t,x0 )Π(t,x0 ) 3 0
Z
ΨA4 (t,x) = G d x, (3.4.19)
|x − x0 |
VA
π kk (t,x0 ) 3 0
Z
ΨA5 (t,x) = G d x, (3.4.20)
|x − x0 |
VA
where
h(t,x) ≡ hii (t,x) = 6U (t,x) . (3.4.21)
Potential χ is determined as a particular solution of the inhomogeneous Poisson equation
4χ(t,x) = −2U (t,x) (3.4.22)
with the right side defined in a whole space, 4 ≡ δ ij ∂ 2 /∂xi ∂xj . Nevertheless, its solution given by
equation (3.4.15), has a compact support inside the volumes of the bodies of N -body system [73, 153].
It is worthwhile to emphasize that all integrals defining the metric tensor in the global coordinates are
taken over the hypersurface of constant coordinate time t. Post-Newtonian spacetime transformations
change the time hypersurface, hence, transforming the corresponding integral.
Notice that the Newtonian gravitational potential U (t,x) has a double camouflage in scalar-tensor
theory of gravity. It appears in the solution of the field equations, first, in (3.4.6) for the scalar field
perturbation φ, and second, in (3.4.7), (3.4.10) for the perturbation of the metric tensor components h00
and hij . It would be wrong, however, to identify the metric tensor components with the scalar field [like
h00 = 2φ] in the scalar-tensor theory of gravity - they are equal only in the Newtonian approximation,
and only in the global coordinates xα . In all other circumstances the metric tensor must be carefully
distinguished from the scalar field to prevent incorrect formulation of post-Newtonian transformation
of potential Ψ which is a functional of both, the metric tensor and the scalar field. This is because the
scalar field and the time-time component of the metric tensor transform differently and cannot be used
to substitute one another. This important point is not emphasized in PPN formalism [153, 155] and has
been overlooked, for example, in [90]. We discuss this issue in more detail in [94, Appendix C].
3.4.2 The Local Post-Newtonian Coordinates
Boundary Conditions and Dynamic Properties

We denote the local coordinates by wα = (w0 ,wi ) = (cu, wi ) where u stands for the local coordinate
time. A local coordinate chart is constructed in a close proximity to the world line Z of the center of
mass of a self-gravitating body in N -body system. There are N local coordinates – one for each body.
The local coordinates are used to describe internal motion of matter inside the body, to define its center
of mass, linear momentum, spin as well as multipolar fine structure of the body’s gravitational field. In
practical applications the local coordinates are used, for example, in geodynamics and satellite geodesy
to determine precession, nutation and Earth Orientation Parameters (EOP) as well as orbital motion of
Moon and satellites, and to build the Global Positioning System (GPS).
The metric in the local coordinates is approximately Minkowskian inside the body. This is because we
have assumed the gravitational field is weak everywhere. In case when the body is a compact relativistic
object, the local metric in the close proximity to the body must be approximated by an exact solution of
the scalar-tensor theory corresponding to the strong gravitational field like the Schwarzschild solution
in general relativity. This case was treated, for example, in [34, 48, 49, 147] but we shall not dwell here
upon further details as it will take us too far aside.
The local coordinates do not cover the entire spacetime manifold. Therefore, the metric in the local
coordinates is not asymptotically approaching the Minkowski metric as the radial distance from the
body is increasing. The local metric diverges at infinity. This is because the local metric is a solution of
the field equations (3.3.31)–(3.3.33) in the local coordinates, wα , which must smoothly match the global
solution of the same equations in the global coordinates xα . The global solution includes the contribution
of gravitational field of other (external) bodies of N -body system which reveals locally as a tidal force.
Newtonian gravitational potential of the tidal force is represented by a harmonic polynomial with respect
to the local spatial coordinates with time-dependent, STF tensor coefficients QL ≡ Qi1 ...il (u) which are
called the external (or tidal) multipoles [46, 101, 146, 147] in contrast to the internal multipoles, I L ≡
I i1 ...il (u) which characterize the structure of the gravitational field of the body under consideration.
In the Newtonian approximation this polynomial is a solution of a homogeneous Laplace equation,
4U (ext) = 0, which has a general polynomial solution
∞
X 1
U (ext) = Q + Qi wi + Q<i2 ...il > w<i2 ...il > , (3.4.23)
l!
l=2
where the external multipoles Q, Qi , Qij ,..., are functions of time, and the angular brackets around
spatial indices indicate the symmetric and trace-free (STF) Cartesian tensor [146].
Since the potential U (ext) enters the metric tensor the monopole, Q, and dipole, Qi , external multipoles
can be eliminated by rescaling of units of measurement and making a transformation to a freely-falling
frame. This effacing of the monopole and dipole moments from the external gravitational field in the
post-Newtonian approximation is a consequence of Einstein’s equivalence principle (EEP) [107, 115,
153]. In particular, EEP suggests that it is always possible to choose the local coordinates in such a way
that the first derivatives of the metric tensor (the Christoffel symbols) vanish along a geodesic world line
of a freely-falling particle [115, 117]. In general relativity EEP is also valid for self-gravitating bodies
[18, 47, 101, 116]. In the latter case it is called the strong principle of equivalence (SEP).
Besides the metric tensor the scalar-tensor theory of gravity has additional long-range gravitational
field caused by the presence of scalar field that can not be eliminated by a coordinate transformation
to a freely-falling frame. This is because the scalar field is a single function which does not change
its numerical value under coordinate transformations and cannot vanish in any coordinates had it been
present, at least, in one. The scalar field couples with the intrinsic gravitational field of an extended
body and affects its internal multipoles like mass, dipole moment, etc. This explains the mathematical
reason behind the mechanism of violation of SEP in scalar-tensor theory of gravity discussed by Dicke
[52, 54] and Nordtvedt [118, 153]. SEP is also violated in any alternative theory of gravity which
violates the local Lorentz invariance [103, 113]. The SEP violation is called Nordtvedt’s effect[153].
Its experimental testing is conducted with Lunar Laser Ranging (LLR)[51, 156, 157] and pulsar timing
observations [44, 108, 152].
The origin of the local coordinates moves along some, yet unspecified, world line, W, which will be
determined later on as a result of matching of the mathematical solutions of the field equations obtained
in the local and global coordinates. We demand that the origin of the local coordinates coincides with
the center of mass of the body under consideration at any instant of time. This requires a precise post-
Newtonian definition of the internal dipole moment, I i , and the center of mass of the body beyond
the Newtonian approximation. Mathematically inadequate or insufficiently precise definition of body’s
center of mass introduces fictitious inertial forces that will causes it to move with respect to the origin of
the local coordinates. Because the scalar-tensor theory of gravity does not violate the law of conservation
of the linear momentum [122, 153] this motion is spurious and must be removed. This problem was
studied and resolved in [93, 94].
In order to keep the center of mass of the body at the origin of the local coordinates we have to take
into account that the linear momentum of the body is subject to the force caused by the coupling of
the internal multipole moments I i1 ...il (l ≥ 2) of the body with the external tidal gravitational field of
other bodies in N -body system characterized by the external multipoles Qi1 ...il . This coupling exists
already in the Newtonian approximation [62, 101, 147] and is shown in the right side of Dixon’s equation
(3.1.13). If one assumes that the origin of the local coordinates moves along a world line W which does
not coincide with worldline Z of the body’s center of mass, the coupling force makes the time derivative
of the linear momentum of the body different from zero which means that the center of mass of the body
moves with acceleration. In order to make the center of mass of the body be always located at the origin
of the local coordinates we have to make the time derivative of the linear momentum nil. This can be
achieved by choosing the external dipole moment Qi in the homogeneous solution of the field equations
in such a way that it compensates the dynamic force stemming from the coupling of the internal and
external multipoles with l ≥ 2 [see equation (3.5.26) for more detail].
We postulate that the spatial axes of the local coordinates are not rotating dynamically about the
origin. It means that the geodesic equations of motion of test particles in the local coordinates do not
contain the Coriolis and centrifugal forces. However, the post-Newtonian nature of the gravitational
interaction suggests that the spatial axes of the dynamically non-rotating local coordinates must slowly
rotate (precess) in the kinematic sense with respect to the spatial axes of the global coordinates. This
precessional motion of the spatial axes of the local coordinates has a pure geometric origin and follows
from the condition of a smooth matching of the local and global coordinates on spacetime manifold
[93]. The relativistic precession includes three terms that are called respectively de-Sitter (geodetic),
Lense-Thirring (gravitomagnetic), and Thomas precession [115]. Exact formulation of this precession
is given below in equation 3.5.25.
Metric Tensor and Scalar Field

We are looking for solution of the field equations (3.3.30)–(3.3.33) inside a world tube covered by the
local coordinates. This world tube spreads out from the body under consideration to another nearest body
from N -body system. Thus, the right side of inhomogeneous equations (3.3.30)–(3.3.33) includes only
the matter of the body. It does not mean that the other bodies are not included into these equations. They
participate implicitly through the solution of the homogeneous equation. Indeed, solution of the field
equations in the local coordinates is a linear combination of a particular solution of the inhomogeneous
equation and a general solution of a homogeneous equation. In order to distinguish solutions in the local
coordinates from the corresponding solutions of the field equations in the global coordinates, we put a
hat over any function of the local coordinates. This is because one and the same mathematical function
has different forms in different coordinates. For example, when we apply a coordinate transformation
x = x(w) to a scalar function F (x), it becomes F̂ (w) = F [x(w)]. It is erroneous to write F (w) instead
of F̂ (w) because F (w) 6= F [x(w)] [3, 63, 69].
Accounting for this remark, the post-Newtonian solution of the scalar field equation (3.3.30) in the
local coordinates is written as a sum of two terms
ϕ̂(u,w) = ϕ̂(int) (u,w) + ϕ̂(ext) (u,w) , (3.4.24)
describing the contributions of the internal matter and external bodies respectively. Perturbation of the
metric tensor, ĝµν (u,w), in the local coordinates is denoted
ĥµν (u,w) = ĝµν (u,w) − ηµν . (3.4.25)
Post-Newtonian solution of the field equations (3.3.31)–(3.3.33) in the local coordinates is given as a
sum of three terms
ĥµν (u,w) = ĥ(int) (ext) (mix)

µν (u,w) + ĥµν (u,w) + ĥµν (u,w) , (3.4.26)
(int) (ext)
where ĥµν describes solution of the inhomogeneous equation generated by internal matter, ĥµν de-
(mix)
scribes solution of the homogeneous equation produced by external bodies, and ĥµν arises due to a
non-linearity of the field equation (3.3.31) for the metric tensor perturbation. In the first post-Newtonian
(mix)
approximation the mixed terms ĥµν appear only in ĝ00 (u,w) component of the metric tensor.
Scalar Field: internal and external solutions The internal, ϕ̂(int) (u,w), and external, ϕ̂(ext) (u,w),
solutions for the scalar field have the following form
ϕ̂(int) (u,w) = ÛB (u,w) , (3.4.27)

∞
X 1
ϕ̂(ext) (u,w) = PL w L . (3.4.28)
l!
l=0
Here, the internal solution ϕ̂(int) (u,w) describes the scalar field which is generated by the matter of the
body (index B) covered by the local coordinates. It is expressed in terms of the Newtonian gravita-
tional potential ÛB (u,w) that is defined in next section by equation (3.4.34). The external solution,
ϕ̂(ext) (u,w), is given in the form of the multipolar expansion with respect to scalar external multipoles
PL ≡ P<i1 ...il > (u) which are STF Cartesian tensors. These multipoles are functions of the local time
u only. Finally, wL ≡ w<i1 ...il > is the STF harmonic polynomial made out of spatial local coordinates.
A subtle point of interpretation of relation between the gravitational potential U and perturbation
of the scalar field ϕ should be discussed here. By definition, the scalar field ϕ is invariant under a
coordinate transformation. On the other hand, the Newtonian potential U is not an independent scalar
field because it appears in time-time component of the metric tensor, g00 , which transforms as a tensor.
This remark indicates that the scalar field ϕ cannot be equal to the Newtonian potential U in all post-
Newtonian approximations. However, the exact relation between ϕ and U is not required in the present
chapter as the scalar field directly perturbs the metric tensor only in the post-Newtonian terms and, thus,
the approximate equations (3.4.6) and (3.4.27) are sufficient in our calculations. Nonetheless, we should
not replace ϕ with U when doing coordinate transformations.
Metric Tensor: internal solution The boundary conditions imposed on the internal solution for
the metric tensor in the local coordinates are identical with those given in equations (3.4.2), (3.4.3). For
this reason the internal solution for the metric tensor has the same form as in the global coordinates, but
all functions refers now only to the body B covered by the local coordinates. We obtain
(int)
ĥ00 (u,w) = 2ÛB (u,w), (3.4.29)
2
(int)
lˆ00 (u,w) = 2Ψ̂B (u,w) − 2(β − 1) ϕ̂(int) (u,w)
2 ∂ 2 χ̂B (u,w)
− 2ÛB (u,w) − , (3.4.30)
∂u2
(int) i
ĥ0i (u,w) = −2(1 + γ)ÛB (u,w), (3.4.31)
(int)
ĥij (u,w) = 2γδij ÛB (u,w), (3.4.32)
where

1 1
Ψ̂B (u,w) = γ+ Ψ̂B1 (u,w) − Ψ̂B2 (u,w) + (1 + γ − 2β)Ψ̂B3 (u,w)
2 6
+ Ψ̂B4 (u,w) + γ Ψ̂B5 (u,w) , (3.4.33)
and the index B indicates the local body. Notice that we have not replaced the scalar field ϕ̂(int) (u,w)
with the Newtonian potential ÛB (u,w) in (3.4.30) to keep track of the scalar field contribution.
All these functions are defined as integrals over a volume VB occupied by matter of the body B:
ρ∗ (u,w0 ) 3 0
Z
ÛB (u,w) = G 0
d w , (3.4.34)
VB |w − w |
ρ∗ (u,w0 )ν i (u,w0 ) 3 0
Z
i
ÛB (u,w) = G d w , (3.4.35)
VB |w − w0 |
ρ∗ (u,w0 )ν 2 (u,w0 ) 3 0
Z
Ψ̂B1 (u,w) = G d w , (3.4.36)
VB |w − w0 |
ρ∗ (u,w0 )ĥ(int) (u,w0 ) 3 0
Z
Ψ̂B2 (u,w) = G d w , (3.4.37)
VB |w − w0 |
ρ∗ (u,w0 )ϕ̂(int) (u,w0 ) 3 0
Z
Ψ̂B3 (u,w) = G d w , (3.4.38)
VB |w − w0 |
ρ∗ (u,w0 )Π(u,w0 ) 3 0
Z
Ψ̂B4 (u,w) = G d w , (3.4.39)
VB |w − w0 |
π kk (u,w0 ) 3 0
Z
Ψ̂B5 (u,w) = G 0
d w , (3.4.40)
VB |w − w |
Z
χ̂B (u,w) = −G ρ∗ (u,w0 )|w − w0 |d3 w0 , (3.4.41)
VB
(int)
where ĥ(int) = ĥii , ν i = dwi /du is a coordinate velocity of the body’s matter element with respect to
the origin of the local coordinates wα . It is worth to emphasize that the integrals given in this section,
are taken over a hypersurface of the coordinate time u. It does not coincide with the hypersurface of
the coordinate time t, which is used in the integrals (3.4.13)–(3.4.20) defining gravitational potentials in
the global coordinates xα . This remark is important for making the post-Newtonian transformations of
the potentials from one frame to another as it requires to use the Lie transport of the integrand function
between the hypersurfaces [93, Section 5.2.3].
The internal terms of the metric tensor in the local coordinates given by equations (3.4.29)–(3.4.32)
are subject to the gauge condition (3.3.22). We shall see later that the external part of the local metric
satisfy this condition by construction. The internal part of the metric must satisfy the gauge condition as
well. It yields
i
∂ ÛB (u,w) ∂ ÛB (u,w)
+ = O(2 ) , (3.4.42)
∂u ∂wi
which relates the potentials of the internal solution of the inhomogeneous field equations in the first post-
Newtonian approximations. We note that equation (3.4.42) is validated by the equation of continuity
(3.3.19).
Metric Tensor: external solution Solution of the homogeneous field equations for the metric
tensor in the local coordinates yields tidal gravitational field of the bodies of N -body system in terms
of the external STF multipoles [93, 94]. It also includes the inertial force exerted on the center of mass
of the body B covered by the local coordinates, caused by acceleration of the body with respect to a
time-like geodesic world line of a freely falling particle. The external solution must converge to a finite
value at the origin of the local coordinates, that is at the point w = 0. The external solution should
also match the tidal gravitational field of other bodies far away from the origin of the local coordinates.
These boundary conditions are typical for constructing local coordinates in curved spacetime [24, 46,
93, 94, 117, 143, 147].
Explicit form of the most general external solution for the linearised metric tensor perturbation in the
local coordinates is given by [93, 94]
∞
(ext)
X 1
QL w<L> + Cp Cq δ pq w2 − wp wq ,

ĥ00 (u,w) = 2 (3.4.43)
l!
l=0
∞ ∞
(ext)
X 1 X 1
ĥ0i (u,w) = εipq Cp wq + εipq CpL−1 w<qL−1> + ZiL w<L>
l! l!
l=2 l=0
∞
X 1
+ SL w<iL> , (3.4.44)
l!
l=0
∞ ∞
(ext)
X 1 X1 1
ĥij (u,w) = 2δij AL w<L> + BL w<ijL> + (δij Cp C p − Ci Cj )w2
l! l! 3
l=0 l=0
∞
X1 sym(ij)
+ DiL−1 w<jL−1> + εipq EpL−1 w<jqL−1>
l!
l=1
∞
X 1
+ FijL−2 w<L−2> + εpq(i Gj)pL−2 w<qL−2> , (3.4.45)
l!
l=2
where the external multipoles QL = QL (u), CL = CL (u), SL = SL (u), etc. are functions of time u,
the multi-index notation L ≡ i1 i2 ...il , L−1 ≡ i1 i2 ...il−1 , L−2 ≡ i1 i2 ...il−2 , the symbol sym(ij) and
the round brackets around tensor indices denote symmetrization with respect to the indices, for instance,
[TijL ]sym(ij) ≡ T(ij)L = (1/2)[TijL + TjiL ], and T(ij) = (1/2) (Tij + Tji ), repeated indices imply
Einstein’s summation rule, spatial indices are raised and lowered with the help of the Kronecker symbol
δij , the angular brackets around indices denote symmetric and trace-free (STF) Cartesian tensors which
vanish if any two indices are contracted [11, 146]. Notice that because the spatial indices are rised and
lowered with δ ij the position of the spatial index (subscript or superscript) does not make any difference
in the local coordinates.
Vector Ci in (3.4.43)–(3.4.45) is the angular velocity of the kinematic rotation of the spatial axes
of the local coordinates with respect to the global coordinates. We keep its contribution only up to
quadratic terms which is sufficient for our goal. We also assume that C i has the post-Newtonian order
of magnitude being comparable with the rate of the geodetic precession. The external solution contains
(ext) (ext)
monopole terms, Q = Q(u) and A = A(u), entering ĥ00 and ĥij respectively. Function Q defines
the unit of measurement of the coordinate time u at the origin of the local coordinates, and function A
defines the unit of measurement of spatial distances on the hypersuface of constant time u. Physical
meaning of the external multipoles QL can be understood if one writes down the Newtonian equation
of motion of freely-falling test particle [93, 94]. It turns out that Qi is an acceleration of the local frame
with respect to the geodesic world line of the particle, and QL (l ≥ 2) describes the tidal force with
multipolarity l (quadrupole, octupole, etc.).
A set of eleven external STF multipole moments AL , BL , CL , DL , EL , FL , GL , PL , QL , SL , ZL
is defined on the world line of the origin of the local coordinates so that these multipoles are functions
of the local coordinate time u only. Furthermore, the external multipole moments are symmetric and
trace-free (STF) Cartesian tensors with respect to any pair of free indices [93, 94]. Imposing four gauge
conditions (3.3.24), (3.3.25) on the external solution of metric tensor (3.4.43)–(3.4.45) reveals that only
7 out of 11 external multipole moments are algebraically independent. Indeed, the gauge conditions can
be satisfied if, and only if, the external multipole moments BL , EL , SL , DL are eliminated from the
local metric [93, 94]. The remaining multipoles: AL , CL , FL , GL , PL , QL , ZL can be constrained by
making use of the residual gauge freedom allowed by differential equation (3.3.37), which excludes four
other multipoles AL , FL , GL , ZL [93, 94]. We conclude that only three families of external moments
PL , QL , and CL have real physical meaning reflecting one degree of freedom for the scalar field and
two degrees of freedom for tidal gravitational field associated with the metric tensor. It is convenient to
keep some gauge freedom by not fixing the external multipoles ZL with l ≥ 2. They can be chosen later
to simplify equations of motion.
After fixing the gauge freedom as indicated above, the external metric tensor assumes in the local
coordinates the following form
∞
(ext)
X 1
ĥ00 (u,w) = 2Q + 2 QL w L , (3.4.46)
l!
l=1
∞
(ext) Q̇ 1−γ X 1
ĥ0i (u,w) = Ȧ + + Ṗ wi + εipq CpL−1 w<qL−1>
3 3 l!
l=1
∞
X 2l + 1
+2 2Q̇L + (γ − 1)ṖL w<iL>
(2l + 3)(l + 1)!
l=1
∞
X 1
+ ZiL wL , (3.4.47)
l!
l=1
∞
(ext)
X 1
ĥij (u,w) = 2δij A + [QL + (γ − 1)PL ]wL , (3.4.48)
l!
l=1
where the dot above the external multipoles denotes a derivative with respect to time u.
The non-linear part lˆ00 of the perturbation of the external metric tensor is determined from (3.3.31)
up to a solution of the homogeneous Laplace equation which is absorbed to the definition of the external
multipoles QL and is not shown for this reason. The particular solution of (3.3.31) is [94]
∞
X 2 ∞
X 2
(ext) 1 1
lˆ00 (u,w) = −2 QL w L − 2(β − 1) PL w L
l! l!
l=1 l=1
∞
X 1
+ Q̈L wL w2 , (3.4.49)
(2l + 3)l!
l=0
where a double dot above QL denotes a second derivative with respect to time u. We have excluded
the monopole, Q, and dipole, Qi terms from the non-linear part of the local metric tensor because
they are absorbed to (yet unknown) Q and Qi in the linear part of the metric (3.4.46). We could also
decompose the product of the two sums in (3.4.49) into irreducible parts and absorb the STF piece of
this decomposition to multipoles QL (l ≥ 2). This procedure is done later in the process of matching
the internal and external solutions.
The Mixed Terms The mixed terms entering the metric tensor in the local coordinates are given as a
particular solution of the inhomogeneous field equation (3.3.31) with the right side taken as a product of
the internal and external solutions found on the previous step of the post-Newtonian iterations. Solving
(3.3.31) yields

(mix)
lˆ00 (u,w) = −2ÛB (u,w) A + (2β − γ − 1)P (3.4.50)
∞
X 1
−4ÛB (u,w) QL + (β − 1)PL wL
l!
l=0
∞ Z ∗
ρ (u,w0 )w0L 3 0
X1
−2G QL + 2(β − 1)PL d w ,
l! |w − w0 |
l=1 VB
where VB denotes a volume of the body B. This completes derivation of the metric tensor in the local
coordinates.
Body’s Internal Multipoles in the Local Coordinates
Had we ignored the tidal gravitational field of all bodies of N -body system but the body B it would be
considered as an isolated object and its gravitational field would be characterized by the sum of the inter-
nal gravitational potentials defined in (3.4.34)–(3.4.41). Multipolar decomposition of the metric tensor
of an isolated gravitating system residing in asymptotically flat spacetime has been well understood and
can be found in papers [12, 42, 146]. This technique has been extended to the case of a self-gravitating
system embedded to a curved background spacetime in [46, 147] and [94].
The body B interacts gravitationally with other bodies of N -body system and this interaction cannot
be ignored. It brings about the mixed terms (3.4.50) to the metric tensor in the local coordinates which
contribute to the numerical values of the body B internal moments in the multipolar decomposition of
the local metric tensor. The presence of the mixed terms raised a question about precise definition of
the internal multipole moments of the body B that is to be used in deriving translational and rotational
equations of motion [147]. There are two options – either to include the contribution of the mixed terms
to the internal multipole moments or to exclude it. Both options look theoretically possible but one of
them is actually more preferred. Straightforward calculations [93, 94] proved that equations of motion
of extended bodies can be significantly simplified if the mixed terms are included to the definition of the
internal multipole moments. In this way the mixed terms do not appear explicitly in the equations of
motion confirming the principle of effacing of internal structure of the bodies in scalar-tensor theory of
gravity
There are three families of the internal mass-type multipole moments in the scalar-tensor theory of
gravity – active, conformal, and scalar multipoles [153]. The active STF mass-type multipoles are
defined by equation [93, 94]
2
Z 2 Z
d
I<L> = σ(u,w)w<L> d3 w + σ(u,w)w<L> w2 d3 w (3.4.51)
2(2l + 3) du2
VB VB
Z
2l + 1 d
− 4(1 + γ) σ i (u,w)w<iL> d3 w
l + 1 du
VB
Z ∞
X 1
− 2 d3 w σ(u,w) A + (2β − γ − 1)P + QK + 2(β − 1)PK w<K> w<L> ,
k!
VB k=1
where VB is the volume occupied by the matter of body B, the matter current density
σ i (u,w) = ρ∗ (u,w)ν i (u,w), (3.4.52)
and the active mass density

1
σ(u,w) = ρ∗ (u,w) 1 + 2 (γ + )ν 2 (u,w) + Π(u,w) (3.4.53)
2

− (2β − 1)ÛB (u,w) + 2 γπ kk (u,w) .
The conformal STF mass-type multipoles of the local system are defined as follows [93, 94]
Z ∞
X 1
I <L> = %(u,w) 1 − 2 A + (1 − γ)P + QK w<K> w<L> d3 w (3.4.54)
k!
VB k=1
2
2 Z
d
+ %(u,w)w<L> w2 d3 w
2(2l + 3) du2
VB
Z
8(2l + 1) d
− σ i (u,w)w<iL> d3 w ,
l + 1 du
VB
with the conformal mass density of matter

3 2
% = ρ∗ (u,w) 1 + 2 ν (u,w) + Π(u,w) − ÛB (u,w) + 2 π kk (u,w). (3.4.55)
2
The conformal mass density does not depend on the PPN parameters β and γ as contrasted to the defi-
nition of the active mass density. The scalar field multipoles, I¯L , are not independent and are related to
the active and conformal multipoles [93, 94]
I¯<L> = 2I<L> − (1 + γ)I <L> . (3.4.56)
Integration in (3.4.51), (3.4.54) is performed over a hypersurface of constant coordinate time u.

In addition to the gravitational mass-type multipoles, IL and I L , there is a set of internal spin-type
multipoles. In the post-Newtonian approximation they are defined by equation [12, 93, 94, 146]
Z
S<L> = εpq<il wL−1>p σ q (u,w)d3 w, (3.4.57)
VB
q
where the matter current density σ has been defined in (3.4.52). All multipole moments are functions
of time u only. They can be considered in the local coordinates as the STF Cartesian tensors attached to
the world line W of the origin of the local coordinates.
3.5 Post-Newtonian Coordinate Transformations by

Asymptotic Matching
3.5.1 General Structure of the Transformation
Post-Newtonian transformations between the global and local coordinate charts are derived by making
use of the mathematical technique known as asymptotic matching of the post-Newtonian expansions
of the scalar field and the metric tensor. This technique was originally proposed in general relativ-
ity by D’Eath [48, 49] as a tool for derivation of equations of motion of black holes [75, 79, 80]. It
proved out to be also efficient in the post-Newtonian theory of reference frames in the solar system
[5, 24, 46, 47, 94, 97, 98, 101]. Perturbations of the metric tensor gαβ and the scalar field ϕ are given
on spacetime manifold as mathematical solutions of the field equations expressed in terms of various
functions which exact form depends on the choice of coordinates. However, these solutions describe
the same physical situation irrespectively of the choice of the coordinates which means that the solu-
tions must match smoothly in the spacetime domain where the coordinate charts overlap. The matching
relies upon the tensor transformation law applied to the post-Newtonian metric tensor and the scalar
field. The matching domain is bounded by the radius of convergence of the post-Newtonian series in the
local coordinates. After the matching is finished and the post-Newtonian transition functions establish-
ing the mapping between coordinates, are found, the domain of validity of the local coordinates can be
analytically extended to a larger radius of convergence if it is required by practical applications [93].
Section 3.5 Post-Newtonian Coordinate Transformations by Asymptotic Matching 91
As we have imposed the conformal harmonic gauge condition (3.3.34), the transition functions of
the post-Newtonian coordinate transformation are constrained and must comply with the homogeneous
differential equation (3.3.37) describing the residual gauge freedom. Solution of this equation must be
convergent at the origin of the local coordinates and consistent with the tidal gravitational field of the
external solution of the local metric. Therefore, we are looking for the post-Newtonian coordinate trans-
formation in the form of a series of harmonic polynomials expanded in powers of the spatial coordinates
of the local coordinates. Because we are solving the homogeneous Laplace equation, the coefficients
of the polynomials are STF Cartesian tensors defined on the world line W of the origin of the local
coordinates [47, 101]. The transition functions are substituted to the equations matching the internal
and external solutions of the field equations expressed in the global and local coordinates. Matching
of the post-Newtonian expansions of the scalar field and the metric tensor allows us to fix all degrees
of the gauge freedom in the final form of the post-Newtonian coordinate transformation except for the
multipoles Q and A defining the units of measurement, and the external dipole moment Qi which is not
constrained by the matching conditions. Notice that we have partially used this gauge freedom in section
3.4.2 to remove the non-physical multipoles in the external solution for the local metric tensor.
The post-Newtonian transformation between coordinate times, t and u, describes the Lorentz (velocity-
dependent) and Einstein (gravitational-field-dependent) time dilations associated with the different si-
multaneity of events in the two coordinate charts [5, 101]. It also includes an infinite series of polynomial
terms [23, 24]. The post-Newtonian transformation between spatial coordinates, xi and wi , is a quadratic
function of spatial coordinates. The linear part of the transformation includes the Lorentz and Einstein
contractions of length as well as a matrix of rotation describing the post-Newtonian precession of the
spatial axes of the local coordinates with respect to the global coordinates due to the translational and
rotational motion of the bodies [35, 97]. The Lorentz length contraction takes into account the kinematic
aspects of the post-Newtonian transformation and depends on the relative velocity of motion of the local
coordinates with respect to the global coordinates. The Einstein (gravitational) length contraction ac-
counts for static effects of the scalar field and the metric tensor [93, 94]. The quadratic part of the spatial
transformation depends on the orbital acceleration of the local coordinates and accounts for the effects
of the affine connection (the Christoffel symbols) of spacetime manifold.
Let us now discuss the mathematical structure of the post-Newtonian transformation between the
local coordinates, wα = (w0 ,wi ) = (cu,w), and the global coordinates, xα = (x0 ,xi ) = (ct,x) in
more detail. This coordinate transformation must be compatible with the weak-field and slow-motion
approximation used in the post-Newtonian iterations. Hence, the coordinate transformation is given as a
post-Newtonian series for time and spatial coordinates:
u = t + 2 ξ 0 (t,x), (3.5.1)
i i 2 i
w = RB + ξ (t,x), (3.5.2)
where ξ 0 and ξ i are the post-Newtonian corrections to the Galilean transformation , u = t, RB i

=
i i i
x − xB (t), and xB (t) is the spatial position of the origin of the local coordinates expressed in terms
of the global coordinates. We denote velocity and acceleration of the origin of the local coordinates
i
as vB ≡ ẋiB and aiB ≡ ẍiB respectively, where here and everywhere else the dot above function is
understood as a derivative with respect to time t. The world line W of the origin of the local coordinates
is decoupled from the worldline Z of the center of mass of the body. Nonetheless, the two world lines
can be superposed and tied up by demanding conservation of the linear momentum of the body [93, 94].
It can be always achieved by introducing the local acceleration Qi which compensates the tidal forces
acting on the body. It makes the local coordinates non-inertial. It is relevant to point out that Dixon’s
translational equations of motion (3.1.13) implicitly assume that the world line W of the origin of local
coordinates is a time-like geodesic of an effective background spacetime manifold because the external
dipole moment Qa ≡ 0 so that only quadrupole, octupole and higher-order external multipoles are
present. Dixon [62] defined the body’s center of mass in terms of the mass internal dipole mα given
in (3.1.5), and defined the world line Z of this center of mass by condition (3.1.6) that is equivalent to
mα = 0. This condition makes the two world lines, W and Z, be identical but it means that Dixon’s
definition of the center of mass does not coincide with the center of mass of the body defined by the
condition that the dipole moment, I α , in the multipolar expansion of gravitational field of the body
vanishes, I α = 0. We discuss this issue in more detail in section 3.6.3.
Matching equations for the scalar field, the metric tensor, and the Christoffel symbols are given by
the general law of coordinate transformations of these geometric objects [63] [63]
ϕ(t,x) = ϕ̂(u,w), (3.5.3)

∂wα ∂wβ
gµν (t,x) = ĝαβ (u, w) µ , (3.5.4)
∂x ∂xν
∂xµ ∂wρ ∂wσ ∂xµ ∂ 2 wν
Γµ
αβ (t,x) = Γ̂νρσ (u,w) ν α β
+ , (3.5.5)
∂w ∂x ∂x ∂wν ∂xα ∂xβ
where

1 µν ∂gνα ∂gνβ ∂gαβ
Γµ
αβ (t,x) = g + − , (3.5.6)
2 ∂xβ ∂xα ∂xν

1 µν ∂ ĝνα ∂ ĝνβ ∂ ĝαβ
Γ̂µ
αβ (u,w) = ĝ + − , (3.5.7)
2 ∂wβ ∂wα ∂wν
are the Christoffel symbols expressed in the global and local coordinates respectively.
Matching equations (3.5.3)–(3.5.5) are valid in the domain covered simultaneously by the local and
global coordinates both inside and outside of the extended body. The scalar field, the metric tensor and
their first derivatives are continuously differentiated functions in this domain. The matching equations
are not identities which are automatically satisfied. We accept that functions in the left side of these
equations are known and given in section 3.4.1 as integrals from matter variables (density, pressure,
etc.) performed over volumes of the bodies of N -body system on hypersurface of constant time t.
The right side of the matching equations contains the known integrals from matter variables of the
body B explained in section (3.4.2) and yet unknown functions which are the external multipoles, P L ,
QL , C L , etc., of the metric tensor in the local coordinate along with function ξ α entering the post-
Newtonian transformation (3.5.1), (3.5.2). All these functions are determined by solving the matching
equations. Matching also allows us to derive equation of motion of the world line of the origin of the local
coordinates in terms of the global coordinates. Matching is necessary but not sufficient for derivation
of equations of motion of the center of mass of the extended bodies of N -body system. Additional
procedure is required that will be explained later.
The starting point for solving matching equations is ĝ0i component of the metric tensor in the local
coordinates. It does not contain the linear terms of the order of O() because the local coordinates are
not dynamically rotating and their time axis is orthogonal to space axes. It eliminates the angular and
linear velocity terms in ĝ0i [46, 94]. This information is used in (3.5.4) which implies that function
ξ 0 (t,x) in (3.5.1) must satisfy the following restriction,
0
ξ,k i
= −vB + O(2 ). (3.5.8)
This is a partial differential equation which can be solved. It yields
ξ 0 (t,x) = −A(t) − vB
k k
RB + 2 κ(t,x) + O(4 ), (3.5.9)
where A(t) and κ(t,x) are analytic but yet arbitrary functions. Notice that function A(t) depends only
on time t while κ(t,x) depends on both time and spatial coordinates.
We now use the differential equation (3.3.37) in order to impose further restrictions of the transition
functions. We substitute ξ 0 from (3.5.9) to w0 = cu in (3.5.1) and use it along with wi from (3.5.2) in
equation (3.3.37) which is reduced to the inhomogeneous Poisson equations
4κ(t,x) = 3vB aB − Ä − ȧkB RB

k k k
+ O(2 ) , (3.5.10)
i
4ξ (t,x) = −aiB + O( ) ,
2
(3.5.11)
where 4 ≡ δ ij ∂ 2 /∂xi ∂xj . General solution of these elliptic-type equations can be written in the form
of the Taylor series which coefficients are the irreducible STF Cartesian tensors. Furthermore, solution
for functions κ(t,x) and ξ i (t,x) in (3.5.10), (3.5.11) consists of two parts – a fundamental solution of
the homogeneous Laplace equation and a particular solution of the inhomogeneous Poisson equation.
We discard the fundamental solution of the Laplace equation that diverges at the origin of the local
coordinates, where wi = 0. This is because we work with a continuous distribution of matter which has
no singular points inside the bodies.
Integrating equations (3.5.10) and (3.5.11) results in

1 k k 1 1 k k 2
κ = vB aB − Ä RB 2
− ȧB RB RB + Ξ(t,x), (3.5.12)
2 6 10
1
ξ i = − aiB RB 2
+ Ξi (t,x), (3.5.13)
6
where functions Ξ and Ξi are the fundamental solutions of the homogeneous Laplace equation which
are convergent at the origin of the local coordinates. These solutions can be written in the form of scalar
and vector harmonic polynomials
∞
X 1 L <L>
Ξ(t,x) = B RB , (3.5.14)
l!
l=0
∞ ∞ ∞
X 1 iL <L> X εipq X 1 L <iL>
Ξi (t,x) = D RB + F pL RB
<qL>
+ E RB , (3.5.15)
l! (l + 1)! l!
l=1 l=0 l=0
where coefficients BL , DL , F L and EL are STF Cartesian tensors [93, 94] which should not be con-
fused with the external multipoles entering the local metric tensor. These coefficients are defined on the
world line W of the origin of the local coordinates and depend only on the coordinate time t of the global
coordinates taken on W. Explicit form of the coefficients BL , DL , F L is obtained in the process of
matching after substituting (3.5.12), (3.5.13) and corresponding expressions for the scalar field and the
metric tensor to equations (3.5.3)–(3.5.5). More details on the process of solving the matching equations
is given in textbook [93] and in our review paper [158]. The matching solution is given in next section.
3.5.2 Matching Solution
Post-Newtonian Coordinate Transformation
We match the local coordinates, wα , built inside and near the extended body B, with the global coor-
dinates, xα . Parametrized post-Newtonian transformation between the local and global coordinates is
given by two equations [93, 101],

1 k k 1 1
u = t − 2 (A + vB k
RBk
) + 4 B + vB aB − Ū˙ (xB ) + Q̇ RB
2
3 6 6
∞
1 k k 2 X 1 L L
− ȧB RB RB + B RB + O(5 ) , (3.5.16)
10 l!
l=1

1 i k
wi = RBi
+ 2 vB vB + δ ik γ Ū (xB ) − δ ik A + F ik RBk
2

1
+ akB RBi
RBk
− aiB RB 2
+ O(4 ) , (3.5.17)
2
where
i
RB = xi − xiB , (3.5.18)
is the (global) coordinate distance taken on the hypersurface of constant time t between the point of
matching, xi , and the origin of the local coordinates, xiB = xiB (t).
Functions A and B depend on the global coordinate time t and obey the ordinary differential equa-
tions,
dA 1 2
= vB + Ū (xB ) − Q , (3.5.19)
dt 2
dB 1 4 1 2 1
= − vB − γ+ vB Ū (xB ) + Ū 2 (xB ) + 2(1 + γ)vBk
Ū k (xB )
dt 8 2 2

1 1 2 1
−Ψ̄(xB ) + χ̄,tt (xB ) + Q − vB + Q − Ū (xB ) , (3.5.20)
2 2 2
that describe the post-Newtonian transformation between the coordinate time u of the local coordinates
and the coordinate time t of the global coordinates. The other functions entering (3.5.16), (3.5.17) are
defined by algebraic relations
1 i 2
Bi = 2(1 + γ)Ū i (xB ) − (1 + 2γ)vB
i
Ū (xB ) − i
vB vB − QvB , (3.5.21)
2
B ij
= Z ij
+ 2(1 + γ)Ū <i,j>
(xB ) − 2(1 + γ)vB Ū (xB ) + 2a j>
B aB , (3.5.22)
B iL
= Z <iL>
+ 2(1 + γ)Ū <i,L>
(xB ) − 2(1 + 
γ)vB Ū (xB ) (l ≥ 2), (3.5.23)
where a comma denotes a partial derivative with respect to spatial global coordinates taken as many
times as the number of indices standing after it, and some residual gauge freedom parametrized by
STF Cartesian tensors Z L has been left out and is explicitly shown. Functions Z L can be chosen, for
example, to make the coefficients BL (l ≥ 2) nil but we do not make this choice and leave Z L arbitrary.
The external (with respect to body B) potentials Ū , Ū i , Ψ̄, χ̄ are obtained by subtracting the local
gravitational potentials of the body B from the total integrals taken over all bodies of N -body system
[94],
X
Ū (xB ) = UA (xB ) (3.5.24a)
A6=B
i
X i
Ū (xB ) = UA (xB ), (3.5.24b)
A6=B
X
Ψ̄(xB ) = ΨA (xB ) (3.5.24c)
A6=B
X
χ̄(xB ) = χA (xB ), (3.5.24d)
A6=B
i
where the potentials UA , UA , ΨA , χA are given by integrals (3.4.13)–(3.4.20) respectively. Each of the
i
potentials, ŪA , ŪA , and χ̄A is a linear functional of mass density ρ∗ of body A alone. On the other
hand, the potential Ψ̄A is split in five parts two of which, ΨA2 and ΨA3 , given by (3.4.17) and (3.4.18)
respectively, depend on the gravitational potential of external (with respect to body A) bodies which
includes the potential of body B. The gravitational potential of body B must be left in the integrands of
ΨA2 and ΨA3 as it describes the back action of the body B on the external gravitational field of body
A. This subtle point in the correct formulation of the external gravitational field for a massive body in
N -body system, was pointed out by Fichtengoltz [70] and emphasized in [5, 85, 107].
The anti-symmetric rotational matrix F ik ≡ εipk F p is a solution of ordinary differential equation
[94, 158]
dF ik [i [i
= −2(1 + γ)Ū [i,k] (xB ) + (1 + 2γ)vB Ū ,k] (xB ) + vB Qk] , (3.5.25)
dt
The first term in the right side of (3.5.25) describes the Lense-Thirring (gravitomagnetic) precession,
the second term describes the de-Sitter (geodetic) precession in the scalar-tensor theory of gravity, and
the third term describes the Thomas precession depending on the local (non-geodesic) acceleration
Qk = δ kp Qp of the origin of the local coordinates with respect to a geodesic world line of a freely-
falling particle [see section 3.6.4]. In the scalar-tensor theory both the Lense-Thirring and the de-Sitter
precession depend on the PPN parameter γ while the Thomas precession does not. The reason is that the
Thomas precession is generically a special relativistic effect [115] that cannot depend on any particular
choice of a specific gravitational theory. The presence of matrix F ik in the spatial part of the post-
Newtonian transformation (3.5.17) means that the spatial axes of the local coordinates has a kinematic
rotation with respect to spatial axes of the global coordinates.
Bootstrap Effect and Self-forces

It is important to emphasize that all internal self-forces inside the bodies cancelled out in the process
of matching of the external and internal solutions of the metric tensor. It means that the internal po-
tentials characterizing the strength of gravitational and/or scalar field inside the bodies do not affect
the relative motion of the bodies with respect to each other. This fact is an important ingredient of the
post-Newtonian theory of motion of extended bodies telling us that there is no body’s bootstrap effect
(internal self-forces cancelled out) that is the third Newton’s law is not violated in the post-Newtonian
approximation 2 . The bootstrap effect is absent in conservative approximations of general relativity and
scalar-tensor theory of gravity. However, in radiative approximations taking into account emission of
gravitational waves the bootstrap effect may be present as the gravitational radiation-reaction force is
basically a self-force – the retarded interaction of the body with its own radiative field [81, 82, 127] –
which may lead to self-accelerated solution of the post-Newtonian equations of motion depending on
higher-order time derivatives. One of us (S.K.) studied the origin of the bootstrap effect with the help
of delay equations that model field retardation effects and predict runaway modes [30]. It was shown
that when retardation effects are small, the physically significant solutions belong to the so-called slow
manifold of the system which is identified with the attractor in the state space of the delay equation.
It was also demonstrated via an example that when retardation effects are no longer small, the motion
could exhibit bifurcation phenomena that are not contained in the local equations of motion. These kind
2 The term bootstrap effect is attributed to R. E. Raspe’s story “The surprising adventures of Baron Münchausen”,
where the main character pulled himself along with his horse out of a swamp by his hair pigtail, thus, surpassing
the third Newton’s law in this tall-tale story.
of effect are absent in the post-Newtonian approximation, and we do not discuss them in the rest of the
chapter.
World Line of the Origin of the Local Coordinates

Matching the metric tensors in the local and global coordinates yields the equations of translational
motion of the origin of the local coordinates, xiB = xiB (t), with respect to the global coordinates
[93, 94]. Namely,
aiB = Ū,i (xB ) − Qi + (3.5.26)

1
Ψ̄,i (xB ) − χ̄,itt (xB ) + 2(γ + 1)Ū˙ (xB ) − 2(γ + 1)vB Ū (xB ) −
2 i k k,i
2
i ˙ 2 i k
(2γ + 1)vB Ū (xB ) − 2(β + γ)Ū (xB )Ū,i (xB ) + γvB Ū,i (xB ) − vB vB Ū,k (xB ) +

1 i k
F + vB vB Qk + [A − 2Q + vB + (γ + 2)Ū (xB )]Qi + O(4 ) ,
ik 2
2
where a dot above function denoted the total derivative with respect to time t, aiB = d2 xiB /dt2 is a
coordinate acceleration of the origin of the local coordinates, Qi is a dipole moment (l = 1) of the
external solution for the metric tensor in the local coordinates that appears in h00 component of the
metric tensor perturbation (3.4.43), and comma denotes a partial derivative with respect to a spatial
global coordinates. It should be noticed that the post-Newtonian terms in (3.5.26) depending on Ū,i and
Qi can be always reshuffled by making use of the Newtonian approximation of this equation, Qi =
Ū,i − aiB .
The acceleration, aiB , is explicitly expressed in terms of the external gravitational potentials, Ū , Ū i ,
etc., and the dipole moment Qi = δ ij Qj . It is remarkable that function Qi = Qi (u) is not limited by
the gauge conditions and can be chosen arbitrary. Physically, it determines the magnitude and direction
of the inertial force acting in the local coordinates on a test particle being in a free fall [115, 117]. Only
after fixing the choice of Qi formula (3.5.26) becomes an ordinary differential equation which can be
solved to find out the world line W of the origin of the local coordinates on the spacetime manifold.
A trivial choice of Qi = 0 looks attractive as it immediately converts (3.5.26) to a fully-determined
differential equations. It is this choice that has been made by Dixon [62] but it means that the origin of
the local coordinates moves along a geodesic world line of a test particle falling freely in the background
spacetime defined by the external part of the local metric tensor. Unfortunately, this choice does not
allow us to keep the origin of the local coordinates at the center of mass of the body B possessing non-
vanishing internal multipole moments IL . This is because the internal moments of the body B interact
with the tidal gravitational field of other bodies from N -body system that forces the body B move along
an accelerated (non-geodesic) world line [101]. Thus, the world line Z of the center of mass of the body
B is not a geodesic, and Qi 6= 0 must be chosen to ensure that the center of mass of the body B is at
the origin of the local coordinates associated with this body at any instant of time. This is equivalent to
solving the internal problem of motion of the body’s center of mass with respect to the local coordinates.
Solution of this problem has been found in [93, 94, 101]) and will be discussed in section 3.6.4.
The External Multipoles

Matching determines the external multipoles in terms of partial derivatives from the gravitational poten-
tials of the external bodies [93, 94]. The external multipoles of the scalar field are
PL = ϕ̄,L (xB ) + O(2 ) , (l ≥ 0) (3.5.27)

Section 3.6 Post-Newtonian Equations of Motion of Extended Bodies in Local Coordinates 97
where the external scalar field ϕ̄ coincides in this approximation with the external Newtonian potential
Ū and is computed at the origin of the local coordinates, xiB (t), at the instant of time t. We remind
that the scalar field perturbation is coupled with the factor γ − 1, so that all physically-observed scalar-
field effects must be proportional to this factor. We also notice that the external monopole (l = 0) and
dipole (l = 1) of the scalar field can not be removed from the observable gravitational effects by making
coordinate transformation to a freely-falling frame because if a scalar field presents in one coordinate
frame it must be present in any other coordinates. In other words, scalar fields do not obey the principle
of equivalence. It was the primary reason why Einstein abandoned a pure scalar theory of gravity.
External mass-type multipoles QL for l ≥ 2 are defined by the following equation 3
QL = Ū ,<L> (xB ) (3.5.28)

1
+ 2 Ψ̄,<L> (xB ) − χ̄,tt<L> (xB ) + 2(1 + γ)Ū˙ <il ,L−1> (xB )
2
2(1 + γ)v k Ū k,<L> (xB ) + (l − 2γ − 2)v l Ū˙ ,L−1> (xB )
k
+ (1 + γ)vB Ū ,<L> (xB )− vB vB Ū (xB ) − lγ Ū (xB )Ū ,<L> (xB )
2
 (xB ) − lF k<il Ū ,L−1>k (xB ) + K L

+ Ż L + lAŪ ,<L> (xB ) + O(2 ) ,
where we have used notations
K ij ≡ 3a
B aB , (3.5.29)
L
K ≡ 0, (l ≥ 3). (3.5.30)
External current-type multipoles C L for l ≥ 2 are given by

4l(1 + γ) [i ,j]L−1
εipj C pL−1 = vB Ū (xB ) − Ū [i,j]L−1 (xB )
l+1

l − 1 il−1 [i ˙ ,j]L−2
− δ Ū (xB ) + O(2 ), (3.5.31)
l
where the dot denotes the time derivative with respect to time t. The external multipole moments QL and
CL are analogues of Dixon’s multipole moments Aα1 ...αl µν and Bα1 ...αl µν respectively [see (3.1.13)
and (3.1.14)]. We shall use the above expressions for the external multipoles in derivation of equations
of motion of extended bodies in next section.
3.6 Post-Newtonian Equations of Motion of Extended

Bodies in Local Coordinates
Matching equations yield equations of motion of the world line W of the origin of the local coordinates
with respect to the global coordinates as a function of yet undetermined, external dipole moment Qi .
This moment is not arbitrary but must be derived from the condition that the world line W coincides
with the world line Z of the body’s center of mass. To solve this problem we will have to find out the
translational equations of motion of the center of mass of the body B. The most natural approach to
3 We remind that the spatial indices are raised and lowered with the Kronecker symbol δ ij .
derive these equations is to employ the local coordinates. Derivation of the equations of motion of the
body’s center of mass in the local coordinates can be done in two different ways which include:
1. method of integration of microscopic equations of motion of matter proposed by Fock [73, 126]
and Papapetrou [123–125] ;
2. method of surface integrals introduced by Einstein, Infeld and Hoffmann (EIH) [68] and signifi-
cantly improved by Thorne and Hartle [147]. This EIH method is thoroughly explained in textbook
[4].
Fock-Papapetrou method is operating with a continuous distribution of matter and will be employed
in this chapter. We define the center of mass and linear momentum of the body B, derive the post-
Newtonian microscopic equations of motion of matter of the body in the local coordinates and, then,
integrate them over the volume of the body B in order to get the law of evolution of the linear momentum
of the body. As soon as the law of evolution is set up, the external dipole moment Qi follows from the
condition that the linear momentum is conserved. We explain this procedure below in more detail.
3.6.1 Microscopic post-Newtonian Equations of Motion

The microscopic post-Newtonian equations of motion of matter include: (1) equation of continuity, (2)
thermodynamic equation relating the elastic energy, Π, to the stress tensor, παβ , and (3) the Navier-
Stokes equation.
The equation of continuity in the local coordinates wα = (cu, w) has the most simple form if we use
the invariant density ρ∗ , defined in (3.3.18). It reads
∂ ρ∗ ν i

∂ρ∗
+ =0, (3.6.1)
∂u ∂wi
where ν i = dwi /du is a coordinate velocity of matter in the local coordinates. This equation is exact
with all post-Newtonian corrections taken into account.
The thermodynamic equation relating the internal elastic energy, Π, and the stress tensor, παβ , is
required only in a linearised approximation where the stress tensor is completely characterized by its
spatial components πij . Hence, one has from Eq. (3.2.10) the following differential equation
dΠ πij ∂ν i
+ ∗ = O(2 ), (3.6.2)
du ρ ∂wj
where the operator of convective derivative is d/du ≡ ∂/∂u + ν i ∂/∂wi .

The Navier-Stokes equation follows from the spatial component of the law of conservation of the
energy-momentum tensor, T iν ;ν = 0. It yields
∂(πij ν j )

d 1 2 1 ĥ i
ρ∗ ν i + 2 ν + Π + ĥ00 + ν + ĥ0i + 2 = (3.6.3)
du 2 2 3 ∂u
1 ∂ lˆ00

1 ∗ ∂ ĥ00 ∂πij 1 ∂ ĥ00
ρ i
− j
+ 2 ρ∗ + (ν 2 + 2Π + ĥ00 ) +
2 ∂w ∂w 2 ∂wi 4 ∂wi

1 2 ∂ ĥ ∂ ĥ0k 1 ∂ ĥ 1 ∂ ĥ00 1 ∂ ĥ
ν + νk + πkk + πik − +
6 ∂wi ∂wi 6 ∂wi 2 ∂wk 3 ∂wk

1 1 ∂πik
ĥ00 − ĥ + O(4 ) ,
2 3 ∂wk
where the metric tensor perturbations ĥ00 , lˆ00 , ĥ0i , ĥij and ĥ = ĥii in the local coordinates have been
defined in section 3.4.2.
3.6.2 Post-Newtonian Mass of an Extended Body
There are two algebraically independent definitions of the post-Newtonian mass in the scalar-tensor
theory – the active mass and the conformal mass which are defined respectively by equations (3.4.51)
and (3.4.54) for multipolar index l = 0.
More specifically, the active mass of body B covered by the local coordinates, is [93, 94]
M = MGR (3.6.4)
Z
1 1
+2 (γ − 1)Ï(2) − η ρ∗ ÛB d3 w − [A + (2β − γ − 1)P ]MGR
6 2
VB
∞
X 1
− [(γl + 1)QL + 2(β − 1)PL ]I<L> + O(4 ) ,
l!
l=1
where Z
1 2 1
MGR = ρ∗ 1 + 2 ν + Π − ÛB d3 w + O(4 ) (3.6.5)
2 2
VB
is general-relativistic definition of the post-Newtonian mass of the body [153], and

Z
I(2) = ρ∗ w 2 d 3 w , (3.6.6)
VB
is the rotational moment of inertia of the body B with respect to the origin of the local coordinates,
I<L> are active multipole moments of the body B defined in (3.4.51), two dots above I(2) denote a
second derivative with respect to time u.
Mass, MGR , depends only on the internal distribution of mass, kinetic, thermal and gravitational
energy densities. In case of a single, isolated body residing in asymptotically-flat spacetime, this mass
coincides with the post-Newtonian expansion of the Tolman mass [97, 148]. If the body is isolated, the
post-Newtonian mass MGR is conserved. However, in N -body system the gravitational interaction of the
body B with external bodies causes tides which change the internal distribution of matter and shape of
the body B, thus, leading to dependence of MGR on time. It is governed by the ordinary differential
equation [93]
∞
X 1
ṀGR = 2 QL İ<L> . (3.6.7)
l!
l=1
The conformal mass of the body B is [93, 94]

∞
X l+1
M = MGR − 2 [A + (1 − γ)P ]MGR + QL I<L> + O(4 ) , (3.6.8)
l!
l=1
Relation between the active and conformal masses is obtained by comparing (3.6.4) and (3.6.8) and
reads [93, 94]
Z
1 1
M = M + 2 η ρ∗ ÛB d3 w − (γ − 1)Ï(2) (3.6.9)
2 6
VB
∞ ∞
X 1 X 1
+2(β − 1) MP + PL I<L> + (γ − 1) QL I<L> + O(4 ) ,
l! (l − 1)!
l=1 l=1
where η = 4β − γ − 3 is called the Nordtvedt parameter [153]. Numerical value of this parameter is
known with the precision better than |η| < 2 × 10−4 from LLR experiment [120].
One can see that in the scalar-tensor theory of gravity the conformal mass of the body differs from
its active mass. It was noticed by Dicke[52, 54] and Nordtvedt [118, 153] who has derived the integral
term being proportional to the Nordvedt parameter η in the right side of (3.6.9). Actual difference is
more complicated and includes the term with the time derivative of the rotational moment of inertia of
the body as well as the tidal contributions. If an extended body were completely isolated from external
gravitational field, the difference between the active and conformal masses would be due to the Nordtvedt
effect caused by η 6= 0, and the time-dependence of the body’s rotational moment of inertia (for example,
because of radial oscillations, etc.). In case when the presence of gravitational field of the external bodies
of N -body system cannot be ignored, we also have to account for the gravitational coupling between the
external gravitational field multipoles, PL and QL , with the internal multipole moments IL of the body
B which yields additional contribution to the difference between the masses.
It is interesting to notice that a rigorous post-Newtonian definition (3.6.8) of conformal mass M of
an extended body in N -body system contains a post-Newtonian contribution of the coupling terms ∼
QL I<L> which can be interpreted as a reminiscence of Mach’s principle stating that mass of each body
originates from its gravitational interaction with the external universe. Our calculations do not confirm
Mach’s principle completely but indicate that Mach’s idea has got a partial support and justification in
general theory of relativity.
3.6.3 Post-Newtonian Center of Mass and Linear Momentum of an Extended Body
Functional form of equations of motion of extended bodies in N -body system depends crucially on the
choice of the reference point inside the body defining its center of mass. Position of the center of mass of
the body is determined by the value of the internal dipole moment of the body. In scalar-tensor theory of
gravity there are two possible definitions of the dipole moment: active dipole moment Ii and conformal
dipole moment I i . It is difficult to foresee which definition is the best. Only after completing direct
calculation of equations of motion it becomes clear that it is the conformal dipole moment yields the
most optimal choice of the post-Newtonian center of mass of each body [93, 94, 158]. This is because
after double differentiation with respect to time, only the conformal dipole moment leads to conservation
of the individual body’s linear momentum, pi , while the post-Newtonian scalar or active dipole moments
do not bear such a property.
Thus, we define the post-Newtonian center of mass of the body by making use of (3.4.54) for the
multipolar index l = 1. After some simplifications and re-arrangements of terms in the integrand, the
conformal dipole moment reads
Z
1 2 1
Ii = ρ∗ wi 1 + 2 ν + Π − ÛB d3 w (3.6.10)
2 2
VB
Z ∞
X l+1
−2 [A + (1 − γ)P ] ρ∗ wi d3 w + QL I<iL>
l!
VB l=1
∞
1 X 1
+ QiL NL + O(4 ) ,
2 (2l + 3)l!
l=0
where Z
NL = ρ∗ w2 w<L> d3 w , (l ≥ 0) (3.6.11)
VB
are the moments generalizing the rotational moment of inertia. Indeed, for l = 0 the scalar function N =
I(2) . Dipole moment is a function of the local coordinate time u only and it defines the displacement of
the cenetr of mass of the body from the origin of the local coordinates. Hence, if the origin of the local
coordinates coincides with the center of mass of the body, the dipole moment vanishes I i = 0.
The post-Newtonian definitions of mass and the center of mass of the body depend not only on
the distribution of density of matter, its velocity and stresses inside the body but also on other terms
describing mutual gravitational coupling of internal and external multipoles. Thorne and Hartle [147]
believed that these terms introduce ambiguity to the definitions of mass and the center of mass as they
can be either included or excluded to these definitions. We have shown [93, 94, 158] that this ambiguity
is fictitious and does not exists. The coupling terms must be included to the definitions of mass, center
of mass and other internal multipoles to achieve the most economic form of the equations of motion
depending only on the mass-type and spin-type internal multipole moments of the bodies. It is also true
that only under such definitions we can get EIH equations of motion in the limiting case of spherically-
symmetric and non-rotating bodies.
The post-Newtonian linear momentum of the body, pi , is defined as the first derivative of the dipole
moment (3.6.10) with respect to the local time u,
pi = I˙i (u) , (3.6.12)
where the dot denotes the time derivative with respect to u. After taking the derivative and using the
local equations of motion of matter (3.6.3), we obtain:
Z
1 2 1
pi = ρ ∗ ν i 1 + 2 ν + Π − ÛB d3 w (3.6.13)
2 2
VB
Z
1 ∗ i 3
+2 πik ν k − ρ ŴB d w
2
VB
∞ ∞
d X l+1 1X 1
−2 [A + (1 − γ)P ]I i + QL I<iL> + QiL NL
du l! 2 (2l + 3)l!
l=1 l=0
∞ Z
X 1 l
+2 QL İ<iL> + QiL−1 ṄL−1 − QL ρ∗ v i w<L> d3 w + O(4 ) ,
l! 2l + 1
l=1 VB
where the new internal potential
ρ∗ (u,w0 )0k (wk − w0k )(wi − w0i ) 3 0

Z
ŴBi = G d w . (3.6.14)
|w − w0 |3
VB
Until now the point xB (t) has represented a location of the origin of the local coordinate system in
the global coordinates taken at the time t. In general, the origin of the local coordinates may not coincide
with the center of mass of the body B which can move with respect to the local coordinates with some
velocity and acceleration. In order to be able to keep the center of mass of the body at the origin of the
local coordinates we have to prove that for any instant of time the dipole moment defined by (3.6.10)
and its time derivative that is, the linear momentum of the body) given by (3.6.13, can be kept equal to
zero. This requirement can be satisfied, if and only if, the second time derivative of the dipole moment
with respect to the local coordinate time u vanishes, that is
ṗi (u) = 0. (3.6.15)
It is remarkable that this equation expressing conservation of the linear momentum, can be satisfied after
making an appropriate choice of the external dipole moment Qi that characterizes a locally measurable
acceleration of the origin of the local coordinates with respect to a geodesic world line of a freely falling
test particle. We prove it in the next section 3.6.4.
3.6.4 Translational Equation of Motion in the Local Coordinates
Translational equations of motion of the center of mass of body B with respect to the local coordinates
wα , which cover the body and its neighbourhood, are derived by making use of definition (3.6.13) of
its linear momentum, pi . Differentiating (3.6.13) one time with respect to the local coordinate time u,
making use of the microscopic equations of motion (3.6.1)-(3.6.3), and integrating by parts to re-arrange
a number of terms, one obtains [93, 158]
∞
X 1
ṗi = MQi + QiL I<L> (3.6.16)
l!
l=1
∞
X
1
−2 [(l2 + l + 4)QL + 2(γ − 1)PL ]Ï<iL>
(l + 1)!
l=2
∞
X 2l + 1
+ [(l2 + 2l + 5)Q̇L + 2(γ − 1)ṖL ]İ<iL>
(l + 1)(l + 1)!
l=2
∞
X 2l + 1
+ [(l2 + 3l + 6)Q̈L + 2(γ − 1)P̈L ]I<iL>
(2l + 3)(l + 1)!
l=2
3
+[3Qk + (γ − 1)Pk ]Ï<ik> + [4Q̇k + (γ − 1)Ṗk ]İ<ik>
2
∞
3 X 1
+ [5Q̈k + (γ − 1)P̈k ]I<ik> + ŻiL I<L>
5 l!
l=2
∞
X 1 l+2
+ εipq ĊpL I<qL> + CpL İ<qL>
(l + 1)! l+1
l=1
∞
X l+1
−2 εipq (2QL + (γ − 1)PpL )Ṡ<qL>
(l + 2)!
l=1
X ∞
l+1 l(l + 2)
+ (2Q̇pL + (γ − 1)ṖpL )S<qL> − CiL S<L>
l+2 (l + 1)(l + 1)!
l=1
1
− εikq [(4Qk + 2(γ − 1)Pk )Ṡq + (2Q̇k + (γ − 1)Ṗk )Sq ]
2 Z
1 1
+(P i − Qi ) η ρ∗ Û (B) d3 w − (γ − 1)Ï(2) +
2 6
VB
∞
! ∞
X 1 X 1
+2(β − 1) MP + PL I<L> + (γ − 1) QL I<L>
l! (l − 1)!
l=1 l=1
where the spatial indices are raised and lowered with the Kronecker symbol, the matching condition
PL = QL + O(2 ) (l ≥ 2) have been used, and all terms which are proportional to I i and/or pi and
vanish when the origin of the local coordinates coincides with the center of mass of body B at any
instant of time have been omitted, thus, making I i = 0 and pi = 0. We shall use these conditions in the
derivation of translational equations of motion of the center of mass of the body B. The omitted terms
can be found in [93, Eq. 6.19].
It is important to notice that the right side of (3.6.16) is the force exerted on the body B due to
the gravitational coupling of its internal active multipole moments, IL , SL with the external multipole
moments QL , PL , CL . The right side of (3.6.16) also contains the inertial force, MQi , due to the non-
geodesic motion of the origin of the local coordinates. Hence, in the most general case, ṗi 6= 0 which
means that the center of mass of the body B accelerates with respect to the origin of local coordinates
and the world line Z of the center of mass does not coincide with the world line W of the origin of
the local coordinates. However, the external dipole moment Qi is yet unconstrained function of time
and we can always reach consensus with the condition (3.6.15) by choosing Qi in such a way that it
compensates the gravitational coupling force. To eliminate acceleration of the world line Z with respect
to W and to fix the center of mass of the body B at the origin of the local coordinates we, first, impose
condition (3.6.15) and solve (3.6.16) with respect to Qi . It constrains the inertial force and yields
∞
X 1
M Qi = (M − M) P i − QiL I<L> (3.6.17)
l!
l=1
∞
X
1
+2 [(l2 + l + 4)QL + 2(γ − 1)PL ]Ï<iL>
(l + 1)!
l=2
∞
X 2l + 1
+ [(l2 + 2l + 5)Q̇L + 2(γ − 1)ṖL ]İ<iL>
(l + 1)(l + 1)!
l=2
∞
X 2l + 1
+ [(l2 + 3l + 6)Q̈L + 2(γ − 1)P̈L ]I<iL>
(2l + 3)(l + 1)!
l=2
3
+[3Qk + (γ − 1)Pk ]Ï<ik> + [4Q̇k + (γ − 1)Ṗk ]İ<ik>
2
∞
3 X 1
+ [5Q̈k + (γ − 1)P̈k ]I<ik> + ŻiL I<L>
5 l!
l=2
∞
X 1 l+2
+ εipq ĊpL I<qL> + CpL İ<qL>
(l + 1)! l+1
l=1
∞
X l+1
−2 εipq (2QL + (γ − 1)PpL )Ṡ<qL>
(l + 2)!
l=1
X ∞
l+1 l(l + 2)
+ (2Q̇pL + (γ − 1)ṖpL )S <qL>
− CiL S<L>
l+2 (l + 1)(l + 1)!
l=1
1
− εikq [(4Qk + 2(γ − 1)Pk )Ṡq + (2Q̇k + (γ − 1)Ṗk )Sq ] ,
2
where M and M are the conformal and active gravitational masses of the body B and we again omitted
terms which are proportional to the dipole moment I i and linear momentum pi of the body like we did it
in (3.6.16). The two masses, M and M, are not equal according to (3.6.9). The difference between them
plays a role of a “charge” of the scalar field φ which couples with the dipole moment P i = Ū,i of the
external scalar field leading to the Dicke-Nordtvedt effect [52, 54, 153]. In general relativity, M = M,
and the first term in the right side of (3.6.17) vanishes.

Equation (3.6.17) is a condition for the fulfilment of the law of conservation of linear momentum
(3.6.15) in local coordinates. It ensures that the world line W of the origin of local coordinates does
not accelerate with respect to the world line Z of the center of mass of body B. Equation (3.6.17)
does not warranty, however, that W and Z coincides. The origin of the local coordinates still can move
uniformly with respect to the center of mass of the body. To eliminate this uniform motion we impose
condition, pi = 0. The freedom which remains is a constant relative displacement of the origin of the
local coordinates with respect to the center of mass of the body. This constant displacement is removed
by additional constrain imposed on the internal (conformal) dipole moment of the body, I i = 0. This
procedure ensures that the world lines W and Z coincide.
The right side of (3.6.17) devided by M, must be substituted for Qi in the equations of motion of the
origin of the local coordinates (3.5.26) to convert it to the equations of motion of the center of mass of
the body B in the global coordinates. These equations still contain the external gravitational potentials
like Ū , Ψ̄, Ū i , etc., which are given in the form of integrals expressed in the global coordinates. These
integrals should be explicitly expanded with respect to the internal multipoles of the bodies in order to
complete the theory. We shall conduct this calculation in next section and derive equations of motion
of extended bodies in N -body system in terms of their internal multipoles as well as coordinates and
velocities of their centers of mass.
3.7 Post-Newtonian Equations of Motion of Extended

Bodies in Global Coordinates
3.7.1 STF Expansions of the External Gravitational Potentials in Terms of the
Internal Multipoles
External gravitational potentials Ū , Ψ̄, Ū i , χ̄ are given in the form of volume integrals (3.5.24) which
are not convenient for practical calculations in relativistic celestial mechanics. The practice is to expand
the gravitational potentials of external bodies in terms of their own internal multipole moments – mass,
dipole, quadrupole, etc. The expansion itself is rather straightforward but the problem here is that the
external potentials are given as integrals in the global coordinates, xα , and the multipolar expansion will
be obtained in terms of the internal multipole moments of the bodies expressed in the global coordinates.
However, the internal multipoles of an extended body makes physical sense only when they are expressed
in the local coordinates, wα , associated with that body. Therefore, we have to convert the internal
multipoles of each body from the global to local coordinates associated with this body.
We have already introduced the local coordinates, wα = (cu,wi ), associated with body B. In a
similar fashion the local coordinates can be introduced near any other extended body C 6= B. We
shall use the letter C as a sub-index to label the bodies as well as the local coordinates and functions
α i
associated with them. The local coordinates of the body C are denoted wC = (cuC ,wC ). Coordinate
α α
transformation between wC and the global coordinates, x , is similar to (3.5.16) and (3.5.17) except
that now we have to pin the label C to all quantities,

1 k k 1 1
uC = t − 2 (AC + vC RC ) + 4 BC +
k k
vC aC − Ū˙ (xC ) + Q̇C RC 2
3 6 6
∞
1 k k 2 X 1 L L
− ȧC RC RC + BC RC + O(5 ) , (3.7.1)
10 l!
l=1
Section 3.7 Post-Newtonian Equations of Motion of Extended Bodies in Global Coordinates 105

i i 1 i k
wC = RC + 2 vC vC + δ ik γ Ū (xC ) − δ ik AC + FCik RC
k
2

1
+ akC RC
i k
RC − aiC RC 2
+ O(4 ) , (3.7.2)
2
where
i
RC = xi − xiC , (3.7.3)
and xiC = xiC (t) marks the global coordinates of the origin of the local coordinates of the body C,
i
vC = dxiC /dt, aiC = dvC
i
/dt. Defining equations for all other functions entering (3.7.1), (3.7.2) remain
similar to corresponding equations shown in section 3.5.2.
Let us introduce the following notations
ij
DC ≡ δ ik γ ŪC (xC ) − δ ik AC , (3.7.4)
ijk 1 j ik
DC ≡ (a δ + akC δ ij − aiC δ jk ) , (3.7.5)
2 C
that will allow us to shorten subsequent equations.
The external gravitational potentials are integrals of the global coordinates which integrands contain a
kernel, 1/|x − x0 |, of Green’s function of the Laplace equation. This kernel is expanded into multipolar
series as follows
∞
(−1)l

1 1 X 1
= = R0<L> , (3.7.6)
|x − x0 | |x − xC − (x0 − xC )| l! RC ,L C
l=0
where
0i
RC = x0i − xiC , (3.7.7)
and (1/RC ),L denotes a partial derivative of l-th order with respect to spatial coordinates (1/RC ),L ≡
∂L (1/RC ) ≡ ∂i1 ...il (1/RC ) where each partial derivative ∂i = ∂/∂xi . Post-Newtonian transformation
0i
of RC to the local coordinates, wC i
, is slightly different from (3.7.2) because the points, xi and x0i are
i 0i
lying on a hypersurface of constant time t, while the points wC and wC are lying on a hypersurface
0i
of constant local time uC . It requires additional Lie transport of wC from one hypersurface to another
0i
which shows that the corresponding post-Newtonian transformation of RC reads [25, 93, 102]

0i 0i 1 i k 0k ijk 0j 0k
RC = wC − 2 vC vC + DC ik
+ FCik wC + DC wC wC (3.7.8)
2

0i k 0k
+ 2 νC vC wC k
− wC + O(4 ) ,
where
0i i
νC = v 0 − vC
i
+ O(2 ) , (3.7.9)
is a relative velocity of matter of the body C with respect to the origin of its own local coordinates.
Equation (3.7.8) allows us to transform the Newtonian counterpart of the internal multipole moments
in the global coordinates to the local coordinates as follows [93, 94, 158]
Z Z
0 0<L> 3 0 l k <il L−1>k
ρ∗0C RC0<L> 3 0
d x = ρ∗C wC d wC + 2 − vC vC IC (3.7.10)
2
VC VC
k<il L−1>k k<il L−1>k i >jk
+ lFC IC − lDC IC − lIjk<L−1
C DCl
Z
0
− vC İC
k k<L>
RC İC
k k <L>
+ vC k
+ vC ρ∗C νC0k wC
0<L> 3 0
d wC + O(4 ),
VC
where we have used the fact that the product of the invariant density, ρ∗ with 3-dimensional coordi-
nate volume is Lie-invariant and does not change when transported from one hypersurface to another
along world lines of matter and transformed from the global to local coordinates, that is ρ∗ (t,x)d3 x =
ρ∗ (uC ,wC )d3 wC ≡ ρ∗C d3 wC exactly [93] .
The Newtonian potential of body C in the global coordinates is transformed with the help of (3.7.6)–
(3.7.10) to the local coordinates as follows [158]
Z ∗
ρ (t,x0 ) 3 0
UC (t,x) = G d x (3.7.11)
|x − x0 |
VC
∞
(−1)l

X 1
= G I<L>
C + 2 AC + (2β − γ − 1)PC I<L>C
l! RC ,L
l=0
0
π kk
Z
0 1 02
− 2 ρ∗C (γ + )νC 0<L> 3 0
+ Π0C + γ C∗ 0 − (2β − 1)ÛC0 wC d wC
2 ρC
VC
2

2l + 1 <L>
− N̈C
<L>
− 4(1 + γ) ṘC
2(2l + 3) l+1
∞ Z
X 1 0 0<K> 0<L> 3 0
+ 2 QKC + 2(β − 1)PC
K
ρ∗C wC wC d wC
k! C
k=1

l k <il L−1>k kk
C − lDC l IL−1>k
C
2

i >jk
− lIjk<L−1
C DCl − vC İC
k k<L>
+ vC RC İC
k k <L>
∞
(−1)l

X 1
+ 2 G vC İC
k <kL>
(l + 1)! RC ,L
l=1
∞
(−1)l l

X 1
− 2 G k
εkpq vC S<qL−1>
(l + 1)! RC ,pL−1 C
l=1
∞
(−1)l (2l − 1) k

X 1
2
+ G vC R<L−1> + O(4 ) ,
(2l + 1)l! RC ,kL−1 C
l=1
where two additional multipole moments

Z
NC
L
= ρ∗C wC
2 <L> 3
wC d wC , (3.7.12)
C
Z
RL
C = ρ∗C νC
k <kL> 3
wC d wC . (3.7.13)
C
They were called "bad" moments in the Ref. [47] because they are not reduced to the “canonical”
multipole moments, IL and SL , which appear in the multipolar expansion of the metric tensor in the
local coordinates. There are also explicit dependence of expansion (3.7.11) on the integrals involving
i ij
interior physical quantities such as velocity of matter νC , potential energy ΠC , stresses πC , and self-
potential ÛC . We have proved that these “interior structure” integrals do not appear in the final equations
of motion, because they cancel out with the same integrals from the multipolar expansions of other
external potentials that are shown below. The “bad” multipoles, NC L

and RL
C , will not show up in the
final form of equations of motion for the same reason – the mutual cancellation with similar terms.
External vector potential is expressed in terms of the internal multipole moments as follows
ρ∗ (t,x0 )v 0i 3 0
Z
UCi (t,x) = G d x (3.7.14)
|x − x0 |
VC
∞ ∞
(−1)l (−1)l

X 1 X 1
= G I<L> vC
i
+ G İ<iL>
l! RC ,L C (l + 1)! RC ,L C
l=0 l=1
∞
(−1)l l

X 1
−G εipq S <qL−1>
(l + 1)! RC ,pL−1 C
l=1
∞
(−1)l (2l − 1) 1
X
+G R<L−1> + O(2 ) .
(2l + 1)l! RC ,iL−1 C
l=1
Multipolar expansion of the other external potentials is given by

Z ∗
ρ (t,x0 )v 02 3 0
ΨC1 (t,x) = G d x (3.7.15)
|x − x0 |
VC
∞
(−1)l
Z
X 1 ∗ 0 02 0<L> 3 0
= G I <L> 2
vC + ρC νC wC d wC
l! RC ,L C
l=0 VC
∞
(−1)l

X 1
+ 2G vC İC
m <mL>
(l + 1)! RC ,L
l=1
∞
(−1)l l

X 1
− 2G m
εmpq vC S<qL−1>
(l + 1)! RC ,pL−1 C
l=1
∞
(−1)l (2l − 1) m 1
X
+ 2G vC R<L−1> + O(2 ),
(2l + 1)l! RC ,mL−1 C
l=1
Z ∗
ρ (t,x0 )ϕ(t,x0 ) 3 0
ΨC3 (t,x) = G d x (3.7.16)
|x − x0 |
VC
∞
(−1)l
Z
X 1 0
= G ρ∗C UC0 wC
0<L> 3 0
d wC
l! RC ,L
l=0 VC
∞ X ∞ X ∞
(−1)l+p

X 1 1
+ G2 I<K>
B
l! k! p! R C ,L RBC ,KP
l=0 k=0 p=0
Z
0 0<L> 0 3 0
× ρ∗C wC wC d wC
VC
∞ X
∞ X∞
(−1)l+k

X X 1 1
+ G2 I<K>
D
l! k! p! R C ,L RCD ,KP
D6=C l=0 k=0
p=0
Z
0
× ρ∗C wC
0<L> 0 3 0
wC d wC + O(2 ) ,
VC
ρ∗ (t,x0 )Π(t,x0 ) 3 0
Z
ΨC4 (t,x) = G d x (3.7.17)
|x − x0 |
VC
∞
(−1)l
Z
X 1 0
= G ρ∗C Π0C wC
0<L> 3 0
d wC + O(2 ) ,
l! RC ,L
l=0 VC
Z kk 0
π (t,x ) 3 0
ΨC5 (t,x) = G d x (3.7.18)
|x − x0 |
VC
∞
(−1)l
Z
X 1 0kk 0<L> 3 0
= G πC wC d wC + O(2 ) ,
l! RC ,L
l=0 VC
where we have also used index D to numerate the bodies in (3.7.16). One more potential
ΨC2 = 6γΨC3 . (3.7.19)
The above expansions contain a number of integrals depending on the internal structure of the bodies
explicitly. They will mutually cancel out by similar terms after substitution of these expansions to
equations of motion (3.5.26).
3.7.2 Translational Equations of Motion

Translational equations of motion of the center of mass of body B in the global coordinates follows
directly from (3.5.26), (3.6.17) and the multipolar decompositions of external potentials provided in
section 3.7.1. The equations have the following form
MB aiB = FN
i
+ 2 FpN
i
+ O(4 ) , (3.7.20)
i
where MB is the inertial (conformal) mass of the body, FN is the Newtonian gravitational force, and
i
FpN is the post-Newtonian gravitational force.
i
The Newtonian gravitational force, FN , is given by a linear superposition of gravitational forces
exerted on the body B by external masses of N body system,
∞ X ∞
i
XX (−1)j (2j + 2l + 1)!! GI<J>
B I<L>
C <iJL>
FN =− 2j+2l+3
RBC , (3.7.21)
j!l! R BC
C6=B j=0 l=0
 <q1 ...ql >

where I<J>
B = IB 1 j are active STF multipole moments of the body B, I<L> C = IC are
i j
active STF multipole moments of the external body C, RBC = |RBC | = δijRBC RBC ,
i i i
RBC = RB − RC , (3.7.22)
is the coordinate distance between the centers of mass of the bodies defined by the condition of van-
ishing conformal dipole moment, I i = 0, of the body in its own local coordinates [see section 3.6.3],
R< iJL >BC = Rip1 ...pj q1 ...ql , and repeated indices mean the Einstein summation rule.
We emphasize that the Newtonian gravitational force (3.7.21) in scalar-tensor theory of gravity de-
pends on the active multipole moments and has a post-Newtonian contribution from the active dipole
moments Ii of the bodies (terms with l = 1 and j = 1) which do not vanish even if the center of mass of
the body is located at the origin of the local coordinates. This is because the center of mass was given in
terms of the conformal dipole moment by condition I i = 0. However, the active dipole moment Ii 6= I i
[93, Eq. 6.26].
Additional notice is that the inertial mass, MB , in the left side of (3.7.20) is the conformal mass of
the body B while the gravitational force (3.7.21) actually depends on the active masses of the bodies, M
which are the terms with l = 0 and j = 0 in (3.7.21). The two masses do not coincide as follows from
(3.6.9). It leads to violation of the strong principle of equivalence in scalar-tensor theory of gravity.
Post-Newtonian Gravitational Force

i
The post-Newtonian gravitational force, FpN , in (3.7.20) can be split in three parts,
I + FpN S + FpN ,
i i i i
FpN = FpN (3.7.23)
i i
where FpN I is caused by the mass-type multipole moments, FpN S is associated with the spin-type mul-
tipole moments, and FpN originates from the post-Newtonian transformation of the internal multipole
i
moments, that appear in calulations of section 3.7.1, from the global to local coordinates coordinates.
i
The mass-type multipole force FpN I consists of various terms describing gravitational coupling be-
tween two, three, four and five internal multipole moments of the extended bodies in N -body system.
The force depends on the first and second time-derivatives of the moments as well. It has the following
structure,
i i
FpN I = FII + FIi İ + FIi Ï + Fİi İ + FIII
i i
+ FIIII + (3.7.24)
i i i
FqII + FqI İ + FqIÏ + Fqiİİ + i
Fq̇II + i
Fq̇Iİ + i
Fq̈II + i
FqIIII ,
where each particular term denotes the number of the moments corresponding to the coupling order. The
terms in the first line of (3.7.24) describe the direct gravitational coupling of the multipole moments.
The terms in the second line of (3.7.24) labelled by a letter ‘q’ (quadrupole) originate from an indirect
coupling of the internal quadrupole moments of a body, I<ij> with the corresponding external dipole
moment Qi , which describes acceleration (3.6.17) of the local coordinates with respect to a geodesic
passing through the origin of the coordinates. Thus, all terms labelled with ‘q’ would be absent had we
chosen the local coordinates moving along time-like geodesics.
Specific expressions for different terms in (3.7.24) are given below in terms of the coordinate distances
(3.7.22) between the bodies and the corresponding multipolar tensor coefficients DII , DIİ , DIÏ , etc.,
which are given in section (3.7.2) below. The force components read
∞ X
∞ <iJL> <kJL>
i
XX JL RBC ikJL RBC
FII = G DII 2j+2l+3
+ DII 2j+2l+3
(3.7.25)
C6=B j=0 l=0
RBC RBC
<ikJL> <kmJL>
kJL RBC ikmJL RBC
+DII 2j+2l+5
+ DII 2j+2l+5
RBC RBC
!
<ikmJL> <ikmJL>
(1) ikmJL RBC (2) ikmJL RBC
+ DII 2j+2l+5
+ DII 2j+2l+7
,
RBC RBC
∞ X
∞ <JL> <iJL>
XX RBC RBC
FIi İ = G DIiJL
İ 2j+2l+1
+ DIJL
İ 2j+2l+3
(3.7.26)
C6=B j=0 l=0
RBC RBC
!
<kJL> <ikJL> <ikJL>
RBC RBC RBC
+DIikJL
İ 2j+2l+3
+ (1) DIkJL
İ 2j+2l+3
+ (2) DIkJL
İ 2j+2l+5
,
RBC RBC RBC
∞ X
∞ <JL>
XX RBC
FIi Ï = G DIiJL
Ï 2j+2l+1
(3.7.27)
C6=B j=0 l=0
RBC
!
<iJL> <iJL>
RBC RBC
+(1) DIJL
Ï 2j+2l+1
+ (2) DIJL
Ï 2j+2l+3
,
RBC RBC
∞ X
∞
!
<JL> <iJL>
XX RBC RBC
Fİi İ = G DİiJL
İ 2j+2l+1
+ DİJL
İ 2j+2l+3
, (3.7.28)
C6=B j=0 l=0
RBC RBC
∞ X
∞ <kmnJL>
i
XX ikmnJL RBC
FqII = G DqII 2j+2l+7
, (3.7.29)
C6=B j=0 l=0
RBC
∞ X
∞ <kmJL>
i
XX ikmJL RBC
FqI İ = G DqI İ 2j+2l+5
, (3.7.30)
C6=B j=0 l=0
RBC
∞ X
∞ <kJL>
i
XX ikJL RBC
FqI Ï = G DqI Ï 2j+2l+3
, (3.7.31)
C6=B j=0 l=0
RBC
∞ X
∞ <kJL>
XX RBC
Fqiİİ = G DqikJL
İİ 2j+2l+3
, (3.7.32)
C6=B j=0 l=0
RBC
∞ X
∞ <kmJL>
i
XX ikmJL RBC
Fq̇II = G Dq̇II 2j+2l+5
, (3.7.33)
C6=B j=0 l=0
RBC
∞ X
∞ <kJL>
i
XX ikJL RBC
Fq̇I İ = G Dq̇I İ 2j+2l+3
, (3.7.34)
C6=B j=0 l=0
RBC
∞ X
∞ <kJL>
i
XX ikJL RBC
Fq̈II = G Dq̈II 2j+2l+3
, (3.7.35)
C6=B j=0 l=0
RBC
∞ X
∞ X
∞ <iJL> <K>
i
X XX (1) RBC RCD
FIII = G2 JLK
DIII 2j+2l+3 2k+1
(3.7.36)
C6=B D6=C j=0 l=0 k=0
RBC RCD
∞ X
∞ X
∞ <iJL> <K>
X XX (2) RBC RBD
+G2 JLK
DIII 2j+2l+3 2k+1
,
C6=B D6=B j=0 l=0 k=0
RBC RBD
∞ X
∞ X
∞ X
∞ <JL> <iKS>
i
X XX (1) RBC RCD
FIIII = G2 JLSK
DIIII 2j+2l+1 2k+2s+3
(3.7.37)
C6=B D6=C j=0 l=0 k=0 s=0
RBC RCD
<imJL> <mKS> <imJL> <mKS>
RBC RCD JLSK RBC RCD
+(2) DIIII
JLSK
2j+2l+3 2k+2s+3
+ (3) DIIII 2j+2l+5 2k+2s+3
RBC RCD RBC RCD
<iJL> <mKS> <mJL> <mKS>
mJLSK RBC RCD JiLSK RBC RCD
+DIIII 2j+2l+3 2k+2s+3
+ DIIII 2j+2l+3 2k+2s+3
RBC RCD RBC RCD
!
JmLSK RBC RCD iJLSK RBC RCD
+ DIIII 2j+2l+3 2k+2s+3
RBC RCD RBC RCD
∞ X
∞ X
∞ X
∞ <JL> <iKS>
X XX RBC RBD
+G2 JSLK
DIIII 2j+2l+1 2k+2s+3
C6=B D6=B j=0 l=0 k=0 s=0
RBC RBD
!
mJSLK RBC RBD iJSLK RBC RBD
+ DIIII 2j+2l+3 2k+2s+3
,
RBC RBD RBC RBD
∞ X
∞ X
∞ X
∞ <pmJL> <mKS>
i
X XX RBC RCD
FqIIII = G2 ipJLSK
DqIIII 2j+2l+5 2k+2s+3
(3.7.38)
C6=B D6=C j=0 l=0 k=0 s=0
RBC RCD
∞ X
∞ X
∞ X
∞ <pqJL> <qKS>
X XX RBC RBD
+G2 ipJSLK
DqIIII 2j+2l+5 2k+2s+3
,
C6=B D6=B j=0 l=0 k=0 s=0
RBC RBD
The spin-type post-Newtonian force has the following structure
FSi = i
FSI i
+ FṠI + FSi İ + FSS
i i
+ FsII i
+ FsI i
İ + FṡII , (3.7.39)
where each component of the force is expressed in terms of the corresponding multipolar tensor coeff-
cients DSI , DṠI , etc. The coefficients are given in section (3.7.2) below. The terms labelled with ‘s’
stems from the cross product, εipq S k Qq , of body’s spin S p and the exterenal multipole moment Qq
after replacing Qq with the right side of (3.6.17) and making use of the rotational equations of motion
for spin of body (B = 1,2,...,N ) [93, see section 6.2.2]
∞
X 1
ṠiB = εijk QkL IjL
B + O( ) .
2
(3.7.40)
l!
l=0
The spin-type force components are

∞ X
∞ <pJL> <ipJL>
XX (1) RBC RBC
i
FSI = G ipJL
DSI 2j+2l+3
+ (2) DSI
pJL
2j+2l+5
(3.7.41)
C6=B j=0 l=0
RBC RBC
!
<kpJL>
(3) ikpJL RBC
+ DSI 2j+2l+5
,
RBC
∞ X ∞ <pJL>
ipJL RBC
i
XX
FṠI = G DṠI 2j+2l+3
, (3.7.42)
C6=B j=0 l=0
RBC
∞ X
∞
!
<pJL> <ipJL>
XX RBC pJL RBC
FSi İ = G ipJL
DS İ 2j+2l+3
+ DS İ 2j+2l+5
, (3.7.43)
C6=B j=0 l=0
RBC RBC
∞ X
∞ <ipqJL>
i
XX pJqL RBC
FSS = G DSS 2j+2l+7
, (3.7.44)
C6=B j=0 l=0
RBC
∞ X
∞ <kpJL>
i
XX ikpJL RBC
FsII = G DsII 2j+2l+5
, (3.7.45)
C6=B j=0 l=0
RBC
∞ X
∞ <pJL>
i
XX ipJL RBC
FsIİ = G DsI İ 2j+2l+3
, (3.7.46)
C6=B j=0 l=0
RBC
∞ X
∞ <pJL>
i
XX ipJL RBC
FṡII = G DṡII 2j+2l+3
, (3.7.47)
C6=B j=0 l=0
RBC
Finally, the force

∞ X
∞ <iJL>
XX (1) RBC
FpN
i
= G JL
DF 2j+2l+3
(3.7.48)
C6=B j=0 l=0
RBC
!
<ikJL> <kmJL>
(1) kJL RBC ikmJL RBC
+ DF 2j+2l+5
+ DF 2j+2l+5
RBC RBC
∞ X
∞
!
<iJL> <ikJL>
XX (2) RBC kJL RBC
+G JL
DF 2j+2l+3
+ (2) DF 2j+2l+5
,
C6=B j=1 l=0
RBC RBC
is also expressed in terms of the corresponding multipolar tensor coeffcients DF given at the end of
section (3.7.2) below.
The post-Newtonian force in translational equations of motion has been calculated in this chapter
for the system of N extended bodies with arbitrary internal structure, shape and density distribution. It
includes the coupling of all internal multipoles of the bodies in the post-Newtonian approximation. The
force converges to Einstein-Infeld-Hoffman (EIH) equations of motion [68, 73, 126] for monopole point-
like particles which can be considered as massive, spherically-symmetric bodies which size is negligibly
small compared with the distances between the bodies. The force (3.7.23) also yields the correct analytic
expression for the Lense-Thirring (gravitomagnetic) force due to the gravitational coupling of the body’s
spin with its orbital angular momentum and spins of the other bodies in the system [19, 32].
We have compared our result for the post-Newtonian gravitational force (3.7.23) with the post-
Newtonian force calculated by Xu, Wu and Schäfer [159] for binary pulsars which are considered as
monopole-spin-quadrupole particles . We found (almost) perfect agreement of the Xu-Wu-Schäfer force
with our expression (3.7.23) in general-relativistic limit of the PPN parameters β = γ = 1. The only
ipJL
difference was found for the force component FSi İ in the second term of the tensor coefficient DS İ
given by equation (3.7.91) in next section. This term is missed in [159].
We also confirmed the post-Newtonian equations of motion derived by Racine and Flanagan [130]
by means of the surface-integral technique [4, 76, 77, 147] in general relativity after taking into account
the omissions pointed out by Vines, Racine and Flanagan in the erratum paper [131]. In particular,
the term omitted in [130, Eq. 6.12c] and recovered in [131, Eq. 1.1] is given by our tensor coefficient
(2) JLSK i
DIIII in equation (3.7.76) which enters our expression (3.7.37) for the post-Newtonian force FIIII .
Notice that we give our tensor coefficients for the gravitational force while Racine, Vines and Flanagan
operates with the coefficients for acceleration. Therefore, our tensor coefficients must be divided by
mass MB of the body in order to get the Racine-Vines-Flanagan coefficients. Our results extend the
general-relativistic calculation of papers [130, 131] to the post-Newtonian realm of scalar-tensor theory
of gravity parametrized with two PPN parameters, β and γ.
We have to make a remark on the post-Newtonian force for spherically-symmetric bodies calculated
by one of us (S.K.) in [93, Section 6.3.4]. According to present chapter the post-Newtonian force (3.7.23)
depends only on STF multipole moments which are supposed to vanish for spherically-symmetric con-
figurations. The reader may think that it should reduce (3.7.23) to EIH force for monopole particles
while the textbook [93, Eqs 6.84 and 6.85] shows that besides the EIH force, the post-Newtonian force
for spherically-symmetric bodies also depends on the rotational moments of inertia of the bodies of the
order four, and higher. There is no contradiction here as the definition of the internal multipole moments
(including mass) depends on the interior mass density distribution which includes the potential energy
of the tidal gravitational field of external bodies [see (3.4.51)]. This makes the internal density distri-
bution depending on the post-Newtonian term being proportional to QK w<K> which is isotropic but
not spatially homogeneous. It contributes to the STF internal mass-type moment of multipolarity l the
σw<K> w<L> d3 w, which should be integrated in the local coordinates [see the
R
following term, QK
VB
last line in 3.4.51)]. This term does not vanish even if the body has a spherical shape and the density dis-
tribution σ is spherically-symmetric. This is why the finite-size effects will appear in the post-Newtonian
equations of motion even for spherically-symmetric bodies as discussed in more detail in textbook [93,
Section 6.3]. This explains why the force (3.7.23) does depend on the rotational moments of inertia of
spherically-symmetric bodies having finite size.
Discussion of the finite size post-Newtonian effects in equations of motion of spherically-symmetric
bodies has been also given by a number of other researchers including Brumberg[19], Spyrou [139–142],
Caporali [26, 27], Dallas[33], Vincent[150] and, more recently, by Arminjon[2]. The post-Newtonian
finite-size effects obtained in these papers depend on the rotational moments of inertia of the second
order which is inconsistent with our derivation of the post-Newtonian force and represent spurious,
coordinate-dependent effect in general relativity which can be removed by the appropriate choice of the
center of mass and a proper definition of body’s quadrupole moments as discussed in [93, Section 6.3.4]
and in [119]. Nonetheless, the gravitational force between spherically-symmetric bodies can depend on
the rotational moments of inertia of the forth order and higher.
Multipolar Tensor Coefficients

The multipolar tensor coefficients are expressed in terms of the internal (active) multipole moments and
i i i
their derivatives. They also depend on velocities vA , vB , vC of the bodies and on their relative velocities,
i i i
vBC = vB − vC . (3.7.49)
The multipolar tensor coefficients are given by the following expressions,
(−1)j (2j + 2l + 1)!! <J> <L>

JL
DII = IB IC × (3.7.50)
j!l!
" #
2 k k 2(2γ + 1)(j + l) + 10γ + 7 2
− γvB + 2(γ + 1)vB vC − vC ,
2(2j + 2l + 5)
(−1)j (2j + 2l + 1)!! <J> <L>
ikJL
DII = IB IC × (3.7.51)
j!l!
" #
i k 2 ik i k
vB vC − vC + 2(γ + 1)vBC vBC ,
2j + 2l + 5
(
(−1)j (2j + 2l + 3)!! 2j + 2l + 3 <J> <mL> k m
kJL
DII = I IC vC vC (3.7.52)
j!l! 2(2j + 2l + 7) B
" #)
<mJ> <L> 1 k m 2
+IB IC m k
vB vB − vB vC + k m
vC vC ,
2 2j + 2l + 7
(
(−1)j (2j + 2l + 3)!! 1
ikmJL
DII = − I<J> I<iL> vCk m
vC (3.7.53)
j!l! 2j + 2l + 7 B C
"
1 2
+I<iJ>
B I<L>
C vBk m
vB − vBk m
vC
2j + 3 2j + 3
#)
2(2j + l + 5) k m 2j 2 + 5j + 4 k m
+ vC vC − vBC vBC ,
(2j + 3)(2j + 2l + 7) 2j + 3
(1) (−1)j (2j + 2l + 3)!! <J> <L> k m

ikmJL
DII = IB IC vC vC , (3.7.54)
2j!l!
(2) (−1)j (2j + 2l + 5)!! <aJ> <aL> k m
ikmJL
DII = IB IC vC vC , (3.7.55)
(2j + 2l + 9)j!l!
(−1)j (2j + 2l − 1)!! <J> <L>
DIiJL
İ = IB İC (3.7.56)
j!l!
( )
i 2[2(γ + 1)(j + l) + 3γ + 2] i
× − (1 + 2γ)vB + vC
(2j + 2l + 3)
(−1)j (2j + 2l − 1)!! <J> <L> i
−2(γ + 1) İB IC vBC ,
j!l!
(
(−1)j (2j + 2l + 1)!!
DIJL
İ = − I<J>
B İ<L>
C
k k
vC vBC (3.7.57)
j!l!
" #
<J> <kL> 2j + 2l + 3 k 2(γ + 1) k
+IB İC vC + vBC
2j + 2l + 5 l+1
" #)
<L> 2j + 2l + 3 k
−I<kJ>
B İC vC + v k
BC
2j + 2l + 5
(−1)j (2j + 2l + 1)!! <kJ> <L> k
−2(γ + 1) İB IC vBC ,
(j + 1)!l!
(−1)j (2j + 2l + 1)!!
DIikJL
İ = × (3.7.58)
j!l!
( " #
2 2(γ + 1) k
− I<J>
B İ<iL>
C
k
vC + vBC
2j + 2l + 5 l+1
" #)
2
+I<iJ>
B İ <L>
C vC
k
+ 2(j + 1)v k
BC
2j + 2l + 5
(−1)j (2j 2 + 3j + 2γ + 3)(2j + 2l + 1)!! <iJ> <L> k
+ İB IC vBC ,
(j + 1)!l!
(1) (−1)j (2j + 2l + 1)!! <J> <L> k
DIkJL
İ = IB İC vC , (3.7.59)
j!l!
(2) 2(−1)j (2j + 2l + 3)!! <mJ> <mL> k
DIkJL
İ = IB İC vC , (3.7.60)
(2j + 2l + 7)j!l!
(1) (−1)j (2j + 2l − 1)!! <J> <L>
DIJL
Ï = IB ÏC , (3.7.61)
2j!l!
(−1)j (2j + 2l − 1)!!
DIiJL
Ï = × (3.7.62)
j!l!
(
4(γ + 1)j + (4γ + 3)l + 6γ + 5 <J> <iL>
IB ÏC
(2j + 2l + 3)(l + 1)
" # )
1 2l
− (j + 2)(2j + 1) + I<iJ> Ï <L>
(2j + 3) 2j + 2l + 3 B C
(−1)j (j 2 + j + 2γ + 2)(2j + 2l − 1)!! <iJ> <L>

− ÏB IC ,
(j + 1)!l!
(2) (−1)j (2j + 2l + 1)!! <kJ> <kL>
DIJL
Ï = IB ÏC , (3.7.63)
(2j + 2l + 5)j!l!
(−1)j (2j + 2l − 1)!!
DİiJL
İ = × (3.7.64)
j!l!
" #
2(γ + 1) <J> <iL> 2j 2 + 3j + 2γ + 3 <iJ> <L>
İB İC − İB İC ,
l+1 j+1
(−1)j (2j + 2l + 1)!! <kJ> <kL>
DİJL
İ = 2(γ + 1) İB İC , (3.7.65)
(j + 1)!(l + 1)!
(−1)j (2j + 2l + 5)!! I<ik>
ikmnJL
DqII = −3 B
I<J> I<L> mn
vBC , (3.7.66)
j!l! MB B C
(−1)j (2j + 2l + 3)!! I<ik>

ikmJL
DqI İ = 6 B
(I<J> İ<L> + İ<J> I<L> m
)vBC , (3.7.67)
j!l! MB B C B C
(−1)j (2j + 2l + 1)!! I<ik>

ikJL
DqI Ï = −3 B
I<J> Ï<L> + Ï<J> I<L> , (3.7.68)
j!l! MB B C B C
(−1)j (2j + 2l + 1)!! I<ik>

DqikJL
İİ = −6 B
İ<J> İ<L> , (3.7.69)
j!l! MB B C
(−1)j (2j + 2l + 3)!! İ<ik>

ikmJL
Dq̇II = 6 B
I<J> I<L> m
vBC , (3.7.70)
j!l! MB B C
(−1)j (2j + 2l + 1)!! İ<ik>

İ q̇Iİ İ<L> + İ<J> I<L>
ikJL
Dq̇I = −6 B
(I<J> ), (3.7.71)
j!l! MB B C B C
j
(−1) (2j + 2l + 1)!! Ï<ik>
ikJL
Dq̈II = −3 I<J> I<L>
B
, (3.7.72)
j!l! MB B C
j
(1) (−1) (l + 1)(2j + 2l + 1)!!(2k − 1)!! <J> <L> <K>
JLK
DIII = γ IB IC ID , (3.7.73)
j!l!k!
j
(2) (−1) (γj + γ − 1)(2j + 2l + 1)!!(2k − 1)!! <J> <L> <K>
JLK
DIII = IB IC ID , (3.7.74)
j!l!k!
(1) JLSK (−1)j+s [4(γ + 1)(j + l) + 6γ + 5](2j + 2l − 1)!!(2k + 2s + 1)!!
DIIII = −
(2j + 2l + 3)j!l!k!s!MC
×I<J>
B I<L>
C I<S>
C I<K>
D , (3.7.75)
(2) (−1)j+s (2j + 2l + 1)!!(2k + 2s + 1)!! <J> <L> <S> <K>
JLSK
DIIII = − IB IC IC ID , (3.7.76)
2j!l!k!s!MC
iJLSK (−1)j+s (2j 2 + 2jl + 7j + 2l + 4)(2j + 2l + 1)!!(2k + 2s + 1)!!
DIIII =
(2j + 2l + 5)j!l!k!s!MC
×I<iJ>
B I<L>
C I<S>
C I<K>
D , (3.7.77)
j+s
(−1) (2j + 2l + 1)!!(2k + 2s + 1)!! <mJ> <L> <S> <K>
mJLSK
DIIII = − IB IC IC ID (,3.7.78)
(2j + 2l + 5)j!l!k!s!MC
(−1)j+s (2j + 2l + 1)!!(2k + 2s + 1)!! <J> <iL> <S> <K>
JiLSK
DIIII = IB IC IC ID , (3.7.79)
(2j + 2l + 5)j!l!k!s!MC
JmLSK (−1)j+s (2l2 + 2jl + 2j + 7l + 4)(2j + 2l + 1)!!(2k + 2s + 1)!!
DIIII = −
(2j + 2l + 5)j!l!k!s!MC
×I<J>
B I<mL>
C I<S>
C I<K>
D , (3.7.80)
j+s
(3) (−1) (2j + 2l + 3)!!(2k + 2s + 1)!! <aJ> <aL> <S> <K>
JLSK
DIIII = − IB IC IC ID (3.7.81)
,
(2j + 2l + 7)j!l!k!s!MC
JSLK (−1)j+s (j + 2γ + 2)(2j + 2l − 1)!!(2k + 2s + 1)!!
DIIII = (3.7.82)
j!l!k!s!MB
×I<J>
B I<S>
B I<L>
C I<K>
D ,
mJSLK (−1)j+s (j + 1)(2j + 2l + 1)!!(2k + 2s + 1)!!
DIIII = − (3.7.83)
j!l!k!s!MB
×I<mJ>
B I<S>
B I<L>
C I<K>
D ,
j+s
(10) iJSLK (−1) j(2j + 2l + 1)!!(2k + 2s + 1)!!
DIIII = − (3.7.84)
j!l!k!s!MB
×I<iJ>
B I<S>
B I<L>
C I<K>
D ,
(−1)j+s (2j + 2l + 3)!!(2k + 2s + 1)!! IB
<ip>
ipJLSK
DqIIII = 3 (3.7.85)
j!l!k!s! MB MC
×I<J>
B I<L>
C I<S>
C I<K>
D ,
(−1)j+s (2j + 2l + 3)!!(2k + 2s + 1)!! IB
<ip>
ipJSLK
DqIIII = −3 (3.7.86)
j!l!k!s! M2B
×I<J>
B I<S>
B I<L>
C I<K>
D ,
(1) (−1)j j(2j + 2l + 1)!! <J> <L>
ipJL
DSI = −2(γ + 1) SB IC εipq vBC
q
, (3.7.87)
(j + 1)2 j!l!
j
(2) (−1) (j + 1)(2j + 2l + 3)!! <qJ> <L>
pJL
DSI = −2(γ + 1) SB IC εkpq vBC
k
(3.7.88)
(j + 2)2 j!l!
(−1)j (2j + 2l + 3)!! <J> <qL>
−2(γ + 1) I B SC k
εkpq vBC ,
(l + 2)j!l!
(3) (−1)j (j + 1)(2j + 2l + 3)!! <qJ> <L>
ikpJL
DSI = 2(γ + 1) SB IC εipq vBC
k
(3.7.89)
(j + 2)2 j!l!
(−1)j (2j + 2l + 3)!! <J> <qL>
+2(γ + 1) I B SC k
εipq vBC ,
(l + 2)j!l!
(−1)j (2j + 2l + 1)!!
ipJL
DṠI = −2(γ + 1) εipq Ṡ<qJ>
B I<L>
C (3.7.90)
(j + 2)j!l!
(−1)j (2j + 2l + 1)!!
−2(γ + 1) εipq I<J>
B Ṡ<qL>
C ,
(l + 2)j!l!
ipJL (−1)j (2j + 2l + 1)!!
DS İ
= 2(γ + 1) εipq (3.7.91)
j!l!
" #
j j + 1 <qJ> <L>
× S<J> İ<qL> − S İC
(j + 1)2 (l + 1) B C
(j + 2)2 B
(−1)j (2j + 2l + 1)!!
−2(γ + 1) εipq İ<J>
B S<qL>
C ,
(l + 2)j!l!
(−1)j (j + 1)(2j + 2l + 3)!!
pJL
DS İ
= 2(γ + 1) εkpq S<qJ>
B İ<kL>
C (3.7.92)
(j + 2)2 j!(l + 1)!
(−1)j (2j + 2l + 3)!!
−2(γ + 1) εkpq İ<kJ>
B S<qL>
C ,
(l + 2)(j + 1)!l!
(−1)j (2j + 2l + 5)!! <pJ> <qL>
pJqL
DSS = 2(γ + 1) SB SC , (3.7.93)
(j + 2)(l + 2)j!l!
(−1)j (2j + 2l + 3)!! SqB
ikpJL
DsII = − I<J> I<L> k
εipq vBC , (3.7.94)
j!l! MB B C
(−1)j (2j + 2l + 1)!! SB

q
ipJL
DsI = εipq I<J> İ<L> + İ<J> I<L> , (3.7.95)
İ j!l! MB B C B C
(−1)j (2j + 2l + 1)!! ṠB

q
ipJL
DṡII = 2 εipq I<J> I<L> , (3.7.96)
j!l! MB B C
(−1)j (2j + 2l + 1)!! <J> <L>

(1) JL
DF = − IB IC (j + 1)AB + (l + 1)AC , (3.7.97)
j!l!
(1) (−1)j (2j + 2l + 3)!! <mJ> <L> km
kJL
DF = − IB IC FB + I<J> B I<mL>
C FCmk (,3.7.98)
j!l!
2(−1)j (2j + 2l + 3)!! <iJ> <L> km
ikmJL
DF = IB IC FB , (3.7.99)
(2j + 3)j!l!
Section 3.8 Covariant Equations of Translational Motion of Extended Bodies 117
(2) (−1)j (2j + 2l + 1)!!

JL
DF = − AB İ<J>
B I<L>
C + I<J>
B İ<L>
C , (3.7.100)
j!l!
(2) (−1)j (2j + 2l + 3)!!
kJL
DF = AB I<J>
B I<L>
C
k
vBC . (3.7.101)
j!l!
This completes derivation of the translational equations of motion of extended bodies in the global
coordinates.
3.8 Covariant Equations of Translational Motion of

Extended Bodies
3.8.1 Effective Background Manifold
Equations of translational motion of extended bodies derived in the previous section, can be re-formulated
in a covariant form that is independent of the choice of coordinates and/or gauge condition. The covari-
ant form of the equations is formulated on the effective background spacetime manifold, M̄, which
emerge naturally in geometric description of world line Z of an extended body B that is considered as
massive particle endowed with mass M , mass-type IL , and spin-type SL , multipoles. In this section
we assume that the body is always at the origin of the local coordinates which wolrd line W coincides
with Z so that they are indistinguishable. We shall also use a convention, c = 1, to avoid appearance of
awkward combinations of symbols.
Manifold M̄ is endowed with the effective background metric tensor, ḡαβ , which is originally defined
in the global coordinates xα in terms of the external potentials Ū , Ū i , Ψ̄, χ̄ given in (3.5.24). The
background metric is determined uniquely by the matching of the external and internal solutions for
the metric tensor and scalar field which effectively cancel out all internal forces and potentials from the
equations of motion of the origin of the local coordinates (3.5.26). These equations of motion can be
represented in the form of equation of geodesic of the effective background metric that is disturbed by
some external force exerted on the “particle” located at the origin of the local coordinates (see section
3.8.2). The effective metric is given by the following equation (cf. [5, 107, 147])

1
ḡ00 (t,x) = −1 + 2Ū + 2 Ψ̄ − β Ū 2 − χ̄,tt + O (5) , (3.8.1)
2
ḡ0i (t,x̄) = −2(1 + γ)Ū i + O (5) , (3.8.2)
δij 1 + 2γ Ū + O (4) ,

ḡij (t,x) = (3.8.3)
where all potentials in the right side of (3.8.1)–(3.8.3) are functions of xα = (ct,x), and the symbol
O(n) = O (n ). The background metric in arbitrary coordinates can be obtained from (3.8.1)–(3.8.3)
by performing a coordinate transformation.
The background metric, ḡαβ , is a starting point of the subsequent covariant development. It has the
Christoffel symbols
1 αβ
Γ̄α
µν = ḡ (ḡβµ,ν + ḡβν,µ − ḡµν,β ) , (3.8.4)
2
α
which can be directly calculated in the global coordinates, x , by taking partial derivatives from the
metric components (3.8.1)–(3.8.3). In what follows, we shall make use of a covariant derivative defined
on the background manifold M̄ with the help of the Christoffel symbols Γ̄α µν . The covariant derivative
on the background manifold, M̄, is denoted with a vertical bar in order to distinguish it from the co-
variant derivative defined on the original spacetime manifold, M, denoted with a semicolon or with ∇.
For example, the parallel transport of any differentiable vector field V α is defined on the background
manifold by the following equation
V α |β = V α ,β + Γ̄α
µβ V
µ
, (3.8.5)
which is naturally extended to tensor fields in a standard way [93]. It is straightforward to define other
geometric objects on the background manifold like the Riemann tensor,
R̄α µβν = Γ̄α α α σ α σ

µν,β − Γ̄µβ,ν + Γ̄σβ Γ̄µν − Γ̄σν Γ̄µβ , (3.8.6)
and its contractions – the Ricci tensor R̄µν = R̄α µαν , and the Ricci scalar R̄ = ḡ µν R̄µν . Tensor indices
on the background manifold are raised and lowered with the help of the metric ḡαβ .
The local coordinates, wα = (u,wi ), of a body B on the effective background manifold are reduced to
the local inertial coordinates of a point-like particle with mass M placed at the origin of the coordinates
and moving along world line Z. The time coordinate u measured at the origin of the local coordinates
on the effective background manifold is now identical to a proper time counted along the world line of
the point-like particle. We shall denote this proper time s ≡ u(x = xB ). The background metric in the
local coordinates is reduced to
∞
X 1
ḡˆ 00 = −1 + 2Q + QL wL + O(4) , (3.8.7)
l!
l=1
ḡˆ 0i = O(3) , (3.8.8)

∞
!
X
ḡˆ ij = δij 1 + 2A + QL wL + O(4) , (3.8.9)
l=1
where the external multipoles QL = Ūi1 ...il (xB ), and the omitted post-Newtonian terms are identical
α α
with ĥext
αβ given in (3.4.46)-(3.4.49). Post-Newtonian transformation from the global, x , to local, w ,
coordinates has been provided in section (3.5.2). It smoothly matches the two forms of the background
metrics in these coordinates. In what follows, we will need a matrix of transformation taken on the world
line of the origin of the local coordinates
α
∂w
Λα β ≡ . (3.8.10)
∂xβ xi =xi
B
The components of this matrix are [93, Section 5.1.3]

1 2
Λ0 0 = 1+ vB − Ū (xB ) − Q + O(4) , (3.8.11)
2
Λ 0
i = −vB + O(3) ,
i
(3.8.12)
Λ i
0 = i
−vB + O(3) , (3.8.13)
1 i j
i ij
vB + F ij + O(4) ,

Λ j = δ 1 − A + γ Ū (xB ) + vB (3.8.14)
2
where F ij is the matrix of relativistic precession and Q and A are defined by the adopted system of units
of time and space measurements in astronomy [89]. The residual terms shown in (3.8.11)–(3.8.14) are
not important in the derivation given below but, if necessary, can be found in [93, Section 5.1.3].
We will also need the inverse matrix of transformation between the local and global coordinates taken
on the world line Z of the origin of the local coordinates. We shall denote this matrix as
α
∂x
Ωα β ≡ . (3.8.15)
∂wβ xi =xi
B
It is obvious that
Λα β Ωβ γ = δγα , (3.8.16)
which complies with the definition of inverse matrix.
Matrices Λα β and Ωα β allows us to transform the internal multipole moments of the bodies and
external gravitational field form the local to global coordinates and opposite. We should be careful about
distinguishing the components of the multipole moments in the two coordinate charts. The convention
is that all multipole moments, both internal and external, are defined in such a way that they have only
spatial components in the local coordinate chart which means that they are orthogonal to four-velocity
of the world line Z. For this reason, only the spatial components (indexed with Roman letters) of
the moments participate in the transformation from local to global coordinates. Thus, the multipole
moments with Roman indices belongs to the local chart and the same object with Greek indices belongs
to the global coordinates.
In order to arrive to covariant formulation of the translational equations of motion we, first, formulate
equations of motion in the local coordinates and, then, transform them to the global coordinates with the
help of the transformation matrices. After having applied (3.8.16) it turned out that all transformation
matrices cancel out at the final form of the equations which acquire a covariant four-dimensional form
without any reference to an original coordinate charts that were used in the intermediate calculations.
We carry out these calculations below.
3.8.2 Geodesic Motion and 4-Force

Let us introduce a four-velocity uα of the center of mass of body B. In the global coordinates, xα , the
world line Z of the body’s center of mass is described parametrically by x0B = ct, and xiB (t). The
four-velocity is defined by
dxα
uα = B
, (3.8.17)
ds
where s is the proper time along the world line Z. The increment ds of the proper time is related to the
increments dxα of the global coordinates by equation,
ds2 = −ḡαβ dxα dxβ , (3.8.18)
which tells us that the four-velocity (3.8.17) is normalized to unity, uα uα = ḡαβ uα uβ = −1. In
the local coordinates the world line Z is given by (u = s, wi = 0), and the four-velocity has com-
ponents uα = (1,0,0,0). In the global coordinates the components of the four-velocity are, uα =
dt/ds, dxiB /ds , which yields three-dimensional velocity of the body’s center of mass, v i = ui /u0 =

i
dxB /dt.
The extended body is treated as a “particle” endowed with mass M , mass-type multipole moments
IL and spin-type multipole moments SL attached to the origin of the local coordinates at any time. This
set of multipoles fully characterize the internal structure of the body. The multipoles can depend on time
including the mass which is not constant in the most general case [see (3.6.7)]. We postulate that the
covariant equations of motion of body B have the following form
duα
M = F α, (3.8.19)
ds
where duα /ds ≡ uβ uα |β is four-acceleration of the center of mass of the body B, and F α is four-
force that causes the world line Z of the center of mass of the body to deviate from geodesic of the
background manifold M̄. We have to introduce this force to equation (3.8.19) because the origin of the
local coordinates experiences acceleration Qi given by (3.6.17). In what follows it is more convenient
to operate with a four-force per unit mass defined by f α ≡ F α /M . The force f α is orthogonal to
four-velocity, uα f α = 0. Hence, in arbitrary coordinates the time (co-vector) component of the force is
i
related to its spatial components: f0 = −vB fi . Contravariant time component of the force relates to its
spatial components as follows
ḡij j i
f0 = − v f . (3.8.20)
ḡ00 B
Our task is to prove that (3.8.19) matches equations of motion (3.5.26) of the center of mass of body B
derived in the global coordinates by making use of matching of the external and internal solutions of the
field equations.
To this end we parametrize (3.8.19) by coordinate time t instead of the proper time s, which yields
aiB = −Γ̄i00 − 2Γ̄i0p vB

p
− Γ̄ipq vB
p q
vB (3.8.21)
ds 2
+ Γ̄000 + 2Γ̄00p vB
p
+ Γ̄0pq vB
p q i
vB vB + f i − f 0 vB
i
,
dt
i
where vB = dxiB /dt and aiB = dvB i
/dt are coordinate velocity and acceleration of the body’s cen-
ter of mass with respect to the global coordinates. We calculate the Christoffel symbols, Γ̄α µν , the
derivative ds/dt, substitute them to (3.8.21) along with (3.8.20), and retain only the Newtonian and
post-Newtonian terms. It yields
1
aiB = Ū,i (xB ) + Ψ̄,i (xB ) − χ̄,itt (xB ) + 2(γ + 1)Ū˙ i (xB )
2
k i ˙
−2(γ + 1)vB Ū k,i (xB ) − (2γ + 1)vB Ū (xB )
2 i k
−2(β + γ)Ū (xB )Ū,i (xB ) + γvB Ū,i (xB ) − vB vB Ū,k (xB )
i i k k 2 i
+f − vB vB f − 2Ū (xB ) + vB − 2Q f . (3.8.22)
This equation matches the equation of motion (3.5.26) if the spatial components of the force are
1 i k k
f i = −Qi − vB vB Q + F ik Qk + A + γ Ū (xB ) Qi ,

(3.8.23)
2
By simple inspection we can prove that the post-Newtonian force (3.8.23) can be written down in a
covariant form
f α = −ḡ αβ Λi β Qi , (3.8.24)
where Λi β is given above in (3.8.11)-(3.8.14), and Qi is a vector of four-acceleration in the local coordi-
nates. The quantity Λi β Qi = Qβ defines components of the four-acceleration in the global coordinates
with the four-acceleration Qα being orthogonal to four-velocity, uα Qα = 0 (see discussion at the end
of section 3.8.1).
Explicit form of Qi in the local coordinates is given in (3.6.17) and should be used in (3.8.24) along
with the covariant form of the external, QL , CL , PL , and internal, IL , SL , moments. This is a matter of
discussion in next section.
3.8.3 Four-dimensional Form of Multipole Moments

Internal Multipole Moments
The mathematical procedure that was used in construction of the local coordinates wα around each
extended body in N body system indicates that all multipole moments defined in the local coordinates
are the STF Cartesian tensors which have only spatial components with their time components being
identically nil. It means that the moments are orthogonal to the four-velocity uα of the world line Z of
the center of mass of the body. Four-dimensional form of the internal multipole moments is defined by
the law of transformation from local to global coordinates,
I<α1 α2 ...αl > ≡ Ωα1 i1 Ωα2 i2 ...Ωαl il I<i1 i2 ...il > , (3.8.25)
S <α1 α2 ...αl >
≡Ωα1
i1 Ω
α2
i2 ...Ω
αl
il S
<i1 i2 ...il >
, (3.8.26)
and the condition of orthogonality to the four-velocity,
uα1 I<α1 α2 ...αl > = 0 , uα1 S<α1 α2 ...αl > = 0 , (3.8.27)
where (3.8.27) is applied to each index. Notice that the matrix of transformation (3.8.15) has been
used in (3.8.25), (3.8.26). It reflects the fact that the internal multipoles were obtained as integrals
from the products of coordinates which increments behave under coordinate transformations as vectors,
dxα = (∂xα /∂wβ )dwβ .
External Multipole Moments

The external multipole moments has been defined by solutions of the field equations in such a way
that they have only spatial components in the local coordinates. It means the external multipoles are
orthogonal to the four-velocity of the world line of the center of mass of the body,
uα1 Q<α1 α2 ...αl > = 0 , uα1 P<α1 α2 ...αl > = 0 , uα1 C<α1 α2 ...αl > = 0 . (3.8.28)
These orthogonality conditions allow us to find out the time components of the multipole moments from
their spatial components in any coordinates other than the local chart. This property makes it possible to
extend the local three-dimensional definition of the external multipole moments to the four-dimensional
tensors by making use of the matrix of transformation (3.8.10). It yields
Q<α1 α2 ...αl > ≡ Λi1 α1 Λi2 α2 ...Λil αl Q<i1 i2 ...il > , (3.8.29)
i1 i2 il
C<α1 α2 ...αl > ≡ Λ α1 Λ α2 ...Λ αl C<i1 i2 ...il > , (3.8.30)
P<α1 α2 ...αl > ≡ Λi1 α1 Λi2 α2 ...Λil αl P<i1 i2 ...il > . (3.8.31)
We have used over here the matrix of transformation (3.8.10) because the external multipole moments
are defined in terms of partial derivatives from the external potentials Ū , Ψ̄, etc., which behave under
coordinate transformations like co-vectors. Four-dimensional form of the gauge-dependent multipoles
ZL is defined similarly
Z<α1 α2 ...αl > ≡ Λi1 α1 Λi2 α2 ...Λil αl Z<i1 i2 ...il > . (3.8.32)
It is known that in general relativity the external multipole moments, QL and CL are defined in
local coordinates by the Riemann tensor, R̄α µβν , of the background metric (3.8.7) and its spatial
derivatives taken at the origin of the local coordinates [127, 143, 147, 161]. This definition remains
valid in the scalar-tensor theory of gravity. In the local coordinates the mass-type multipole moments
QL = Q<i1 ...il > , are expressed in terms of the Riemann tensor R0i1 0i2 for l = 2 and its spatial
derivatives R̄0i1 0i2 ,i3 ...il for l ≥ 2. The spin-type multipoles CL = C<i1 i2 ...il > are given in the local
coordinates by the contraction of the Levi-Civita symbol and the Riemann tensor, R̄pq0<i1 εi2 >pq for
l = 2, and by partial derivatives R̄pq0<i1 ,i2 ...il−1 εil >pq for l ≥ 2. The scalar field external multipoles,
PL , are not related in any way to the Riemann tensor because they are expressed in terms of the partial
derivatives of the scalar field ϕ̄.
Four-dimensional formulation of the external multipole moments is achieved by contracting the Rie-
mann tensor with a vector of four-velocity, uα , and taking covariant derivatives projected on the hy-
perplane being orthogonal to the four-velocity. The projection is fulfilled with the help of the tensor
operator of projection
Pα α α
β ≡ δβ + u uβ , (3.8.33)
which satisfies to Pαγ Pβ = Pβ . The operator of projection has only three independent components
γ α
and reduces to a three-dimensional Kronecker symbol on the world line Z of the body’s center of mass
(which coincides with the origin of the local coordinates).
Four-dimensional scalar field external multipoles are defined by (3.5.27) and (3.8.31) and reads
β
P<a1 ...αl > = Pβ<α
1
1
· · · Pαll > ϕ̄|β1 ···βl + O(2) , (3.8.34)
where ϕ̄ is the scalar field perturbation caused by external with respect to the body B masses. In the
global coordinates ϕ̄ = Ū defined in (3.5.24a).
Tedious but straightforward calculations prove that the four-dimensional form of the mass-type exter-
nal multipoles QL and CL are
Q<a1 ...al > = E<a1 ...al > + Ż<a1 ...al > (3.8.35)
l−2
X (l − 2)!
+3 E<α1 ...αs+1 Eαs+2 ...αl >
s=0
s!(l − 2 − s)!
l−3
X (l − 2)!
+2 E<α1 ...αs+2 Eαs+3 ...αl >
s=0
s!(l − 2 − s)!
l−3 Xk
X (l − 2 − k)k!
+2 E<α1 ...αs+1 Eαs+2 ...αl >
s=0
s!(k − s)!
k=0
l−2
X (l − 2)!
+2 E<α1 ...αs+1 Φαs+2 ...αl >
s=0
s!(l − 2 − s)!
l−3 Xk
X (l − k − 1)k!
+2 Φ<α1 ...αs+1 Eαs+2 ...αl >
s=0
s!(k − s)!
k=0
l−2
X (l − 2)!
+2 Θ<α1 ...αs+2 Φαs+3 ...αl >
s=0
s!(l − 2 − s)!
l−2
X (l − 2)!
+2 Θ<α1 ...αs+1 Φαs+2 ...αl > + O(4) ,
s=0
s!(l − 2 − s)!
l
C<a1 ...al > = B<a1 ...al > + O(2) , (3.8.36)
l+1
where a dot denotes a covariant derivative defined later in (3.8.41) and we have introduced the following
abbreviations
Φ<a1 ...αl > ≡ (γ − 1)P<a1 ...αl > , (3.8.37)

β−1
Θ<a1 ...αl > ≡ P<a1 ...αl > , (3.8.38)
γ−1
β
E<a1 ...αl > ≡ −Pβ<α
1
1
· · · Pαll > uµ uν R̄µβ1 νβ2 |β3 ···βl , (3.8.39)
β
B<a1 ...αl > ≡ −εµβ1 ρσ P P ρδ σγ
Pβ<α
1
1
· · · Pαll > uµ uν R̄δγβ2 ν|β3 ···βl , (3.8.40)
with εαβγδ being a four-dimensional Levi-Civita tensor. It can be checked that in the local coordinates
the right sides of (3.8.35) and (3.8.36) are reduced to QL and CL respectively.
Four-diemnsional definitions of the multipole moments given in this section allows us to transform
the products of the moments given in the local coordinates to their covariant counterparts, for exam-
ple, QL IL = Q<i1 ...il > I<i1 ...il > = Q<α1 ...αl > I<α1 ...αl > , etc. In all such products the matrices
of transformation cancel out giving rise to covariant expressions being independent of the coordinate
choice.
3.8.4 Covariant Translational Equations of Motion
The covariant equations of translational motion of the center of mass of an extended body B are given
by (3.8.19) with the force f α defined in (3.8.24). We use (3.6.17) to replace Qi in the expression
(3.8.24) for the force, and a covariant version of equation (3.7.40) to replace the time derivative of
body’s spins. We also replace all products of the multipole moments in the local coordinates to their four-
dimensional forms in accordance with the definitions given above in section 3.8.3. The time derivatives
of the multipole moments in the local coordinates are transformed to covariant derivatives taken on
the background manifold along the direction of the four-velocity vector uα . For instance, for a tensor
multipole I (the indices are suppressed) the first time derivative mapping is
dI
İ = = uα I|α . (3.8.41)
ds
and the second time derivative becomes
d dI
Ï = = uα uα I|αβ + uα uα |β I|α = uα uα I|αβ + f α I|α , (3.8.42)
ds ds
where we have used the equation of motion (3.8.19) in order to replace the covariant derivative from the
four-velocity.
Linear Momentum
We have noticed that a number of terms in the covariant expression for the four-force, f α , can be
grouped together to form a total time derivative. Following Fock [73], it is more natural to put all total
time derivative terms to the left side of the translational equations of motion and combine it with the
four-acceleration uα uµ |β . This procedure introduces a four-momentum of the extended body B in the
following form (envisaged by Dixon [62, Eq. 83])
pµ = M nµ , (3.8.43)
where the dynamical velocity, nα of the body is an algebraic sum of the kinematic velocity, uα and
post-Newtonian corrections:
nµ = uµ (3.8.44)
∞
1 X (l2 + l + 4)
+ E<α1 ...αl > uν I<µα1 ...αl > |ν
M (l + 1)!
l=2
∞
1 X (2l + 1)(l2 + 3l + 6)
+ E<α1 ...αl >|ν uν I<µα1 ...αl >
M (2l + 3)(l + 1)!
l=2
∞
1 X 1
+ εµρ νσ B<ρα1 ...αl > uν I<σα1 ...αl >
M (l + 2)l!
l=1
∞
4 X (l + 1)2
− εµρ νσ E<ρα1 ...αl > uν S<σα1 ...αl >
M (l + 2)(l + 2)!
l=1
∞
2 X 1
+ Φ<α1 ...αl > uν I<µα1 ...αl > |ν
M (l + 1)!
l=1
∞
2 X (2l + 1)
+ Φ<α1 ...αl >|ν uν I<µα1 ...αl >
M (2l + 3)(l + 1)!
l=1
∞
2 X (l + 1)2
− εµρ νσ Φ<ρα1 ...αl > uν S<σα1 ...αl > ,
M (l + 2)(l + 2)!
l=0
∞
3 X1
− 2 E<σα1 ...αl >|ν uν I<α1 ...αl > I<µσ>
M l!
l=1
∞
3 X1
− 2 E<σα1 ...αl > uν I<α1 ...αl > |ν I<µσ>
M l!
l=1
∞
3 X1
− 2 E<σα1 ...αl > uν I<α1 ...αl > I<µσ> |ν
M l!
l=1
∞
1 X 1 µρ
+ 2 ε νσ E<ρα1 ...αl > uν I<α1 ...αl > Sσ ,
M l!
l=1
which provides the post-Newtonian momentum-velocity relation that has not been established in Dixon’s
paper [62] to full extent. Partial progress in finding this relation has been made by Ehlers and Rudolph
[67] but their momentum-velocity relation is incomplete and, furthermore, goes beyond the post-Newtonian
approximation without justification.
It is straightforward to check that the dynamical velocity (3.8.44) is orthogonal to the kinematic ve-
locity uµ ,
nα uα = −1 , (3.8.45)
which is a consequence of the orthogonality conditions (3.8.27) and (3.8.28). The dynamical velocity is
normalized nα nα = −1 which yields for the linear momentum
p2 = ḡµν pµ pν = −M 2 . (3.8.46)
We notice that the body’s four-momentum is not conserved, p2 6= const., because of non-conservation
of mass (3.6.7).
Gravitational Force
Reshuffling all total time derivative terms to the definition of linear momentum pµ , we find out that all
second time derivative terms are eliminated from the four-force. It brings equation (3.8.19) of transla-
tional motion of body B to the following covariant form
dpµ µ µ
= FM + FD µ
+ FEµ + FBµ + FΦµ + FEE µ
+ FEB µ
+ FEΦ µ
+ FΦΦ (3.8.47)
ds
where the first term in the right side of (3.8.47) is caused by the change in the mass of the body
µ
FM = Ṁ uµ , (3.8.48)
where Ṁ is derived by taking a time derivative from the definition of mass (3.6.8) and making use of
(3.6.7). It yields
∞ ∞
X l+1
X 1
Ṁ = − Q<a1 ...al > İ<a1 ...al > − Q̇<a1 ...al > I<a1 ...al > + O(4) , (3.8.49)
(l − 1)! l!
l=1 l=1
where the factor 1/c2 in front of the two sums was omitted since we work in this section under as-
sumption that c = 1. We emphasize that formula (3.8.49) is valid in general relativity as well as in
scalar-tensor theory of gravity.
The second term in the right side of (3.8.47) describes the post-Newtonian Dicke’s force caused by
violation of the strong principle of equivalence (SEP)
µ
FD = (M − M ) P µ , (3.8.50)
where
P µ = ḡ βγ Pµ
β ϕ̄|γ , (3.8.51)
is an external scalar field dipole moment. The difference between the active, M, and conformal, M ,
masses is given in (3.6.9). It plays a role of scalar “charge” of body B which interacts with external
scalar field like an electric charge interacts with an external electric field in electrodynamics.
The other forces in the right side of (3.8.47) correspond to the gravitational interaction between in-
ternal multipole moments of the body B and their derivatives with the external multipole moments
describing tidal gravitational field,
∞
X 1 µν
FEµ = ḡ E<να1 ...αl > I<α1 ...αl > (3.8.52)
l!
l=1
∞
X 1 ν µ
+ u u E<α1 ...αl > I<α1 ...αl > |ν (3.8.53)
l!
l=2
∞
X (l2 + 2l − 3)
− uν uβ E<α1 ...αl >|ν I<µα1 ...αl > |β
(2l + 3)(l + 1)(l + 1)!
l=2
∞
X (l + 1)
+4 uν uβ εµρ βσ E<ρα1 ...αl > S<σα1 ...αl > |ν
(l + 2)(l + 2)!
l=1
∞
1 X 1 ν β µρ
− u u ε βσ E<ρα1 ...αl > I<α1 ...αl > Sσ |ν
M l!
l=1
∞
X l
FBµ = ḡ µν B<να1 ...αl > S<α1 ...αl > (3.8.54)
(l + 1)!
l=1
∞
X 1
+ uν uβ εµβρσ B<ρα1 ...αl > I<σα1 ...αl > |ν
(l + 2)!
l=1
∞
X 1
FΦµ = 2 uν uβ Φ<α1 ...αl >|ν I<µα1 ...αl > |β (3.8.55)
(2l + 3)(l + 1)(l + 1)!
l=1
∞
X (l + 1)
+2 uν uβ εµρ βσ Φ<ρα1 ...αl > S<σα1 ...αl > |ν ,
(l + 2)(l + 2)!
l=0
∞ X
l−1
X 1
µ
FEE = 3 ḡ µν E<α1 ...as+1 Eναs+2 ...αl > I<α1 ...αl > (3.8.56)
s!(l − 1 − s)!l
l=1 s=0
∞ X
l−2
X 1
+2 ḡ µν E<α1 ...as+2 Eναs+3 ...αl > I<α1 ...αl >
s!(l − 1 − s)!l
l=1 s=0
∞ X
l−2 Xk
X (l − 1 − k)k! µν
+2 ḡ E<α1 ...as+1 Eναs+2 ...αl > I<α1 ...αl >
s=0
l!s!(k − s)!
l=1 k=0
∞ X ∞
4 X (l + 1)2
+ εµν ρσ E<ρα1 ...αl > E<νβ1 ...βk > I<β1 ...βk > S<σα1 ...αl >
M (l + 2)(l + 2)!k!
l=1 k=1
∞ ∞
1 X X 1 µρν
+ 2 ε σ E<νβ1 ...βk > E<ρα1 ...αl > I<β1 ...βk > I<α1 ...αl > Sσ
M l!k!
l=1 k=1
∞ ∞
1 XX 1
µ
FEB = εµρν σ E<νβ1 ...βk > B<ρα1 ...αl > I<β1 ...βk > I<σα1 ...αl > , (3.8.57)
M (l + 2)l!k!
l=1 k=1
∞
X γ − 1 µν
µ
FEΦ = 2 ḡ ΘE<να1 ...αl > I<α1 ...αl > (3.8.58)
l!
l=1
∞ X
l−1
X 1
+2 ḡ µν E<α1 ...αs+1 Φναs+2 ...αl > I<α1 ...αl >
s!(l − 1 − s)!l
l=1 s=0
∞ X
l−2 Xk
X (l − k)k! µν
+2 ḡ Φ<α1 ...αs+1 Eναs+2 ...αl > I<α1 ...αl >
s=0
l!s!(k − s)!
l=1 k=0
∞ ∞
2 XX (l + 1)2
+ εµνρ σ E<νβ1 ...βk > I<β1 ...βk >
M (l + 2)(l + 2)!k!
l=0 k=1
×Φ<ρα1 ...αl > S<σα1 ...αl > ,

∞
X 1 µν
µ
FΦΦ = −2 ḡ ΘΦ<να1 ...αl > I<α1 ...αl > (3.8.59)
l!
l=1
∞ X
l−1
X 1
+2 ḡ µν Θ<α1 ...αs+2 Φναs+3 ...αl > I<α1 ...αl >
s!(l − 1 − s)!l
l=1 s=0
∞ X
l−1
X 1
+2 ḡ µν Θ<α1 ...αs+1 Φναs+2 ...αl > I<α1 ...αl >
s!(l − 1 − s)!l
l=1 s=0
Herein, functions Φ and Θ without indices correspond to definitions (3.8.37) and (3.8.38) with indices
omitted,
β−1
Φ = (γ − 1)ϕ̄ , Θ= ϕ̄ . (3.8.60)
γ−1
3.8.5 Comparison with Dixon’s Translational Equations of Motion

µ
The force FM is absent at Dixon’s paper [62] because he assumed that the mass M of the body is con-
served but this is a rather strong constrain on the time dependence of the internal and external multipoles
of the body which cannot be satisfied in the most general case. Dixon’s paper [62] does not contain
µ
forces FD , FΦµ , FEΦ
µ µ
, and FΦΦ because he worked in the framework of general relativity where these
forces are nil by definition because there is no a long-range scalar field in general relativity. Dixon has
µ
also neglected general-relativistic forces FEE and an essential number of terms in FEµ and FEBµ
. Dixon’s
equations of translational motion (3.1.13) correspond to the following, severely truncated version of our
equations (3.8.47)
∞
dpµ 1 X 1 µν
= ḡ µν B<να> Sα + ḡ E<να1 ...αl > I<α1 ...αl > , (3.8.61)
ds 2 l!
l=2
where we have kept only the spin-dipole term in FBµ and the first term in the right side of (3.8.52) in FEµ .
Comparison with Dixon’s equation (3.1.13) allows us to make the following identifications between
our and Dixon’s notations for multipole moments (l ≥ 2),
S µν ≡ εµναβ uα Sβ , (3.8.62a)
l + 1 [µ <α1 ]α2 ...αl−1 [αl > ν]
J α1 ...αl µν
≡ u I u , (3.8.62b)
l−1
l−1
Aα1 ...αl µν ≡ 2 R̄µ<α1 /ν/α2 |α3 ...αl > . (3.8.62c)
l+1
which agrees with Dixon’s definitions and a forward backslashes around index ν means that it is excluded
from STF symmetrization indicated by the angular brackets.
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4 On the DSX-framework
Michael Soffel
Lohrmann Observatory TU Dresden, Planetary Geodesy, Mommsenstr 13, 01062
Dresden, Germany
4.1 Introduction
In the beautiful city of Tübingen in the south-west part of Germany I worked with my good old friend
Chongming Xu on the problem of relativistic celestial mechanics for several years. We carefully stud-
ied the literature on that subject and thought about possible improvements. When we first saw the
paper by Brumberg and Kopeikin [1] we got really excited and realized that it presented a major break-
through. During the 5th Marcel Grossmann meeting 1988 in Perth, Australia, I met Thibault (Damour)
and learned that he had good ideas to push our subject forward. We agreed that he would visit us
in Tübingen. Finally he came with a huge pile of notes on a new formulation of relativistic celestial
mechanics. It was not too different from the Brumberg-Kopeikin approach but contained important im-
provements in various directions. Moreover, Thibault’s approach was constructive in the sense that very
little was assumed from the very beginning and one proceeds with a series of Lemmas and Theorems.
The whole structure had a great intrinsic beauty and mathematical rigor which, I think, was really new in
that field. So we started working out details of this new approach by proving more Lemmas and Theo-
rems and formulating a new and improved theory of relativistic reference systems. Finally we published
a series of papers (Damour, Soffel and Xu, [2-5]), on what later was called the DSX-framework.
This contribution focuses on the first DSX paper [2], where the foundations of this new approach
was worked out and, as a first application, a new and improved derivation of the well-known post-
Newtonian Einstein-Infeld-Hoffmann equations for a system of mass-monopoles was presented. The
following major part of the Introduction was taken literally from [2]. I would like to apologize for that
but improving this text, that was essentially formulated by Thibault, is not an easy job.
The problem of describing the dynamics of N gravitationally interacting extended bodies is the car-
dinal problem of any theory of relativity. Within the framework of Newton’s theory this problem, called
"celestial mechanics", has been thoroughly investigated (see, e.g., [6]). Very shortly after the discov-
ery of Einstein’s theory of gravity, Einstein [7], Droste [8], de Sitter [9]), and Lorentz and Droste [10]
devised an approximation method (called "post-Newtonian") which allowed them to compare general
relativity with Newton’s theory of gravity, and to predict several "relativistic effects" in celestial me-
chanics, such as the relativistic advance of the perihelion of planets, and the relativistic precession of
the Moon’s orbit. This post-Newtonian approach to general-relativistic celestial mechanics was sub-
sequently developed by many authors, notably by Fock [11] (for a review of the development of the
problem of motion in general relativity, see, e.g., [12]).
However, to match the high precision of modern observational techniques such as satellite laser rang-
ing, lunar laser ranging, very long baseline interferometry, radar ranging to spacecraft and planets etc.,
one needs a correspondingly accurate relativistic theory of celestial mechanics able to describe both the
global gravitational dynamics of a system of N extended bodies, the local gravitational structure of each,
and the way each of these N local structures meshes into the global one. The traditional post-Newtonian
approach to relativistic celestial mechanics fails, for both conceptual and technical reasons, to bring a
140 Chapter 4 On the DSX-framework
satisfactory answer to this problem. This traditional post-Newtonian approach uses only one global co-
ordinate system xµ = (ct,x,y,z), to describe an N -body system, and models itself on the Newtonian
approach to celestial mechanics consisting of decomposing the problem into two subproblems [6,11]:
(i) the external problem, to determine the motion of the centers of mass of the N bodies; (ii) the internal
problem, to determine the motion of each body around its center of mass. However, the treatments of
both subproblems in the traditional post-Newtonian approach are unsatisfactory.
The external problem is attacked by introducing some collective variable, say z i (t), i = 1,2,3, gen-
eralizing the Newtonian center of mass, i.e., describing the overall motion of each body as seen in the
global coordinate system xµ . Then, one attempts to derive some (translational) "equations of motion"
for z i (t) by integrating over each considered body the local law of balance of energy and momentum,
i.e., the covariant conservation of the stress-energy tensor,
∇ν T µν = 0 .
However, the various definitions of the position in the global coordinate system of the center of mass
z i used in post-Newtonian investigations have never been quite satisfactory, especially when consider-
ing rotating bodies. Moreover, the final equations of motion for z i (t) contain various other collective
variables ("spin" and higher "multipole moments") describing the gravitational structure of each body
as seen in the global system xµ , which are not related in a simply way to any physical "local" multipole
moments, defined, say, in an operational way by the motion of artificial satellites around each body.
Concerning the treatment of the internal problem in the usual post-Newtonian approach, it is even
more unsatisfactory for the following reasons. In Newtonian celestial mechanics the introduction of
nonrotating accelerated "mass-centered frames" associated with each body, i.e., of local coordinates
X i = xi − z i (t) , (4.1.1)
where i = 1,2,3 and where z i denotes the global coordinates of the center of mass, serves both a
kinematical and a dynamical purpose. The kinematical usefulness of the local coordinates X i stems from
the fact that they are "comoving" with the considered body, while their dynamical usefulness comes from
the fact that they succeed in decoupling, to a large degree, the "internal" from the "external" problem.
Indeed, with respect to these local frames X i the external gravitational field is greatly "effaced" [12] in
the sense that the effective external gravitational potential acting locally on the body and its environment,
d2 z i i
U eff (X i ) = U ext (z i + X i ) − U ext (z i ) − X ,
dt2
is essentially reduced to tidal forces.
For a long time, the relativistic internal problem has been given only little attention, and many authors
working in the global post-Newtonian framework have, more or less implicitly, assumed that the usual
Newtonian formula (4.1.1) was sufficient for defining a useful "mass-centered frame" in Einsteinian
gravity. In principle, this view is admissible because the coordinate systems are arbitrary in general
relativity, and the definition (4.1.1) is as kinematically useful as in Newtonian gravity. However, the
formula (4.1.1) does not define a dynamically useful mass-centered frame in general relativity, in the
sense that it does not efface the external gravitational field down to tidal effects, but, instead introduces
in the description of the internal dynamics of the body many external "relativistic" effects proportional
to the square of the orbital velocity or the external gravitational potential. As discussed in [11] the
latter effects come from the fact that the external description (xµ -coordinate representation) of each
body contains many "apparent deformations" which are not intrinsic to the body (notably the "Lorentz
contraction", linked with the orbital velocity, and the "Einstein contraction", linked with the external
gravitational potential).
As emphasized by Damour [12], those technical defects of the usual global post-Newtonian approach
are partly rooted in, and certainly further enhanced by, the conceptual defect of surreptitiously introduc-
ing a kind of "neo-Newtonian" [13] interpretation of general relativity, by which the global coordinates
t ≡ x0 /c,(x,y,z) ≡ (xi ), i = 1,2,3 are implicitly identified with the absolute time and space of New-
tonian theory. This implicit conceptual reduction of Einstein’s theory to the Procustean bed of Newton’s
framework is liable to cause technical mistakes when one forgets the existence of the "apparent defor-
mations" alluded to above. In recent years, several authors have tried to remedy some of the defects of
the traditional post-Newtonian approach to the N -body problem. For instance, Martin et al., [14] and
Hellings [15] have tried, in an essentially heuristic manner, to explicitly include the main apparent de-
formations due to the use of an external coordinate representation. A more ambitious approach consists
of defining a local comoving frame by using, not a kinematical criterion [like in Eq. (4.1.1)], but a dy-
namical one: i.e., to find a useful relativistic definition of an accelerated frame of reference with respect
to which the external gravitational effects are strongly effaced. In the simple case of negligibly self-
gravitating test bodies moving in a background gravitational field (e.g., an artificial satellite around the
Earth) such "external-gravitational-field-effacing" frames are the well-known "locally inertial frames"
which can be explicitly constructed by means of Fermi coordinates based on the center-of-mass world
line (see, e.g., [16,17]).
In the more subtle case of (possibly strong) self-gravitating test bodies (i.e., of mass much smaller
than the masses of the other bodies) it has been argued as early as 1921 by Weyl [18] that such frames
should exist, and be the locally inertial frames (or Fermi frames) of the "external space-time" generated
by the masses of the other bodies only. Weyl [18] used this argument to conclude that test bodies (even
self-gravitating ones) follow geodesics of the "external space-time". This heuristic reasoning has been
later taken up [19-21] although it never became clear what could be rigorously proven with its help
(because of the lack of mathematical control on the limiting process which defines what one means by
"test-body" and "external universe").
Concerning nontest bodies (of mass comparable to the masses of the other bodies), some authors (in
particular Bertotti [22]) remarked that, at the first post-Newtonian approximation, the orbital motion,
according to the Lorentz-Droste-Einstein-Infeld-Hoffmann equations of motion, of one self-gravitating
body member of an N -body system, could be interpreted as the motion of a test body in some effective
external gravitational field. This remark, together with the previous results for test bodies, suggests that
it should be always possible to define good "external-gravitational-field-effacing frames" around any
body A, abstracted from an N -body system, by constructing some "locally inertial coordinate system"
in some "effective external gravitational field". At the heuristic level, such a construction has been more
or less explicitly assumed by many authors [17,23-27]. More explicit results on such local external-
field-effacing frames have been obtained in the study of the motion of strongly self-gravitating bodies
(neutron stars or black holes), because this was a problem where the standard only global-frame approach
was definitively inadequate to derive results needed for astrophysical applications. In particular, D’Eath
[20] and Damour [28], in their studies of binary systems of gravitationally condensed bodies, have made
an explicit use of local external-field-effacing coordinate systems X α = (cT,X a ) (one for each body),
linked with the global coordinate system xµ , covering the binary system, by transformation formulas of
the type (a = 1,2,3)
a (T )X + O((X ) ) + · · · ,
xµ (T,X a ) = z µ (T ) + eµ a a 2
and have derived the constraints on the functions z µ (T ), eµ

a (T ), imposed by the requirement of a suitable
effacement in the X α system. Other explicit results about such good local frames were also obtained in
the study of weakly self-gravitating bodies, treated at the first post-Newtonian approximation, notably
through the introduction of "generalized Fermi coordinates" by Ashby and Bertotti [29,30] (see also
[31], and the contributions of Bertotti, of Boucher, of Fukushima, Fujimoto, Kinoshita, and Aoki, and
of others in [32]).
More recently a notable progress in the theory of such local relativistic frames (at the post-Newtonian
approximation, relevant to systems of N weakly self-gravitating bodies) has been achieved by Brum-
berg and Kopeikin in a series of publications [1,33-34]; see also Voinov [35]). Their approach combines
the usual post-Newtonian-type expansions with the multipole expansion formalisms for internally gen-
erated [36-38], and externally generated [27,39,40] gravitational fields, and with asymptotic matching
techniques [20,28].
A further step forward was made by Damour, Soffel and Xu in a series of papers [2-5]. Improvements
of the Brumberg-Kopeikin formalism are:
- the form of the energy-momentum tensor of matter composing the N bodies is left completely
open;
- whereas the spatial coordinates in the global and the N local coordinate systems are taken to be
harmonic the gauge of the various time coordinates is left open within the post-Newtonian approx-
imation;
- a compact notation for the metric tensor in the N + 1 coordinate systems; in each of these systems
the metric tensor is completely specified by two metric potentials only, a scalar potential describing
gravito-electric type effects, and a vector potential related with gravito-magnetic type effects;
- the use of post-Newtonian physically meaningful mass- and spin-multipole moments (Blanchet-
Damour moments) in each of the local systems.
This contribution to the Festschrift discusses the most important features of this DSX (Damour-Soffel-
Xu) formalism.
4.2 The post-Newtonian formalism

4.2.1 The general form of the metric
The idea of the post-Newtonian formalism is to employ the fact that in the solar system velocities of
astronomical bodies are small and gravitational fields are weak. The PN-formalism is a slow motion,
weak field approximation to Einstein’s theory of gravity. The small parameter of this expansion is
v
=
c
and everywhere in the solar system we have
2
U p 1
∼ 2 ∼Π∼ ∼ 2 < 10−5 .
c2 ρc c
Here, p denotes the pressure, ρ the density of matter and Π the specific internal energy density (internal
energy density divided by the rest energy density). An upper limit for U/c2 will be given by GM/(c2 R),
where M is the mass of some gravitating body and R its radius. For example, (E: Earth, S: Sun)
GME GMS
' 6.9 × 10−10 , ' 2.1 × 10−6 .
c2 RE c2 RS
Section 4.2 The post-Newtonian formalism 143
For the orbital velocity of the Earth about the Sun we have
v 2 2
E 30
' ' 10−8 .
c 300 000
We will now assume the existence of certain coordinates xµ = (ct,xi ) such that the following PN-
assumptions hold
g00 = −1 + O(c−2 ), g0i = O(c−3 ), gij = δij + O(c−2 ) . (4.2.1)
In the following we will frequently encounter such order symbols that we will often abbreviate by
On ≡ O(c−n ) .
Moreover, we will assume, in accordance with the results from Special Relativity, that
T 00 = O(c2 ), T 0i = O(c+1 ), T ij = O(c0 ) (4.2.2)
and
∂
∂0 ≡ = O1 · ∂i .
∂ct
With assumptions (4.2.1) we can write the metric tensor in the form

2
g00 = − exp − 2 w
c
4
g0i = − 3 wi (4.2.3)
c
2
gij = γij exp + 2 w
c
with
γij = δij + O2 .
Below we will show that the spatial field equations
Gij = κ Tij (4.2.4)
will be satisfied by
γij = δij + O4 . (4.2.5)
Our canonical form of the metric therefore reads
2w2

2 2w
g00 = − exp − 2 w = −1 + 2 − 4 + O6
c c c
4
g0i = − 3 wi (4.2.6)
c
2
gij = = δij 1 + 2 w + O4 .
c
Note, that the metric tensor is completely specified by two potentials: a (gravito-electric) scalar potential
w(t,x) and a vector potential wi (t,x) just as in case of Maxwell’s theory of electromagnetism. From the
’Newtonian’ limit one finds that the scalar potential w generalizes the Newtonian gravitational potential
U . As will become clear from the following a split of w into a Newtonian and some post-Newtonian
part is not meaningful; it only gives rise to confusion. The potential w determines the time-time and the
space-space part of the metric tensor. The time-space component, g0i , that is determined by the vector
potential wi describes gravito-magnetic type effects, i.e., effects that arise from matter-currents (moving
or rotating masses).
We will now compute the differential geometrical quantities to post-Newtonian accuracy for celestial
mechanical problems:
2
g ≡ − det gµν = e4w/c + O4
√ 2
g = e2w/c + O4 . (4.2.7)
The inverse metric g µν is given by
2w2

2 2w
g 00 = − exp + 2 w = −1 − 2 − 4 + O6
c c c
4
g 0i = − 3 wi + O 5 (4.2.8)
c
2 2
g ij = = δij exp − 2 w + O4 = δij 1 − 2 w + O4 .
c c
For the Christoffel-symbols

1 αλ
Γαβγ = g (gλβ,γ + gλγ,β − gβγ,λ )
2
one finds
w,0
Γ000 = − + O5
c2
w,i
Γ00i = − 2 + O6
c
w,0 4
Γ0ij = δij 2 + 3 w(i,j) + O5
c c
w,i w w,i 4
Γi00 = − 2 + 4 4 − 3 wi,0 + O6
c c c
4 w,0
Γi0j = − 3 w[i,j] + 2 δij + O5
c c
w,k w,j w,i
Γijk = δij 2 + δik 2 − δjk 2 + O4 (4.2.9)
c c c
where
1
w(i,j) ≡ (wi,j + wj,i )
2
1
w[i,j] ≡ (wi,j − wj,i ) .
2
For the components of the Riemann curvature tensor one obtains
1 1
R0i0j = − w,ij − 4 [−2ww,ij + δij w,tt + 2∂t (wi,j + wj,i )
c2 c
− 3w,i w,j + δij w,k w,k ] + O6
1
R0ijk = [δij w,tk − δik w,tj + 2∂i (wj,k − wk,j )] + O5
c3
Section 4.3 Field equations and the gauge problem 145
1
Rijkl = [δil w,jk − δjl w,ik + δkj w,il − δki w,jl ] + O4 (4.2.10)
c2
and
R0 ijk = −R0ijk + O5 = Ri 0jk + O5

R i
jkl = Rijkl + O4
w,ij 1
R0 i0j = + 4 [δij w,tt + 2∂t (wi,j + wj,i )
c2 c
− 3w,i w,j + δij w,k w,k ] + O6
1 1
Ri 0j0 = − 2 w,ij − 4 [−4ww,ij + δij w,tt + 2∂t (wi,j + wj,i )
c c
− 3w,i w,j + δij w,k w,k ] + O6 . (4.2.11)
From this one derive the components of the Ricci-tensor
Rµν = Rσ µσν
1 1
R00 = − ∆w − 4 [−4w ∆w + 3w,tt + 4∂ti wi ] + O6
c2 c
1
R0i = [−2w,ti + 2(∆w − ∂ij wj )] + O5
c3
1
Rij = − 2 δij ∆w + O4 (4.2.12)
c
and
1 3 4
R00 = − ∆w − 4 w,tt − 4 ∂ti wi + O6
c2 c c
2
R0i = − 3 [∆wi − ∂ij wj − ∂ti w] + O5
c
1
Rij = − 2 δij ∆w + O4 . (4.2.13)
c
From this we find the curvature scalar
R = g µν Rµν = g 00 R00 + 2g 0i R0i + g ij Rij ,
i.e.,
2
R=− ∆w + O4 . (4.2.14)
c2
4.3 Field equations and the gauge problem

We will now show that the spatial field equations are satisfied to O4 by γij = δij + O4 . According to
(4.2.2) Tij = O0 and we have to prove that
1 8πG
Rij − gij R = 4 Tij = O4 .
2 c
The left hand side of this equation reads
1 1 2
− δij ∆w + O4 + δij 2 ∆w + O4 = O4
c2 2 c
as was to be shown. This implies that only the field equations
G00 = κ T00 , G0i = κ T0i
or

8π G 1 00
R00 = g gαβ T αβ
T 00 −
c4 2

8π G 1
R0i = T 0i
− g 0i
gαβ T αβ
c4 2
remain to be solved. Taking into account the order of T αβ one finds
4πG 00
R00 = (T + T ss ) + O6
c4
8πG 0i
R0i = T + O5 .
c4
Inserting expressions (4.2.13) we get the field equations in the form:
3 4
∆w + w,tt + 2 ∂ti wi = −4πGσ + O4
c2 c (4.3.1)
∆wi − ∂ij wj − ∂ti w = −4πGσ i + O2
with
T 00 + T ss T 0i
σ≡ 2
; σi ≡ .
c c
Here, σ acts as active gravitational mass-energy density generalizing the density ρ in Newton’s theory of
gravity. σ i is the active gravitational mass-current density that does not act as a field generating source
in Newton’s theory. The field equations (4.3.1) have a very remarkable property: they are linear in the
metric potentials
wµ ≡ (w,wi ) .
This results from the special form of the metric tensor and the various approximations involved. Note that
Einstein’s field equations are non-linear in general and the achieved linearity simplifies the formalism
tremendously. The choice of spatially isotropic coordinates with γij = δij + O4 corresponds to three
gauge conditions for the three spatial coordinates xi .
A function f (xµ ) is called harmonic if
1 √
g f ≡ √ ( gg µν f,ν ),µ = 0.
g
Note, that in three dimensional Euclidean space, IR3 , the operator g reduces to the usual Laplacian ∆
thus g might be called the covariant Laplacian.
By direct calculation one can show that the spatial coordinates xi of the metric (4.2.6) are harmonic
up to terms of O4 , i.e.,
g xi = O4 . (4.3.2)
Because of this only the gauge freedom of the time coordinate is left. The field equations simplify further
if also the time coordinate is chosen to be harmonic, i.e., if
g x0 = O5 .
Section 4.3 Field equations and the gauge problem 147
We have
√ √ √ √
gg x0 = ( gg µ0 ),µ = ( gg 00 ),0 + ( gg i0 ),i
4w,0 4wi,i
= − 2 + 3 + O5 .
c c
Hence we can write the condition for the time coordinate t to be harmonic in the form
4
0=− (∂t w + ∂i wi ) + O5 . (4.3.3)
c3
With this harmonic gauge the field equations take the form
w = −4πGσ + O4 (4.3.4)
∆wi = −4πGσ + O2 ,
i
(4.3.5)
where is the flat space d’Alembertian
1 ∂2
≡− + ∆.
c2 ∂t2
We can even combine the source- and the field-variables
σµ ≡ (σ,σi ); wµ ≡ (w, wi )
and write in obvious notation

wµ = −4πGσµ + O(4,2) .
If we consider one isolated system (e.g., an idealized solar system) with no gravitational sources outside
this system we can require
gµν −→ ηµν , |x| → ∞
i.e., we consider our space-time manifold to be asymptotically flat. For our potentials this implies
(w,wi ) −→ 0 |x| → ∞ .
With such a condition for asymptotic flatness the retarded solution of our field equations reads
σ µ (tret ; x0 )
Z
µ
wret (t,x) = G d3 x0 , (4.3.6)
|x − x0 |
where tret is retarded time

|x − x0 |
tret ≡ t − .
c
Another possible solution is the advanced one with tret being replaced by tadv with
|x − x0 |
tadv ≡ t + .
c
Still another solution is
µ 1 µ µ
wmixed (t,x) = [w (t,x) + wadv (t,x)] .
2 ret
This mixed solution that might also be called time-symmetric solution is in fact usually used in post-
Newtonian theories. The reason for this is the following: if we expand σ µ around the coordinate time t
we encounter a sequence of time derivatives and the first time derivative is related with irreversible pro-
cesses such as the emission of gravity waves that do not occur in the first post-Newtonian approximation
to Einstein’s theory of gravity. With r ≡ |x − x0 | we get
σ µ (t ∓ r/c; x0 ) σ µ (t; x0 )
Z Z
µ
wret/adv (t,x) = G d3 x0 = G d3 x0
r r
Z 2 Z
G ∂ G ∂
∓ d3 x0 σ µ (t; x0 ) + 2 2 d3 x0 σ µ (t; x0 )r .
c ∂t 2c ∂t
If we take the time symmetric solution then in the expansion the first time derivative terms cancel auto-
matically, i.e.,
σ µ (t; x0 ) G ∂2
Z Z
µ
wmixed (t,x) = G d 3 x0 0
+ 2 2 d3 x0 σ µ (t; x0 )|x − x0 | . (4.3.7)
|x − x | 2c ∂t
Note that the retarded, advanced or mixed solutions of the harmonic field equations are not the only
ones. If w,wi solve these equations then also
1 1
w0 = w − ∂t λ; wi0 = wi + ∂i λ
c2 4
corresponding to a change of the time variable of the form
t −→ t0 : t0 = t − c−4 λ .
If ∆λ = O2 then wµ0 will be another harmonic solution of the PN field equations.
4.4 The gravitational field of a body

As first application of the post-Newtonian formalism we will discuss the gravitational field of some
isolated matter distribution.
4.4.1 Post-Newtonian multipole moments
Let us consider first the case of a single static body. From the PN field equations, (4.3.4) and (4.3.5), we
get
σ(t,x0 )
Z
w(t,x) = G d3 x0 + O5 ; wi (t,x) = 0 .
|x − x0 |
Usually the term |x − x0 |−1 is expanded in terms of spherical harmonics. In relativity, partly due to the
important role of Lorentz-transformations, usually another expansion, related with Cartesian tensors, is
employed.
A Cartesian l-tensor is a set of real or complex numbers Ti1 i2 ...il with l different indices i1 to il , each
taking the values 1,2,3 or equivalently (x,y,z). For the sake of compactness often a set of l Cartesian
indices is abbreviated by a multi-index, e.g., L ≡ i1 i2 . . . il etc. Usually Einstein’s summation conven-
tion is assumed, i.e., if some index appears twice a summation over that index is implied automatically,
e.g.,
3
X
AL BL ≡ Ai1 i2 ...il Bi1 i2 ...il ≡ Ai1 i2 ...il Bi1 i2 ...il .
i,j=1
Section 4.4 The gravitational field of a body 149
Given a Cartesian tensor TL , we denote its symmetric part by parentheses

1X
T(L) ≡ T(i1 ...il ) ≡ Tiσ(1) ...iσ(l) ,
l! σ
where σ runs over all l! permutations of (12 . . . l). If TL is a Cartesian l-tensor; each quantity where
we put two arbitrary indices equal with a subsequent summation is called a trace of TL . If every trace
of TL vanishes it is called trace-free. Of great importance are symmetric and trace-free (STF) Cartesian
tensors. The STF-part of TL is denoted indifferently by T̂L ≡ T<L> ≡ T<i1 ...il > . The explicit
expression of the STF part reads (e.g., [36])
[ 21 l]
X
T̂L = alk δ(i1 i2 . . . δi2k−1 i2k Si2k+1 ...il )a1 a1 ...ak ak
k=0
where
SL = T(L) ,
l! (−1)k (2l − 2k − 1)!!
alk = ,
(2l − 1)!! (l − 2k)!(2k)!!
1
l denoting the integer part of 12 l. For instance, T̂ij = T(ij) − 13 δij Taa ; T̂ijk = T(ijk) − 15 δij T(kaa) + δjk T(iaa) + δki T(jaa) .

2
0 −1
A Taylor-expansion of |x − x | in the gravito-electric potential w(t,x) then yields:
X (−1)l Z 3 0 0 0i
1
w(t,x) = G d x σ x 1 · · · x0il ∂i1 ···il ,
l! r
l≥0
where r ≡ |x|. Let

∂l

1 1
ϕi1 ...il ≡ ∂i1 ···il ≡ .
r ∂x 1 · · · x l r
i i
Then because of
1
∆ =0
r
the symmetric Cartesian (every index takes the values x,y,z) tensor ϕi1 ...il is trace-free, i.e., a Cartesian
STF-tensor. By induction one can prove that
n̂L
ϕL = (−1)l (2l − 1)!! ,
rl+1
where
xi1 xil
ni1 ...il ≡ ··· ,
r r
so that
X (2l − 1)!! n̂L
w(t,x) = G ML l+1 (4.4.1)
l! r
l≥0
with Z
ML = M̂L = d3 x0 σ 0 x̂0L .
The following Theorem is not restricted to the static case.

Theorem 4.1: (Blanchet and Damour [41]) Outside of some isolated matter distribution the functions
σµ (t± ; x0 )
Z
wµ (t,x) = G d3 x0
|x − x0 |
with
1
t± ≡ tmixed = (tret + tadv ) .
2
admit an expansion of the form (r = |x|)
X (−1)l 1
∂L r−1 ML (t± ) + 2 ∂t Λ + O4

w(t,x) = G (4.4.2)
l! c
l≥0
X (−1)l
d

wi (t,x) = −G ∂L−1 r−1 MiL−1
l! dt
l≥1

l 1
ijk ∂jL−1 r−1 SkL−1 − ∂i Λ + O2 .

+ (4.4.3)
l+1 4
Here,
X (−1)l 2l + 1
Λ ≡ 4G ∂L (r−1 µL (t± ))
(l + 1)! 2l + 3
l≥0
Z
µL ≡ d3 x x̂iL σ i (t,x)
and
d2
Z Z
1
ML (t) ≡ d3 x x̂L σ + d3 x x̂L x2 σ
2(2l + 3)c2 dt2
Z
4(2l + 1) d
− d3 x x̂iL σ i (l ≥ 0) (4.4.4)
(l + 1)(2l + 3)c2 dt
Z
SL (t) ≡ d3 x ij<kl x̂L−1>i σ j . (l ≥ 1) (4.4.5)
The Cartesian STF-tensors ML and SL are the post-Newtonian mass- and spin multipole moments
that characterize the gravitational action of the matter distribution in the outside vacuum region. They
have been first introduced in a paper by Blanchet and Damour [41] and are called BD-moments. M is
the post-Newtonian mass of the matter distribution, Mi is its mass dipole, Mij its mass quadrupole etc.
S i with Z Z
S i = d3 x jki xj σ k = d3 x (x × σ)i
is the spin-vector (total angular momentum) of the matter distribution (note that for an ideal fluid σ i =
ρv i + O2 ).
The proof of the BD-theorem can be taken from [41].
It is not difficult to see that Λ is a harmonic function, i.e., ∆Λ = 0. For that reason we can remove
the Λ-terms by means of a gauge transformation; in such a skeletonized harmonic gauge the potentials
w and wi also satisfy our harmonic field equations.
For many applications we can neglect the time derivatives of ML . That is the case for bodies with
approximate axial symmetry rotating approximately around their symmetry axis. We will also neglect
the higher spin-moments SL for l > 1. With these approximations in the skeletonized gauge the metric
Section 4.5 Geodesic motion in the PN-Schwarzschild field 151
potentials read [42]:

X (−1)l
w(t,x) ' G ML ∂L (r−1 ) + O4
l!
l≥0
G (x × S)i
wi (t,x) ' − + O2 . (4.4.6)
2 r3
This result for w is remarkable indeed because formally it agrees with the Newtonian one. Note, how-
ever, that here we are dealing with solutions of the post-Newtonian field equations! In other words: the
post-Newtonian BD-mass moments ML have been introduced in such a clever way that the solution w
of a relativistic field equation practically looks Newtonian. If one thinks about a parameter formalism
whose parameters are determined from observational data then the BD-moments are the quantities that
parametrizes the gravitational field outside some matter distribution in our post-Newtonian framework;
they are directly measurable (e.g., by an analysis of satellite orbits). Especially it makes no sense to split
the ML ’s into some ’Newtonian-part’ plus c−2 -corrections.
4.5 Geodesic motion in the PN-Schwarzschild field

We now come to the problem of motion of some satellite or planet or Moon in the gravitational field of
some spherically symmetric central mass that is described by wi = 0 and
GM µ
w= ≡ .
r r
There are many possibilities to formulate such a post-Newtonian orbit [43]; here we concentrate on the
Brumberg [34] form.
As we will see later the motion of a test body in some external gravitational field is along a geodesic
determined by the external metric, that satisfies an equation of the form
d2 xµ dxν dxσ
2
+ Γµ
νσ = 0, (4.5.1)
dλ dλ dλ
where λ parametrizes the geodetic curve xµ (λ). Replacing λ by the coordinate time t by using the
geodetic equation (4.5.1) with µ = 0, the geodesic equation takes the form
d2 xi vj vj vk vj vj vk vi

2
= −c2 Γi00 + 2Γi0j + Γijk − Γ000 + 2Γ00j + Γ0jk .
dt c c c c c c c
The relevant non-vanishing Christoffel-symbols read to PN-order
µ xi µ
Γi00 = 1−4 2 ,
c2 r 2 r c r
µ
Γijk = − xk δij + xj δik − xi δjk ,
c2 r3
j
µ x
Γ00j = ,
c2 r 2 r
so that the equation of motion takes the form
d2 xi xi xi xi vi

µ
2
= −µ 3 + 2 4µ 3 − 2 v2 + 4 2 (x · v) . (4.5.2)
dt r c r r r r
It has several advantages to derive this equation of motion from a Lagrangian L. The geodesic equation
satisfies an extremal principle of the form
Z Z
ds
0 = δ ds = δ dt
dt
and instead of ds we can equally well use the proper time interval dτ . For that reason one usually defines
a Lagrangian L via
dτ
= 1 − c2 L .
dt
For the PN Schwarzschild field the Lagrangian takes the form
µ 1 1 µ 2 3 µ v 2 1 v 4
L= + v2 − 2 + + (4.5.3)
r 2 2c r 2r c 8 c2
up to terms of O4 .
The main advantage of the Lagrangian is the possibility to construct explicitly first integrals of motion
connected with the conservation of (specific) energy E and angular momentum J (e.g., [34,43]). These
two conserved quantities are given by
∂L
E = v −L
∂v
1 2 µ 3 v 2 µ µ
= v − + v2 + 2 + 3v2 (4.5.4)
2 r 8 c 2c r r
and
∂L
J = x×
∂v
1 v2

3µ
= (x × v) 1 + 2
+ 2 . (4.5.5)
2c c r
Note, that the conservation of angular momentum implies that the orbit is confined to a coordinate plane.
Using polar coordinates in the orbital plane with
x = r er , ẋ = ṙ er + rφ̇ eφ
and
|x × ẋ| = r2 φ̇ , v2 = ṙ2 + r2 φ̇2
the specific energy E and the absolute value of the orbital angular momentum J = |J| can be written as
1 2 µ 3
E = (ṙ + r2 φ̇2 ) − + 2 (ṙ2 + r2 φ̇2 )2
2 r 8c
µ µ
+ 2 + 3(ṙ2 + r2 φ̇2 )
2cr r
1 3µ
J = r φ̇ 1 + 2 (ṙ2 + r2 φ̇2 ) + 2
2
.
2c c r
This leads us to first order differential equations of motion

1 3µ
r2 φ̇ = J 1 − 2 (ṙ2 + r2 φ̇2 ) − 2
2c c r
Section 4.5 Geodesic motion in the PN-Schwarzschild field 153
E

4µ
= J 1− 2 − 2 (4.5.6)
c c r
and
2µ 3 3µ
ṙ2 = −r2 φ̇2 + + 2E − 2 (ṙ2 + r2 φ̇2 )2 − 2 (ṙ2 + r2 φ̇2 )
r 4c c r
µ2
−
c2 r2
2µ E2 µE µ µ
= −r2 φ̇2 + + 2 E − 3 2 − 12 − 10 . (4.5.7)
r c r c2 r c2 r
Eliminating the φ̇2 term from the last equation we get
2B C D
ṙ2 = A + + 2 + 3 (4.5.8)
r r r
with
E2
A = 2E − 3
c2
E
B = µ − 6µ 2
c
µ2

2E
C = −J2 1 − 2 − 10 2
c c
2 µ
D = 8J 2 .
c
Using
2 2
dr(φ(t)) dr
ṙ2 = = φ̇
dt dφ
2
E

d(1/r) µ
= J2 1−8 2 + 2
dφ c r c
the radial equation can be written in the form

2
2B 0 C0

d(1/r)
= A0 + + 2 (4.5.9)
dφ r r
with
1E

2E
A0 = 1+
J2 2 c2
E

µ
B0 = 1 + 4
J2 c2
µ2
C0 = −1 + 6 .
c2 J2
Let us now write the radial equation as
2
d(1/r) 1 1 1 1
=C − − (4.5.10)
dφ r a(1 + e) a(1 − e) r
with
µ2
C=1−6 . (4.5.11)
c2 J2
From this representation we see that r± = a(1 ± e) represent the minimal and maximal values for r,
i.e. a and e have the usual meaning as semi-major axis and numerical eccentricity of the post-Newtonian
orbit and might be considered as integration constants alternative to E and J. We have
1
A0 = − C
a2 (1 − e2 )
and
C
B0 =
a(1 − e2 )
from which we derive

µ 6 µ
J2 = µa(1 − e2 ) 1 − 2 2 +
c a 1 − e2 c2 a

µ 7µ
E = − 1− 2 .
2a 4c a
The solution of (4.5.10) is then simply given by
a(1 − e2 )
r= (4.5.12)
1 + e cos f
where the ’true anomaly’ f obeys the relation

2
df µ
=C=1−6 ,
dφ a(1 − e2 )c2
i.e.,
µ
f = 1−3 (φ − φ0 ) . (4.5.13)
a(1 − e2 )c2
This implies that the post-Newtonian orbit is that of a precessing ellipse, the secular drift of the argument
of periastron per revolution being given by
3µ
∆φ = 2π . (4.5.14)
a(1 − e2 )c2
Eliminating E and J from (4.5.6) we obtain

p 1 3 a µ
r2 φ̇ = µa(1 − e2 ) 1 + − + −4 (4.5.15)
2 (1 − e )
2 2
r c a
or together with (4.5)

p µ 1 µ
µa(1 − e2 )dt = r2 df 1 + 4 2 + . (4.5.16)
c r 2 c2 a
For a circular orbit with r = a, e = 0 we have
µ µ
φ̇2 ≡ n2 = 1−3 2 , (4.5.17)
a3 c a
Section 4.6 Astronomical Reference Frames 155
defining the mean motion n of the post-Newtonian orbit and the mean anomaly M via the relation
M = n(t − t0 ) = nt + M0 .
If one introduces as in the usual Newtonian Kepler theory an eccentric anomaly E via
(1 − e2 )1/2 sin E cos E − e

sin f = ; cos f =
1 − e cos E 1 − e cos E
so that √
df 1 − e2
=
dE 1 − e cos E
and
r = a(1 − e cos E)
then the integration of (4.5.16) leads to a corresponding Kepler equation in the form
µ h µ i
M = 1+3 2 E− 1− 2 e sin E . (4.5.18)
c a c a
The time dependence of the orbital point is then obtained by means of the Kepler equation via t →
M → E → f.
4.6 Astronomical Reference Frames

4.6.1 Transformation between global and local systems: first results
For the description of the gravitational N -body problem we will consider a total of N + 1 different
coordinate systems: one global coordinate system xµ = (ct, xi ) in which all N bodies are contained and
a
in which the global dynamics of the system can be described and N local charts XA = (cTA , XA ), A =
α
1, . . . ,N , where the system XA is assumed to move with body A of the system.
In the following we will often speak about two coordinates systems, a global one, Σglob , with coordi-
nates xµ = (ct,xi ) and a local one, Σloc , with coordinates X α = (cT, X a ). Mostly the global one will
be the Barycentric Celestial Reference System (BCRS) and the local one will be the Geocentric Celes-
tial Reference System (GCRS). We now assume the corresponding metric tensors to be of the following
canonical form
g00 = − exp(−2w/c2 )
4
g0i = − 3 wi
c
gij = δij exp(+2w/c2 ) + O4
in Σglob and
G00 = − exp(−2W/c2 )
4
G0a = − 3 Wa
c
Gab = δab exp(+2W/c2 ) + O4
in Σloc , i.e., we assume the metric tensors to be of the same form but with metric potentials wµ ≡ (w,wi )
in the global system and different one W α ≡ (W,W a ) in the local system. Moreover, we assume the
usual conditions (4.2.2) for the energy-momentum tensor in all N+1 coordinate systems. We write the
transformation X α → xµ in the general form
xµ (X α ) = z µ (T ) + eµ a µ a
a (T )X + ξ (T,X ) , (4.6.1)
where ξ µ is at least quadratic in X a . Here, z µ (T ) describes the world-line of some suitably selected
point associated with the body under consideration. This world-line will be called the central world-line
of the corresponding body; later it will be chosen as the body’s post-Newtonian center of mass.
Lemma 4.1: Let Aµ µ α

α ≡ ∂x /∂X and assume
A00 = 1 + O2 , Ai0 = O1 , A0a = O1 , Aia = O0 .
Then from the form of the metric tensors in the two coordinate systems we have
1 dz 0
e00 (T ) ≡ = 1 + O2 ,
c dT
1 dz i i
e0a (T ) = e a + O3 ,
c dT

1 1
e00 (T )eia (T ) = 1 + 2 v2 δij + 2 v i v j Raj (T ) + O4 ,
2c 2c
ξ 0 (T,X a ) = O3 ,

1 i 1
ξ i (T,X a ) = ea (T ) Aa X2
− X a
(A · X) + O4 ,
c2 2
where
dz i dz i
vi = = + O2 ,
dt dT
d2 z i d2 z i
Aa = eia + O2 = eia 2 + O2
dT 2 dt
and Rai (T ) is a slowly time-dependent rotation matrix with
Rai Raj = δij , Rai Rbi = δab
and
dRai
= O2 .
dT
For a proof of Lemma 1.1 see [2].
Let is now consider the transformation law for the metric potentials.
Theorem 4.2: From the transformation of the metric tensors one finds
2V2 c2 0 0

4
W + 2 V a Wa + ln A0 A0 − A0a A0a + O4

w = 1+ 2
c c 2
(4.6.2)
c3 0 i
wi = Rai W a + v i W + (A0 A0 − A0a Aia ) + O2 ,
4
where
v i ≡ Rai V a or V a ≡ Ria v i .
The proof follows directly from g µν (x) = Aµ ν αβ
α Aβ G (X).
Note, that the post-Newtonian transformation of metric potentials (W,W a ) → (w,wi ) is linear, i.e., of
the form
wµ (x) = Aµ α (T )W (X) + B (X) .
α µ
Here,
2V2 4 a
A00 = 1 + ; A0a = V ; Ai0 = v i ; Aia = Rai
c2 c2
and
c2 0 0 c3
B0 = ln A0 A0 − A0a A0a ; Bi = (A00 Ai0 − A0a Aia ) .

2 4
For several purposes one also needs the inverse of (4.6.2) namely
Wα = A−1
αµ (wµ − Bµ ) ,
which reads explicitly (Bµ = Bµ )

2 4
W = 1 + 2 V2 (w − B) − 2 v i (wi − Bi ) + O4
c c (4.6.3)
Wa = −V a (w − B) + Rai (wi − Bi ) + O2 .
4.6.2 Split of local potentials, multipole-moments
Let us consider the metric tensor Gαβ in the local system defined by the metric potentials Wα ≡
(W,Wa ). In the gravitational N -body system these local potentials can be split into two parts
W (T,X) = Wself (T,X) + Wext (T,X)

(4.6.4)
W a (T,X) = Wself
a a
(T,X) + Wext (T,X) .
In the following we will often use the notation

a a
W ≡ Wext ; W ≡ Wext .
If the local system is associated with body E (e.g., the Earth) then the self parts of the metric potentials,
α
Wself , result from the gravitational action of body E itself whereas the external parts of the potentials,
α
Wext , result from the action of all other bodies of the system and inertial terms that appear in the local
E-system. In mathematical terms the self-parts of the local metric potentials are defined by
Σ(T,X0 ) G ∂2
Z Z
Wself (T,X) = G d3 X 0 + d3 X 0 Σ(T,X0 )|X − X0 | ,
E |X − X0 | 2c2 ∂T 2 E
(4.6.5)
Σa (T,X0 )
Z
a
Wself (T,X) = G d3 X 0 ,
E |X − X0 |
where the integrals extend over the support of body E only.
Theorem 4.3: (Extended Blanchet-Damour Theorem) In the local system of body E, outside of E, the
α
self-parts of the metric potential, Wself , admit a convergent expansion of the form (R = |X|)
X (−1)l 1
Wself (T,X) = G ∂L [R−1 MLE (T± )] + 2 ∂T ΛE + O4
l! c
l≥0
X (−1)l
d

a
Wself (T,X) = −G ∂L−1 R−1 E
MaL−1 (4.6.6)
l! dT
l≥1

l 1
+ abc ∂bL−1 (R−1 ScL−1
E
) − ∂a ΛE + O2 .
l+1 4
Here,
X (−1)l 2l + 1
ΛE ≡ 4G ∂L (R−1 µEL (T± ))
(l + 1)! 2l + 3
l≥0
Z
µEL ≡ d3 X X̂ bL Σb (T,X)
E
and
d2
Z Z
1
MLE (T ) = d3 X X̂ L Σ + d3 X X̂ L X2 Σ
E 2(2l + 3)c dT E
2 2
Z
4(2l + 1) d
− d3 X X̂ aL Σa (l ≥ 0) (4.6.7)
(l + 1)(2l + 3)c dT E
2
Z
E
SL (T ) = d3 X ab<cl X̂ L−1>a Σb , (l ≥ 1) ,
E
where f (T± ) = [f (T + R/c) + f (T − R/c)]/2.

The proof is analogous to the one for the Blanchet-Damour Theorem.
MLE and SL E
are the BD mass- and spin multipole moments of body E that reduce to the corresponding
moments in case that there is only one single body. Note, that a post-Newtonian center of mass of body
E can be introduced by the vanishing of the BD mass-dipole, i.e., by
MaE = 0 .
Theorem 4.4: Let

σ(t,x0 ) G ∂2
Z Z
wE (t,x) = G d 3 x0 + d3 x0 σ(t,x0 )|x − x0 | ,
E |x − x0 | 2c2 ∂t2 E
σ i (t,x0 )
Z
wEi (t,x) = G d 3 x0
E |x − x0 |
be the metric potentials in the global system induced by body E, then
wEµ = Aµα (T )Wself,E

α
+ O(4,2) ,
or explicitly
2v2

4 a a
wE = 1+ Wself,E + V Wself,E + O4
c2 c2 (4.6.8)
wEi = Rai Wself,E
a
+ v i Wself,E + O2 ,
α
where Wself,E are the self-parts of the metric potentials in the local E-system and the velocity v refers
to the central point of that system (e.g., the barycentric velocity of the geocenter).
Damour, Soffel and Xu have found several independent proofs for this central Theorem. One of them
might be called the physicist’s proof, where the transformation rules for the energy-momentum tensor
a
and the coordinate transformations are used to transform the integrals defining Wself and Wself into
the global system (see also [1]). Another proof, that might be called the proof of a field theoretician,
can be found in Damour et al., [2].
4.6.3 Tetrad induced local coordinates
So far the quantities eµα have been constrained but not fully specified. This will now be done by requiring
these quantities to represent an orthonormal tetrad along the worldline LE of the origin of the local E-
system given by X a = 0 with respect to the external metric, i.e.,
ext µ ν
gµν eα eβ |LE = ηαβ . (4.6.9)
ext
Here, the external metric tensor, gµν , is defined by
2
ext
g00 = −e−2w/c ,
4
ext
g0i = − 3 wi ,
c
2
ext
gij = δij e2w/c + O4
with X
wµ = µ
wA .
A6=E
Thus, considering the external part of the metric only, the local coordinates X α will be chosen as tetrad
induced coordinates. One consequence of this is
α
Wext (T, 0) = 0 (4.6.10)
and the external part of the metric is Minkowskian at the spatial coordinate origin, i.e.,
Gext
αβ |X a =0 = ηαβ .
The gravitational influence of external bodies is effaced and relation (4.6.10) is sometimes called weak
effacement condition.
Theorem 4.5: From the tetrad condition (4.6.9) one infers that

1 1 2
e00 (T ) = 1 + 2 vE + w(zE )
c 2

1 3 4 1 5
+ 4 vE + (w(zE ))2 + w(zE )vE2 − 4wi (zE )vEi + O6
c 8 2 2
i (4.6.11)
v 1 1 4
e0a (T ) = Rai E
1+ 2 vE2 + 3w(zE ) − 3 wi (zE ) + O5
c c 2 c

1 1
eia (T ) = 1 − 2 w(zE ) δ ij + 2 vEi vEj Raj + O4 .
c 2c
4.6.4 The standard transformation between global and local coordinates
According to (4.6.1) the transformation between global coordinates xµ = (ct, xi ) and local ones X α =
(cT,X a ) for one and the same event is given by
ct = z 0 (T ) + e0a (T )X a + ξ 0 ,
(4.6.12)
xi = z i (T ) + eia (T )X a + ξ i .
The tetrad components, eµ i

a (T ), are given by Theorem 1.5 and ξ is given by Lemma 1.1.
0 4
The quantity ξ , that fixes the 1/c -part in the T ↔ t transformation is left open in Damour et al.,
[2]. If the harmonic gauge is chosen both in the global as well as in the local system the condition for ξ 0
reads:
1 de00 1 d2 e0a a
∆ξ 0 = + 2 X + O5 .
c dT c dT 2
0
Taking any solution of this equation we can then always add a function ξL which satisfies the Laplace
i 0
equation ∆X ξL = 0. From this one finds that ξ for harmonic coordinates might be chosen in the form
1
c3 ξ 0 (T,X) = −2wa,b X a X b − w,t X2 + v · X(w,a X a ) + v · X(w,a − aa )X a
2
1 1
+ (v · a − v a w,a )X2 + (ȧ · X)X2 . (4.6.13)
2 10
Since solar-system ephemerides are given in global coordinates for practical applications it is useful
to invert these relations in the form X α = X α (xµ ). To this end let us consider the three events in
Fig.4.2 denoted by eX , eT and et . In the local E-system eX has coordinates (cT,X a ) related to (ct,xi )
by the general transformation rule (4.6.1), eT (et ) denotes the intersection of the T = const. (t = const.)
hyper-surface through eX with the world-line of the origin of the local E-system (e.g., the geocenter),
given by X a = 0. These two events have coordinates
eT : (tsim , zEi (tsim ) (T,0) ,

et : (t,zEi (t)) (Tsim , 0) .
Figure 4.1. One global and N local coordinate systems are used for the description of the gravitational
N -body problem (from [44]).
Figure 4.2. Three events of importance for the inversion of the coordinate transformation, eX , et and
eT .
Using the general transformation rule (4.6.1) one finds that
1 e0a a
Tsim = T + X + O4
c e00
1
tsim = t − e0a X a + O4
c
and with zsim (t) ≡ z(Tsim )
ei0 e0a

xi − zsim
i
(t) = eia − X a + ξ i (T, X a ) + O4 . (4.6.14)
e00 T
Now, ei0 (T ) = dz i (T )/(cdT ) and, therefore, ei0 /e00 = dz i /(c dt), we get by inserting the expressions
for eia , ei0 and ξ i and solving for X a

1 1 i 1
X a = Rai ri + 2 v (v · r) + w(zE )ri + ri (a · r) − ai r2 + O4 , (4.6.15)
c 2 2
where r(t) ≡ x − zE (t) and a(t) is the acceleration of zE in global coordinates.

The derivation of the T = T (t) relation is more complicated. First we will derive this relation for an
event on the central world-line, i.e., for X a = 0 where we have t = z 0 (T )/c and therefore
dt 1 dz 0 (T )
= = e00 (T )
dT c dT
or
dT
= (e00 (T ))−1 .
dt
Let f be some function defined at the central world-line, LE , then for some event on LE we have
f (t) = f (T ) though the values for t and T will differ to post-Newtonian order. From the last relation
we therefore get for events at X a = 0

dT 1 dA(t) 1 dB(t)
=1− 2 + 4 + O5
dt LE c dt c dt
with
d 1
A(t) = vE2 + w(zE ) ,
dt 2
d 1 3 1
B(t) = − vE4 − w(zE ) + 4vEi wi (zE ) + w2 (zE ) .
dt 8 2 2
Next we consider some event outside the central world-line where we have
ct = z 0 (T ) + e0a (T )X a + ξ 0 (T, X a ) .
If f is again some function on the central world-line we have
f (T ) ' f (t − v · r/c2 ) = f (t) − f,t · (v · r/c2 ) + O3
and ∂T ' ∂t + v i ∂i . We then find that

1 1 1
e0a (T )X a = v · r + 3 (v · r)v2 + 4w(v · r) − (v · a)r2 − 4wi ri + O5 ,
c c 2
where again the indices E have been dropped and the metric potentials have to be taken at LE . Inserting
expression (4.6.13) for ξ 0 we finally get [42]
1
T =t− [A(t) + v · r]
c2 (4.6.16)
1
+ 4 [B(t) + B i (t)ri + B ij (t)ri rj + C(t,x)] + O5 ,
c
where
1
B i (t) = − v2 v i + 4wi − 3v i w ,
2
∂ ∂ 1
B ij (t) = −v i Raj Qa + 2 j wi − v i j w + ẇδ ij .
∂x ∂x 2
The dot on w indicated the total time derivative, i.e.,
ẇ ≡ w,t + v i w,i
and
1 2
C(t,x) = − r (ȧ · r) ,
10
∂
Qa (t) = Rai i
w − ai .
∂x
4.6.5 The description of tidal forces
Post-Newtonian tidal moments
We will now introduce a useful post-Newtonian generalization of the Newtonian tidal expansion of the
effective potential describing the gravitational action of external bodies in a local system co-moving with
some body E together with the inertial forces appearing in that system. As it will become obvious later in
the discussion of the equations of motion of astronomical bodies the following external gravito-electric
and gravito-magnetic fields defined by
4
E a (T,x) ≡ ∂a W + ∂T W a , B a (T,X) = −4abc ∂b W c ,
c2
will play a central role. For that reason they will be considered as post-Newtonian analogues of ∇U tidal .
In a more compact notation we write
4
E = ∇W + ∂T W , B = −4∇ × W .
c2
It is easy to see that the gravito-electric and gravito-magnetic external fields, E and B, are invariant
under gauge transformations of the external metric potentials of the form
1 1
W0 = W − ∂T Λ ; W 0a = W a + ∂Λ .
c2 4
Under such gauge transformations the E and B fields transform according to
0 4 1 4 1
E a = ∂a W 0 + ∂T W 0a = ∂a W − 2 ∂T a Λ + 2 ∂T W a + 2 ∂T a Λ = E a
c2 c c c
and
0 1
B a = abc ∂b (−4W 0c ) = −4abc ∂b W c + ∂c Λ = −abc ∂b W c = B a ,
4
i.e., they are gauge invariant in the sense defined above.
Lemma 4.2: In virtue of these field equations and the definitions of the external ’gauge invariant’ E and
B fields, they satisfy the following homogeneous equations
1
∇×E = − ∂T B ,
c2
∇×B = 4∂T E + O2 ,
3 2
∇·E = − ∂T W + O4 ,
c2
∇·B = 0,
1 2
∇2 E = ∂T E + O4 ,
c2
∇2 B = O2 .
The proof follows by direct calculation.
Next we characterize these fields by two corresponding (gravito-electric and gravito-magnetic) sets
of post-Newtonian tidal moments
GL (T ) ≡ [∂<L−1 E al > (T,X)]X a =0 (l ≥ 1) ,

HL (T ) ≡ [∂<L−1 B al > (T,X)]X a =0 (l ≥ 1) .
Theorem 4.6: The external gravito-electric and gravito-magnetic fields, E and B, are completely deter-
mined by the corresponding sets of tidal moments, GL and HL to post-Newtonian order. Explicitly
they are given by
X1 L 1 d2
E a (T,X) = X̂ GaL (T ) + X2 X̂ L 2 GaL (T )
l! 2(2l + 3)c2 dT
l≥0
7l − 4 d2
− X̂ aL−1 2 GL−1 (T )
(2l + 1)c 2 dT
(4.6.17)
l d
+ abc X̂ bL−1 HcL−1 (T ) + O4 ,
(l + 1)c 2 dT
X L 4l d

B a (T,X) = X̂ HaL (T ) − abc X̂ bL−1 GcL−1 (T ) + O2 ,
l+1 dT
l≥0
where, by convention, we are assuming that any term which contains an undefined tidal moment has
to be replaced by zero. E.g., the term containing the factor (7l − 4)/(2l + 1) vanishes for l = 0.
The proof is indicated in [2]. It involves the relation
l(l − 1) 2 (L−2 al−1 al )

X L = X̂ L + X X̂ δ + O(δδ) ,
2(2l − 1)
where O(δδ) denotes any term with (at least) two Kronecker-deltas and the STF decomposition of a
tensor Ta<L> that is STF only with respect to the multi-index L:
(+) (0) (−)
Ta<L> = T̂aL + ca<al T̂L−1>c + δa<al T̂L−1> ,
in which each tensor T̂ ±,0 is STF:

(+)
T̂L+1 = STFL+1 (TLal+1 ) ,

(0) l
T̂L = STFL al bc TbcL−1 ,
l+1
(−) 2l − 1
T̂L−1 = TccL−1 .
2l + 1
With similar techniques one can show that apart from simple gauge-terms, given by a single function Λ,
also the external potentials, W and W a , can be expressed in terms of the tidal-moments GL and HL
(remember that W (X a = 0) = W a (X a = 0) = 0).
Lemma 4.3: The external metric potentials W and W a can be written in terms of post-Newtonian tidal
series in the form (DSX II, eqs. (4.15))
X1 L 1

W (T,X) = X̂ GL + X2 X̂ L G̈L
l! 2(2l + 3)c 2
l≥1
1
+ Λ,T + O4 ,
c2
(4.6.18)
X1 2l + 1
W a (T,X) = − X̂ aL ĠL
l! (l + 1)(2l + 3)
l≥1

l 1
+ abc X̂ bL−1 HcL−1 − Λ,a + O2 .
4(l + 1) 4
The tidal moments, GEL and HLE , can be split into two parts: a part resulting from the gravitational
action of some external body A6= E, and some inertial part related with the inhomogeneous part in the
transformation of metric potentials:
X A/E
GEL = GL + G00L
A6=E
X
+ HL00
A/E
HLE = HL
A6=E
The parts resulting from body A, as seen in the local E-system, result from the self-parts of body A in
its own local system, W +A and Wa+A , that are directly given in terms of the mass- and spin-moments of
A.
Lemma 4.4: In the local E-system all non-vanishing inertial terms G00L and HL00 are given by (DSX I,
eqs. (6.30)):
G00a = −Aa + O4
3
G00ab = 2 A<a Ab> + O4
c (4.6.19)
dRi

Ha00 = abc Vb Ac + c2 b Rci + O2 ,
dT
where
d2 zEν
Aa = ηµν eµ
a
dτf2
with
dτf2 = −c−2 ηµν dzEµ dzEν .
Explicitly, G00a is given by
d2 zEi w(zE ) a 1 v2
G00a = −Rai 2
+ 2
aE − 2 (vE · aE )vEa − 2E aaA . (4.6.20)
dt c 2c c
Thus the inertial gravito-electric tidal dipole moment, G00a contains the global coordinate acceleration of
the center of mass of body E.
4.7 The gravitational N-body problem

4.7.1 Local evolution equations
The global equations of motion in the gravitational N-body problem result from the local evolution
equation
ν ν
Tµ;ν = Tµ,ν + Γννσ Tµσ − Γσµν Tσν = 0 . (4.7.1)
Lemma 4.5: In any local system the local evolution equations (4.7.1) take the following form (DSX I,
eqs. (5.6)):
∂ ∂ a 1 ∂ bb 1 ∂
Σ+ Σ = 2 T − 2Σ W + O4 . (4.7.2)
∂T ∂X a c ∂T c ∂T
This equation for µ = 0 is the energy-equation. The µ = a equation reads

∂ 4 ∂ 4
1 + 2 W Σa + 1 + W T ab
= F a + O4 . (4.7.3)
∂T c ∂X b c2
This is the post-Newtonian Euler-equation. Here,

1
F a = ΣEa + Bab Σb
c2
is the gravitational Lorentz-force, where
4
Ea = ∂a W + ∂T Wa ,
c2
Bab = abc Bc = ∂a (−4Wb ) − ∂b (−4Wa ) .
Section 4.7 The gravitational N-body problem 167
We will show (4.7.3). The post-Newtonian energy equation (4.7.2) follows similarly. Since (G ≡
− det(Gαβ ))
1 ∂ √
Γννσ = √ G
G ∂X σ
the µ = a equation reads
√

ν ∂ ν 1 σ
Ta;ν = Ta + √ ∂X G Taσ − Γσaν Tσν
∂X ν G
1 ∂ √
= √ ( GTaν ) − Γσaν Tσν = 0
G ∂X ν
or
∂ √ √
( GTaν ) = GGσν Γσaµ T µν .
∂X ν
Since
1 µν
Gσν Γσaµ T µν = T ∂a Gµν
2
we obtain:
∂ √ 1√
( GTaν ) = GT µν ∂a Gµν .
∂X ν 2
√
Since G = 1 + 2W/c2 + O4 , the right hand side of the last equation reads:

1 2W h 00 i
1+ 2 T ∂a G00 + 2T 0b ∂a G0b + T bc ∂a Gbc
2 c
T 0b b

1 2W W
= 1+ 2 2(T 00 + T ss )∂a 2 − 8 3 W,a + O4
2 c c c
4
= ΣW,a − 2 Σb ∂a W b + O4 .
c
From this we get the Blanchet-Damour-Schäfer (BDS) form [45] of the post-Newtonian Euler-equation:
∂ a ∂ √ 4
Π + ( GTab ) = ΣW,a − 2 Σb ∂a W b + O4 , (4.7.4)
∂T ∂X b c
where
√

4 4W
Πa = c−1 GTa0 = − 2 W a Σ + 1 + 2 Σa + O4 .
c c
Relation (4.7.3) then follows from the BDS-form with
√

4W 4
GTab = 1 + 2 T ab − 2 W a Σb
c c
and
∂
(W a Σ) = ΣW,T
a
+ W a Σ,T = ΣW,T
a
− W a Σb,b + O2 .
∂T
4.7.2 The translational motion
We will now fix the central world-line of a body A, member of gravitational N -body system, by choosing
the origin of the local system, X a = 0, to coincide with the post-Newtonian center of mass by
MaA (T ) = 0 . (4.7.5)
A d’Alembert criterion [46] will lead to the global translational equations of motion for the N -body
problem.
Theorem 4.7: From the local evolution equations (4.7.2) and (4.7.3) one obtains (DSX II, (4.20),(4.21)):
dM A
= F0 + O4
dT
d2 MaA
= Fa + O4
dT 2
dSaA
= Da + O2
dT
with

1 X1 dGL dML
F0 = − (l + 1)M L + l GL + O4
c2 l! dT dT
l≥0
X1 l 1 dHcL
Fa = ML GaL + 2 SL HaL + 2 abc MbL
l! c (l + 1) c (l + 2) dT
l≥0
1 dMbL 4(l + 1) dGcL 4 dSbL

+ abc HcL − 2 abc SbL − 2 abc GcL
c2 (l + 1) dT c (l + 2)2 dT c (l + 2) dT
2l3 + 7l2 + 15l + 6 d2 GL 2l3 + 5l2 + 12l + 5 dMaL dGL

− MaL −
c2 (l + 1)(2l + 3) dT 2 c2 (l + 1)2 dT dT
l2 + l + 4 d2 MaL

− GL + O4 ,
c2 (l + 1) dT 2
X1
Da = abc MbL GcL + O2 .
l!
l≥0
In the following we will study the consequences of Theorem 4.7 for a system of N mass-monopoles,
i.e., we will assume
MLA = SLA
= 0, λ ≥ 1,
for all bodies A from the system. Since G(T ) = 0, one finds that for the mass-monopole model
dM
= O4 ,
dT
i.e., the masses of the N bodies are conserved to post-Newtonian order. From (4.7.5) and Theorem 1.7
we then get from the d’Alembert (1743) equation
d 2 Ma
0= = Fa = MA Ga ,
dT 2
Section 4.7 The gravitational N-body problem 169
or
Ga = 0 . (4.7.6)
Implicitly the translational equations of motion is given by the vanishing of the external gravito-electric
tidal dipole-moment Ga . It is not difficult to show that equation (4.7.6) is equivalent to the geodesic
equation
duλ λ
+ Γµν uµ uν = O4 (4.7.7)
dτ
in the external metric. Here,
dzEµ dz µ
uµ = uµE = = E ,
dτ dT
where we have assumed, according to the weak effacement condition, that T = τ and −c2 dτ 2 =
g µν dxµ dxν |X=0 .
Inserting the Christoffel-symbols of the global external metric as seen by body A, given by wA and wAi ,
the translational equation of motion for the center of mass of body A in the mass-monopole model reads:
d2 zAi

4 1 2 4 A 4 A A j
= 1 − w A + v A ∂i w A + 2 ∂t w i − 2 (∂i w j − ∂j w i )vA
dt2 c2 c2 c c (4.7.8)
1
− 2 (3∂t wA + 4vAj ∂j wA )vAi + O4 ,
c
where vAi is the global coordinate velocity of body A. Finally we need the external metric potentials
related with body A explicitly X B
wAµ = wµ .
B6=A
In the local B-system we simply have
B GMB
Wself = ; WaB,self = 0 .
RB
Transformation according to Theorem 1.4 yields

2 GMB GMB i
wB = 1 + 2 vB2 ; wiB = vB . (4.7.9)
c RB RB
At this point we need the transformation rule for the inverse distance from the center of body B, that
follows from (4.6.14) (rB (t) ≡ x − zB (t), nB (t) ≡ rB (t)/|rB (t)|, aB ≡ d2 zB /dt2 ):

1 1 w(zB ) 1 2 1
= 1− − (v B · nB ) − a B · r B
RB rB c2 2c2 2c2
we get
v2

GMB w(zB ) 1 2 1
wB = 1 + 2 2B − − (v B · nB ) − aB · r B
rB c c2 2c2 2c2
(4.7.10)
GM B i
wiB = vB .
rB
Inserting these potentials into (4.7.8) we finally end up with the Einstein-Infeld-Hoffmann (EIH) equa-
tions of motion for body A:
d2 zAi i(LD)
= aA (zA , vA ) + O4 , (4.7.11)
dt2
(LD)
where the Lorentz-Droste acceleration, aA , is given by
X GMB
(LD) 1 2 2 3 2
aA = − 2
n AB 1 + v A + 2v B − 4vA · v B − (nAB · vB )
rAB c2 2
B6=A

X GMC X GMC 1 rAB

−4 2
− 2
1+ nAB · nCB 
c rAC c rBC 2 rCB
C6=A C6=B
(4.7.12)
7 XX G2 MB MC
− nBC 2 2
2 c rAB rBC
B6=A C6=B
X GMB
+ (vA − vB ) 2 2 (4nAB · vA − 3nAB · vB ) ,
c rAB
B6=A
where
rAB ≡ |zA (t) − zB (t)| , nAB ≡ [zA (t) − zB (t)]/rAB .
4.8 Further developments

So far the central elements of Damour et al., [2] were exhibited and complemented with some so far
unpublished material devoted to certain aspects of the whole issue. The DSX-formalism was further
developed by three following papers [3-5].
In the second paper [3] the problem of translational laws of motion for a system of N arbitrarily
composed and shaped, weakly self-gravitating, rotating deformable bodies are obtained at the first post-
Newtonian approximation to Einstein’s theory of gravity. The full set of mass- and spin- multipole
moments, ML and SL , was taken into account for each of the bodies. First complete and explicit results
for the laws of motion of each body of the system were presented as an infinite series exhibiting the
coupling of all the BD-moments to the post-Newtonian tidal moments, GL and HL , felt by this body.
Finally explicit expressions of these tidal moments in terms of BD-moments of the other bodies are
derived. For a derivation of corresponding equations of motion assumptions about the time dependence
of BD-moments for each of the bodies have to be made. A rigidly rotating multipole model, that leads
to a closed set of dynamical equations of motion, was presented in Klioner et al., [47].
The third paper of this series [4] deals with rotational equations of motion. It is shown how to associate
with each body, in its own local frame, a post-Newtonian spin vector, whose local time evolution is
entirely determined by the coupling between the BD-moments of that body and the tidal moments it
experiences. The leading relativistic effects in the spin motion (geodetic-, Lense-Thirring- and Thomas
precession) are discussed in detail as are the dominant relativistic contributions to the torque acting on a
body in its local frame.
Finally the 4th paper of the series [5] formulates the basic translational equations of motion for ar-
tificial Earth satellites. These equations are given both in a local, geocentric system (the Geocentric
Celestial Reference System, GCRS) and in the global, barycentric one (the Barycentric Celestial Refer-
ence System, BCRS).
Of course many other scientists have contributed important publications to the field of relativistic ce-
lestial mechanics (especially to the dynamics of compact binaries because of its relevance to the problem
of gravity wave emission). For these developments the reader is referred to the corresponding literature
(see e.g., [48] and the contributions by Thibault Damour by Gerhard Schäfer in this Festschrift and ref-
erences cited therein). A generalization of the DSX-formalism by including the PPN-parameters β and
γ was presented in [49].
Section 4.8 Further developments 171
An extended version of the Brumberg-Kopeikin formalism with PPN parameters included has been
worked out in the review paper by Kopeikin and Vlasov [50] (see also the discussion in this volume,
chapter 3).
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5 General Relativistic Theory of Light Propaga-
tion in Multipolar Gravitational Fields
Pavel Korobkov1 and Sergei Kopeikin2
1
Solovetsky Township, Arkhangelsk Region 164070, Russia
2
Department of Physics & Astronomy, University of Missouri, 322 Physics Bldg., Columbia,
Missouri 65211, USA
5.1 Introduction
5.1.1 Statement of the problem
Direct experimental detection of gravitational waves is a fascinating but yet unsolved problem of mod-
ern fundamental physics Enormous efforts have been undertaken to make progress in its solution both
by theorists and experimentalists [7–11]. The main theoretical efforts are presently focused on calcula-
tion of templates of the gravitational waves emitted by coalescing binary systems comprised of neutron
stars and/or black holes [12–17] as well as creation of improved filtering technique for gravitational
wave detectors [18, 19] which will enable the extraction of the gravitational wave signal from all kind
of interferences present in the noisy data collected by the gravitational wave observatories . Direct
experimental efforts have led to the construction of several ground-based optical interferometers with
the length of arms reaching a few miles [20–24]. Certain work is under way to build super-sensitive
cryogenic-bar gravitational-wave detectors of Weber’s type [25–28]. Spaceborne laser interferometric
detectors like LISA [29], NGO [30] or ASTROD [31, 32] may significantly increase the sensitivity of
the gravitational-wave detectors and revolutionize the field of gravitational physics.
There is no doubt, the detection of gravitational waves by the specialized gravitational wave antennas
would provide the most direct evidence of the existence of these elusive ripples in the fabric of spacetime.
On the other hand, there exist a number of astronomical phenomena which might be used for an indi-
rect detection of gravitational waves and understanding gravitational physics of astrophysical systems
emitting gravitational waves. It is worth emphasizing that the present-day ground-based gravitational
wave detectors are sensitive to a rather narrow band of the gravitational wave spectrum ranging between
1000÷1 Hz [33, 34]. Spaceborne gravitational wave detectors may bring the sensitivity band down to
1 mHz [30, 32]. In order to explore the ultra-long gravitational wave phenomena much below 1 mHz
we have to rely upon other astronomical techniques for example, pulsar timing [35, 36] or radiometry of
polarization of cosmic microwave background radiation [37, 38]. The short gravitational waves can be
generated by coalescing binary systems of compact astrophysical objects like neutron stars and/or black
holes [39–42]. The ultra-long gravitational waves can be generated by localized gravitational sources
and/or topological defects in the early universe [43–47].
Astronomical observations are conducted with electromagnetic waves (photons) of different frequen-
cies over the spectrum spreading from very long radio waves to gamma rays. Therefore, in order to
detect the effects of the gravitational waves by astronomical technique one has to solve the problem of
propagation of electromagnetic waves through the field generated by a localized gravitationally-bounded
distribution of masses which motion is determined by general relativity. Notice that gravitational wave
176 Chapter 5 General Relativistic Theory of Light Propagation in Multipolar Gravitational Fields
detectors are optical interferometers making use of light propagation. Therefore, correct understanding
of the process of physical interaction of the laser beam of the interferometer with incoming gravitational
waves is important for their unambiguous recognition and detection. Equations of propagation of elec-
tromagnetic signals must be derived in the framework of the same theory of gravity in order to keep
description of gravitational and electromagnetic phenomena on the same theoretical ground. We draw
attention of the reader that the parametrized post-Newtonian (PPN) formalism [135] does not comply
with this requirement. The PPN formalism is constructed on the ground of plausible physical hypothesis
and assumptions about alternatives to general relativity but it does not demand their overall conformity.
Therefore, straightforward application of PPN formalism to discuss gravitational physics may eventu-
ally lead to incorrect results. As an example, we point out that PPN formalism does not produce the
consistent post-Newtonian equations of motion of extended bodies even in a simplified case of two PPN
parameters, β and γ, when more subtle effects of body’s gravitational multipoles are taken into account
[136]. PPN formalism applied to interpret gravitational light-ray deflection experiments by major plan-
ets of the solar system [137–139] created a notable “speed-of-light versus speed-of-gravity” controversy
[140] which originates in the inability of PPN formalism to distinguish between physical effects of grav-
ity and electromagnetism due to the non-covariant nature of PPN parametrization limited merely by the
metric manifolds [61, 142].
In the present chapter we rely upon the Einstein theory of general relativity and assume that light
propagates in vacuum that is the interstellar medium has no impact on the speed of light propagation.
General relativity is a geometrized theory of gravity and it assumes that both gravity and electromag-
netic field propagate locally in vacuum with the same speed which is equal to the fundamental speed c
in special theory of relativity [1]. Gravitational field is found as a solution of the Einstein field equa-
tions. The electromagnetic field is obtained by solving the Maxwell equations on the curved spacetime
manifold derived at previous step from Einstein’s equations. In the approximation of geometric optics
the electromagnetic signals propagate along null geodesics of the metric tensor [1–3].
The problem of finding solutions of the null ray equations in curved spacetime attracted many re-
searchers since the time of discovery of general relativity. Exact solutions of this problem were found
in some, particularly simple cases of symmetric spacetimes like Schwarzschild’s or Kerr’s black hole,
homogeneous and isotropic Friedman-Lemeître-Robertson-Walker (FLRW) cosmology, plane gravita-
tional wave , etc. [1–3, 48]. For quite a long time these exact description of null geodesics was sufficient
for the purposes of experimental gravitational physics. However, real physical spacetime has no symme-
tries and the solution of such current problems as direct detection of gravitational waves, interpretation
of an anisotropy and polarization of cosmic microwave background radiation (CMBR), exploration of
inflationary models of the early universe, finding new experimental evidences in support of general rel-
ativity and quantum gravity, development of higher-precision relativistic algorithms for space missions,
and many others, can not fully rely upon the mathematical techniques developed mainly in the symmet-
ric spacetimes. Especially important is to find out a method of integration of equations of light geodesics
in spacetimes having time-dependent gravitational perturbations of arbitrary multipole order.
In the present chapter we consider a case of an isolated astronomical system embedded to an asymptotically-
flat spacetime. This excludes Friedmann-Lemeître-Robertson-Walker spacetime which is not asymptotically-
flat (see chapter 7). We assume that gravitational field of this system is characterized by an infinite set
of gravitational multipoles emitting gravitational waves. Precise and coordinate-free mathematical def-
inition of the asymptotic flatness is based on the concept of conformal infinity that was worked out in a
series of papers [49–54] and we recommend the textbook of R. Wald [3] for a concise but comprehensive
mathematical introduction to this subject. The asymptotic flatness implies the existence of a comformal
compactification of the manifold but it may not exist for a particular case of an astrophysical system.
There is also coordinate-free definition of gravitational multipoles of an isolated system given by Hansen
[55]. However, they make sense only in stationary spacetimes but are not applicable for time-dependent
gravitational fields which make them useless for practical analysis of real astronomical observations
and for interpretation of relativistic effects in propagation of electromagnetic signals. We use neither
Hansen’s multipoles in the present chapter nor the comformal compactification technique. Moreover,
because observable effects of gravitational waves are weak we shall consider only linear effects of grav-
itational multipoles so that relativistic effects produced by non-linearity of the gravitational field will be
ignored. The linearized approximation of general theory of relativity represents a straightforward and
practically useful approach to description of the multipole structure of the gravitational field of a local-
ized astronomical system. The multipolar gravitational formalism was developed by several scientists,
most notably by Kip Thorne [5] and Blanchet & Damour [56–60], and we use their results in the present
chapter.
Exact formulation of the problem under discussion in this chapter is as follows (see Fig 5.1). We
assume (in a less restrictive sense than the comformal compactification technique does) that spacetime
under consideration is asymptotically flat and covered globally by a single coordinate chart, xα =
(ct, x), where t is coordinate time and x are spatial coordinates. An electromagnetic wave (photon) is
emitted by a source of light at time t0 at point x0 towards an observer which receives the photon at the
instant of time t and at the point x. Photon propagates through the time-dependent gravitational field
of the isolated astronomical system emitting gravitational waves. The structure of the gravitational field
is described by a set of Blanchet-Damour multipole moments of the system which are functions of the
retarded time t − r/c, where r is the distance from the isolated system to the field point and c is the
fundamental speed. The retardation is physically due to the finite speed of gravity which coincides in
general relativity with the speed of light in vacuum. Some confusion may arise in the interpretation of
the observational effects because both gravity and light propagates in general relativity with the same
speed. The discrimination between various effects is still possible because gravity and light propagate
to observer from different sources and along different spatial directions (characteristics of the null cone
in spacetime) [61].
Our task is to find out the relations connecting various physical parameters (direction of propagation,
frequency, polarization, intensity, etc.) of the electromagnetic signal at the point of emission with those
measured by the observer. Gravitational field affects propagation of the electromagnetic signal and
changes its parameters along the light ray. Observing these changes allow us to study various properties
of the gravitational field of the astronomical system and to detect gravitational waves with astronomical
technique. The rest of the chapter is devoted to the mathematical solution of this problem.
5.1.2 Historical background

Propagation of electromagnetic signals through stationary gravitational field is a rather well-known sub-
ject having been originally discussed in classic textbooks by [120, 121]. Presently, almost any standard
textbook on relativity describes solution of the null geodesic equations in the field of a spherically-
symmetric and rotating body or black hole. This solution is practical since it is used, for example, for
interpretation of gravitational measurements and light-propagation experiments in the solar system [61].
Another application is gravitational lensing in our galaxy and in cosmology [62, 63, 65]. Continually
growing accuracy of astronomical observations demands much better treatment of secondary effects
in the propagation of light produced by the perturbations of the gravitational field associated with the
higher-order gravitational multipoles of planets and Sun[122]. Time-dependent multipoles emit gravita-
tional waves which perturb propagation of light with the characteristic period of the gravitational wave.
Influence of these periodic perturbations on light propagation parameters is important for developing
correct strategy for understanding the principles of operation of gravitational wave detectors as well as
for searching gravitational waves by astronomical techniques.
+
I
J+ (t,x) J+
0 0
I I
(t0 ,x0 )
Q
k
J− J−
k
P
I−
Figure 5.1. The Penrose diagram shows a worldline of the isolated system (binary star) originating at
past timelike infinity I − and ending at the future timelike infinity I + . Both light and gravity propagate
along null geodesics going from the past null infinity J − to the future null infinity J + . A particular light
geodesic is emanating from the event (t0 ,x0 ) and ending at the event (t,x). Extrapolation of this light
geodesic to the past null infinity defines the null wave vector kα = (k0 ,k) of the electromagnetic wave
under consideration.
Among the most interesting sources of periodic gravitational waves with a well-predicted behaviour
are binary systems comprised of two stars orbiting each other around a common barycenter (center of
mass). Indirect evidence in support of existence of gravitational waves emitted by binary pulsars was
given by Joe Taylor with collaborators [66, 67]. However, direct observation of gravitational waves
remains a challenging problem for experimental gravitational physics. The expected spectrum of gravi-
tational waves extends from ∼ 104 Hz to 10−18 Hz [33, 34]. Within that range, the spectrum of periodic
waves from known binary systems extends from about 10−3 Hz – the frequency of gravitational radiation
from a contact white-dwarf binary [40], through the 10−4 to 10−6 Hz – the range of radiation from the
main-sequence binaries [41], to the 10−7 to 10−9 Hz – the frequencies emitted by binary supermassive
black holes presumably lurking in active galactic nuclei (AGN) [42]. The dimensionless strain of these
waves at the Earth, h, may be as great as 10−21 at the highest frequencies, and as great as 3 × 10−15 at
the lowest frequencies in the spectrum of gravitational waves [33, 34].
Sazhin [68] was the first who suggested the method of detection of gravitational waves emitted from a
binary system by using timing observations of a background pulsar, more distant than the binary lying on
the line of sight which passes sufficiently close in the sky to the binary. He had shown that the integrated
time delay for the propagation of an electromagnetic pulse near the binary is proportional to 1/d2 where
d is the impact parameter of the unperturbed trajectory of the signal. Similar idea was independently
proposed by Detweiler [69] who has focused on discussing an application of pulsar timing for detection
of a stochastic cosmological background of gravitational waves. More recently, Sazhin & Saphonova
[70] have made estimates of the probability of observation of such an effect for pulsars in globular
clusters and showed that the probability can be high, reaching 97%. Sazhin-Detweiler idea is currently
used in pulsar timing arrays to detect gravitational waves in nano-Hertz frequency band [71–73].
Wahlquist [74] proposed another approach to the detection of periodic gravitational waves based
on Doppler tracking of spacecraft travelling in deep space. His approach is restricted by the plane
gravitational wave approximation developed earlier by Estabrook & Wahlquist [75]. Tinto ([76], and
references therein) made the most recent theoretical contribution in this research area. The Doppler
tracking technique is used in deep space missions for detection of gravitational waves by seeking for
the characteristic triple signature in the continuously recorded phase of radio waves in the radio link
between the ground station and spacecraft. The presence of this specific signature would indicate to the
influence of the Doppler signal by a gravitational wave crossing the line of sight from the spacecraft to
observer [77].
Braginsky et al. [79, 80] raised a question about a possibility of using Very-Long Baseline Interfer-
ometry (VLBI) as a detector of stochastic gravitational waves produced in the early universe. This idea
had been also investigated by Kaiser & Jaffe [81] and, the most prominently, by Pyne et al. [82] and
Gwinn et al. [83] who showed that the overall effect in the time delay of VLBI signal is proportional
to the strain of the metric perturbation, h, caused by the plane gravitational wave. They also calculated
the pattern of proper motions of quasars over all the sky as an indicator of the presence of quadrupole
and high-order harmonics of ultra-long gravitational wave and set an observational limit on the energy
density of such gravitational waves present in the early universe. Montanari [84] studied the perturba-
tions of polarization of electromagnetic radiation propagating in the field of a plane gravitational wave
and found that the effects are exceedingly small, unlikely to be observable.
Fakir ([85, 86], and references therein) has suggested to use astrometry to detect the periodic varia-
tions in apparent angular separations of appropriate light sources, caused by gravitational waves emitted
by isolated sources of gravitational radiation. He was not able to develop a self-consistent approach to
tackle the problem with a necessary completeness and rigour. For this reason, his estimate of the effect
of the deflection of light caused by gravitational wave perturbation, is too optimistic. Another attempt
to work out a more consistent approach to the calculation of the light deflection angle by the radiation
field of an arbitrary source of gravitational waves has been undertaken by Durrer [87]. However, the cal-
culations have been done only for the plane wave approximation. Nonetheless, the result obtained was
extrapolated to the case of the localized source of gravitational waves without convincing justification.
For this reason the magnitude of the periodic changes of the light deflection angle was largely overes-
timated. The same misinterpretation of the effect of gravitational waves from localized sources can be
found in the paper by Labeyrie [88] who studied a photometric modulation of background sources of
light (stars) by gravitational waves emitted by fast-orbiting binary stars. Because of the erroneous pre-
dictions, the expected detection of gravitational waves from VLBI observations of a radio source GPS
QSO 2022+171 undertaken by Pogrebenko et al. [89] was not based on firm theoretical ground and did
not lead to success.
Damour & Esposito-Farèse [90] have studied the deflection of light and integrated time delay caused
by the time-dependent gravitational field generated by a localized astrophysical source lying in the sky
close to the line of sight to a background source of light. They worked in a quadrupole approximation and
explicitly calculated the effects of the retarded gravitational field of the astrophysical source in its near,
intermediate, and wave zones by making use of the Fourier-decomposition technique. Contrary to the
claims of Fakir [85, 86] and Durrer [87] and in agreement with Sazhin’s [68] calculations, they found that
the contribution of the wave-zone and intermediate-zone fields to the deflection angle vanish exactly due
to some remarkable mutual cancellations of different components of the gravitational field. The leading,
total time-dependent deflection of light is created only by the quasi-static, near-zone quadrupolar part of
the gravitational field.
Damour and Esposito-Farese [90] analyzed propagation of light under a simplifying condition that
the impact parameter of the light ray is small with respect to the distances from observer and the source
of light to the isolated system. We have found [6, 91] another way around to solve the problem of
propagation of electromagnetic waves in the quadrupolar field of the gravitational waves emitted by
the system without making any assumptions on mutual disposition of observer, source of light, and the
system, thus, significantly improving and extending the result of Damour’s paper [94]. At the same
time the paper [91] did not answer the question about the impact of the other, higher-order gravitational
multipoles of the isolated system on the process of propagation of electromagnetic signals. This might
be important if the effective gravitational wave emission of an octupole and/or higher-order multipoles
is equal or even exceeds that of the quadrupole as it may be in case of, for example, highly-asymmetric
stellar collapse [95], nearly head-on collision of two stars, or break-up of a binary system caused by a
recoil of two black holes [96].
In the present chapter we work out a systematic approach to the problem of propagation of light rays
in the field of arbitrary gravitational multipole. While the most papers on light propagation consider both
a light source and an observer as being located at infinity we do not need these assumptions. For this
reason, our approach is generic and applicable for any mutual configuration of the source of light and
observer with respect to the source of gravitational radiation. The integration technique which we use
for finding solution of the equations of propagation of light rays was worked out in series of our papers
[6, 91–93].
The metric tensor and coordinate systems involved in our calculations are described in section 5.2
along with gauge conditions imposed on the metric tensor. The equations of propagation of electromag-
netic waves in the geometric optics approximation are discussed in section 5.3 and the method of their
integration is given in section 5.4. Exact solution of the equations of light propagation and the exact
form of relativistic perturbations of the light trajectory and the coordinate speed of light are obtained in
section 5.5. Section 5.6 is devoted to the derivation of the primary observable relativistic effects - the in-
tegrated time delay, the deflection angle, the frequency shift, and the rotation of the plane of polarization
of an electromagnetic wave. We discuss in sections 5.7 and 5.8 two limiting cases of the most interesting
relative configurations of the source of light, the observer, and the source of gravitational waves – the
gravitational-lens configuration (section 5.7) and the case of a plane gravitational wave (section 5.8).
5.1.3 Notations and Conventions
We consider a spacetime manifold which is asymptotically flat at infinity [106]. Metric tensor of the
spacetime manifold is denoted by gαβ and its perturbation
hαβ = gαβ − ηαβ . (5.1.1)
The determinant of the metric tensor is negative, and is denoted as g = det[gαβ ]. A four-dimensional,
fully antisymmetric Levi-Civita symbol αβγδ is defined in accordance with the convention 0123 = +1.
In the present chapter we use a geometrodynamic system of units [2] such that the fundamental speed,
c, and the universal gravitational constant, G, are equal to unity, that is c = G = 1. spacetime is assumed
to be globally covered by a Cartesian-like coordinate system (xα ) ≡ (x0 ,x1 ,x2 ,x3 ) ≡ (t,x,y,z) , where
t and (x,y,z) are time and space coordinates respectively. This coordinate system is reduced at infinity to
the inertial Lorentz coordinates defined up to a global Lorentz-Poincare transformation [4]. Sometimes
we shall use spherical coordinates (r,θ,φ) related to (x,y,z) by a standard transformation
x = r sin θ cos φ , y = r sin θ sin φ , z = r cos θ . (5.1.2)
Spatial coordinates (xi ) ≡ (x,y,z) in some equations will be denoted with a boldface font, x ≡ (xi ).
We shall operate with various geometric objects which have tensor indices. We agree that Greek
(spacetime) indices α,β,γ, . . . range from 0 to 3, and Latin (space) indices a,b,c, . . . run from 1 to
3. If not specifically stated the opposite, the Greek indices are raised and lowered by means of the
Minkowski metric ηαβ ≡ diag(−1,1,1,1), for example, Aα = αβ Aβ , Bαβ = αµ βν B µν , and so
on. The spatial indices are raised and lowered with the help of the Kronecker symbol (a unit matrix),
δij ≡ diag(1,1,1). Regarding this rule the following conventions for the Cartesian coordinates hold:
xi = xi and x0 = −x0 .
Repeated indices are summed over in accordance with Einstein’s rule [2], for example,
Aαβγ Bαµ ≡ A0βγ B0µ + A1βγ B1µ + A2βγ B2µ + A3βγ B3µ . (5.1.3)
In the linearized (with respect to G) approximation of general relativity used in the present chapter,
there is no difference between spatial vectors and co-vectors nor between upper and lower space indices.
Therefore, we do not distinguish between the superscript and subscript spatial indeses. For example, for
a dot (scalar) product of two space vectors we have
Ai Bi = Ai Bi ≡ A1 B1 + A2 B2 + A3 B3 . (5.1.4)
In what follows, we shall commonly use the spatial multi-index notations for three-dimensional, Carte-
sian tensors [5] like this
IAl ≡ Ia1 ...al . (5.1.5)
A tensor product of l identical spatial vectors ki will be denoted as a three-dimensional tensor having l
indices
ka1 ka2 . . . kal ≡ ka1 ...al . (5.1.6)
Full symmetrization with respect to a group of spatial indices of a Cartesian tensor will be denoted with
round brackets embracing the indices

1X
Q(a1 ...al ) ≡ Qσ(1)...σ(l) , (5.1.7)
l! σ
where σ is the set of all permutations of (1,2,...,l) which makes Qa1 ...al fully symmetric in a1 . . . al .
It is convenient to introduce a special notation for symmetric trace-free (STF) Cartesian tensors by
making use of angular brackets around STF indices. The explicit expression of the STF part of a tensor
Qa1 ...al is [5, 56]
[l/2]
X
Q<a1 ...al > ≡ alk δ(a1 a2 · · · δa2k−1 a2k Sa2k+1 ...al )b1 b1 ...bk bk , (5.1.8)
k=0
where [l/2] is the integer part of the number l/2,
Sa1 ...al ≡ Q(a1 ...al ) , (5.1.9)
and the numerical coefficients
(−1)k l! (2l − 2k − 1)!!

alk = . (5.1.10)
(2k)!! (2l − 1)!! (l − 2k)!
We also assume that for any integer l ≥ 0
l! ≡ l(l − 1) . . . 2 · 1 , 0! ≡ 1 , (5.1.11)
and
l!! ≡ l(l − 2)(l − 4) . . . (2 or1) , 0!! ≡ 1 . (5.1.12)
One has, for example,
1
T<ab> = T(ab) − δab Tcc , (5.1.13)
3
1 1 1
T<abc> = T(abc) − δab T(cjj) − δbc T(ajj) − δac T(bjj) , (5.1.14)
5 5 5
and so on.
Cartesian tensors of the mass-type (mass) multipoles I<Al > and spin-type (spin) multipoles S<Al >
entirely describing gravitational field outside of an isolated astronomical system are always STF objects
that can be checked by inspection of the definition following from the multipolar decomposition of the
metric tensor perturbation hαβ [5, 56]. For this reason, to avoid the appearance of overcomplicated
index notations we shall omit the angular brackets around the spatial indices of these (and only these)
Cartesian tensors, that is we adopt: IAl ≡ I<Al > and SAl ≡ S<Al > .
We shall also use transverse (T) and transverse-traceless (TT) Cartesian tensors in our calculations
[2, 5, 56]. These objects are defined by making use of the operator of projection
Pjk ≡ δjk − kjk , (5.1.15)
onto the plane orthogonal to a unit vector kj . This operator plays a role of a Kroneker symbol in the
two dimensional space in the sense that Pij Pjk = Pik , and Pii = 2. Definitions of the transverse and
transverse-traceless tensors is [5, 6]
QTa1 ...al ≡ Pa1 b1 Pa2 b2 ...Pal bl Qb1 ...bl , (5.1.16)

[l/2]
X
QTT
a1 ...al ≡ blk P(a1 a2 · · · Pa2k−1 a2k Wa2k+1 ...al )b1 b1 ...bk bk , (5.1.17)
k=0
where again [l/2] is the integer part of l/2, Wa1 ...al ≡ QT(a1 ...al ) , and the numerical coefficients
(−1)k l(l − k − 1)!!

blk = . (5.1.18)
4k k!(l − 2k)!
For instance,
1
QTT
ab ≡ Pi(a Pb)j Qij − Pab Pjk Qjk . (5.1.19)
2
We shall also use the polynomial coefficients Cl (p1 , . . . ,pn ) in some of our equations . They are
defined by
l!
Cl (p1 , . . . ,pn ) ≡ , (5.1.20)
p1 ! . . . pn !
where l and pi are positive integers such that n
P
i=1 pi = l. We introduce a Heaviside step function ,
H(p − q), such that on the set of whole numbers
(
0, if p ≤ q ,
H(p − q) = (5.1.21)
1, if p > q .
Partial derivatives of any differentiable function, f = f (t,x), are denoted as follows: f,0 = ∂f /∂t
and f,i = ∂f /∂xi . In general, comma standing after a function denotes a partial derivative with respect
to a corresponding coordinate: f,α ≡ ∂f (x)/∂xα . A dot above function denotes a total derivative of
the function with respect to time
f˙ ≡ df /dt = ∂f /∂t + ẋi ∂f /∂xi , (5.1.22)
where ẋi denotes velocity along the integral curve xi = xi (t) parametrized with coordinate time t.
In this chapter the integral curves are light rays, and the derivatives ẋi are taken along the light ray
trajectory (xi ) = x(t). Sometimes the partial derivatives with respect to space coordinate xi will be
also denoted as ∂i ≡ ∂/∂xi , and the partial time derivative will be denoted as ∂t ≡ ∂/∂t. A covariant
derivative with respect to the coordinate xα will be denoted as ∇α .
We shall introduce and distinguish notations for integrals taken with respect to time at a fixed spatial
point, from those taken along a light-ray trajectory. Specifically, the time integrals from a function
F (t, x), where x is a fixed point in space, are denoted as
Zt Zt
F (−1) (t, x) ≡ F (τ, x)dτ , F (−2) (t, x) ≡ F (−1) (τ, x)dτ . (5.1.23)
−∞ −∞
The time integrals from a function F (t,x) taken on a light ray suggest that the spatial coordinate x is a
function of time x ≡ x(t), taken along the light ray. These integrals are denoted as
Zt Zt
F [−1] (t,x) ≡ F (τ,x(τ ))dτ, F [−2] (t,x) ≡ F [−1] (τ,x(τ ))dτ , (5.1.24)
−∞ −∞
where x ≡ x(t) in the right side of these definitions. The integrals in (5.1.23) represent functions of
time, t, and spatial, x, coordinates. The integrals in (5.1.24) are functions of time, t, only.
Partial time derivative of the order p from a function F (t,x) is denoted by
∂ p F (t,x)
F (p) (t,x) = , (5.1.25)
∂tp
so that its action on the time integrals eliminates integration in the sense that
∂ p+1 F (−1) (t,x) ∂ p+2 F (−2) (t,x)

F (p) (t,x) = p+1
= . (5.1.26)
∂t ∂tp+2
Total time derivative of the order p from a function F (t,x) is denoted by
dp F (t,x)
F [p] (t,x) = . (5.1.27)
dtp
The reader can easily confirm that
dp+1 F [−1] (t,x) dp+2 F [−2] (t,x)

F [p] (t,x) = p+1
= . (5.1.28)
dt dtp+2
In what follows, we shall denote spatial vectors by the bold italic letters, for instance, Ai ≡ A,
i
k ≡ k, etc. The Euclidean dot product between two spatial vectors, for example a and b, is denoted
with a dot between them: ai bi = a · b. The Euclidean wedge (cross) product between two spatial
vectors is denoted with a symbol ×, that is ijk aj bk = (a × b)i . Other particular notations will be
introduced as soon as they appear in text.
5.2 The Metric Tensor, Gauges and Coordinates

5.2.1 The canonical form of the metric tensor perturbation
We consider an isolated astronomical system emitting gravitational waves and assume that gravitational
field is weak everywhere so that the metric tensor can be expanded in a Taylor series with respect to
the powers of gravitational constant G which labels the order of products of the metric tensor perturba-
tions that are kept in the solution of the Einstein equations. We shall consider only a linearised post-
Minkowskian approximation of general relativity and discard all terms of the order of G2 and higher.
The metric tensor is a linear combination of the Minkowski metric, ηαβ , and a small perturbation hαβ
gαβ = ηαβ + Ghαβ + O(G2 ) , (5.2.1)

αβ αβ 2
g = ηαβ − Gh + O(G ) , (5.2.2)
where hαβ 1 and we use ηαβ to rise and lower indices so that, for example, hαβ = η αµ η βν hµν . In
many cases the origin of the coordinates is placed to the center of mass of the astrophysical system. It
eliminates the dipole component of the gravitational field which is associated with a coordinate degree
of freedom. In some cases, however, it is necessary to keep the dipole component unrestricted in order
to determine position of the center of mass of the system under consideration with respect to another
coordinate chart which is introduced independently for solving some other astronomical problems. This
is important for unambiguous interpretation of gravitational experiments done with astrometric instru-
ments [61, 93]. Fact of the matter is that the displacement of the center of mass of an astrophysical
system from the origin of the coordinates induces translational deformations of the higher-order multi-
Section 5.2 The Metric Tensor, Gauges and Coordinates 185
pole moments of the gravitational field which introduce a bias to the physical values of the multipoles.
Therefore, physical interpretation of the observed values of the multipoles requires identification of the
dipole moment and subtraction of the coordinate deformations caused by it. We shall discard the dipole
component of the gravitational field in our solution.
The most general expression for the linearised perturbation of the metric tensor outside of the astro-
nomical system emitting gravitational radiation was derived by Blanchet and Damour [56] by solving
Einstein’s equations. The perturbation is given in terms of the symmetric and trace-free (STF) mass and
spin multipole moments (similar formulas were derived by Thorne [5]) and is described by the following
expression
hαβ = hcan.
αβ + wα,β + wβ,α , (5.2.3)
where wα are, the so-called, gauge functions describing the freedom in the choice of coordinates cover-
ing the manifold. The canonical perturbation, hcan.
αβ , obeys the homogeneous wave equation in vacuum
hcan.
αβ = 0 , (5.2.4)
which solution is chosen as

∞
X (−1)l IAl (t − r)

2M
hcan.
00 (t,x) = +2 , (5.2.5)
r l! r ,Al
l=2
∞
2ipq Sp (t − r)Nq (−1)l l ipq SpAl−1 (t − r)
X
hcan.
0i (t,x) = − − 4 + (5.2.6)
r2 (l + 1)! r ,qAl−1
l=2
∞
" #
X (−1)l İiAl−1 (t − r)
4 ,
l! r
l=2 ,Al−1
hcan.
ij (t,x) =δij hcan. can.
00 (t,x) + qij (t,x) , (5.2.7)
(−1) l pq(i Ṡj)pAl−2 (t − r)
∞
" # ∞
" #
can.
X (−1)l ÏijAl−2 (t − r) X l
qij (t,x) =4 −8 .
l! r (l + 1)! r
l=2 ,Al−2 l=2 ,qAl−2
(5.2.8)
Here M and Si are the total mass and spin (angular momentum) of the system, and IAl and SAl
are two independent sets of mass-type and spin-type multipole moments, N i = xi /r is a unit vector
directed from the origin of the coordinate system to the field point. Because the origin of the coordinate
system has been chosen at the center of mass, the expansions (5.2.5) – (5.2.8) do not depend on the
mass-type dipole moment, Ii , which is equal to zero by definition. We emphasize that in the linearised
approximation the total mass M and spin Si of the astronomical system are constant while all other
multipoles are functions of time which temporal behaviour obeys the equations of motion derived from
the law of conservation of the stress-energy tensor of the system [1, 2]. Gravitational waves emitted by
the system reduce its energy, linear and angular momenta. This effect does not appear in the linearised
general relativity but in higher order approximations we would obtain where the mass, spin, and linear
momentum of the system must be considered as functions of time like any other multipole. Higher-order
gravitational perturbations in the metric going beyond (5.2.5)–(5.2.8) are shown in a review paper by
Blanchet [97]. They are not of concern in the present chapter.
The canonical metric tensor (5.2.5)–(5.2.8) depends on the multipole moments IAl (t−r) and SAl (t−
r) taken at the retarded instant of time. The retardation is explained by the finite speed of propagation
of gravity (light propagation will be considered below). In the near zone of the isolated system the
retardation due to the propagation of gravity is small and all functions of time in the metric tensor can
be expanded in Taylor series around the present time t [5, 56]. This near-zone expansion of the metric
tensor is called the post-Newtonian expansion leading to the post-Newtonian successive approximations
[61]. The post-Newtonian expansion can be smoothly matched to the solution of the linearized Einstein
equations in the domain of space being occupied by matter of the isolated system. The matching allows
us to express the multipole moments in terms of matter variables [57]
σ ≡ T 00 + T kk , σ i ≡ T 0i , (5.2.9)
where T αβ is the stress-energy tensor of matter bounded in space. In the first post-Newtonian approxi-
mation the multipole moments have a matter-compact support [57]
|x|2 x<Al > 2

4(2l + 1)x<iAl >
Z
I1PN
Al = d 3
x x<Al > σ + ∂ t σ − ∂ σ
t i + O(c−4 )(5.2.10)
,
V 2(2l + 3) (l + 1)(2l + 3)
Z
S1PN
Al = d3 xpq<al xAl−1 p> σq + O(c−2 ) , (5.2.11)
V
where notations have been explained in section 5.1.3. In the higher post-Newtonian approximations
the multipole moments have contributions coming directly from the stress-energy tensor of gravitational
field (Landau-Lifshitz pseudotensor) which have non-compact support. Therefore, the multipole mo-
ments are expressed by more complicated functionals [97]. Radiative approximation of the canonical
metric tensor reveals that contribution of the tails of gravitational waves must be added to the definitions
of the multipole moments (5.2.10), (5.2.11) so that the multipole moments in the radiative zone of the
isolated system read [58, 59, 98]
Z+∞ " l−2

#
ζ 2l2 + 5l + 4 X 1
I Al = I1PN
Al + 2M dζ ÏAl (t − r − ζ) ln
1PN
+ + ,(5.2.12)
2b l(l + 1)(l + 2) k
0 k=1
Z+∞ " l−1

#
ζ l−1 X 1
SAl = S1PN
Al + 2M dζ S̈Al (t − r − ζ) ln
1PN
+ + , (5.2.13)
2b l(l + 1) k
0 k=1
where b is a normalization constant which value is supposed to be absorbed to the definition of the origin
of time scale in the radiative zone but this statement has not been checked so far.
5.2.2 The harmonic coordinates

Equation (5.2.3) holds in an arbitrary gauge imposed on the metric tensor. The harmonic gauge is defined
by the condition [61]
2hαβ,β − h,α = 0 , (5.2.14)
where h ≡ hµµ . The gauge condition (5.2.14) reduces the Einstein vacuum field equations to the wave
equation (5.2.4) for the gravitational potentials hαβ . Harmonic coordinates xα are defined as solutions
of the homogeneous wave equation xα = 0 up to the gauge functions wα . In particular, the harmonic
canonical coordinates are defined by the condition that all gauge functions wα = 0. The canonical
metric tensor (5.2.5) – (5.2.8) depends on two sets of multipole moments [5, 56] which reflects the
existence of only two degrees of freedom of a free (detached from matter) gravitational field in general
relativity [1–3]. At the same time one can obtain a generic expression for the harmonic metric tensor by
making use of infinitesimal coordinate transformation
x0α = xα − wα (5.2.15)
Section 5.2 The Metric Tensor, Gauges and Coordinates 187
from the canonical harmonic coordinates xα to arbitrary harmonic coordinates x0α with the harmonic
gauge functions wα which satisfy to a homogeneous wave equation
wα = 0 . (5.2.16)
The most general solution of this vector equation contains four sets of STF multipoles [5, 56]
∞
WAl (t − r)
X
w0 = , (5.2.17)
r ,Al
l=0
∞ ∞
XAl (t − r) YiAl−1 (t − r)
X X
wi = + + (5.2.18)
r ,iAl r ,Al−1
l=0 l=1
∞
ZqAl−1 (t − r)
X
ipq ,
r ,pAl−1
l=1
where WAl , XAl , YiAl−1 , and ZqAl−1 are Cartesian tensors depending on the retarded time. Their
specific form is a matter of computational convenience (or the boundary conditions) for derivation and
interpretation of observable effects but it does not affect the invariant quantities like the phase of elec-
tromagnetic wave propagating through the field of the multipoles.
The most convenient choice simplifying the structure of the metric tensor perturbations, is given by
the following gauge functions
" (−1)
(−1)l IAl (t − r)
∞
#
0
X
w = , (5.2.19)
l! r
l=2 ,Al
" (−2)
(−1)l IAl (t − r)
∞
#
i
X
w = − (5.2.20)
l! r
l=2 ,iAl
∞
(−1)l IiAl−1 (t − r)
X
4 +
l! r ,Al−1
l=2
 
(−1)
(−1)l l  iba SbAl−1 (t − r) 
X∞
4 .
(l + 1)! r
l=2
,aAl−1
These functions, after they are substituted to equation (5.2.3), transform the canonical metric tensor
perturbation to a remarkably simple form
2M
h00 = , (5.2.21)
r
2ipq Sp Nq
h0i = − , (5.2.22)
r2
hij = δij h00 + hTijT , (5.2.23)
hTijT = can.
Pijkl qkl , (5.2.24)
where the TT-projection differential operator Pijkl , applied to the symmetric tensors depending on both
time and spatial coordinates, is given by
1
Pijkl = (δik − ∆−1 ∂i ∂k )(δjl − ∆−1 ∂j ∂l ) − (δij − ∆−1 ∂i ∂j )(δkl − ∆−1 ∂k ∂l ) ,(5.2.25)
2
and ∆ and ∆−1 denote the Laplacian and the inverse Laplacian respectively.
When comparing the canonical metric tensor with that given by equations (5.2.21)–(5.2.24) it is
instructive to keep in mind that (−2) ÏAl (t − r) = IAl (t − r) and ∆(IAl (t − r)/r) = ÏAl (t − r)/r for
r 6= 0. This is a consequence of the
h fact that functioni
(−2)
ÏAl (t − r) is a solution of the homogeneous
d’Lambert’s equation, that is, (−2) IAl (t − r)/r = 0 for r 6= 0. We also notice that IAl (t −
r)/r = ∆−1 ÏAl (t − r)/r and (−2) IAl (t − r)/r = ∆−1 [IAl (t − r)/r]. The metric tensor harmonic

perturbation (5.2.21)–(5.2.24) is similar to the Coulomb gauge in electrodynamics [99, 100].
5.2.3 The ADM coordinates

The Arnowitt-Deser-Misner (ADM) gauge condition in the linear approximation is given by two equa-
tions [53]
2h0i,i − hii,0 = 0 , 3hij,j − hjj,i = 0 , (5.2.26)
where the second equation holds exactly, and for any function f = f (t,x) we use notations: f,0 =
∂f /∂t and f,i = ∂f /∂xi . For comparison, the harmonic gauge condition (5.2.14) in the linear approx-
imation reads:
2h0i,i − hii,0 = h00,0 , 2hij,j − hjj,i = −h00,i . (5.2.27)
The ADM gauge condition (5.2.26) brings the space-space component of the metric to the following
form
1
gij = δij (1 + hkk ) + hTijT , (5.2.28)
3
where hTijT denotes the transverse-traceless part of hij and hkk = 3h00 . The ADM and harmonic gauge
conditions are not compatible inside the regions occupied by matter. However, outside of matter they
can co-exist simultaneously. Indeed, it is straightforward to check out that the metric tensor (5.2.21)–
(5.2.24) satisfies both the harmonic and the ADM gauge conditions in the linear approximation along
with the assumption that Ṁ = 0. This was first noticed in [91]. We call the coordinates in which the
metric tensor is given by equations (5.2.21)–(5.2.24) as the ADM-harmonic coordinates.
The experimental problem of detection of gravitational waves is reduced to the observation of motion
of test particles in the field of the incident or incoming gravitational wave. These test particles are
photons in the electromagnetic wave used in observations and mirrors in ground-based gravitational-
wave detectors or pulsars and Earth in case of using a pulsar timing array. The gravitational wave affects
propagation of photons and perturbs motion of the mirrors or pulsars and Earth. These perturbations
must be explicitly calculated and clearly separated from noise to avoid possible misinterpretation of
observable effects due to the gravitational wave. It turns out that the canonical form of the metric
tensor (5.2.5) – (5.2.8) is well-adapted for performing an analytic integration of equations of light rays.
At the same time, freely-falling mirrors (or pulsars and Earth) experience influence of gravitational
waves emitted by the isolated astronomical system and move with respect to the coordinate grid of the
canonical harmonic coordinates in a complicated way. For this reason, the perturbations produced by
Section 5.3 Equations of Propagation of Electromagnetic Signals 189
the gravitational waves on the light propagation get mixed up with the motion of massive test particles
in these coordinates.
Arnowitt, Deser and Misner [53] showed that there exist canonical ADM coordinates which have a
special property such that freely-falling massive particles are not moving with respect to this coordi-
nates despite that they are perturbed by the gravitational waves. This means that the ADM coordinates
themselves are not inertial and, although have an advantage in treating motion of massive test particles,
should be used with care in the interpretation of gravitational wave experiments. Making use of the
canonical ADM coordinates simplifies analysis of the gravitational wave effects observed at gravita-
tional wave observatories (LIGO, LISA, NGO, etc.) or by astronomical technique because the motion
of observer (proof mass) is excluded from the equations. However, the mathematical structure of the
metric tensor in the canonical ADM coordinates does not allow us to directly integrate equations for
light rays analytically because it contains terms that are instantaneous functions of time. Integrals from
these instantaneous functions of time cannot be performed explicitly [91].
The ADM-harmonic coordinates have the advantages of both harmonic and ADM coordinates. Thus,
the ADM-harmonic coordinates allow us to get a full analytic solution of the light-ray equations and to
eliminate the effects produced by the motion of observers with respect to the coordinate grid caused by
the influence of gravitational waves. In other words, all physical effects produced by gravitational waves
are contained merely in the solution of the equations of light propagation. This conclusion is, of course,
valid in the linear approximation of general relativity and is not extended to the second approximation
where gravitational-wave effects on light and motion of observers can not be disentangled and have to
be analysed together.
Similar ideology based on the introduction of TT coordinates , has been earlier applied for analysis
of the output signal of the gravitational-wave detectors with freely-suspended masses [1, 2, 25] placed
to the field of a plane gravitational wave, that is at the distance far away from the localized astronomical
system emitting gravitational waves where the curvature of the gravitational-wave front is negligible.
Our ADM-harmonic coordinates are an essential generalization of the standard TT coordinates because
they can be constructed at an arbitrary distance from the astronomical system, thus, covering the near,
intermediate and radiative zones.
5.3 Equations of Propagation of Electromagnetic Signals

5.3.1 Maxwell equations in curved spacetime
In this section we treat gravitational field exactly without approximation. Therefore, all indices are
raised and lowered by means of the metric tensor gαβ with g αβ defined in accordance with the standard
rule g αβ gβγ = δβα . The general formalism describing the behavior of electromagnetic radiation in
an arbitrary gravitational field is well known and can be found, for example, in textbooks [1, 2, 108]
or in reviews [107, 109]. Electromagnetic field is defined in terms of the (complex) electromagnetic
tensor Fαβ as a solution of the Maxwell equations. In the high-frequency limit one can approximate the
electromagnetic tensor Fαβ as [1, 2]
Fαβ = {Aαβ exp(iϕ)} , (5.3.1)
where Aαβ is a slowly varying (complex) amplitude and ϕ is a rapidly varying phase of the electro-
magnetic wave which is called eikonal [1, 102], and i is the imaginary unit, i2 = −1. In the most
general case of propagation of light in a transparent medium the eikonal is a complex function which
real and imaginary parts are connected by the Kramers-Kr´’onig dispersion relations [99]. We shall con-
sider propagation of light in vacuum and neglect the imaginary part of the eikonal that is associated
with absorption. Of course, the amplitude, Aαβ , and phase, ϕ, are functions of both time and spatial
coordinates.
The source-free (vacuum) Maxwell equations are given by [1, 2]
∇α Fβγ + ∇β Fγα + ∇γ Fαβ = 0 , (5.3.2)
∇β F αβ = 0 , (5.3.3)
where ∇α denotes covariant differentiation. Taking a covariant divergence from equation (5.3.2), using
equation (5.3.3) and applying the rule of commutation of covariant derivatives of a tensor field of a
second rank, we obtain the covariant wave equation for the electromagnetic field tensor
g Fαβ + Rαβµν F µν + Rµα Fβ µ − Rµβ Fα µ = 0 , (5.3.4)
where g ≡ g αβ ∇α ∇β , Rαβµν is the Riemann curvature tensor, and Rαβ = Rγ αγβ is the Ricci tensor
(definitions of the Riemann and Ricci tensors in this chapter are the same as in the textbook [3]). We
consider the case of propagation of light in vacuum where the stress-energy tensor of matter, T αβ , is
absent. Due to the Einstein equations it yields Rαβ = 0. Hence, in our case (5.3.4) is reduced to a more
simple form
g Fαβ + Rαβµν F µν = 0 . (5.3.5)
Differential operator g in (5.3.4) taken along with the Riemann and Ricci tensors is called de Rham’s
operator for the electromagnetic field [2, 101].
5.3.2 Maxwell equations in the geometric optics approximation

Let us now assume that the electromagnetic tensor Fαβ shown in (5.3.1) can be expanded with respect to
a small dimensionless parameter ε = λem /L where λem is a characteristic wavelength of the electromag-
netic wave and L is a characteristic radius of spacetime curvature. The parameter ε is a bookkeeping
parameter of the high-frequency approximation in expansion of the electromagnetic field beyond the
limit of the geometric optics. More specifically, we assume that the expansion of the electromagnetic
field given by equation (5.3.1) has the following form [2]

iϕ
Fαβ = aαβ + εbαβ + ε2 cαβ + ... exp

, (5.3.6)
ε
where aαβ , bαβ , cαβ , etc. are functions of time and spatial coordinates.
Substituting expansion (5.3.6) into equation (5.3.2), taking into account a definition of the electro-
magnetic wave vector, lα ≡ ∂ϕ/∂xα , and arranging the terms with similar powers of ε, lead to the
chain of equations
lα aβγ + lβ aγα + lγ aαβ = 0, (5.3.7)

∇α aβγ + ∇β aγα + ∇γ aαβ = −i (lα bβγ + lβ bγα + lγ bαβ ) , (5.3.8)
where we have neglected the effects of spacetime curvature which are of the order of O(ε2 ) that are too
small to measure.
Similarly, equation (5.3.3) gives a chain of equations
lβ aαβ = 0, (5.3.9)
αβ
∇β a + ilβ bαβ = 0, (5.3.10)
where we again neglected the effects of spacetime curvature.

Equation (5.3.9) implies that the amplitude, aαβ , of the electromagnetic field tensor is orthogonal in
the four-dimensional sense to a wave vector lα , at least, in the first approximation. Contracting equation
(5.3.7) with lα and accounting for (5.3.9), we find that the wave vector lα is null, that is
lα l α = 0 . (5.3.11)
Taking a covariant derivative from this equation and using the fact that
∇[β lα] = 0 , (5.3.12)
because lα = ∇α ϕ, one can show that the vector lα obeys the null geodesic equation
l β ∇β l α = 0 . (5.3.13)
It means that the null vector lα is parallel transported along itself in the curved spacetime. Equation
(5.3.13) can be expressed more explicitly as
dlα
+ Γα β γ
βγ l l = 0 , (5.3.14)
dσ
where σ is an affine parameter along the light-ray trajectory, and
1 αµ
Γα
βγ = g (∂γ gµβ + ∂β gµγ − ∂µ gβγ ) , (5.3.15)
2
are the Christoffel symbols.
Finally, contracting equation (5.3.8) with lγ , and using (5.3.7), (5.3.9), (5.3.10) along with (5.3.12)
we can show that in the first approximation
lγ ∇γ aαβ + ϑaαβ = 0 , (5.3.16)
where
ϑ ≡ (1/2)∇α lα , (5.3.17)
is the expansion of the light-ray congruence defined at each point of spacetime by the derivative of the
wave vector lα .
Equation (5.3.16) represents the law of propagation of the tensor amplitude of electromagnetic wave
along the light ray. In the most general general case, when the expansion ϑ 6= 0, the tensor amplitude of
the electromagnetic wave is not parallel-transported along the light rays. It can be shown that the expan-
sion ϑ of the light-ray congruence is defined only by the stationary components of the gravitational field
of the isolated astronomical system determined by its mass M, and spin Si , but it does not depend on the
higher-order multipole moments. It means that gravitational waves do not contribute to the expansion
of the light-ray congruence in the linearised approximation of general relativity and their impact on ϑ
is postponed to the terms of the second order of magnitude with respect to the universal gravitational
constant G.
5.3.3 Electromagnetic eikonal and light-ray geodesics

The unperturbed congruence of light rays
We have assumed that geometric optics approximation is valid and electromagnetic waves propagate
in vacuum. We also assume that each electromagnetic wave has a wavelength λem much smaller than
the characteristic wavelength λgw of gravitational waves emitted by the isolated astronomical system.
Physical speed of light in vacuum, measured locally, is equal to the speed of propagation of gravitational
waves, and is equal to the fundamantal speed c in tangent Minkowski spacetime. We have neglected
all relativistic effects associated with the curvature tensor of spacetime in equations of light propaga-
tion. In accordance with the consideration given in previous section 5.3.2, a kinematic description of
propagation of each electromagnetic wave can be given by tracking position of its phase ϕ, which is a
null hypersurface in spacetime, as a function of time or by following the congruence of light rays that
are orthogonal to the phase. Quantum electrodynamics tells us that the light rays are tracks of massless
particles of the quantized electromagnetic field (photons) which are moving along light-ray geodesics
defined by equation (5.3.14).
Particular solution of these equations can be found after imposing the initial-boundary conditions
dx(−∞)
x(t0 ) = x0 , =k. (5.3.18)
dt
These conditions determine the spatial position, x0 , of an electromagnetic signal (a photon) at the time
of its emission, t0 , and the initial direction of its propagation given by the unit vector, k, at the past
null infinity , that is at the infinite spatial distance and at the infinite past [2, 3] where the spacetime
is assumed to be flat (see Fig. 5.1). We imply that vector k is directed towards observer. Notice that
the initial-boundary conditions (5.3.18) have been chosen as a matter of convenience only. Instead of
them, we could chose two boundary conditions when both the point of emission and that of observation
of the electromagnetic signal are fixed in time and space. It is always possible to convert solution of
equations of the null geodesics given in terms of the initial-boundary conditions to that given in terms of
the boundary conditions. We discuss it in section 5.5.3.
In the next sections we will derive an explicit form of equations of null geodesics and solve them by
iterations with the initial boundary conditions (5.3.18). At the first iteration we can neglect relativistic
perturbation of photon’s motion and approximate it by a straight line
xi = xiN (t) ≡ xi0 + ki (t − t0 ) , (5.3.19)
where t0 is the time of emission of electromagnetic signal, xi0 are spatial coordinates of the source of the
electromagnetic signal taken at the time t0 , and ki = k is the unit vector along the trajectory of photon’s
motion defined in (5.3.18).
The bundle of light rays makes 2+1 split of space by projecting any point in space onto the plane being
orthogonal to the bundle (see Figure 5.2). This allows to make a transformation to new independent
variables, τ and ξ i , defined as follows
τ = ki xiN , ξ i = P ij xjN , (5.3.20)
where
P ij = δji − ki kj , (5.3.21)
is the operator of projection on the plane being orthogonal to k. It is easy to see that the parameter τ is
equivalent to time
τ ≡ k · x = t − t∗ , (5.3.22)
where
t ∗ ≡ k · x0 − t 0 , (5.3.23)
is the time of the closest approach of the electromagnetic signal to the origin of the spatial coordinates
which is taken, in our case, coinciding with the center of mass of the isolated astronomical system.
Because for each light ray the time t∗ is fixed, we conclude that the time differential dτ = dt on the
light ray. The reader may expect that the results of our calculation of observable quantities are to depend
on the parameters ξ i and t∗ . This is, however, not true since ξ i and t∗ depend on the choice of the
origin of the coordinates and direction of its spatial axes that is they are coordinate-dependent. The
observable quantities have nothing to do with the choice of coordinates and, thus, ξ i and t∗ cannot enter
the expressions for observable quantities. Inspection of the resulting equations in the sections which
follow, shows that parameters ξ i and t∗ do vanish from the observed quantities.
The unperturbed light-ray trajectory (5.3.19) written in terms of the new variables (5.3.20) reads
xiN (τ ) = ki τ + ξ i , (5.3.24)
so that the new variable ξ i ≡ ξ = k × (x × k) should be understood as a vector drawn from the origin
of the coordinate system towards the point of the closest approach of the ray to the origin. For vectors
√
ki and ξ i are orthogonal, the unperturbed distance rN = xi xi between the photon and the origin of
the coordinate system √
rN = τ 2 + d2 , (5.3.25)
where d = |ξ| is the impact parameter of the unperturbed light-ray trajectory with respect to the coordi-
nate origin.
We introduce two other operators of partial derivatives with respect to τ and ξ i determined for any
smooth function taken on the congruence of light rays. These operators will be denoted with a hat above
them and are defined as
∂ ∂ ∂
∂ˆτ ≡ , ∂î ≡ Pi j j , , ∂ˆt∗ ≡ ∗ (5.3.26)
∂τ ∂ξ ∂t
so that, for example,

ki ∂î = 0 . (5.3.27)
An important consequence of the projective structure of the bundle of the light rays is that for any
smooth function F (t, x) defined on the light-ray trajectories, one has

∂ ∂ ∂ ∂
+ k i F (t , x) = + k i F (t∗ + τ , ξ + kτ ) , (5.3.28)
∂xi ∂t x=x0 +k(t−t0 ) ∂ξ i ∂τ

∂ ∂ d
+ ki i F (t , x) = F (t∗ + τ , ξ + kτ ) , (5.3.29)
∂t ∂x x=x0 +k(t−t0 ) dτ

∂ ∂
F (t , x) = ∗ F (t∗ + τ , ξ + kτ ). (5.3.30)
∂t x=x0 +k(t−t0 ) ∂t
Here, in the left sides of Eqs. (5.3.28)–(5.3.30) one must, first, calculate the partial derivatives and only
after that substitute the unperturbed trajectory of the light ray, x = x0 + k(t − t0 ), while in the right
side of these equations one, first, substitute the unperturbed trajectory parameterized by the variables τ
and ξ i and, then, differentiate. Equations (5.3.28)–(5.3.30) define the commutation rule of interchanging
the operations of substitution of the light ray trajectory to a function defined on spacetime manifold and
the calculation of the partial derivatives from the function. It turns out to be very effective for analytic
integration of the light-ray geodesic equations.
It is worth noticing that (5.3.30) allows us to re-write (5.3.28) as follows

∂F (t,x) ∂ ∂ ∂
= + k i − k i F (t∗ + τ , ξ + kτ ) . (5.3.31)
∂xi x=x0 +k(t−t0 ) ∂ξ i ∂τ ∂t∗
This equation will be used later for decomposing STF spatial derivatives from the potentials of gravita-
tional field depending on retarded time.
The eikonal equation

The eikonal ϕ is related to the wave vector lα of the electromagnetic wave as lα = ∂α ϕ. This definition
along with equation (5.3.11) immediately gives us a differential Hamilton-Jacobi equation for the eikonal
propagation [1]
∂ϕ ∂ϕ
g αβ α =0. (5.3.32)
∂x ∂xβ
The unperturbed solution of this equation is a plane electromagnetic wave
ϕN = ϕ0 + ωkα xα , (5.3.33)
where ϕ0 is a constant, ω = 2πν∞ , ν∞ is a constant frequency of the electromagnetic wave at infinity,

and kα is the unperturbed direction of the co-vector lα . Equation (5.3.32) assumes that kα is a null
co-vector with respect to the Minkowski metric in the sense that
η αβ kα kβ = 0 . (5.3.34)
We postulate that the co-vector kα = (−1,k) where the unit Euclidean vector k is defined at past null
infinity by equation (5.3.18).
In the linearized approximation of general relativity the eikonal can be decomposed in a linear com-
bination of unperturbed, ϕN , and perturbed, ψ, parts
ϕ = ϕN + ωψ , (5.3.35)
so that the wave co-vector

∂ψ
lα = ω k α + , (5.3.36)
∂xα
Making use of equations (5.2.2) and (5.3.32)–(5.3.36) yield a partial differential equation of the first
order for the perturbed part of the eikonal
∂ψ 1
kα = hαβ kα kβ . (5.3.37)
∂xα 2
This equation can be solved in all space by the method of characteristics [103] which, in the case under
consideration, are the unperturbed light-ray geodesics given by Eq. (5.3.24). Hence, after making use of
relation (5.3.29) one gets an ordinary differential equation for finding the eikonal perturbation
dψ 1 α β

i i 0

= hcan. ˆ
αβ (τ,ξ)k k + ∂τ k w − w , (5.3.38)
dτ 2
where both the gauge functions wα and the canonical metric tensor perturbation hcan. αβ are taken on the
unperturbed light-ray trajectory. In particular, the components of the metric tensor perturbation have the
following form
∞ l p
2M X X X lpq ∗
hcan.
00 = +2 h 00 (t ,τ,ξ) , (5.3.39)
r p=0 q=0
(M )
l=2
∞ X
l−1 Xp
2iba Sb N a X
∗ lpq ∗
hcan.
0i = − 2
+4 h lpq
0i (t ,τ,ξ) + h 0i (t ,τ,ξ) , (5.3.40)
r p=0 q=0
(S) (M )
l=2
hcan. can. can.

ij =δij h00 + qij , (5.3.41)
∞ X p
l−2 X ∞ X p
l−2 X
X ∗
X ∗
can.
qij =4 q lpq
ij (t ,τ,ξ) − 8 q lpq
ij (t ,τ,ξ) , (5.3.42)
(M ) (S)
l=2 p=0 q=0 l=2 p=0 q=0
where
" (p−q)
IAl (t − r)
#
∗ (−1)l+p−q
h lpq
00 (t ,τ,ξ) = q
Cl (l − p,p − q,q)k<a1 ...ap ∂âp+1 ...al > ∂ˆτ , (5.3.43)
(M ) l! r
∗ (−1)l+p−q
h lpq
0i (t ,τ,ξ) = Cl−1 (l − p − 1,p − q,q)× (5.3.44)
(M ) l!
 
(p−q+1)
q
IiAl−1 (t − r)
k<a1 ...ap ∂âp+1 ...al−1 > ∂ˆτ  ,
r
∗ (−1)l+p−q l
h lpq
0i (t ,τ,ξ) = Cl−1 (l − p − 1,p − q,q)× (5.3.45)
(S) (l + 1)!
 
(p−q)
q
iab SbAl−1 (t − r)
ˆ ˆ ˆ ˆ
(∂a + ka ∂τ − ka ∂t∗ )∂τ  ,
r
∗ (−1)l+p−q
q lpq
ij (t ,τ,ξ) = Cl−2 (l − p − 2,p − q,q)× (5.3.46)
(M ) l!
 
(p−q+2)
q
IijAl−2 (t − r)
k<a1 ...ap ∂âp+1 ...al−2 > ∂ˆτ   ,
r
∗ (−1)l+p−q l
q lpq
ij (t ,τ,ξ) = Cl−2 (l − p − 2,p − q,q)(∂â + ka ∂ˆτ − ka ∂ˆt∗ )× (5.3.47)
(S) (l + 1)!
 
(p−q+1)
q
ba(i Sj)bAl−2 (t − r)
k<a1 ...ap ∂âp+1 ...al−2 > ∂ˆτ   .
r
All quantities in the right side of (5.3.43)–(5.3.47), which are explicitly shown as functions of xi , r = |x|
and t, must be understood as taken on the unperturbed light-ray trajectory and expressed in terms of ξ i ,
(p−q)
d = |ξ|, τ and t∗ in accordance with equations (5.3.22), (5.3.25). For example, the ratio IAl (t − r)/r
in equation (5.3.43) must be understood as
(p−q) (p−q) √
IAl (t − r) IAl (t∗ + τ − τ 2 + d2 )
≡ √ , (5.3.48)
r τ 2 + d2
and the same replacement rule is applied to the other equations. After accounting for (5.3.39)–(5.3.47),
equation (5.3.38) can be solved analytically with the mathematical technique shown in section 5.4.
Light-ray geodesics
The geodesic equation for light rays is given in (5.3.14). It is reduced to a more explicit form after
making use of the linearized post-Minkowskian expressions for the Christoffel symbols
1
Γ000 = − ∂t h00 (t, x) , (5.3.49)
2
1
Γ00i = − ∂i h00 (t, x) , (5.3.50)
2
1
Γ0ij = − [∂i h0j (t, x) + ∂j h0i (t, x) − ∂t hij (t, x)] , (5.3.51)
2
1
Γi00 = ∂t h0i (t, x) − ∂i h00 (t, x) , (5.3.52)
2
1
Γi0j = [∂j h0i (t, x) − ∂i h0j (t, x) + ∂t hij (t, x)] , (5.3.53)
2
1
Γijp = [∂j hip (t, x) + ∂p hij (t, x) − ∂i hjp (t, x)] . (5.3.54)
2
The affine parameter σ in this equation is an implicit function of the coordinate time t. Relation
between σ and t is derived from the time component of (5.3.14)
2
d2 t

dt
+ Γ000 + Γ00p ẋp + Γ0pq ẋp ẋq

=0, (5.3.55)
dσ 2 dσ
where the dot above coordinates denote a derivative with respect to the coordinate time, ẋp ≡ dxp /dt.
Effectively, there is no need to solve (5.3.55) explicitly as we are not interested in the parameter σ
because the coordinate time t is more practical parameter which can be measured with clocks. Therefore,
we express the spatial components of the geodesic equation (5.3.14) in terms of the Christoffel symbols
(5.3.49)–(5.3.54), and replace differentiation with respect to the canonical parameter σ by differentiation
with respect to the coordinate time t. With the help of equation (5.3.55) the spatial components of the
geodesic equation for light ray propagation becomes
1 1
ẍi (t) = h00,i − h0i,0 − h00,0 ẋi − hik,0 ẋk − (h0i,k − h0k,i )ẋk − (5.3.56)
2 2
1 1
h00,k ẋ ẋ − hik,j − hkj,i ẋk ẋj +
k i
hkj,0 − h0k,j ẋk ẋj ẋi ,
2 2
where h00 , h0i , hij are components of the metric tensor taken on the unperturbed light-ray trajectory as
shown in equations (5.3.39)–(5.3.47), that is hαβ ≡ hαβ (t, xN (t)).
Equation (5.3.56) can be further simplified after substituting the unperturbed light-ray trajectory
(5.3.24) to the right side of Eq. (5.3.56) and making use of equation (5.3.28). Working in arbitrary
coordinates one obtains
d2 xi (τ )

1 1 i j p can.
2
= kα kβ ∂î hcan.
αβ − ∂τ
ˆ kα hcan.
iα − k k k qjp − ∂ˆτ τ (wi − ki w0 ). (5.3.57)
dτ 2 2
where all functions in equation (5.3.57) are taken (before any differentiation) on the unperturbed light-
ray trajectory given by equation (5.3.24) and the gauge functions wα (they are explained in (5.2.3)) have
not yet been specified which means that equations (5.3.57) are gauge-invariant. We discuss the choice
of the gauge functions later on in section 5.3.3.
The main advantage of the form (5.3.57) to which we have reduced the light ray propagation equation
(5.3.14) is the convenience of its analytic integration. Indeed, when we integrate along the light-ray path
the following rules, applied to any smooth function F (τ,ξ), can be used
Z
∂
F (τ,ξ) dτ =F (τ,ξ) + C(ξ) , (5.3.58)
∂τ
Z Z
∂ ∂
F (τ,ξ) dτ = i F (τ,ξ) dτ . (5.3.59)
∂ξ i ∂ξ
This means that terms which are represented as partial derivatives with respect to the time parameter τ
can be immediately integrated out by making use of (5.3.58). At the same time (5.3.59) shows that one
can change the order of integration and differentiation with respect to the parameter ξ i . It allows us to
calculate the integral along the light ray from a more simple scalar expression instead of integrating a
vector function. This technique will be demonstrated explicitly in next sections.
Equation (5.3.57) is linear with respect to the perturbation of the metric tensor, hαβ . Hence, it can
be linearly decomposed in the equation for perturbations of the light-ray trajectory caused separately by
mass and spin multipole moments. Substitution of the metric tensor (5.2.5) – (5.2.8) to Eq. (5.3.57) and
replacement of spatial derivatives with respect to xi with those with respect to parameters ξ i and τ by
making use of (5.3.30) yield the following linear superposition
ẍi = ẍi(G) + ẍi(M ) + ẍi(S) , (5.3.60)
where ẍi(M ) and ẍi(S) are the components of photon’s coordinate acceleration caused by mass and spin
multipoles of the metric tensor respectively, and ẍi(G) is the gauge-dependent acceleration. These com-
ponents read h i
ẍi(G) = ∂ˆτ τ (ki φ0 − φi ) + (ki w0 − wi ) , (5.3.61)
M
ẍi(M ) =2(∂î − ki ∂ˆτ )
+ (5.3.62)
r
∞ X p
l X
X (−1)l+p−q
2∂î Cl (l − p,p − q,q)H(2 − q)×
l!
l=2 p=0 q=0
q IAl (t − r)

p − q p−q
1− 1− k<a1 ...ap ∂âp+1 ...al > ∂ˆtp−q
∗
ˆ
∂ τ −
l l−1 r
∞ X l
X (−1)l+p p
2∂ˆτ Cl (l − p,p) 1 − ×
l! l
l=2 p=0
IAl (t − r)

p
1+ ki<a1 ...ap ∂âp+1 ...al > ∂ˆtp∗ −
l−1 r
IiAl−1 (t − r)

2p
k<a1 ...ap−1 ∂âp ...al−1 > ∂ˆtp∗ ,
l−1 r
and
jba Sb
ẍi(S) =2 kj ∂îa − δij ∂âτ − (5.3.63)
r
∞ l−1 p
X X X (−1)l+p−q l
4kj ∂îa Cl−1 (l − p − 1,p − q,q) H(2 − q)×
(l + 1)!
l=2 p=0 q=0
jba SbAl−1 (t − r)

p−q
1− k<a1 ...ap ∂âp+1 ...al−1 > ∂ˆtp−q
∗ ∂ˆτq +
l−1 r
∞ X l−1
X (−1)l+p l
4 ∂â − ka ∂ˆt∗ ∂ˆτ Cl−1 (l − p − 1,p) ×
(l + 1)!
l=2 p=0
iba SbAl−1 (t − r)

p
1− k<a1 ...ap ∂âp+1 ...al−1 > ∂ˆtp∗ +
l−1 r
∞ X l−1
X (−1)l+p l
4kj ∂ˆτ Cl−1 (l − p − 1,p) (5.3.64)
(l + 1)!
l=2 p=0
jbal−1 ṠîbAl−2 (t − r)
" #
k<a1 ...ap ∂âp+1 ...al−1 > ∂ˆtp∗ ,
r
where a dot above any function denotes a partial time derivative with respect to the parameter t∗ [104],
wα are the gauge functions coming from (5.2.3), and φα are the gauge functions which appear as a result
of integration of the light ray propagation equation. These terms enter the equations of motion in the
form of combination ki φ0 − φi given in next section. It is worth emphasizing that we do not intend to
separate ki φ0 − φi in two functions, φi and φ0 because such a separation is not unique while the linear
combination ki φ0 − φi is unambiguous. We have not combined functions φα with the gauge functions
wα for two reasons:
1. to indicate that the solution of equations of light-ray geodesic, performed in one specific coordinate
system, leads to generation of terms which can be eliminated by gauge transformation,
2. to simplify the final form of the result of the integration as all terms with second and higher order
time derivatives are immediately integrated in accordance with (5.3.58).
Gauge functions wα are still arbitrary which makes our equations gauge-invariant. However, for the
sake of physical interpretation of the result of integration of equations of light-ray geodesic, we shall
choose a specific form of functions wα to make our coordinate system both harmonic and ADM which
makes the coordinate description of motion of free-falling particles in these coordinates simple. Specific
form of the gauge functions wα at arbitrary field point is shown in equations (5.2.17), (5.2.18) and their
form at any point on the light-ray trajectory is given in section 5.3.3.
It is important to notice that all terms depending on mass-type multipoles of the order l in the right
sides of (5.3.62) have a numerical factor 1 − p/l where p is the summation index. Such terms vanish
when p = l. It means that (5.3.62) does not contain terms with the time derivatives of the order l
from the multipoles of the l-th order which, actually, describe gravitational wave emission from the
astrophysical system because they decay slowly as 1/r. The same is true with regard to the spin-type
multipoles of the order l − 1 in (5.3.63) – the time derivatives of the order of l − 1 from the spin-type
multipoles (which decay as 1/r) vanish in the right side of (5.3.63). Explicit analytic integration of
such terms would be impossible but they simply do not present in the solution of general relativistic
equations of light propagation for the reason mentioned above. It is this property of the null geodesic in
general relativity which prevents the amplification of the gravitational wave perturbation for a light ray
propagating closely to an astrophysical system emitting multipolar gravitational radiation (binary star,
etc.). This fact was established in [91] in a quadrupole approximation and extended to any multipole in
[92]. Present chapter confirms this result. The reader should notice that the cancellation of the
Gauge freedom of equations of propagation of light

Gauge functions wα , generating the coordinate transformation from the canonical harmonic coordinates
to the ADM-harmonic ones, are given by equations (5.2.19), (5.2.20). They transform the metric tensor
as follows
hcan.
αβ = hαβ − ∂α wβ − ∂β wα , (5.3.65)
where hcan.
αβ is the canonical form of the metric tensor in harmonic coordinates given by equations (5.2.5)–
(5.2.8) and hαβ is the metric tensor given in the ADM-harmonic coordinates by equations (5.2.21)–
(5.2.24).
The gauge functions taken on the light-ray trajectory and expressed in terms of the variables ξ and τ
can be written down in the next form
∞ X
X p Z
l X τ +t∗
w0 = du h lpq
00 (u,τ,ξ), (5.3.66)
(M )
l=2 p=0 q=0 −∞
∞ X
X p Z
l X τ +t∗ Z v
wi = (∂î + ki ∂ˆτ − ki ∂ˆt∗ ) dv du h lpq
00 (u,τ,ξ) (5.3.67)
(M )
l=2 p=0 q=0 −∞ −∞
∞ X
X p Z
l−1 X τ +t∗
−4 du h lpq
0i (u,τ,ξ) + h lpq
0i (u,τ,ξ) ,
(M ) (S)
l=2 p=0 q=0 −∞
where h lpq lpq lpq

00 (u,τ,ξ), h 0i (u,τ,ξ) and h 0i (u,τ,ξ) are defined by the Eqs. (5.3.43), (5.3.44) and (5.3.45)
(M ) (M ) (S)
after making use of the substitution t∗ → u. It is worth noticing the following relationships
∞ l p
∂w0 X X X lpq ∗
= h 00 (t ,τ,ξ), (5.3.68)
∂t∗ p=0 q=0
(M )
l=2
∞ l p τ +t∗
∂wi X X X
Z
= du(∂î + ki ∂ˆτ ) h lpq
00 (u,τ,ξ) (5.3.69)
∂t∗ p=0 q=0 −∞ (M )
l=2
∞ X p
l−1 X
X ∗ lpq ∗
−4 h lpq
0i (t ,τ,ξ) + h 0i (t ,τ,ξ) ,
(M ) (S)
l=2 p=0 q=0
and
∞ p τ +t∗
l
"Z #
∂wi ∂w0 X X X ∗
ki ∗ − = du∂ˆτ h lpq
00 (u,τ,ξ) − h lpq
00 (t ,τ,ξ) (5.3.70)
∂t ∂t∗ p=0 q=0 −∞ (M ) (M )
l=2
∞ X p
l−1 X
X ∗ i lpq ∗
−4 ki h lpq
0i (t ,τ,ξ) + k h 0i (t ,τ,ξ) ,
(M ) (S)
l=2 p=0 q=0
which are helpful in calculation of the gravitational shift of the frequency of light.
A linear combination, ki φ0 − φi , of the gauge-dependent functions φα that appear in (5.3.61), is
given by the expressions
ki φ0 − φi = (ki φ0(M ) − φi(M ) ) + (ki φ0(S) − φi(S) ) , (5.3.71)
∞ Xl X p
X (−1)l+p−q
ki φ0(M ) − φi(M ) = 2∂î Cl (l − p,p − q,q)× (5.3.72)
p=2 q=2
l!
l=2
" (p−q)
IAl (t − r)
#
p − q p−q ˆ ˆ q−2
1− 1− k<a1 ...ap ∂ap+1 ...al > ∂τ +
l l−1 r
∞ Xl X p
X (−1)l+p−q
2 Cl (l − p,p − q,q)×
p=1 q=1
l!
l=2
" (p−q)
IAl (t − r)
#
p − q p−q ˆ ˆ q−1
1− 1+ ki<a1 ...ap ∂ap+1 ...al > ∂τ −
l l−1 r
 
(p−q)
p−q q−1
IiAl−1 (t − r) 
2 k<a1 ...ap−1 ∂âp ...al−1 > ∂ˆτ   ,
l−1 r 
iab ka Sb
ki φ0(S) − φi(S) = 2 + (5.3.73)
r
∞ Xl−1 Xp
X (−1)l+p−q l
4kj ∂îa Cl−1 (l − p − 1,p − q,q) ×
(l + 1)!
l=3 p=2 q=2
 
(p−q)

p−q

q−2 
jab SbAl−1 (t − r)
1− ˆ ˆ
k<a1 ...ap ∂ap+1 ...al−1 > ∂τ −
l−1 r
∞ Xl−1 X p
X (−1)l+p−q l
4 ∂â − ka ∂ˆt∗ Cl−1 (l − p − 1,p − q,q) ×
(l + 1)!
l=2 p=1 q=1
 
(p−q)

p−q

q−1
1− k<a1 ...ap ∂âp+1 ...al−1 > ∂ˆτ  −
l−1 r
∞ Xl−1 Xp
X (−1)l+p−q l
4ka Cl−1 (l − p − 1,p − q,q) ×
(l + 1)!
l=2 p=0 q=0
 
(p−q)

p−q

q
1− ˆ ˆ
k<a1 ...ap ∂ap+1 ...al−1 > ∂τ +
l−1 r
∞ X l−1 X p
X (−1)l+p−q l
4kj Cl−1 (l − p − 1,p − q,q) ×
(l + 1)!
l=2 p=1 q=1
(p−q+1)
 
jbal−1 SîbA (t − r)
q−1
k<a1 ...ap ∂âp+1 ...al−1 > ∂ˆτ  l−2

r
5.3.4 Polarization of light and the Stokes parameters

Reference tetrad
Propagation of electromagnetic fields in vacuum and evolution of their physical parameters in curved
spacetime can be studied with various mathematical techniques. One of the most convenient techniques
was worked by Newman and Penrose [105, 106] and is called the Newman-Penrose formalism [107].
This formalism introduces at each point of spacetime a null tetrad of four vectors associated with the
bundle of light rays defined by the electromagnetic wave vector field lα . The Newman-Penrose tetrad
consists of two real and two complex null vectors – (lα , nα , mα , m̄α ) – where the bar above function
indicates complex conjugation. The null tetrad vectors are normalized in such a way that nα lα = −1
and mα m̄α = +1 are the only non-vanishing products among the four vectors of the tetrad.
The vectors of the null tetrad are not uniquely determined by specifying lα . Indeed, for a fixed direc-
tion lα the normalization conditions for the tetrad vectors are preserved under the linear transformations
(null rotation) [105, 107]
l0α = Alα , (5.3.74)

0α
A−1 nα + B̄mα + B m̄α + B B̄lα ,

n = (5.3.75)
m0α e−iΘ mα + B̄lα ,

= (5.3.76)
0α iΘ α α
m̄ = e (m̄ + Bl ) , (5.3.77)
where A,Θ are real scalars and B = B1 + iB2 is a complex scalar. These transformations form a
four-parameter (A,B1 ,B2 ,Θ) subgroup of the homogeneous Lorentz group which is equivalent to the
point-like Lorentz transformations [108].
For doing mathematical analysis of the intensity and polarization of electromagnetic waves it is useful
to introduce a local orthonormal reference frame of observer moving with a four velocity uα who is
seeing the electromagnetic wave travelling in the positive direction of z axis of the reference frame. It
means that at each point of spacetime the observer uses a tetrad frame eα(β) = eα(0) , eα(1) , eα(2) , eα(3)

defined in such a way that

h i
eα(0) = uα , eα(3) = (−lα uα )−1 lα + (lβ uβ )uα , (5.3.78)
and two other vectors of the observer’s tetrad, eα(1) and eα(2) , are the unit space-like vectors being orthog-
onal to each other as well as to eα(0) and eα(3) . In other words, vectors of the observer tetrad are subject
to the following normalization conditions
gαβ eα(µ) eβ(ν) = ηµν , η µν eα(µ) eβ(ν) = g αβ . (5.3.79)
It is worth noticing that the observer’s tetrad eα

β has two group of indices. The indices without round
brackets run from 0 to 3 and are associated with time and space coordinates. The indices enclosed in the
round brackets numerate vectors of the tetrad and also run from 0 to 3. The coordinate-type indices of the
tetrad have no relation to the tetrad indices. If one changes spacetime coordinates (passive coordinate
transformation) it does not affect the tetrad indices while the coordinate indices of the tetrad change
in accordance with the transformation law for vectors. On the other hand, one can change the tetrad
vectors at each point in spacetime by doing the Lorentz transformation (active coordinate transformation)
without changing the coordinate chart [106].
Let us define at each point of spacetime a coordinate basis of static observers

1
E α(0) = 1 + h00 , 0 , 0 , 0 , (5.3.80)
2

1
E α(1) = h0j aj , ai − hij aj , (5.3.81)
2

1
E α(2) = h0j bj , bi − hij bj , (5.3.82)
2

1
E α(3) = h0j kj , ki − hij kj , (5.3.83)
2
which is written down for the case of weak gravitational field, gαβ = ηαβ + hαβ . Here the unit spatial
vectors a = (a1 ,a2 ,a3 ), b = (b1 ,b2 ,b3 ), and k = (k1 ,k2 ,k3 ) are orthonormal in the Euclidean sense
(δij ai bj = δij ai kj = δij bi kj = 0 and δij ai aj = δij bi bj = δij ki kj = 1) with vector k pointing
to the direction of propagation of the light ray at infinity as given in (5.3.18). These basis vectors are
convenient to track the changes in the parameters of the electromagnetic wave as it travels from the point
of emission of light to the point of its observation.
The local tetrad eα(β) of observers moving with four-velocity uα with respect to the static observers
relates to the tetrad E α(β) by means of the Lorentz transformation
eα(β) = Λγβ E α(γ) , E α(β) = λγβ eα(γ) , (5.3.84)

where the matrix of the Lorentz transformation is [2]
Λ00 = u0 ≡ γ (5.3.85)
Λi0 = Λ0i 0
=u , (5.3.86)
ui uj
Λij = ij
δ + , (5.3.87)
1+γ
and the inverse matrix of the Lorentz transformation λαβ is obtained from Λαβ by replacing ui → −ui
that complies with the definition of the inverse matrix Λαβ λβγ = δγα .
The connection between the null tetrad and the observer’s tetrad frame, eα(β) , is given by equations
lα −(lγ uγ ) eα(0) + eα(3) ,

= (5.3.88)
1
nα − (lγ uγ ) eα(0) − eα(3) ,

= (5.3.89)
2
1
mα √ eα(1) + ieα(2) ,

= (5.3.90)
2
1
m̄α √ eα(1) − ieα(2) .

= (5.3.91)
2
Vector pairs eα(0) , eα(3) and eα(1) , eα(2) split spacetime at the point in two sub-spaces. In particular,
vectors eα(1) , eα(2) defines the plane of polarization in spacetime. If vectors eα(0) and eα(3) are fixed, then,
vectors eα(1) and eα(2) are defined up to an arbitrary rotation in the plane of polarization. Transformations
(5.3.76), (5.3.77) with B = 0 yield
e0α
(1) = cos Θ eα(1) + sin Θ eα(2) , (5.3.92)
e0α
(2) = − sin Θ eα(1) + cos Θ eα(2) , (5.3.93)
where Θ is the rotation angle of the vectors in the plane of polarization. We notice that, since vectors
eα(1) , eα(2) are orthogonal to eα(0) = uα , the two null vectors, mα and m̄α , are also orthogonal to the
four-velocity,
mα uα = 0 , m̄α uα = 0 . (5.3.94)
The null vector nα is also orthogonal to uα , nα uα = 0. On the other hand, the scalar product of lα with
four-velocity yields the angular frequency of electromagnetic wave, ω ≡ −lα uα .
Propagation laws for the reference tetrad
Discussion of the rotation of the polarization plane and the change of the Stokes parameters of electro-
magnetic radiation is inconceivable without understanding of how the local reference frame propagates
along the light-ray geodesic from the point of emission of light to the point of its observation. To this
end we construct the reference tetrad frame of observer at the point of observation of light and render a
parallel transport of it backward in time along the light-ray geodesic. By definition, vectors of the tetrad
frame of the observer, eα(β) , and those of the null tetrad (lα , nα , mα , m̄α ) do not change in a covariant
sense as they are parallel transported along the light ray. The propagation equation for the tetrad vectors
are, thus, obtained by applying the operator D ≡ lα ∇α of the parallel transport along the null vector lα .
Explicit form of the parallel transport of the reference tetrad is
deα(µ)
+ Γα β γ
βγ l e (µ) = 0 , (5.3.95)
dσ
where σ is an affine parameter along the light ray. Using definition of the Christoffel symbols (5.3.15)
and changing parameter σ to the proper time τ of the observer we recast (5.3.95) to

d 1 1
eα(µ) + hαβ eβ(µ) = η αν (∂ν hγβ − ∂γ hνβ ) kβ eγ(µ) . (5.3.96)
dτ 2 2
The propagation of the null vectors mα and m̄α along the direction of the null vector lα is given by
dmα
+ Γα β γ
βγ l m = 0 , (5.3.97)
dλ
dm̄α
+ Γα β γ
βγ l m̄ = 0 , (5.3.98)
dλ
and the same laws are valid for nα and lα (see, for example, (5.3.14)). Equations (5.3.96)–(5.3.98) are
the main equations for the discussion of the rotation of the plane of polarization and variation of the
Stokes parameters.
Relativistic description of polarized electromagnetic radiation

We consider propagation of a bundle of plane electromagnetic waves from the point of emission to the
point of observation. Each of these waves have an electromagnetic tensor Fαβ = Fαβ + O(ε) defined
in the first approximation by equation [107, 109]

iϕ
Fαβ = aαβ exp , (5.3.99)
ε
aαβ = Φ m[α lβ] + Φ̄ m̄[α lβ] (5.3.100)
where Φ is a complex scalar amplitude of the wave with a real and imaginary components which are
independent of each other in the most general case of incoherent radiation. In the proper frame of
the observer with 4-velocity uα the components of the electric and magnetic field vectors are defined
√
respectively as E α = −F αβ uβ and H α = (−1/2 −g)αβγδ Fγδ uβ [3]. The electric field is a product
of slowly-changing amplitude Eα = −aαβ u and fast-oscillating phase exponent
β

iϕ
Eα = Eα exp . (5.3.101)
ε
The polarization properties of electromagnetic radiation consisting of an ensemble of the waves with
equal frequencies but different phases are defined by the components of the electric field measured by
observer. In the rest frame of the observer with 4-velocity uα , the intensity and polarization properties
of the electromagnetic radiation are described in terms of the polarization tensor [1, 99]
Jαβ =< Eα Ēβ >=< Eα Ēβ > , (5.3.102)
where the angular brackets represent an average with respect to an ensemble of the electromagnetic
waves with randomly distributed phases. This averaging eliminates all fast-oscillating terms from Jαβ .
One has to notice [1] that the polarization tensor Jαβ is symmetric only for a linearly polarized radia-
tion. In all other cases, the polarization tensor is not symmetric. The polarization tensor Jαβ is purely
spatial at the point of observation which means it is orthogonal to the four-velocity of observer, Jαβ uβ .
Furthermore, because the polarization tensor is defined in the sub-space of the polarization plane, it is
orthogonal to the wave vector Jαβ lβ = 0. This equality follows directly from its definition (5.3.102)
and (5.3.9).
The vector amplitude Eα of the electric field can be decomposed in two independent components in
the plane of polarization. Two vectors of the null tetrad, mα ,m̄α , form the circular-polarization basis,
and vectors eα(1) , eα(2) form a linear polarization basis. The decomposition reads
Eα = EL mα + ER m̄α , (5.3.103)
E α
= E(1) eα(1) + E(2) eα(2) , (5.3.104)
where
1 1
EL = ωΦ , ER = ω Φ̄ , (5.3.105)
2 2
are left and right circularly-polarized components of the electric field,
1 i
E(1) = √ (EL + ER ) , E(2) = √ (EL − ER ) , (5.3.106)
2 2
are linearly polarized components, ω = −lα uα is the angular frequency of the electromagnetic wave,
and we have taken into account the condition of orthogonality (5.3.94).
There are for electromagnetic Stokes parameters Sα = (S0 ,S1 ,S2 ,S3 ). They are defined by pro-
jecting the polarization tensor Jαβ on four independent combination of the tensor products of the two
vectors, eα(1) , eα(2) making up the polarization plane. More specifically, [99, 110]
h i
S0 = Jαβ eα(1) eβ(1) + eα(2) eβ(2) , (5.3.107)
h i
S1 = Jαβ eα(1) eβ(1) − eα(2) eβ(2) , (5.3.108)
h i
S2 = Jαβ eα(1) eβ(2) + eα(2) eβ(1) , (5.3.109)
h i
S3 = iJαβ eα(1) eβ(2) − eα(2) eβ(1) , (5.3.110)
where S0 ≡ I is the intensity, S1 ≡ Q and S2 ≡ U characterize the degree of a linear polarization, and
S3 ≡ V is the degree of a circular polarization of the electromagnetic wave.
Making use of equation (5.3.102) in (5.3.107)–(5.3.110) allows us to represent the Stokes parameters
in the linear polarization basis as follows [99]
S0 = < |E(1) |2 + |E(2) |2 > , (5.3.111)

2 2
S1 = < |E(1) | − |E(2) | > , (5.3.112)
S2 = < E(1) Ē(2) + Ē(1) E(2) > , (5.3.113)
S3 = i < E(1) Ē(2) − Ē(1) E(2) > , (5.3.114)
where E(n) = Eα eα(n) for n = 1,2.

It is important to emphasize that though the Stokes parameters have four components, they do not
form a 4-dimensional spacetime vector because they do not behave like a vector under transformations
of the Lorentz group [1, 99]. Indeed, if we perform a pure Lorentz boost all four Stokes parameters
remain invariant [1]. However, for a constant rotation of angle Θ in the polarization plane, the Stokes
parameters transform as [1]
S00 = S0 , (5.3.115)
S10 = S1 cos 2Θ + S2 sin 2Θ , (5.3.116)
S20 = S1 cos 2Θ − S2 sin 2Θ , (5.3.117)
S30 = S3 . (5.3.118)
This is what would be expected for a spin-1/2 field. That is, under a duality rotation of Θ = π/4,
one linear polarization state turns into the other, while the circular polarization state remains the same.
The transformation properties (5.3.115)–(5.3.118) of the Stokes parameters point out that the Stokes
parameters S1 , S2 represent a linearly polarized components, and S3 represents a circularly polarized
component.
The polarization vector P = (P1 ,P2 ,P3 ) and the degree of polarization P = |P | of the electromag-
netic radiation can be defined in terms of the normalized Stokes parameters by P = (S1 /I,S2 /I,S3 /I).
Any partially polarized wave may be thought of as an incoherent superposition of a completely polar-
ized wave with the degree of polarization P and the polarization vector P , and a completely unpolarized
wave with the degree of polarization 1 − P and nil polarization vector, P = 0, so that for an arbitrary
polarized radiation one has: (S0 ,S1 ,S2 ,S3 ) = I(P,P1 ,P2 ,P3 ) + I(1 − P,0,0,0). For completely po-
larized waves, vector P describes the surface of the unit sphere introduced by Poincaré [1]. The center
of the Poincaré sphere corresponds to an unpolarized radiation and the interior to a partially polarized
radiation. Orthogonally polarized waves represent any two conjugate points on the Poincaré sphere.
In particular, (P1 = ±1,P2 = 0,P3 = 0), and (P1 = 0, P2 = 0,P3 = ±1) represent orthogonally
polarized waves corresponding to the linear and circular polarization bases, respectively.
Propagation law of the Stokes parameters

Taking definition (5.3.100) of the electromagnetic tensor and accounting for the parallel transport of the
null vectors lα , mα , m̄α along the light ray and the laws of propagation of the electromagnetic tensor
given by equations (5.3.16), yield the law of propagation of the complex scalar functions Φ and Φ̄
dΦ
+ ϑΦ = 0, (5.3.119)
dσ
dΦ̄
+ ϑΦ̄ = 0, (5.3.120)
dσ
where σ is an affine parameter along the ray and ϑ is the expansion of the light-ray congruence defined
in (5.3.17).
Let us consider a sufficiently small, two-dimensional area A in the cross-section of the congruence
of light rays lying on a null hypersurface of constant phase ϕ that is in the polarization plane. The law
of transportation of the cross-sectional area is [2, 62, 107]
dA
− 2ϑA = 0 . (5.3.121)
dσ
Thus, the product, A|Φ|2 = AΦΦ̄, remains constant along the congruence of light rays:
lα ∇α A|Φ|2 = 0 .

(5.3.122)
This law of propagation for the product A|Φ|2 corresponds to the conservation of photon’s flux [2, 62].
The law of conservation of the number of photons propagating along the light ray, corresponds to the
propagation law of vector |Φ|2 lα . Indeed, taking covariant divergence of this quantity and making use
of the equations (5.3.119), (5.3.120) along with definition (5.3.17) for the expansion ϑ of the bundle of
light rays, yields [2, 62]
∇α |Φ|2 lα = 0 .

(5.3.123)
This equation assumes that the scalar amplitude Φ of the electromagnetic wave can be interpreted in
terms of the number density, N, of photons in phase space and the energy of one photon, Eω = ~ω, as
follows 1/2
√ N

|Φ| = 8π~ , (5.3.124)
Eω
where the reduced Planck constant ~ = h/2π and the normalizing factor were introduced for consis-
tency between classical and quantum definitions of the energy of an electromagnetic wave [2].
Each of the Stokes parameter is proportional to the square of frequency of light, ω = −uα lα , as
directly follows from equations (5.3.111)–(5.3.114) and (5.3.106). Therefore, the variation of the Stokes
parameters along the light ray can be obtained directly from their definitions (5.3.107)–(5.3.110) along
the light ray and making use of the laws of propagation (5.3.119), (5.3.120). However, the set of the
Stokes parameters Sα (α = 0,1,2,3) is not directly observed in astronomy and we do not discuss their
laws of propagation. Instead the set of four other polarization parameters (Fω ,P1 ,P2 ,P3 ) is practically
measured [62, 111] and we focus on the discussion of the laws of propagation for these parameters.
Here P = (P1 ,P2 ,P3 ) is the polarization vector as defined at the end of the preceding section, and Fω
is the specific flux of radiation (also known as the monochromatic flux of a light source [111]) entering
a telescope from a given source. The specific flux is defined as an integral of the specific intensity (also
known as the surface brightness [111]) indexsurface brightness of the radiation, Iω ≡ Iω (ω,l), over the
total solid angle (assumed 4π) subtended by the source on the observer’s sky:
Z
Fω = Iω (ω,l)dΩ(l̂) , (5.3.125)
where l̂ = (sin θ̂ cos φ̂, sin θ̂ sin φ̂, cos θ̂) is the unit vector in the direction of the radiation flow and
dΩ̂(l) = sin θ̂dθ̂dφ̂ is the element of the solid angle formed by light rays from the source and measured
in the observer’s local Lorentz frame.
The specific intensity Iω of radiation at a given frequency ω = 2πν, flowing in a given direction, l̂,
as measured in a specific local Lorentz frame, is defined by
d(energy)
Iω = . (5.3.126)
d(time)d(area)d(frequency)d(solid angle)
A simple calculation (see, for instance, the problem 5.10 in [112]), reveals that
8π 3 Iω
N= , (5.3.127)
h4 ω 3
where h is the Planck’s constant. The number density N is invariant along the light ray and does not
change under the Lorentz transformation. Invariance of N is a consequence of the kinetic equation
for photons (radiative transfer equation) which in the case of gravitational field and without any other
scattering processes, assumes the following form [2]
dN
=0. (5.3.128)
dλ
Equations (5.3.127), (5.3.128) tell us that the ratio Iω /ω 3 is invariant along the light-ray trajectory, that
is 3
Iω ω
= , (5.3.129)
Iω0 ω0
where ω0 , ω are frequency of light at the point of emission and observation respectively, Iω0 is the
surface brightness of the source of light at the point of emission, and Iω is the surface brightness of the
source of light at the point of observation.
Section 5.4 Mathematical Technique for Analytic Integration of Light-Ray Equations 207
Equations (5.3.119)–(5.3.129) make it evident that in the geometric optics approximation the gravita-
tional field does not mix up the linear and circular polarizations of the electromagnetic radiation but can
change its surface brightness Iω due to gravitational (and Doppler) shift of the light frequency caused
by the time-dependent part of the gravitational field of the isolated system emitting gravitational waves.
Furthermore, the monochromatic flux from the source of radiation changes due to the distortion of the
domain of integration in equation (5.3.125) caused by the gravitational light-bending effect. Taking
into account that the gravitationally-unperturbed solid angle dΩ(k) = sin θdθdφ, and introducing the
Jacobian, J(θ,φ), of transformation between the spherical coordinates (θ̂,φ̂) and (θ,φ) at the point of
observation, one obtains that the measured monochromatic flux is
Z Z
Fω = dφ Iω0 (θ,φ)J(θ,φ)(ω/ω0 )3 sin θdθ . (5.3.130)
Equation (5.3.130) tells us that the monochromatic flux of the source of light can vary due to:
1. the gravitational Doppler shift of the electromagnetic frequency of light when it travels from the
point of emission to the point of observation;
2. the change in the solid angle at the point of observation caused by the gravitational light deflection.
The "magnification" matrix is the Jacobian of the transformation of the null directions on the ce-
lestial sphere generated by the bending of the light-ray trajectories by the gravitational field of the
isolated system.
Spatial orientation of two components, P1 and P2 , of the polarization vector P at the point of ob-
servation differs from that taken at the point of emission of the electromagnetic wave due to the parallel
transport of the polarization vector. The reference tetrad with resepct to which the orientation of the
polarization vector is measured is subject to the same law of the parallel transport. Therefore, in order to
understand the change in the orientation of the polarization vector it is sufficient to consider the change
in the orientation of the reference tetrad as it propagates from the point of emission of light to the point
of observation. The change in the polarization can be easily extracted then from the law of transforma-
tion of the Stokes parameters under the change of the reference tetrad. As we will see later in section
5.6.4, only two vectors of the tetrad, eα(1) and eα(2) , will undergo the change in their orientation as they
propagate along the light ray. Consequently, only two components of the polarization vector, P1 and P2
will change. This can be seen after taking into account equations (5.3.92), (5.3.93) where the rotational
angle Θ is determined as a solution of the parallel transport equation (5.3.96) for the reference tetrad
(see section 5.6.4). Thus, the gravitational field changes the tilt angle of the polarization ellipse as light
propagates along the light-ray trajectory. This gravitationally-induced rotation of the polarization ellipse
is called the Skrotskii effect [113, 114]. It was also predicted and discussed independently in [115, 116].
Its observation would play a significant role for detection of gravitational waves of cosmological origin
by CMBR-radiometry space missions [38, 143]. The third component of the polarization vector, P3 , re-
mains the same along the light-ray trajectory because it represents the circularly polarized component of
the radiation and is not affected by the rotation of the reference tetrad as it propagates along the light-ray
path.
5.4 Mathematical Technique for Analytic Integration of

Light-Ray Equations
This section provides mathematical technique for performing integration of equations of propagation of
various characteristics of electromagnetic wave from the point of emission of light x0 to the point of its
observation x. It has been worked out in [6, 91–93]. The basic function, which is to be integrated, is
the metric tensor perturbation, hαβ (t,x), generated by the localized astronomical system and taken at
the points lying on the light-ray trajectory. The metric tensor depends on the multipole moments and/or
their derivatives taken at the retarded instant of time s = t − r divided by the radial distance from the
system, r, taken to some power. The most difficult integrals are those taken from the functions like this,
F (s)/r, with F (s) being a multipole moment of the gravitating system depending on the retarded time
s. Calculation of the integrals goes slightly differently for stationary and non-stationary components of
the metric tensor and these issues are treated in the next two subsections.
5.4.1 Monopole and dipole light-ray integrals
The monopole and dipole parts of the metric tensor perturbation (5.2.5)–(5.2.8) are formed by terms be-
ing proportional to the mass M and spin Si of the isolated system. These terms are given in the solution
of the light-ray equations by integrals [M(s)/r][−1] , [M(s)/r][−2] and [Si (s)/r][−1] , [Si (s)/r][−2] . In
this section we shall assume for simplicity that mass M and spin Si of the isolated system are constant
during the time of propagation of light from the source of light to observer. The assumption of constancy
of the mass M and spin Si of the isolated system is valid as long as one neglects the energy emitted in
the form of the gravitational waves by the isolated system under consideration. For light ray propagating
from the sources at the edge of our visible universe (quasars) the characteristic interval of time of emis-
sion of gravitational waves by an isolated system is comparable with the interval of time the light takes
to travel from the point of emission to observer. In this case the time evolution of the mass-monopole
and spin-dipole is to be taken into account for correct calculation of the perturbations of the trajectory
of light ray (see section 5.5.1 for more detail). Mass and spin of the isolated system can also change due
to a catastrophic disruption of the isolated system resulting in supernova explosion. Specific details of
how this process affects the light-ray propagation are not considered in the present chapter but, perhaps,
are worthwhile to study.
In the case when mass and spin are constant, the integrals that we need to carry out are reduced to
[1/r][−1] and [1/r][−2] which are formally divergent at the lower limit of integration at the past null
infinity when time t → −∞. However, one must bear in mind that these integrals do not enter equa-
tions of light-ray geodesics (5.3.57) alone but appear in these equations after taking at least one partial
derivative with respect to either ξ i or τ parameters. This differentiation effectively eliminates divergent
parts of the integrals from the final result. Hence, in what follows, we drop out the formally divergent
terms so that the integrals under discussion assume the following form
[−1] Z t Z
1 dτ dτ r(τ ) − τ
≡ = √ = − ln , (5.4.1)
r −∞ r d2 + τ 2 rE
[−2] Z t [−1]
1 1 r(τ ) − τ
≡ dτ = −τ ln − r(τ ) , (5.4.2)
r −∞ r rE
where τ = t − t∗ and we used equation (5.3.25) explicitly while expressing the distance r = rN (τ ) as
a function of time τ . The constant distance rE was introduced to make the argument of the logarithmic
function dimensionless. This constant is not important for calculations as it always cancel out in final
formulas. However, in the case of gravitational lensing it is convenient to identify the scale constant rE
with the radius of the Einstein ring [62–65]
1/2
4GM rr0
rE = , (5.4.3)
c2 R
where r, r0 are distances from the isolated system (the deflector of light) to observer and to the source of
light respectively, and R = τ + τ0 . The radius of the Einstein ring is a characteristic distance separating
naturally the case of weak gravitational lensing (d > rE ) from the strong lensing (d < rE ). The Einstein
ring at the observer’s point has the angular size given by
1/2
rE 4GM r0
θE = = . (5.4.4)
r c2 Rr
The angular radius θE defines the angular scale for a lensing situation. For example, in cosmology
typical mass of an isolated system is M ' 1012 M and distances r, r0 , R are of the order of 1 Gpc
(Gigaparsec). Consequently, the angular Einstein radius θE is of the order of one arcsecond, and the linear
radius rE is of the order of 1 Kpc (Kiloparsec). A typical star within our galaxy has mass M ' M and
distances r, r0 , R are of the order 10 Kpc. It yields an angular Einstein radius θE of the order of one
1µas (one milli-arcsecond) and the linear radius rE is of the order of 10 au (10 astronomical units).
5.4.2 Light-ray integrals from quadrupole and higher-order multipoles

Time-dependent terms in the metric tensor (5.2.5) – (5.2.8) result from the multipole moments which can
be either periodic (a binary system) or aperiodic (a supernova explosion) functions of time. The most
straightforward way to calculate the impact of the gravitational field of such a source on the propagation
of light would be to decompose its multipole moments F (s) := {IL (s),SL (s)} in the Fourier series
[91] Z +∞
1
F (t − r) = √ F̃ (ω̃)eiω̃(t−r) dω̃ , (5.4.5)
2π −∞
where ω̃ is a Fourier frequency, then to substitute this decomposition to the light-ray propagation equa-
tions and to integrate them term by term. This method makes an impression that in order to obtain the
final result of the integration the (complex-valued) Fourier image F̃ (ω̃) of the multipole moments of the
isolated system must be specified explicitly, otherwise neither explicit integration nor the convolution of
the integrated Fourier series will be possible. However, this impression is misleading, at least in general
relativity. We shall show below that the explicit structure of F̃ (ω̃) is irrelevant for general-relativistic
calculation of the light-ray integrals. This may be understood better, if one recollects that in general
relativity gravitational field propagates with the same speed as light in vacuum. In alternative theories
of gravity the speed of gravity and light may be different [117], so it can lead to the appearance of terms
being proportional to the difference between the speed of gravity and the speed of light that will dras-
tically complicates calculations which will do require to know the explicit form of the Fourier images
of gravitational-wave sources as functions of ω̃. All such terms, which might depend on the difference
between the speed of gravity and the speed of light are cancelled out in general relativity, thus, making
integration of the light ray propagation equations manageable and applicable for any source of gravita-
tional waves without particular specification of its temporal behaviour. It may be worth mentioning that
the experimental limit on the difference between the speed of gravity and the speed of light in general
relativity can be measured in the solar system experiments with major planets as predicted in [118].
It was experimentally tested with the precision 20% in the jovian light-ray deflection experiment with
VLBI network [119]. No deviation from general relativity was detected.
We notice that the integration of the light-ray equations is effectively reduced to the calculation of
only two types of integrals along the light ray: [F (s)/r][−1] and [F (s)/r][−2] , where F (s) = F (t − r)
denotes any type of the time-dependent multipole moments of the gravitational field of the localized
astronomical system. These integrals can be performed after introducing a new variable [91]
√
y ≡ s − t∗ = τ − r(τ ) = τ − d2 + τ 2 , (5.4.6)
which is the interval of time between two spacetime events: position of photon xα = (τ,x) on the light
ray and that of the center of mass of the isolated system z α = (y,0) taken at the retarded time y 1 .
Equation (5.4.6) is a retarded solution of the null cone equation in flat spacetime
ηαβ (xα − z α )(xβ − z β ) = 0 , (5.4.7)
giving the time of propagation of gravity from the isolated system to the photon along the null cone
characteristic of Einstein’s equations. This retardation of gravity effect presents in the time argument of
the solution (5.2.5)–(5.2.8) of the linearized Einstein equations. The replacement of the time argument τ
with the retarded time y = τ − r(τ ) allows us to perform integration of equations of light-ray geodesics
completely without making specific assumptions about the time dependence of the multipole moments.
It is worth noting that while the parameter τ runs from −∞ to +∞, the retarded time y runs from −∞
to 0, that is, y is always negative, y ≤ 0.
Equation (5.4.6) leads to other useful transformations
y 2 − d2 √ 1 d2 + y 2
τ = , d2 + τ 2 = − , (5.4.8)
2y 2 y
and
1 d2 + y 2
dτ = dy . (5.4.9)
2 y2
Making use of the new variable y and relationships (5.4.8), (5.4.9) the integrals under discussion can be
explicitly displayed as follows:
[−1] Zy
F (t∗ + ζ)

F (t − r)
= − dζ , (5.4.10)
r ζ
−∞
[−2] Zy Zη Zy Zη
F (t∗ + ζ) d2 F (t∗ + ζ)

F (t − r) 1 1
= − dζdη − dζdη , (5.4.11)
r 2 ζ 2 η2 ζ
−∞ −∞ −∞ −∞
where ζ, η are the dummy variables of the integration replacing the integration along the light-ray
trajectory by that along a null characteristic of the gravitational field, and t∗ is the time of the closest
approach of photon to the origin of the coordinate chart coinciding with the center of mass of the isolated
gravitational system. The time t∗ has no physical meaning in the most general case and appears in
calculations as an auxiliary (constant) parameter which vanishes in the final result. The reason is simple,
by choosing the origin of the coordinate chart at a different point we change the numerical value of t∗ but
it cannot change physical observables – the angle of light-ray deflection, frequency shift, etc. The time
of the closest approach t∗ can be used as an approximation of the retarded time s = t − r in the case of a
small impact parameter d of the light ray with respect to the isolated astronomical system, that is s ' t∗ +
O(d2 /cr) (see equation (5.7.6)). This approximation makes an erroneous impression that the physical
effect of propagation of gravity is unimportant for calculation of observable effects caused by variable
gravitational field of the isolated system. This misinterpretation of the role of gravity’s propagation in
the interpretation of the observable astronomical effects did happen in some papers [140, 141] which
used the simplifying approximation s = t∗ from the very beginning of calculations but it precludes to
spatially disentangle the null cone characteristics of gravity and light. The only case when the time t∗
makes physical sense is when observer is located in infinity. Indeed, in this case the radial distance
r = ∞, and the retarded time s = t∗ exactly, because the residual terms O(d2 /r) = 0 irrespectively of
1 The retarded time y should not be confused with the Cartesian coordinate y.
the choice of the coordinate origin [90].

A remarkable property of the integrals in the right side of equations (5.4.10), (5.4.11) is that they
depend on the parameters
√ ξ i and τ only through either the upper limit of the integration, which is the
variable y = τ − d + τ 2 , or the square of the impact parameter, d2 = (ξ · ξ) standing in front of
2
the second integral in the right side of equation (5.4.11). For this reason, differentiation of the integrals
in the left part of equations (5.4.10), (5.4.11) with respect to either ξ i or τ will effectively eliminate the
integration along the light ray trajectory. For example,
( [−1] )
ˆ F (t − r) F (t∗ + y) ˆ ξi
∂i =− ∂i y = F (t − r) . (5.4.12)
r y yr
Similar calculation can be easily performed in case of differentiation of integral [F (t − r)/r][−1] with
respect to τ . It results in
( [−1] )
F (t − r) F (t∗ + y) ˆ F (t∗ + y) τ F (t − r)
∂ˆτ =− ∂τ y = − 1− = , (5.4.13)
r y y r r
as it could be expected because the partial differentiation with respect to τ keeps ξ i and t∗ fixed and,
hence, is equivalent to taking a total time derivative with respect to t along the light ray. Since all terms
in the solution of the light geodesic equation are represented as partial derivatives from the integrals
[F (t−r)/r][−1] and [F (t−r)/r][−2] with respect to the parameters ξ i and/or τ , it is clear from equations
(5.4.12), (5.4.13) that the solution will not contain any single integral like [F (t − r)/r][−1] at all – only
the derivatives of this integral will appear which are ordinary functions.
Dealing with the double integrals of type [F (t−r)/r][−2] is more sophisticated. We notice that taking
the first and second derivatives from the double integral [F (t − r)/r][−2] do not eliminate the integration
along the light ray trajectory
 
( [−2] ) [−1] Zy Zη
F (t∗ + ζ)

F (t − r) 1 F (t − r) 1

ˆ k
∂k = ξ − dζdη (5.4.14)
,
r y r η2 ζ 
−∞ −∞
( [−2] ) k j
F (t − r) ξ ξ F (t − r)
∂ˆjk = (5.4.15)
r y2 r
 
[−1] Zy Zη
F (t∗ + ζ)

1 F (t − r) 1

jk
+ P − 2
dζdη ,
y r η ζ 
−∞ −∞
However, taking one more (a third) derivative eliminates all integrals from equation (5.4.15) completely.
More specifically, we have
( [−2] ) [−1]
ξ ij ˆ

F (t − r) 1 F (t − r)
∂îjk = P ij + ∂k + P jk ∂î + ξ j ∂îk , (5.4.16)
r y yr r
and making use of equations (5.4.12), (5.4.15) for explicit calculation of partial derivatives in the right
part of equation (5.4.16) proves that all integrations disappear, that is
( [−2] )
F (t − r) P ij ξ k + P jk ξ i + P ik ξ j F (t − r)
∂îjk = (5.4.17)
r y2 r
ξiξj ξk

2 1
+ − F (t − r) − Ḟ (t − r) ,
y2 r2 y r
where Ḟ (t − r) ≡ ∂t F (t − r). The same kind of reasoning works for the third derivatives with respect
to the parameter τ and to the mixed derivatives taken with respect to both ξ i and τ .
We shall obtain solution of the light geodesic in terms of (STF) partial derivatives taken with respect
to the impact parameter vector, ξ i , of the light ray acting on the single and double integrals having a
symbolic form ∂ˆ<a1 ...ak > [F (t − r)/r][−1] and ∂ˆ<a1 ...ak > [F (t − r)/r][−2] where the angular brackets
around the spatial indices indicate the STF symmetry. Explicit expression for this STF derivative can
be obtained directly by applying the differentiation rules shown in equations (5.4.14) – (5.4.16). In
what follows, we shall see that the solution of the equations of propagation of light rays will always
contain three or more derivatives acting on the integrals from the higher-order multipole moments (with
the multipole index l ≥ 2) which have the same structure as that shown in equations (5.4.10), (5.4.11).
It means that the final result of the integration of the light-ray propagation equations depending on the
higher-order multipoles of the localized astronomical system can be expressed completely solely in terms
of these multipoles taken at the retarded instant of time s = t − r. In other words, the observed effects
of light deflection, etc. does not depend in general relativity directly on the past history of photon’s
propagation that is the effects of temporal variations of the gravitational field (gravitational waves) do
not accumulate. On the other hand, the past history of the isolated system can affect the propagation of
light ray, at least in principle, through the tails of the gravitational waves which contribute to the present
value of multipole moments of the system as shown in equations (5.2.12), (5.2.13).
We would like to point out that if an electromagentic wave propagated through a dispersive medium,
the physical speed of light would be different from the speed of gravity. In such case the effects of the
temporal variations of the gravitational field can do accumulate and the motion of the electromagnetic
signal depends on its past history. This effect was studied by Bertotti and Catenacci [146] in order to set
an upper limit on the light scintillations caused by the stochastic gravitational wave background. They
concluded that such a limit is not very interesting but it might be a good time now to reconsider this
conclusion in application to the more advanced level of observational technologies..
5.5 Gravitational Perturbations of the Light Ray
5.5.1 Relativistic perturbation of the electromagnetic eikonal
Perturbation of eikonal (phase) of electromagnetic signal propagating from the point x0 to the point
x are obtained by solving equation (5.3.38). This solution is found by integrating the metric tensor
perturbation along the light-ray trajectory and can be written down in the linearized approximation as an
algebraic sum of three separate terms
ψ = ψ(G) + ψ(M ) + ψ(S) , (5.5.1)
where ψ(G) represents the gauge-dependent part of the eikonal, and ψ(M ) , ψ(S) are eikonal’s perturbations
caused by the mass and spin multipole moments correspondingly. Their explicit expressions are as
follows
ψ(G) = (ki φi − φ0 ) + (ki wi − w0 ) , (5.5.2)

Section 5.5 Gravitational Perturbations of the Light Ray 213
[−1] ∞ X l−1
M(t − r) (−1)l+p
X p
ψ(M ) = 2 +2 Cl (l − p,p) 1 − (5.5.3)
r l! l
l=2 p=0
 [−1]
(p)
IAl (t − r)
" #

p
× 1+ k<a1 ...ap ∂âp+1 ...al >
 l−1 r
 [−1] 
i (p)
k I (t − r)

2p

iAl−1
− k<a1 ...ap−1 ∂âp ...al−1 >  
l−1 r 

∞ X l−1
X (−1)l+p l
ψ(S) = 2abi ka Sb ∂î ln (r − τ ) + 4∂â Cl−1 (l − p − 1,p) (5.5.4)
(l + 1)!
l=2 p=0
 [−1]
(p)

p
ki iba SbAl−1 (t − r)
× 1− ˆ
k<a1 ...ap ∂ap+1 ...al−1 >   ,
l−1 r
where H(p − q) is the Heaviside function defined by the expression (5.1.21), Cl (p,q) are the polynomial
coefficient given by equation (5.1.20) and F (p) (t − r) denotes ∂ p F (t − r)/∂tp where r is considered
as constant.
We note that the gauge-dependent part of the eikonal, ψ(G) , contains combination of terms ki φi −
0
φ defined by equations (5.3.71)–(5.3.73) which can be, in principle, eliminated with an appropriate
choice of the gravitational field gauge functions w0 and wi . However, such a procedure will introduce a
reference frame in the sky with a coordinate grid being dependent on the direction to the source of light
rays that is, to the direcion of the unit vector ki . The coordinate frame obtained in this way will have
the direction-dependent distortions which can change as time goes on because of the proper motion of
stars. For this reason the elimination of functions φ0 and φi from equation (5.5.2) may be not practically
justified. The ADM-harmonic coordinate chart admits a more straightforward treatment of observable
relativistic effects. Thus, we leave the gauge functions φ0 and φi in equation (5.5.2) along with the
gauge functions w0 and wi that are defined by formulas (5.3.66), (5.3.67).
Expressions (5.5.3), (5.5.4) for the eikonal contain derivatives from the retarded integrals of the mass-
and spin-multipoles. After taking the derivatives one can prove that the integrals from all high-order
multipoles (l ≥ 2) are eliminated. Indeed, scrutiny inspection of equations (5.5.3), (5.5.4) elucidates
that all the integrals from the multipole moments enter the equation in combination with at least one
derivative with respect to the impact parameter vector ξ i . Hence, the differentiation rule (5.4.12) is
applied which eliminates the integration.
The only integral which must be performed, is that from the mass monopole in equation (5.5.3). It
can be taken by parts as follows
[−1]
M(t − r) y
Z
= −M(t − r) ln(r − τ ) + Ṁ(t∗ + ζ) ln ζdζ , (5.5.5)
r −∞
where Ṁ ≡ ∂M/∂t∗ . If one assumes that the mass M is constant, then the second term in the right side
of equation (5.5.5) vanishes and the eikonal does not contain any integral dependence on the past history
of the light propagation. This assumption is usually implied, for example, in the theory of gravitational
lensing [62, 63, 65] and other applications of the relativistic theory of light propagation. Here, we extend
our approach to take into account the case of time-dependent mass of the isolated system, Ṁ 6= 0. The
mass may change because stars are losing mass in the form of the stellar wind. We shall not consider
this case but focus on another process of the lost of mass by the isolated system due to the emission of
gravitational radiation. The rate of the mass loss is, then, given by [1, 2]
1 (3) (3)
Ṁ(t) = − I (t)Iij (t) + O c−9 ,

(5.5.6)
c7 ij
(3)
where Iij represents a third time derivative from the mass quadrupole moment and terms of order
O c−9 describe contribution of higher-order mass and spin multipoles which we neglect here. Making

use of equation (5.5.6) in (5.5.5) yields

[−1]
M(t − r)

= −M(t − r) ln(r − τ ) (5.5.7)
r
Z y
1 (3) (3)
I (t∗ + ζ)Iij (t∗ + ζ) ln ζdζ + O c−9 .

−
c7 −∞ ij
It is instructive to evaluate contribution of the second term in the right side of equation (5.5.5) in the
case of light propagating in gravitational field of a binary system consisting of two stars with masses m1
and m2 orbiting each other on a circular orbit. The total mass M of the system is defined in accordance
with equation (5.2.10) (for l = 0) which takes into account the first post-Newtonian correction
1 µM
M(t) = m1 + m2 − + O c−4 ,

2
(5.5.8)
2c a(t)
where µ = m1 m2 /M is the reduced mass of the system and a = a(t) is the orbital radius of the
system. Assuming that the lost of the orbital energy is due to the emission of gravitational waves from
the system in accordance with the quadrupole formula approximation (5.5.6), the time evolution of the
orbital radius is defined by equation [1, 2]
64 µM2
ȧ = − . (5.5.9)
5c5 a3
It has a simple solution [2]
τ 1/4
a(t) = a0 1 − , (5.5.10)
T
5c5 a40
T = , (5.5.11)
256 µM2
where τ = t − t∗ , t∗ is the time of the closest approach of light ray to the binary system, a0 = a(t∗ ) is
the orbital radius of the binary system at the time of the closest approach, and T is the spiral time of the
binary. The lost of the total mass due to the emission of the gravitational waves is
1 µM τ −5/4
Ṁ = − 1 − . (5.5.12)
8c2 a0 T T
Substituting equation (5.5.12) to the right side of equation (5.5.5) and performing integration yields
Z y
1 µM
Ṁ(t∗ + ζ) ln ζdζ = − 2 ln(r − τ ) (5.5.13)
−∞ 2 c a(s)
( )
1 m1 m2 a(s) − a0 a(s)
+ ln + 2 arctan ,
2 c2 a0 a(s) + a0 a0
where a(s) ≡ a(t−r) denotes the orbital radius of the binary system taken at the retarded time s = t−r.
Putting all terms in equations (5.5.5), (5.5.8), (5.5.13) together yields

[−1] ( )
M(t − r)

1 m1 m2 a(s) − a0 a(s)
= −(m1 + m2 ) ln(r − τ ) + ln + 2 arctan ,
r 2 c2 a0 a(s) + a0 a0
(5.5.14)
where the first term in the right side of this equation is the standard Shapiro time delay in the gravitational
field of the binary system with constant total mass m1 + m2 , and the second term represents relativistic
correction due to the emission of gravitational waves by the system causing the overall loss of its orbital
energy.
Eikonal describes propagation of a wave front of the electromagnetic wave. Light rays are orthogonal
to the wave front and their trajectories can be easily calculated as soon as the eikonal is known. In
the present chapter we do not use this technique and obtain solution for the light rays directly from the
light-ray geodesic equations that gives identical results.
5.5.2 Relativistic perturbation of the coordinate velocity of light
Integration of the light-ray propagation equations (5.3.60) – (5.3.63) is fairly straightforward. Perform-
ing one integration of these equations with respect to time yields
ẋi (τ ) = ki + Ξ̇i (τ,ξ) , (5.5.15)

Ξ̇i (τ,ξ) = Ξ̇i (τ,ξ) + Ξ̇i (τ,ξ) + Ξ̇i (τ,ξ) , (5.5.16)
(G) (M ) (S)
where the relativistic perturbations of photon’s trajectory are given by

h i
Ξ̇i (τ,ξ) = ∂ˆτ (φi − ki φ0 ) + (wi − ki w0 ) , (5.5.17)
(G)
[−1]
M

Ξ̇i (τ,ξ) = 2(∂î − ki ∂ˆτ ) (5.5.18)
(M ) r
∞ Xl X p
X (−1)l+p−q
+ 2∂î Cl (l − p,p − q,q)H(2 − q)×
p=0 q=0
l!
l=2
" (p−q) #[−1]
IAl (t − r)

p − q p−q
1− 1− k<a1 ...ap ∂âp+1 ...al > ∂ˆτq −
l l−1 r
" (p)
∞ X
IAl (t − r)
l−1
( #
(−1)l+p

X p p ˆ
2 Cl (l − p,p) 1 − 1+ ki<a1 ...ap ∂ap+1 ...al > −
l! l l−1 r
l=2 p=0
 )
(p)
2p IiAl−1 (t − r)
k<a1 ...ap−1 ∂âp ...al−1 >   ,
l−1 r
[−1]
jba Sb iba Sb

Ξ̇i (τ,ξ) = 2kj ∂îa − 2∂â − (5.5.19)
(S) r r
∞ X
l−1 Xp
X (−1)l+p−q l
4kj ∂îa Cl−1 (l − p − 1,p − q,q) H(2 − q)×
p=0 q=0
(l + 1)!
l=2
 [−1]
(p−q)

p−q
1− k<a1 ...ap ∂âp+1 ...al−1 > ∂ˆτq   +
l−1 r
∞ X l−1
X (−1)l+p l
4 ∂â − ka ∂ˆt∗ Cl−1 (l − p − 1,p) ×
(l + 1)!
l=2 p=0
 
(p)

p
1− k<a1 ...ap ∂âp+1 ...al−1 >  +
l−1 r
(p+1)
 
∞ X
X l−1
(−1)l+p l jbal−1 SîbA (t − r)
4kj ˆ
Cl−1 (l − p − 1,p) k<a1 ...ap ∂ap+1 ...al−1 >  l−2
 .
p=0
(l + 1)! r
l=2
Here H(p−q) is a Heaviside function defined by the expression (5.1.21) and Cl (p,q) are the polynomial
coefficients (5.1.20). The gauge functions are given in section 5.3.3.
Mass monopole and spin dipole terms are written down in equations (5.5.18), (5.5.19) in symbolic
form and after taking derivatives are simplified
[−1]
M ξi

2M
2(∂î − ki ∂ˆτ ) = − ki , (5.5.20)
r r y
[−1]
jba Sb Sb ξ i

2kj ∂îa = 2kj jba ∂â . (5.5.21)
r yr
Remaining integrals shown in the right side of equations (5.5.18), (5.5.19) are convenient for presenta-
tion of the result of integration in the symbolic form. The integration is actually not required because
the integrals are always appear in combination with, at least, one operator of a partial derivative ∂î
with respect to the impact parameter vector of the light ray. The derivative operator acts on the inte-
grals in accordance with equation (5.4.12) converting the integrals into functions of the retarded time
y = τ − r(τ )
(p−q)
#[−1]
IAl
"
(t − r) (p−q)
∂î = I Al (t∗ + y)∂î ln(−y) , (5.5.22)
r
 [−1]
(p−q)
jba SbAl−1 (t − r) (p−q)
∂î   = jba SbAl−1 (t∗ + y)∂î ln(−y) . (5.5.23)
r
We conclude that at each point of the wave front of the electromagnetic wave the relativistic perturbation
of the direction of propagation of light ray (wave vector) caused by the time-dependent gravitational field
of the isolated system depends only on the value of its multipole moments taken at the retarded instant
of time and it does not depend on the past history of the light propagation.
5.5.3 Perturbation of the light-ray trajectory
Integration of equation (5.5.15) with respect to time yields the relativistic perturbation of the trajectory
of the light ray
xi (τ ) = xiN + ∆ Ξi +∆ Ξi +∆ Ξi , (5.5.24)
(G) (M ) (S)
where
∆ Ξi ≡ Ξi (τ,ξ) − Ξi (τ0 ,ξ) , ∆ Ξi ≡ Ξi (τ,ξ) − Ξi (τ0 ,ξ) , ∆ Ξi ≡ Ξi (τ,ξ) − Ξi (τ0 ,ξ) .

(G) (G) (G) (M ) (M ) (M ) (S) (S) (S)
(5.5.25)
Here, the term
Ξi (τ,ξ) = (φi − ki φ0 ) + (wi − ki w0 ) , (5.5.26)
(G)
is the gauge-dependent part of the trajectory’s perturbation, and the physically meaningful perturbations
due to the mass and spin multipoles
[−2]
M

Ξi (τ,ξ) = 2(∂î − ki ∂ˆτ ) + (5.5.27)
(M ) r
∞ Xl X p
X (−1)l+p−q
2∂î Cl (l − p,p − q,q)H(2 − q)×
p=0 q=0
l!
l=2
" (p−q) #[−2]
IAl (t − r)

p − q p−q
1− 1− k<a1 ...ap ∂âp+1 ...al > ∂ˆτq −
l l−1 r
" (p) #[−1]
∞ X
IAl (t − r)
l−1
(
(−1)l+p

X p p
2 Cl (l − p,p) 1 − 1+ ki<a1 ...ap ∂âp+1 ...al > −
l! l l−1 r
l=2 p=0
 [−1] )
(p)
2p IiAl−1 (t − r)
k<a1 ...ap−1 ∂âp ...al−1 >   ,
l−1 r
[−2] [−1]
jba Sb iba Sb

Ξi (τ,ξ) = 2kj ∂îa − 2∂â − (5.5.28)
(S) r r
∞ Xl−1 Xp
X (−1)l+p−q l
4kj ∂îa Cl−1 (l − p − 1,p − q,q) H(2 − q)×
(l + 1)!
l=2 p=0 q=0
 [−2]
(p−q)

p−q

q
1− k<a1 ...ap ∂âp+1 ...al−1 > ∂ˆτ   +
l−1 r
∞ Xl−1
X (−1)l+p l
4 ∂â − ka ∂ˆt∗ Cl−1 (l − p − 1,p) ×
(l + 1)!
l=2 p=0
 [−1]
(p)

p
1− k<a1 ...ap ∂âp+1 ...al−1 >   +
l−1 r
 (p+1)
[−1]
∞ X
X l−1
(−1)l+p l jbal−1 SîbA (t − r)
4kj Cl−1 (l − p − 1,p) k<a1 ...ap ∂âp+1 ...al−1 >  l−2
 .
p=0
(l + 1)! r
l=2
Here, again H(p−q) is a Heaviside function defined by equation (5.1.21) and Cl (p,q) are the polynomial
coefficients (5.1.20).
Relativistic perturbations (5.5.27), (5.5.28) contain two types of integrals along the light ray which
symbolic form is [F (s)/r][−1] and [F (s)/r][−2] where s = t − r is the retarded time. In section 5.4
we have shown that taking a derivative from an integral [F (s)/r][−1] reduces the integral to an ordinary
function of the retarded time s with no integral dependence on the past history of the propagation of
light ray. In order to eliminate the integration in integrals [F (s)/r][−2] one needs to take three or more
derivatives. One can notice that in (5.5.27), (5.5.28) there are terms with the summation index p = l − 1
or p = l in which the number of the derivatives is less than three that is not sufficient to eliminate the
integrals. However, all such integrals are multiplied with a numerical coefficients like 1 − p/l, etc.
which gets nil for those values of the summation index p. It effectively eliminates all the integrals in
(5.5.27), (5.5.28) in which the integration cannot be eliminated by the partial derivatives. As an example
let us examine equation (5.5.27) for Ξi (τ,ξ). In this expression the term with the double integral
(M )
h i[−2]
(p−q)
IAl (s)/r has l − p + 1 derivatives ∂â and q derivatives ∂ˆτ . After performing the differentiation,
h i[−2]
(p−q)
the expression ∂ˆ<a1 ...al−p+1 > ∂ˆτq IAl (s)/r can contain the explicit integrals in the following
cases: q = 0 and p = l − 1; q = 0 and p = l; q = 1 and p = l. However, these integrals are
multiplied with [1 − (p − q)/l][1 − (p − q)/(l − 1)] and in any case of p and q values mentioned above
either the factor 1 − (p − q)/l or 1 − (p − q)/(l − 1) will vanish. Similar reasoning is applied to the
spin-dependent perturbation Ξi (τ,ξ) which shows that there is no need to explicitly integrate the spin
(S)
multipole moments in order to calculate the perturbation of light ray trajectory.
The only integral dependence of the light-ray trajectory perturbation depending on the past history of
the propagating photon remains in equation (5.5.26) which contains the gauge functions φα and wα used
later in this chapter for interpretation of observable effects caused by the gravitational waves from the
isolated system. The past-history dependence of the light-ray trajectory may come into play also through
the integrals from mass monopole and spin-dipole terms: [M/r][−2] and [S/r][−2] , if mass and/or spin
of the isolated system are not conserved and change as time passes on. The non-conservation can be
caused, for example, by emission of gravitational waves carrying away the orbital energy and angular
momentum of the system. The past-history contribution of these integrals can be calculated similarly to
the eikonal perturbation considered in equations (5.5.6)–(5.5.14).
Solution of the light-ray equation with the initial-boundary conditions (5.3.18) depends on the unit
vector k defining direction of the light-ray propagation extrapolated backward in time to the past null
infinity. In real practice the light ray is emitted at the point x0 where the source of light is located at
time t0 , and it arrives to observer at time t to the point x separated from the source of light by a finite
coordinate distance R = |x − x0 |. Therefore, solution of the light-ray equations must be expressed
in this case in terms of the integrals of the boundary-value problem. It is formulated in terms of the
coordinates of the initial, x0 , and the final, x, positions of the photon
x(t) = x , x(t0 ) = x0 , (5.5.29)
and a unit vector
xi − xi0
Ki = − , (5.5.30)
|x − x0 |
that defines a coordinate direction from the observer towards the source of light and can be interpreted
as a unit vector in the Euclidean space in the sense that δij K i K j = 1.
In what follows it is convenient to make use of the astronomical coordinates x ≡ xi = (x1 ,x2 ,x3 )
based on a triad of the Euclidean unit vectors (I0 ,J0 ,K0 ) as shown in Fig. 5.2. Vector K0 points from
observer towards the isolated system emitting gravitational waves and deflecting light rays. Vectors I0
and J0 lie in the plane being orthogonal to vector K0 . The unit vector I0 is directed to the east, and
that J0 points towards the north celestial pole . The origin of the coordinate system is chosen to lie at
the center of mass of the isolated system.
North
τ To observer
ξi I0
K Vector field of light rays
θ Ω x i = k iτ + ξi
ξi
K0 I East
τ To observer
Plane of the sky
Figure 5.2. Astronomical coordinates used in calculation of light propagation. The origin of the coor-
dinates is at the center-of-mass of the source of gravitational waves. The bundle of light rays is defined
by a vector field ki . Vector K i = −ki + O(c−2 ) is directed from observer towards the source of
light. Vector K0i is directed from the observer towards the source of gravitational waves. We define
K0i = −N i = −xi /r, where xi are the coordinates of observer with respect to the source of gravita-
tional waves, and r = |x|. The picture shows the plane of the sky being orthogonal to vector K i .
We will need in our discussion an another reference frame based on a triad of the Euclidean unit
vectors (I,J,K) that are turned at some angle with respect to vectors (I0 ,J0 ,K0 ). Vector K points
from the observer towards the source of light, and vectors I and J lie in the plane of the sky defined to
be orthogonal to vector K. Mutual orientation of the triads is defined by the orthogonal transformation
(rigid rotation)
I0 = I cos Ω + J sin Ω , (5.5.31)

J0 = −I cos θ sin Ω + J cos θ cos Ω + K sin θ , (5.5.32)
K0 = I sin θ sin Ω − J sin θ cos Ω + K cos θ , (5.5.33)
where the rotational angles, Ω and θ, are constant.

It is rather straightforward to obtain solution of the boundary value problem (5.5.29) for light propa-
gation in terms of the unit vector K instead of the unit vector k of the initial-boundary value problem.
All what we need, is to convert the unit vector k to K written in terms of spatial coordinates of the
points of emission, x0 , and observation, x, of the light ray. From formula (5.5.24) one has
ki = −K i − β i (τ,ξ) , (5.5.34)
where the relativistic correction β i (τ,ξ) to the vector K i is derived from the solution of the initial-
boundary value problem, It is explicitly defined as follows

P ij ∆ Ξj +∆ Ξj +∆ Ξj
(G) (M ) (S)
β i (τ,ξ) = , (5.5.35)
|x − x0 |
with P ij = δ ij − ki kj being the operator of projection on the plane being orthogonal to vector ki .
Denominator of equation (5.5.35) contains the distance R = |x − x0 | between observer and source
of light which can be very large, thus, making an impression that the relativistic correction β i (τ,ξ) is
probably negligibly small. However, the difference ∆Ξj (τ,ξ) in the numerator of (5.5.35) is proportional
either to the distance r = |x| between the observer and the isolated system or to the distance r0 =
|x0 | between the source of light and the isolated system. Either one of them or both distances can be
comparable with R so that the relativistic correction β i (τ,ξ) cannot be neglected in general case, and
must be taken into account for calculation of relativistic perturbations of light-ray trajectory. Only in
a case where observer and source of light reside at extremely large distances on opposite sides of the
source of gravitational waves can the relativistic correction β i be neglected.
5.6 Observable Relativistic Effects

In this section we derive general expressions for four observable relativistic effects – the time delay, the
light bending, the frequency shift, and the rotation of the polarization plane.
5.6.1 Gravitational time delay of light
Relativistic time delay for light propagating through time-dependent gravitational field can be obtained
either directly from expression (5.5.24) or from the electromagnetic eikonal (5.3.35), (5.5.1) by observ-
ing that the eikonal is constant not only on the null hypersurface of the wave front of electromagnetic
Section 5.6 Observable Relativistic Effects 221
wave but also along the light rays citepfrolov. Both derivations lead, of course, to the same result
t − t0 = |x − x0 | + ∆(τ,τ0 ) , (5.6.1)
∆(τ,τ0 ) = ∆ (τ,τ0 ) + ∆ (τ,τ0 ) + ∆ (τ,τ0 ) , (5.6.2)
(G) (M ) (S)
where xα0 = (t0 ,x0 ) are four-coordinates of the point of emission of light, x
α
= (t,x) are four-
coordinates of the point of observation, ∆ (τ,τ0 ), ∆ (τ,τ0 ) and ∆ (τ,τ0 ) are functions describing
(G) (M ) (S)
correspondingly the delay of the electromagnetic signal due to the gauge-freedom, mass and spin multi-
poles of the gravitational field of the isolated system. These functions are expressed as follows

∆ (τ,τ0 ) = −ki Ξi (τ,ξ) − Ξi (τ0 ,ξ) , (5.6.3)
(G) (G) (G)

∆ (τ,τ0 ) = −ki Ξi (τ,ξ) − Ξi (τ0 ,ξ) , (5.6.4)
(M ) (M ) (M )

∆ (τ,τ0 ) = −ki Ξi (τ,ξ) − Ξi (τ0 ,ξ) . (5.6.5)
(S) (S) (S)
There exists a relation between the relativistic perturbations of the eikonal and the light-ray trajectory
ψ(τ,ξ) = −ki Ξi (τ,ξ) , (5.6.6)
where the eikonal perturbation, ψ, reads off the equations (5.5.1)–(5.5.4). From equations (5.6.3)–
(5.6.5) one can infer that in the linear approximation the functions describing the time delay are just the
projections of vector functions describing the coordinate perturbation of the light-ray trajectory onto the
unperturbed direction ki from the source of light to the observer.
Equation (5.6.1) defines the light time delay effect in the global time t of the ADM-harmonic reference
frame. In order to convert it to observable proper time T of the observer, we assume for simplicity that
the observer is in a state of free fall and moves with velocity V i with respect to the reference frame of
the isolated system. Transformation from the ADM-harmonic coordinate time t to the proper time T is
made with the help of the standard formula [1, 2]
dT 2 = −gαβ dxα dxβ . (5.6.7)
Substituting the metric tensor expansion gαβ = ηαβ + hαβ , where hαβ is given by equations (5.2.21)–
(5.2.24), to equation (5.6.7) yields
dT
q
= 1 − V 2 − h00 (1 + V 2 ) − 2h0i V i − hTijT V i V j . (5.6.8)
dt
In the most simple case, when observer is at rest (V = 0) with respect to the ADM-harmonic reference
frame equation (5.6.8) is drastically simplified and depends only on h00 component of the metric tensor.
Implementation of formula (5.2.21) for h00 and subsequent integration of (5.6.8) with respect to time
yields then
M

T = 1− (t − ti ) , (5.6.9)
r
where ti is the initial epoch of observation. Another simple case of equation (5.6.8) is obtained for
observer located at the distance r so large that one can neglect h00 and h0i quasi-static perturbations of
the metric tensor. Then, the difference between the observer’s proper time T and coordinate time t is
dT
q
= 1 − V 2 − hTijT V i V j , (5.6.10)
dt
leading in the case of a small velocity to
t
V iV j
p Z
T = 1 − V 2 (t − ti ) − √ hTijT dt . (5.6.11)
1−V2 ti
The last term in the right side of this equation describes the effect of the gravitational waves on the
rate of the proper time of observer which may be important under some specific circumstances but is
enormously small in real practice and, as a rule, can be neglected. Nonetheless, it would be interesting
to study this effect in more detail as a possible tool to detect gravitational waves emitted, for example,
in galactic supernova explosions or by powerful gamma-ray bursts (GRB) in distant galaxies.
5.6.2 Gravitational deflection of light
The coordinate direction to a source of light measured at the point of observation x is defined by the
spatial components of four-vector of photon lα incoming to an observer from the source of light. This
vector depends on the energy (frequency) of photon which is irrelevant for the present section. Nor-
malization of lα to its frequency eliminates the frequency-dependence and brings about a four-vector
pα = (1,pi ) where pi = −ẋi and the dot denotes derivative with respect to coordinate time t. Vector
Φa is null, pα pα = 0, and is only direction-dependent. The spatial part of this vector can be presented
as a small deviation from the direction ki of the unperturbed photon’s trajectory
pi = −ki − Ξ̇i , (5.6.12)
where the minus sign indicates that the tangent vector pi is directed from the observer to the source of
light.
The coordinate direction pi is not a directly observable quantity as it is defined with respect to the
chosen coordinate grid on the curved spacetime manifold. A real observable vector towards the source
of light, sα = (1,si ), is defined with respect to the local inertial frame co-moving with the observer
[147]. In this frame si = −dX i /dT , where T is the observer’s proper time, and X i are the spatial
coordinates of the local inertial frame with the observer at its origin. We shall assume for simplicity
that the observer is at rest with respect to the global ADM-harmonic coordinates (t,xi ). The case of an
observer moving with respect to the ADM-harmonic system with velocity V i can be treated with the
help of the Lorentz transformation which is a straightforward procedure so that we do not discuss it. In
case of a static observer, transformation from the global ADM-harmonic coordinates (t,xi ) to the local
coordinates (T,X i ) is given by the formulas
dT = Λ00 dt + Λ0j dxj , dX i = Λi0 dt + Λij dxj , (5.6.13)
where the matrix of transformation Λα

β is defined by the requirements of orthonormality
gαβ = ηµν Λµα Λνβ . (5.6.14)
In particular, the orthonormality condition (5.6.14) assumes that the spatial angles and lengths at the
point of observations are measured with the Euclidean metric δij . Because the vector sα is null (sα sα =
0) with respect to the Minkowski metric ηαβ , we conclude that the Euclidean length, |s|, of vector si is
equal to 1. Indeed, one has
ηαβ sα sβ = −1 + s2 = 0 . (5.6.15)
Hence, |s| = 1.
In the linear approximation with respect to the universal gravitational constant G, the matrix of the
transformation is as follows [120, 121]
1
Λ00 = 1− h00 (t,x) ,
2
Λ0i = −h0i (t,x) ,
Λi0 = 0,

1 1
Λij = 1 + h00 (t,x) δij + hTijT (t,x) . (5.6.16)
2 2
Using transformation (5.6.13) we obtain relationship between the observable vector si and the coordinate
direction pi
Λi0 − Λij pj
si = − . (5.6.17)
Λ00 − Λ0j pj
In the linear approximation this takes the form

1
si = 1 + h00 − h0j pj pi + hTijT pj . (5.6.18)
2
Remembering that vector |s| = 1, we find the Euclidean norm of the vector pi from the relationship
1 TT i j
|p| = 1 − h00 + h0j pj − hij p p , (5.6.19)
2
which brings equation (5.6.18) to the form
1 ij q T T
si = mi + P m hjq (t,x) , (5.6.20)
2
where P ij = δ ij − ki kj is the operator of projection onto the plane being orthogonal to ki , and the
Euclidean unit vector mi = pi /|p|.
Let us now denote by αi the dimensionless vector describing the total angle of deflection of the light
ray measured at the point of observation, and calculated with respect to vector ki given at past null
infinity. It is defined according to [122]
αi (τ,ξ) = ki k · Ξ̇(τ,ξ) − Ξ̇i (τ,ξ) , (5.6.21)
or
αi (τ,ξ) = − P ij Ξ̇j (τ,ξ) . (5.6.22)
As a consequence of definitions (5.6.12) and (5.6.22), we conclude that
mi = −ki + αi (τ,ξ) . (5.6.23)
Accounting for expressions (5.6.20), (5.6.23), and (5.5.34) we obtain the observed direction to the source
of light
si (τ,ξ) = K i + αi (τ,ξ) + β i (τ,ξ) + γ i (τ,ξ) , (5.6.24)
where the unit vector K i is given by equation (5.5.30), relativistic correction β i is defined by equation
(5.5.35), and the perturbation
1
γ i (τ,ξ) = − P ij kq hTjqT (t,x) (5.6.25)
2
describes a deformation of the local coordinates of observer with respect to the global ADM-harmonic
frame caused by the transverse-traceless part of the gravitational field of the isolated system at the point
of observation.
Let two sources of light (quasars, stars, etc.) be observed along the directions s1 and s2 corre-
sponding to two different light rays passing near the isolated gravitating system at the minimal distances
corresponding to the (vector) impact parameters, ξ1 and ξ2 . The angle Ψ between them, measured in
the local inertial frame is
cos Ψ = s1 · s2 , (5.6.26)
where the dot between the two vectors denotes the usual Euclidean scalar product. It is worth empha-
sizing that the observed direction si to each source of light includes relativistic deflection of the light
ray in the form of three perturbations. Two of them, αi and γ i , depend only on the quantities taken at
the point of observation, but β i , according to equation (5.5.35), is also sensitive to the strength of the
gravitational field taken at the point of emission of light. This remark reveals that according to relation
(5.6.24) a single gravitational wave signal may cause different angular displacements and/or time delays
for different sources of light located at different distances from the source of gravitational waves even if
the directions to the light sources are the same.
Without going into further details of the observational procedure we give an explicit expression for
the total angle of the light deflection αi . We have
αi (τ,ξ) = αi (τ,ξ) + αi (τ,ξ) + αi (τ,ξ) , (5.6.27)

(G) (M ) (S)
where

αi (τ,ξ) = −P ij ∂ˆτ φj + wj , (5.6.28)
(G)
2M ξ i
αi (τ,ξ) = − (5.6.29)
(M ) r y
∞ Xl X p
X (−1)l+p−q
−2∂î Cl (l − p,p − q,q)H(2 − q) ×
p=0 q=0
l!
l=2
" (p−q) #[−1]
IAl (t − r)

p − q p−q q
1− 1− k<a1 ...ap ∂âp+1 ...al > ∂ˆτ
l l−1 r
∞ Xl−1
X (−1)l+p
−4P ij Cl−2 (l − p − 1,p − 1) ×
l!
l=2 p=0
 
(p)
IjAl−1 (t − r)
ˆ
k<a1 ...ap−1 ∂ap ...al−1 >   ,
r
Sb ξ i jba Sb

αi (τ,ξ) = −2kj jba ∂â + 2P ij ∂â (5.6.30)
(S) yr r
∞ Xl−1 Xp
X (−1)l+p−q l
+4∂îa Cl−1 (l − p − 1,p − q,q) H(2 − q) ×
(l + 1)!
l=2 p=0 q=0
 [−1]
(p−q)
j

p−q
k
1− k<a1 ...ap ∂âp+1 ...al−1 > ∂ˆτq  
l−1 r
∞ Xl−1
X (−1)l+p l
−4 ∂â − ka ∂ˆt∗ Cl−2 (l − p − 2,p) ×
(l + 1)!
l=2 p=0
 
(p)
ij
k<a1 ...ap ∂âp+1 ...al−1 > P 
r
∞ X l−1
X (−1)l+p l
−4 Cl−1 (l − p − 1,p) ×
(l + 1)!
l=2 p=0
 
(p+1)
P iq kj jbal−1 SqbAl−2 (t − r)
k<a1 ...ap ∂âp+1 ...al−1 >   .
r
These expressions do not contain any explicit integration along the light ray trajectory because all ex-
plicit integrals are either eliminated after taking partial derivatives with respect to the upper limit of the
integrals or they are vanish because the numerical coefficient in front of them become nil.
5.6.3 Gravitational shift of frequency
Exact calculation of the gravitational shift of frequency of electromagnetic wave travelling from the
point of emission to observer plays a crucial role for the adequate interpretation of spectral astronomical
investigations of high resolution including the astronomical measurements of radial velocities of stars
[123], anisotropy of cosmic microwave background radiation (CMBR), and others. In the last decade the
technique for measuring the radial velocity of stars had reached an unprecedented precision of 1 m/sec
[124, 125]. At this level the post-Newtonian effects in the orbital motion of spectroscopic binary stars
can be measured [144, 145]. Gravitational shift of frequency of light affect the apparent brightness of
the observed sources according to equation (5.3.125). Therefore, it can be important in highly accurate
photometric measurements of faint radio sources with large radio telescope like SKA [148]
Let a source of light move with respect to the ADM-harmonic coordinate frame (t,xi ) with velocity
V0 = ẋ0 (t0 ) and emit continuous electromagnetic radiation at frequency ν0 = 1/(δT0 ), where t0 and
T0 are the coordinate and proper time of the source of light, respectively. We denote by ν = 1/(δT ) the
observed frequency of the electromagnetic radiation measured at the point of observation by an observer
moving with velocity V = ẋ(t) with respect to the ADM-harmonic coordinate frame. In the geometric
optics approximation we can consider the increments of the proper time, δT0 and δT , as infinitesimally
small which allows us to operate with them as with differentials [120, 121]. Time delay equation (5.6.1)
can be considered as an implicit function of the emission time t0 = t0 (t) having the time of observation
t as its argument. Because the coordinate and proper time at the points of emission of light and its
observation are connected through the metric tensor we conclude that the observed gravitational shift of
frequency 1 + z = ν/ν0 can be defined through the consecutive differentiation of the proper time of the
source of light, T0 , with respect to the proper time of the observer, T , [120, 121]
dT0 dT0 dt0 dt

1+z = = . (5.6.31)
dT dt0 dt dT
Synge calls relation (5.6.31) the Doppler effect in terms of frequency [126, p. 122]. It is fully con-
sistent with the definition of the Doppler shift in terms of energy [126, p. 231] when one compares the
energy of photon at the points of emission and observation of light. The Doppler shift in terms of energy
is given by
ν lα u α
1+z = = , (5.6.32)
ν0 l0α uα0
where uα α
0 , u and l0α , lα are 4-velocities of the source of light and observer and 4-momenta of photon
at the points of emission and observation respectively. It is quite easy to see that both mentioned formu-
lations of the Doppler shift effect are equivalent. Indeed, taking into account that uα = dxα /dT and
lα = ∂ϕ/∂xα , where ϕ is the phase of the electromagnetic wave (eikonal), we obtain lα uα = dϕ/dT ,
and l0α uα0 = dϕ0 /dT0 respectively. Thus, equation (5.6.32) yields
dϕ dT0
1+z = . (5.6.33)
dϕ0 dT
The phase of electromagnetic wave remains constant along the light ray trajectory. For this reason,
dϕ/dϕ0 = 1 and, hence, equation (5.6.33) is reduced to equation (5.6.31) as expected on the ground of
physical intuition. Detailed comparison of the two definitions of the Doppler shift and the proof of their
identity in general theory of relativity is thoroughly discussed in [61, 127].
We emphasize that in equation (5.6.31) the time derivative
dT0 h i1/2
= 1 − V02 − (1 + V02 )h00 (t0 ,x0 ) − 2V0i h0i (t0 ,x0 ) − V0i V0j hTijT (t0 ,x0 ) ,
(5.6.34)
dt0
is taken at the time t0 at the point of emission of light x0 along the world line of the emitter of light
while the time derivative
dt h i−1/2
= 1 − V 2 − (1 + V 2 )h00 (t,x) − 2V i h0i (t,x) − V i V j hTijT (t,x) , (5.6.35)
dT
is calculated at the time of observation t at the position of observer x along the world line of the observer.
The time derivative dt0 /dt is taken along the light-ray trajectory and calculated from the time delay
equation (5.6.1) where we have to take into account that function ∆(τ,τ0 ) depends on times t0 and t
indirectly through the retarded times s0 = t0 − r0 and s = t − r that are arguments of the multipole
moments of the isolated system and wherein r0 ≡ |x0 | = |x(t0 )|, r = |x| = |x(t)| are functions of
time t0 and t respectively. Function ∆(τ,τ0 ) also depends on the time of the closest approach of light
ray, t∗ , through variables τ = t − t∗ , τ0 = t0 − t∗ , and on the unit vector k. Both t∗ and k should
be considered as parameters depending on t0 and t because of the relative motion of the observer with
respect to the source of light which causes variation in the relative position of the source of light and
observer and, consequently, to the corresponding change in the parameters characterizing trajectory of
the light ray, that is in t∗ and k. Therefore, the function ∆(τ,τ0 ) must be viewed as parametrically-
dependent on four variables ∆ = ∆(s,s0 ,t∗ ,k). Accounting for these remarks the derivative along the
light ray reads as follows
∂t∗ ∂ ∂ki ∂

∂s ∂ ∂s0 ∂
1+K ·V − + + + ∆(s,s0 ,t∗ ,k)
dt0 ∂t ∂s ∂t ∂s0 ∂t ∂t∗ ∂t ∂ki
= ,
(5.6.36)
∂t∗ ∂ ∂ki ∂

dt ∂s ∂ ∂s0 ∂ ∗
1 + K · V0 + + + + ∆(s,s0 ,t ,k)
∂t0 ∂s ∂t0 ∂s0 ∂t0 ∂t∗ ∂t0 ∂ki
where the unit vector K is defined in (5.5.30) and where we explicitly show the dependence of function
∆(τ,τ0 ) on all parameters which implicitly depend on time.
The time derivative of vector k is calculated using the approximation k = −K and formula (5.5.30)
where the coordinates of the source of light, x0 (t0 ), and of the observer, x(t), are considered as func-
tions of time. These derivatives are
∂ki (k × (V × k))i ∂ki (k × (V0 × k))i

= , =− , (5.6.37)
∂t R ∂t0 R
where R = |x − x0 | is the coordinate distance between the observer and the source of light.
Time derivatives of the retarded times s and s0 with respect to t and t0 are calculated from their
definitions, s = t − r and s0 = t0 − r0 , where we have to take into account that the spatial position of
the point of observation is connected to the point of emission of light by the unperturbed trajectory of
light, x = x0 + k (t − t0 ). More explicitly, we use for the calculations the following relations
s = t − |x0 (t0 ) + k (t − t0 )| , (5.6.38)

s0 = t0 − |x0 (t0 )| , (5.6.39)
where the unit vector k = k(t,t0 ) must be also considered as a function of two arguments t, t0 with
its derivatives given by equation (5.6.37). It is instructive to notice that relation (5.6.38) combines two
characteristics of the null cone – the first one is related to the propagation of gravitational field from the
isolated system to an observer, and the second one is related to the propagation of light from the source
of light to the same observer. Equation (5.6.39) describes a null cone characteristic corresponding to the
propagation of gravitational field from the isolated system to the point of emission of light. Calculation
of the infinitesimal variations of equations (5.6.38), (5.6.39) immediately gives a set of partial derivatives
∂s
= 1 − k · N − (k × V ) · (k × N ) , (5.6.40)
∂t
∂s
= (1 − k · V0 )(k · N ) , (5.6.41)
∂t0
∂s0
= 1 − V0 · N0 , (5.6.42)
∂t0
∂s0
= 0, (5.6.43)
∂t
where N i = xi /r and N0i = xi0 /r0 are the unit vectors directed from the isolated system to the observer
and to the source of light respectively..
Time derivatives of the parameter t∗ = t0 − k · x0 (t0 ), where again k = k(t,t0 ), read
∂t∗ V0 · ξ ∂t∗ V ·ξ
= 1 − k · V0 + , =− . (5.6.44)
∂t0 R ∂t R
Terms of the order |ξ|/R in both formulas are produced by the time derivatives of vector k.
In what follows we shall restrict ourselves with a static case of an observer and a source of light,
that is we shall assume velocities V = V0 = 0. Taking into account this restriction in equations
(5.6.37)–(5.6.44) we expand denominator of (5.6.36) leaving only the linear with respect to the universal
gravitational constant G terms. Reducing, then, similar terms allows us to simplify (5.6.36) to
dt0 n∂ ∂ ∂ o
= 1− + + ∗ ∆(s,s0 ,t∗ ,k) . (5.6.45)
dt ∂s ∂s0 ∂t
Function ∆ = ∆(s,s0 ,t∗ ,k) is defined by (5.6.2) which (in the case of the static observer and the source
of light) depends on the retarded times s, s0 and the time of the closest approach t∗ through the argu-
ments (see (5.4.6)) y = s − t∗ , y0 = s0 − t∗ , and t∗ , that is
∆(s,s0 ,t∗ ,k) ≡ ∆(y,y0 ,t∗ ) = ψ(y,t∗ ) − ψ(y0 ,t∗ ) , (5.6.46)
where ψ = −ki Ξ is the relativistic perturbation of the eikonal defined in (5.6.6). This particular depen-
dence of ψ on its arguments means that the partial derivative of ψ(y,t∗ ) with respect to y and and that
of ψ(y0 ,t∗ ) with respect to y0 vanish in (5.6.45) which is reduced to a simpler form
dt0 ∂ψ(y,t∗ ) ∂ψ(y0 ,t∗ )

= 1− + . (5.6.47)
dt ∂t∗ ∂t∗
The partial time derivative from ψ(y,t∗ ) is found by differentiating relations (5.5.2)–(5.5.4)
∂ψ(y,t∗ ) ∂ψ(G) (y,t∗ ) ∂ψ(M ) (y,t∗ ) ∂ψ(S) (y,t∗ )

∗
= + + (5.6.48)
∂t ∂t∗ ∂t∗ ∂t∗
where
∂ψ(G) (y,t∗ ) ∂φi ∂φ0 ∂wi ∂w0

= (ki − ∗ ) + (ki ∗ − ), (5.6.49)
∂t∗ ∂t ∗ ∂t ∂t ∂t∗
∞ Xl−1
∂ψ(M ) (y,t∗ ) X (−1)l+p p
= 2 Cl (l − p,p) 1 − (5.6.50)
∂t∗ l! l
l=2 p=0
 " (p+1) #[−1]
 IAl (t − r)

p
× 1+ k<a1 ...ap ∂âp+1 ...al >
 l−1 r
 [−1] 
(p+1)
ki IiAl−1 (t − r)

2p

− ˆ
k<a1 ...ap−1 ∂ap ...al−1 >  
l−1 r 

∞ Xl−1
∂ψ(S) (y,t∗ ) X (−1)l+p l
= 4∂â Cl−1 (l − p − 1,p) (5.6.51)
∂t∗ (l + 1)!
l=2 p=0
 [−1]
(p+1)

p
ki iba SbAl−1 (t − r)
× 1− k<a1 ...ap ∂âp+1 ...al−1 >   .
l−1 r
When deriving these equations we assumed that the time evolution of both the mass M and the angular
momentum Si of the isolated system can be neglected. In opposite case, the right side of equations
(5.6.50) and (5.6.51) would contain also time derivatives from the mass and the angular momentum.
Partial derivative from function ψ(y0 ,t∗ ) is obtained from equations (5.6.48)–(5.6.51) after replacements
y → y0 , t → t0 , and r → r0 . Equations (5.6.49)–(5.6.51) do not contain explicit integrals along the
light ray trajectory because all of the explicit integrals vanish after taking the partial derivatives in these
equations.
Frequency shift, z, is given by equation (5.6.31). In case when both observer and source of light are at
rest with respect to the ADM-harmonic reference frame, the frequency (energy) of a photon propagating
through the gravitational potential of the isolated astronomical system changes in accordance with
s
∂ψ(y,t∗ ) ∂ψ(y0 ,t∗ )

ν 1 − h00 (t0 ,x0 )
= 1− + , (5.6.52)
ν0 1 − h00 (t,x) ∂t∗ ∂t∗
In the linear with respect to G approximation, equation (5.6.52) is simplified
δν 1 1 ∂ψ(y,t∗ ) ∂ψ(y0 ,t∗ )

= h00 (t,x) − h00 (t0 ,x0 ) − + , (5.6.53)
ν0 2 2 ∂t∗ ∂t∗
where δν ≡ ν − ν0 . The time of the closest approach t∗ enters equation (5.6.53) explicitly. At the
first glance, the reader may think that it must be known in order to calculate the gravitational shift of
frequency. However, the partial derivative with respect to t∗ have to be understood in the sense of
equation (5.3.30) which makes it evident that the time derivative with respect to t∗ taken on the light-ray
path is, in fact, the time derivative with respect to time t taken before the using of the light-ray trajectory
substitution. Thus, the gravitational shift of frequency is actually not sensitive to the time of the closest
approach t∗ , and can be recast to the form which is more suitable for practical applications,
δν 1 1 ∂ψ(t,x) ∂ψ(t0 ,x0 )

= h00 (t,x) − h00 (t0 ,x0 ) − + , (5.6.54)
ν0 2 2 ∂t ∂t0
where ψ(t,x) and ψ(t0 ,x0 ) are relativistic perturbations of the electromagnetic eikonal taken at the
points of observation and emission of light respectively. These time derivatives are given by the same
equations (5.6.48)–(5.6.51) after making use of (5.3.30).
Equation (5.6.54) has been derived by making use of the definition (5.6.31). It is straightforward to
prove that the definition (5.6.32) brings about the same result. Indeed, in the case of the static observer
and the source of light their 4-velocities are, uα = (dt/dT,0,0,0) and uα 0 = (dt0 /dT0 ,0,0,0), and
the wave vector lα = ω(kα + ∂α ψ) (see equation (5.3.36)) where ω is a constant frequency of light
wave which is conserved along the light ray because of the equation of the parallel transport (5.3.14).
Substituting these relations to (5.6.32) leads immediately to equation (5.6.54) as expected.
Physical interpretation of the relativistic frequency shift given by equation (5.6.54) is straightforward
although the calculation of different components of the time derivatives from the eikonal are tedious.
The first two terms in the left side of (5.6.54) reads
1 1 GM GM
h00 (t,x) − h00 (t0 ,x0 ) = − , (5.6.55)
2 2 r r0
and represents the difference between the values of the spherically-symmetric part of the Newtonian
gravitational potential of the isolated system taken at the point of observation and emission of light. The
time derivatives of the eikonal depending on the mass and spin multipole moments of the isolated system
are given in (5.6.50), (5.6.51). Scrutiny examination of these equations reveal that these components
of the frequency shift depend on the first and higher order time derivatives of the multipole moments
which vanish in the stationary case. The gravitational frequency shift contains the gauge-dependent
contribution as well. This contribution is given by equation (5.6.49) and can be calculated by making
use of equations from section 5.3.3. The result is as follows

" (−1)
(−1)l IAl (t − r)
∞
# ∞
i
∂w0 (−1)l IAl (t − r)

i ∂w i
X X
k − = k ∇ i − (5.6.56)
∂t∗ ∂t∗ l! r l! r ,Al
l=2 ,Al l=2
∞
" i #
X (−1)l k İiAl−1 (t − r)
− 4 (5.6.57)
l! r
l=2 ,Al−1
∞
" i #
X (−1)l l k iba SbAl−1 (t − r)
+ 4 ,
(l + 1)! r
l=2 ,aAl−1
∞ X p
l X
∂φi ∂φ0 X (−1)l+p−q p − q
ki − ∗ = −2 Cl (l − p,p − q,q) 1 − × (5.6.58)
∂t∗ ∂t l! l
l=2 p=1 q=1
" (p−q+1)
IAl
( #
(t − r)

p−q ˆ ˆ q−1
1+ k<a1 ...ap ∂ap+1 ...al > ∂τ −
l−1 r
 )
(p−q+1)
p−q q−1 
ki IiAl−1 (t − r)
2 ˆ ˆ
k<a1 ...ap−1 ∂ap ...al−1 > ∂τ  −
l−1 r
∞ Xl−1 Xp
X (−1)l+p−q l
4∂â Cl−1 (l − p − 1,p − q,q) ×
(l + 1)!
l=2 p=1 q=1
 
(p−q+1)

p−q

q−1
ki iab SbAl−1 (t − r)
1− k<a1 ...ap ∂âp+1 ...al−1 > ∂ˆτ   .
l−1 r
Time derivatives ∂ˆt∗0 (ki wi − w0 ) and ∂ˆt∗0 (ki ϕi − ϕ0 ) can be obtained from equations (5.6.56), (5.6.58)
after making replacements t → t0 , r → r0 , and τ → τ0 .
5.6.4 Gravity-induced rotation of the plane of polarization of light

Any kind of axisymmetric gravitational field induces a relativistic effect of the rotation of the polarization
plane of an electromagnetic wave propagating through this field. To some extent this effect is similar
to Faraday’s effect in electrodynamics [100]. The Faraday effect is caused by the presence of magnetic
field along the trajectory of propagation of electromagnetic wave while the gravity-induced rotation
of the plane of polarization of light is caused by the presence of the, so-called, gravitomagnetic field
associated with the angular momentum and spin-type multipoles of the isolated system [128, 149]. This
gravitomagnetic effect was first discussed by Skrotskii ([113, 114] and a number of other researches
[115, 116, 129–133]. Recently, we have studied the Skrotskii effect caused by a spinning body moving
arbitrarily fast and derived the Lorentz-invariant expression for this effect [134]. In the present chapter
we further generalize the Skrotskii effect to the case of an isolated system emitting gravitational waves
of arbitrary multipolarity.
We consider the parallel transport of the reference polarization tetrad eα(β) defined by equations
(5.3.84) along the light ray. We assume for simplicity that at the past null infinity the spatial vectors
of the tetrad, e0(i) , coincide with the spatial unit vectors of the coordinate tetrad defined by equation
(5.3.80). The parallel transport of the tetrad along the light ray is defined by equation (5.3.96). We are
interested only in solving equation (5.3.96) for vectors eα(n) (n = 1,2) that are used in description of
polarization of light. The propagation equation for the spatial components ei(n) can be written in the
following form

d 1
ei(n) + hij ep(n) + ijp ej(n) Ωp = 0 , (n = 1,2.) (5.6.59)
dτ 2
where the angular velocity vector

1
Ωi = − ijp ∂j (hpα kα ) , (5.6.60)
2
describes the rate of the rotation of the plane of polarization of electromagnetic wave caused by the
presence of the gravitomagnetic field. As soon as equation (5.6.59) is solved, the time component e0(n)
of the tetrad is obtained from the orthogonality condition, lα eα(n) = 0, which implies that
e0(n) = ki ei(n) + h0i ei(n) + hij ki ej(n) + δij Ξ̇i ej(n) , (5.6.61)
where the relativistic perturbation Ξ̇i of the light-ray trajectory is given in equation (5.5.15).
Let us decompose vector Ωi into three components that are parallel and perpendicular to the unit
vector ki . We can use in the first approximation the well-known decomposition of the Kroneker symbol
in the orthonormal basis
δ ij = ai aj + bi bj + ki kj , (5.6.62)
where (a,b,k) are three orthogonal unit vectors of the reference tetrad at infinity. Decomposition of Ωi
is, then, given by
Ωi = (a · Ω)ai + (b · Ω)bi + (k · Ω)ki . (5.6.63)
Taking into account that at any point on the light-ray trajectory, vectors ei(1) = ai + O(hαβ ), ei(2) =
bi + O(hαβ ), one obtains equations of the parallel transport of these vectors (5.6.59) in the following
form

d 1
ei(1) + hij ej(1) − (k · Ω)ei(2) = 0 , (5.6.64)
dτ 2

d 1
ei(2) + hij ej(2) + (k · Ω)ei(1) = 0 , (5.6.65)
dτ 2
where equalities εijl ei(1) kl = −ei(2) + O(hαβ ), and εijl ei(2) kl = ei(1) + O(hαβ ) have been used.
Solutions of equations (5.6.64), (5.6.65) in the linear approximation with respect to the (post-Newtonian)
angular velocity Ωi read
Z τ
1
ei(1) = ai − hij aj + k · Ω(σ) dσ bi , (5.6.66)
2 −∞
Z τ
1
ei(2) = bi − hij bj − k · Ω(σ) dσ ai , (5.6.67)
2 −∞
where the second term in the right side of (5.6.66), (5.6.67) preserve orthogonality of vectors ei(1) and
ei(2) in the presence of gravitational field while the last term describes the Skrotskii effect which is a
small rotation of each of the vectors at the angle
Z τ
Φ(τ ) = k · Ω dσ (5.6.68)
−∞
about the direction of light propagation, k, in the local plane of vectors e(1) and e(2) . It is worth noting
that the Euclidean dot product k · Ω can be expressed in terms of partial differentiation with respect to
the vector ξ i of the impact parameter. This can be done by making use of equation (5.3.31) and noting
that εijp kj kp ≡ 0, so that
1
k · Ω = kα kj εjpq ∂ˆq hαp . (5.6.69)
2
Hence, the transport equation for the angle Φ assumes the following form
dΦ 1
= kα kj jpq ∂ˆq hαp . (5.6.70)
dτ 2
This equation can be split in three, linearly-independent parts corresponding to the contributions of
the gauge, Φ(G) , the mass, Φ(M ) , and the spin, Φ(S) , multipoles of the gravitational field to the Skrotskii
effect. Specifically, we have
Φ = Φ(G) + Φ(M ) + Φ(S) + Φ0 , (5.6.71)
where Φ0 is a constant angle characterizing the initial orientation of the polarization ellipse of the elec-
tromagnetic wave in the reference plane formed by the e(1) and e(2) vectors.
The gauge-dependent part of the Skrotskii effect is easily integrated so that we obtain
1 j
Φ(G) = k jpq ∂ˆq (wp + χp ) , (5.6.72)
2
where the gauge vector functions wi = w(M
i i
) + w(S) are given in equations (5.2.20). The gauge functions
i
χ appear in the process of integration of the equation of the parallel transport. They can be linearly
decomposed in two parts corresponding to the mass and spin multipoles:
χi = χi(M ) + χi(S) , (5.6.73)
where
∞ Xl−1 Xp
(−1)l+p−q

X p−q
χi(M ) = 4 Cl−1 (l − p − 1,p − q,q) 1 − (5.6.74)
l! l−1
l=2 p=1 q=1
 
(p−q+1)
q−1
IiAl−1 (t − r)
× k<a1 ...ap ∂âp+1 ...al−1 > ∂ˆτ   ,
r
∞ Xl−1 Xp
(−1)l+p−q l

X p−q
χi(S) = − 4 Cl−1 (l − p − 1,p − q,q) 1 − H(l − 1) (5.6.75)
(l + 1)! l−1
l=1 p=0 q=0
 
(p−q)
× H(q)(∂â − ka ∂ˆt∗ ) + ka ∂ˆτ k<a1 ...ap ∂âp+1 ...al−1 > ∂ˆτq−1 


r
∞ Xl−1 Xp
(−1)l+p−q l

X p q
+4 Cl−1 (l − p − 1,p − q,q) 1 − 1−
(l + 1)! l p
l=3 p=1 q=1
 
(p−q)
baal−1 SibAl−2 (t − r)
× k<a1 ...ap ∂âp+1 ...al−1 >a ∂ˆτq−1   .
r
The gauge-dependent equation (5.6.72) has the following, more explicit form:
∞ l p
1 j X X X (−1)l+p−q l − p + q p − q
k jmn ∂ˆn χm
(M ) =2 Cl (l − p,p − q,q) (5.6.76)
2 l! (l − 1) p
l=2 p=1 q=1
" (p−q)
#[−1]
jbal IbL−1 (t − r)
× k<a1 ...ap ∂âp+1 ...al >j ∂ˆτq−1 ,
r
∞ l p
1 j X X X (−1)l+p−q l
k jmn ∂ˆn χm
(S) =2 Cl (l − p,p − q,q) (5.6.77)
2 (l + 1)!
l=1 p=0 q=0

p l−p−1 p−q
× H(q) 1 − 1 + H(l − 1) − ∂ˆt2∗
l l−1 l−1

l−p p−q p−q l−p p−q−1
− +2 − H(l − 1) +2 ∂ˆt∗ τ
l l l l−1 l−1

p−q p − q − 1 ˆ2
+ 1 − H(l − 1) ∂τ
l l−1
" #[−1]
(p−q−1)
ˆ ˆ q−1 SL (t − r)
× k<a1 ...ap ∂ap+1 ...al > ∂τ ,
r
∞ l p
1 j m
X X X (−1)l+p−q p − q
k jmn ∂ˆn w(M ) = − 2 Cl (l − p,p − q,q) (5.6.78)
2 l! p
l=2 p=0 q=0
(p−q−1)
" #
ˆ ˆ q jbal IbL−1 (t − r)
× k<a1 ...ap ∂ap+1 ...al >j ∂τ ,
r
∞ l p
1 j m
X X X (−1)l+p−q l
k jmn ∂ˆn w(S) =−2 Cl (l − p,p − q,q) (5.6.79)
2 (l + 1)!
l=1 p=0 q=0

p ˆ2 l−p p−q ˆ p − q ˆ2
× 1− ∂t∗ − +2 ∂t∗ τ + ∂τ
l l l l
" #
(p−q−2)
ˆ ˆ q SL (t − r)
× k<a1 ...ap ∂ap+1 ...al > ∂τ .
r
The reader can notice the presence of integrals in the right side of equations (5.6.76) and (5.6.77).
The integrals are actually not supposed to be calculated explicitly since there is sufficient number of
partial derivatives in front of them which cancels the integration in correspondence with the rules of
differentiation of such integrals which have been explained in section 5.4.
The Skrotskii effect due to the mass-type multipoles of the isolated system is given by
 [−1]
(p)
∞ X
X l
(−1)l+p l−p jbal IbAl−1 (t − r)
Φ(M ) (τ ) = 2 Cl (l − p,p) k<a1 ...ap ∂âp+1 ...al >j   .
p=0
l! l−1 r
l=2
(5.6.80)
The gravitational field of the spin-type multipoles rotates the polarization plane of the electromagnetic
wave at the following angle

∞ Xl
X (−1)l+p l p
Φ(S) (τ ) = 2 Cl (l − p,p) 1 − × (5.6.81)
p=0
(l + 1)! l
l=1
" (p+1) #[−1]
SAl (t − r)

2p
1 + H(l − 1) 1 − k<a1 ...ap ∂âp+1 ...al >
l−1 r
Integrals in equations (5.6.80) and (5.6.81) are eliminated after taking at least one partial derivative so
that we do not need to integrate.
5.7 Light Propagation through the Field of Gravitational

Lens
This section considers propagation of light in a special case of gravitational lens approximation when
the impact parameter d of the light ray with respect to an isolated system, is much smaller than both the
distance r0 from the isolated system to the source of light and distance r from the isolated system to
observer, as illustrated in Fig. 5.3.
5.7.1 Small parameters and asymptotic expansions
In the case of a small impact parameter of the light ray with respect to the isolated system the near-zone
gravitational field of the system strongly affects propagation of the light ray only when the light particle
(photon) moves in the close proximity to the system where the effects on the light ray propagation caused
by gravitational waves are suppressed [90, 91]. In what follows, we assume that the impact parameter
d of the light ray is small as compared with both distances r and r0 , that is, d min[r,r0 ] (see
Fig. 5.3). This assumption allows us to introduce two small parameters: ε ≡ d/r and ε0 ≡ d/r0 .
If the light ray is propagated through the near-zone of the isolated system one more small parameter
can be introduced, ελ ≡ d/λ, where λ is the characteristic wavelength of the gravitational radiation
emitted by the system proportional to the product of the speed of gravity (= c in general relativity) and
the characteristic period of matter oscillations in the isolated system. Parameters ε, ε0 and ελ are not
physically correlated. Parameter ελ is used for the post-Newtonian expansion of the equations describing
the observed effects – deflection of light, time delay, etc. This expansion can be also viewed as a Taylor
expansion with respect to the parameter v/c, where v is the characteristic speed of motion of matter
composing the isolated system and c is the speed of propagation of gravity. If the light ray does not enter
the near zone of the system the parameter ελ is not small and the post-Newtonian expansion can not be
performed.
The small-impact-parameter expansions for the retarded time variables y and y0 yield:
∞
!
p ε X
y = r2 − d2 − r = −d − Ck ε2k−1 , (5.7.1)
2
k=2
∞
!
ε0 X
q
2
y0 = − r0 − d − r0 = −2r0 + d
2 − Ck ε0
2k−1
, (5.7.2)
2
k=2
Section 5.7 Light Propagation through the Field of Gravitational Lens 235
and
∞
!
1 1 X
= − 2+ Ck ε2k , (5.7.3)
yr d2
k=1
∞
1 1 X
= 2
Ck ε2k
0 , (5.7.4)
y0 r0 d
k=1
where the numerical coefficients entering the expansions are
(−1)k 1 1

1 (2k − 1)!
Ck = − 1 · ... · −k+1 = . (5.7.5)
k! 2 2 2 (2k)!!
Retarded time variables, s = t − r and s0 = t0 − r0 , are expanded as follows

∞
!
∗ ε X
s = t −d − Ck ε 2k−1
, (5.7.6)
2
k=2
∞
!
ε0 X
s0 = t∗ − 2r0 + d − Ck ε2k−1
0 , (5.7.7)
2
k=2
where Ck is given by equation (5.7.5) and t∗ is the time of the closest approach of the light ray to the
barycenter of the isolated system.
Using (5.7.6) and (5.7.7) we can write down the post-Newtonian expansions for functions of the
retarded time t − r as follows
∞ ∞
!k
X (−1)k dk ε X
F (t − r) = − Ck ε2k−1 F (k) (t∗ ) (5.7.8)
k! 2
k=0 k=2
d
F (t ) − ε Ḟ (t∗ ) + O ε2 ε2λ ,
∗
=
2
∞ ∞
!k
X d k ε0 X
F (t0 − r0 ) = − Ck ε2k−1
0 F (k) (t∗ − 2r0 ) (5.7.9)
k! 2
k=0 k=2
d
F (t∗ − 2r0 ) + ε0 Ḟ (t∗ − 2r0 ) + O ε20 ε2λ ,

=
2
where the dot above functions denote the total time derivative. We notice that convergence of the post-
Newtonian time series for light propagation depends, in fact, not just on a single parameter, ελ ∼ v/c,
that is typical in the post-Newtonian celestial mechanics of extended bodies [61, 120, 121], but on the
product of two parameters. Therefore, the convergence requires satisfaction of two conditions
εελ 1, (5.7.10)
ε0 ελ 1, (5.7.11)
where the numerical value of the parameter ελ must be taken for the smallest wavelength, λmin , in the
spectrum of the gravitational radiation emitted by the isolated system. Conditions (5.7.10), (5.7.11) en-
sure convergence of the post-Newtonian series in (5.7.8) and (5.7.9) respectively. If the source of light
rays and observer are at infinite distances from the isolated system then ε ' 0 and ε0 ' 0, and the
requirements (5.7.10), (5.7.11) are satisfied automatically, irrespective of the structure of the Fourier
spectrum (5.4.5) of the gravitational radiation emitted by the isolated system. In a real astronomical
practice such an assumption may not be always satisfied. In such cases it is more natural to avoid the
post-Newtonian expansions of the metric tensor and/or observable effects and operate with the func-
tions of the retarded time. It is important to notice that the retarded time s = t − r which enters the
result of the calculation of the is a characteristic of the Einstein equations of gravitational field, not the
Maxwell equations. Therefore, measuring the effect of the gravitational deflection of light caused by
time-dependent gravitational field allows us to measure the speed of gravity with respect to the speed
of light. This type of experiments have been proposed in our paper [118] and successfully performed
in 2003 [119]. There were several publications (see review [150]) arguing that the speed of gravity is
irrelevant in the light-ray deflection experiments by moving bodies. Unfortunately, all the authors of
those publications operated with the post-Newtonian expansion of the gravitational field which replaces
the retarded time s = t − r of the gravitational field with the time t∗ of the closest approach of light to
the light-ray deflecting body like in equation (5.7.6). This explains why the effect of the retardation of
gravity was confused with the retardation of light in [150].
If we assume that the mass and angular momentum of the isolated system are conserved the asymp-
totic expansions of integrals (5.4.1), (5.4.2) of the stationary part of the metric tensor have the following
form
[−1] ∞
1 r X (2k − 1)! 2k
= − ln − 2 ln ε − ε (5.7.12)
r 2rE 22k (k!)2
k=1
= −2 ln d + ln r + ln(2rE ) + O(ε2 ),
[−1] X ∞
1 2r0 (2k − 1)! 2k 2r0
= − ln − ε 0 = − ln + O(ε20 ), (5.7.13)
r0 rE 22k (k!)2 rE
k=1
[−2] ( ∞
!" ∞
#)
ε2 r
X
1 X (2k − 1)! 2k
= −r 1 + 1 + Ck ε 2k
ln + ε (5.7.14)
r 2rE 22k (k!)2
k=1 k=1
2
d 1 d
= −r − 2r ln d + r ln (2rrE ) − ε − ln + O(ε2 ),
2 2 2rrE
[−2] ( ∞
!" X ∞
#)
1 X 2r0 (2k − 1)! 2k
= −r0 1 − 1 + Ck ε2k
0 ln + ε 0 (5.7.15)
r0 rE 22k (k!)2
k=1 k=1

2r0 d 1 2r0
= −r0 + r0 ln − ε0 + ln + O(ε20 ).
rE 2 2 rE
Next several equations yield the estimates for the partial derivatives with respect to the parameters ξ i
and τ from functions of the retarded time (of gravity). In these estimates the numbers m and n depend
on l; we give the estimates from below for m and n for l ≥ 1.

F (t − r) F
∂ˆ<a1 ...al > =O ε2 l+1 , (5.7.16)
r d

F (t0 − r0 ) F
∂ˆ<a1 ...al > =O ε2 l+1 , (5.7.17)
r0 d

F (t − r) F
∂ˆτl =O ε2 l+1 , (5.7.18)
r d

F (t0 − r0 ) F
∂ˆτl =O ε l+1 , (5.7.19)
r0 d
[−1]
F (t0 − r0 ) F
∂ˆ<a1 ...al > =O ε2 l , (5.7.20)
r0 d
Section 5.7 Light Propagation through the Field of Gravitational Lens 237
[−2]
F (t0 − r0 ) F
∂ˆ<a1 ...al > =O ε3 l−1 . (5.7.21)
r0 d
Two asymptotic expansions will be also useful.

[−1]
F (t − r) F
∂ˆ<a1 ...al > = − 2F (t − r)∂ˆ<a1 ...al > ln d + O εελ l , (5.7.22)
r d
[−2]
F (t − r) F
∂ˆ<a1 ...al > = − 2rF (t − r)∂ˆ<a1 ...al > ln d + O ελ l−1 , (5.7.23)
r d
These estimates and asymptotic expansions will be used for obtaining the observable relativistic effects
in the gravitational lens approximation.
5.7.2 Asymptotic expressions for observable effects
This section provides the reader with the asymptotic expressions for potentially observable relativistic
effects by taking into account only the leading terms and neglecting all terms which are proportional to
the parameter ε.
The relativistic time delay is given by
∆ (τ,τ0 ) = ∆ (τ,τ0 ) + ∆ (τ,τ0 ) , (5.7.24)

(M ) (S)
where
∆ (τ,τ0 ) = − 4M ln d + 2M ln(4rr0 )− (5.7.25)

(M )
∞ X
l−2
(−1)l+p

X p p
4 Cl (l − p,p) 1 − 1− ×
p=0
l! l l−1
l=2
∂ˆtp∗ IAl (t − r)k<a1 ...ap ∂âp+1 ...al > ln d ,
∆ (τ,τ0 ) = − 4iba ki Sb ∂â ln d+ (5.7.26)

(S)
∞ X
l−1
(−1)l+p l

X p
8iba ki ∂â Cl−1 (l − p − 1,p) 1 − ×
p=0
(l + 1)! l−1
l=2
∂ˆtp∗ SbAl−1 (t − r)k<a1 ...ap ∂âp+1 ...al−1 > ln d .
The observable unit vector in the direction from the observer to the source of light is given by the
following expression
si (τ,ξ) =K i + αi (τ,ξ) + β i (τ,ξ) , (5.7.27)
where we have dropped off the quantities β i (τ0 ,ξ) and γ i (τ,ξ) as being negligibly small. Vector
αi (τ,ξ), characterizing the deflection of light, is given by expression
αi (τ,ξ) =α(M
i i
) (τ,ξ) + α(S) (τ,ξ) , (5.7.28)
where
i ˆ
α(M ) (τ,ξ) =4M∂i ln d+ (5.7.29)
∞ X
l−2 l+p

X (−1) p p
4∂î Cl (l − p,p) 1 − 1− ×
l! l l−1
l=2 p=0
∂ˆtp∗ IAl (t − r)k<a1 ...ap ∂âp+1 ...al > ln d ,

i
α(S) (τ,ξ) =4jba kj Sb ∂îa ln d− (5.7.30)
∞ X
l−1 l+p

X (−1) l p
8iba ki ∂îa Cl−1 (l − p − 1,p) 1 − ×
(l + 1)! l−1
l=2 p=0
∂ˆtp∗ SbAl−1 (t − r)k<a1 ...ap ∂âp+1 ...al−1 > ln d .
The corresponding relativistic correction to the light-ray deflection is
β i (τ,ξ) =β(M
i i
) (τ,ξ) + β(S) (τ,ξ) , (5.7.31)
where
i r i
β(M ) (τ,ξ) = − α(M ) , (5.7.32)
R
i r i
β(S) (τ,ξ) = − α(S) . (5.7.33)
R
We can use (5.7.8) and (5.7.9) to rewrite functions of the retarded time s = t − r in (5.7.25), (5.7.26),
(5.7.29) and (5.7.30) as functions taken at the moment of the closest approach t∗ of photon to the
gravitating system. This is accomplished by formal replacing t − r → t∗ because the corrections
will be of the higher order with respect to the parameter ε (see Eq. (5.7.8)). It should not confuse the
reader about the physical meaning of the retardation, which is due to the finite speed of propagation of
gravity.
The time delay (5.7.24) and the light-ray deflection (5.7.28) can be written in a very short and concise
form by making use of the gravitational lens potential, ψ, which is just the eikonal perturbation. More
specifically,
∆ (τ,τ0 ) = − 4ψ + 2M ln(4rr0 ), (5.7.34)

αi (τ,ξ) =4∂î ψ, (5.7.35)
where
ψ = ψ(M ) + ψ(S) , (5.7.36)
and
ψ(M ) = M ln d + (5.7.37)
∞ X
l−2
(−1)l+p

X p p
Cl (l − p,p) 1 − 1− ×
p=0
l! l l−1
l=2
∂ˆtp∗ IAl (t∗ )k<a1 ...ap ∂âp+1 ...al > ln d ,
ψ(S) = jba kj Sb ∂â ln d − (5.7.38)

Section 5.8 Light Propagation through the Field of Plane Gravitational Waves 239
∞ X
l−1
(−1)l+p l

X p
2jba kj ∂â Cl−1 (l − p − 1,p) 1 − ×
p=0
(l + 1)! l−1
l=2
∂ˆtp∗ SbAl−1 (t∗ )k<a1 ...ap ∂âp+1 ...al−1 > ln d .
This expression takes into account the multipole moments of the isolated gravitating system of all or-
ders and presents a generalization of our previous results obtained in [6] (stationary gravitational field)
and [91] (the quadrupolar gravitational field).
The angle of rotation of the polarization plane of electromagnetic wave is simplified if one uses the
approximation of gravitational lens. More specifically, the total angle of the rotation is given by
∆Φ = ∆Φ(M ) + ∆Φ(S) , (5.7.39)
where
∞ Xl
X (−1)l+p l−p
∆Φ(M ) = −4 Cl (l − p,p) × (5.7.40)
p=0
l! l−1
l=2
∂ˆtp∗ jbal IbAl−1 (t∗ )k<a1 ...ap ∂âp+1 ...al >j ln d ),
∞ Xl
(−1)l+p l

X p 2p
∆Φ(S) = −4 Cl (l − p,p) 1 − 1 + H(l − 1) 1 − ×
p=0
(l + 1)! l l−1
l=1
∗
∗ SAl (t )k<a1 ...ap ∂ap+1 ...al > ln d .
∂ˆtp+1 ˆ (5.7.41)
5.8 Light Propagation through the Field of Plane

Gravitational Waves
5.8.1 Plane-wave asymptotic expansions
We consider the source of light and observer located respectively at radial distances r0 = |r0 | and
r = |r| from the isolated system emitting gravitational waves. The distance between the source of light
and observer is R = |r − r0 |. In the plane gravitational-wave approximation we assume that
(a) the characteristic wavelength of gravitational waves, λ min[r,r0 ] ,
(b) the distance between the source of light and observer, R min[r,r0 ] .
Condition (a) tells us that both the source of light and observer are lying in the wave zone of the isolated
system. Condition (b) allows us to consider the gravitational waves emitted by the isolated system as
plane waves when they propagate from the source of light to observer. This approximation is vizualized
in Fig. 5.4.
We introduce small parameters δλ = max[λ/r,λ/r0 ], δ = R/r and δ0 = R/r0 The coordinate
relation between r, r0 and R = |r − r0 | can be written as follows
r02 = r2 − 2rR cos θ + R2 = r2 (1 − 2δ cos θ + δ 2 ) , (5.8.1)
where θ – is the angle between the directions "observer – the source of light" and "observer – the
gravitating system" (see Fig. 5.4). From (5.8.1) it follows that
r0 = r(1 − δ cos θ) + O(δ 2 ) , (5.8.2)

1 1
= (1 + δ cos θ) + O(δ 2 ) . (5.8.3)
r0 r
For the variables τ and τ0 we have exact relations,
τ = r cos θ, τ0 = τ − R = r cos θ − R. (5.8.4)
Quantities d and y are given in terms of the distance r and the angle τ as follows
d = r sin θ , (5.8.5)
y = τ − r = −r(1 − cos θ) . (5.8.6)
The retarded instants of time s = t − r and s0 = t0 − r0 are related to each other as follows
t0 − r0 = t − r − R(1 − cos θ) . (5.8.7)
Using this expression we can write the Taylor expansion for the functions of the retarded time s = t − r
F (t0 − r0 ) = F (t − r) − R(1 − cos θ)Ḟ (t − r) + O R2 /λ2 .

(5.8.8)
The impact parameter vector, ξ i , can be decomposed as follows

ξ i = r N i − ki cos θ . (5.8.9)
This form of the impact parameter vector ξ i is useful in subsequent approximations.

Let us now write out the asymptotic expressions for the derivatives with respect to ξ i and τ of func-
tions depending on the retarded time s = t − r. We have

F (t − r) F (t − r)
∂ˆτn = (1 − cos θ)n ∂ˆtn∗ + O(δ 2 ) , (5.8.10)
r r

F (t − r) ξa . . . ξa F (t − r)
∂â1 ...an = (−1)n 1 n n ∂ˆtn∗ + O(δ 2 ) , (5.8.11)
r r r
[−1]
F (t − r) F (t − r)
∂ˆτn = (1 − cos θ)n−1 ∂ˆtn−1
∗ + O(δ 2 ) , (5.8.12)
r r
[−1]
(−1)n ξa1 . . . ξan ˆ n−1 F (t − r)

F (t − r)
∂â1 ...an = ∂t∗ + O(δ 2 ) . (5.8.13)
r 1 − cos θ rn r
One more asymptotic expression is given for the integral taken from an STF partial derivative
( )[−1] p
l X
F (t − r) X
= (−1)p−q Cl (l − p,p − q,q) × (5.8.14)
r ,<Al > p=0 q=0
[−1]
F (t − r)
k<a1 ...ap ∂âp+1 ...al > ∂ˆtp−q
∗ ∂ˆτq .
r
Taking into account only the leading order terms in (5.8.15), we obtain the asymptotic expansion
( )[−1] (−1)
F (t − r) 1 F (t − r)
= − (5.8.15)
r ,<Al > 1 − cos θ r ,<Al >
(−1)l F (t − r)
k<Al > ∂ˆtl−1
∗ +
1 − cos θ r
[−1]
F (t − r)
(−1)l k<Al > ∂ˆtl∗ + O(δλ ) ,
r
where N i = xi /r. Equations (5.8.10)–(5.8.15) can be checked by induction.

Corresponding expressions for the functions taken at the retarded instant of time s0 = t0 − r0 can be
obtained from (5.8.10)–(5.8.15) by replacing the arguments t, r and θ with t0 , r0 and θ0 respectively.
5.8.2 Asymptotic expressions for observable effects
In this section we give expressions for the relativistic effects of the time delay, bending of light and
the Skrotskii effect in the gravitational plane-wave approximation. In this approximation we neglect all
terms of the order of δ 2 , δ02 and δλ2 , and higher. For the time delay we have
∆ = ∆ (τ,τ0 ) + ∆ (τ,τ0 ) , (5.8.16)

(M ) (S)
where
R
∆ (τ,τ0 ) = 2M + (5.8.17)
(M ) r
" #T T #T T 
∞
"
 İ İijAl−2 (t0 − r0 )
2ki kj X (−1)l ijAl−2 (t − r) 
− ,
1 − cos θ l!  r r0 
l=2 ,Al−2 ,Al−2
iba ki ξa Sb 1

1 4ki kj
∆ (τ,τ0 ) = −2 − − × (5.8.18)
(S) 1 − cos θ r r0 1 − cos θ
" #T T #T T 
∞  Ṡ ba(i Ṡj)bAl−2 (t0 − r0 )
"
(−1)l l ba(i j)bAl−2 (t − r)
X 
− ,
(l + 1)!  r r0 
l=2 ,aAl−2 ,aAl−2
Here the transverse - traceless (TT) part of the tensors depending on the multipole moments of the
isolated astronomical system is taken with respect to the direction N i . Taking into account expression
(5.2.8) for the components of the metric tensor hij we can re-write expressions (5.8.16) – (5.8.18) for
the time delay as follows
iba ki ξa Sb 1

R 1
∆ = 2M − 2 − − (5.8.19)
r 1 − cos θ r r0
Zt0
 t 
Z
ki kj TT TT
 hij (τ,x)dτ − hij (τ,x0 )dτ  .
2(1 − cos θ)
−∞ −∞
The observed astrometric direction from observer to the source of light in the plane gravitational-wave
approximation is
si (τ,ξ) = K i + αi (τ,ξ) + γ i (τ,ξ) , (5.8.20)
where
1 kp kq h i
αi (τ,ξ) = (cos θ − 2)ki + N i hTpqT (t,x) + kq hTipT (t,x), (5.8.21)
2 1 − cos θ
1
γ i (τ,ξ) = − P ij kq hTjqT (t, x).
2
In expression (5.8.20) we have dropped off the quantities β i (τ,ξ) which are negligibly small. Truncated
expressions (5.8.19) – (5.8.21) were obtained in [91] in the spin-dipole, mass-quadrupole approximation.
Current expressions (5.8.19) – (5.8.21) include all multipole moments of arbitrary order.
In the case when the distance between observer and the source of light is much smaller than the
wavelength of the gravitational waves R λ, expression (5.8.19) is reduced to a well known result [2]
for gravitational wave detectors located in a wave-zone of an isolated system

∆R 1 R
= kij hTijT (t,x) + O(δ 2 ) + O , (5.8.22)
R 2 λ
where ∆R = c ∆ (τ,τ0 ).
Relativistic rotation of the polarization plane of light (the Skrotskii effect) in the gravitational plane-
wave approximation assumes the next form
1 (k × N )i kj T T 1 (k × N0 )i kj T T
∆Φ = hij (t,x) − hij (t0 ,x0 ) . (5.8.23)
2 1−k·N 2 1 − k · N0
O τ E τ0 S
d
Observer r0 Source of light
r
D
Source of gravitational waves
Figure 5.3. Relative configuration of an observer (O), a source of light (S), and a localized system emit-
ting gravitational waves (D). Static part of gravitational field of the localized system and gravitational
waves deflect light rays which are emitted at the moment t0 at the point S and received at the moment
t at the point O. The point E on the line OS corresponds to the moment t∗ of the closest approach of a
light ray to the deflector D. We denoted the distances as follows: OS = R, DO = r, DS = r0 , the
impact parameter DE = d, OE = τ > 0, ES = τ0 = τ − R < 0. The impact parameter d is much
smaller as compared with all other distances.
Source of gravitational waves

D
r
d
r0
θ k θ0 τ0
O R S E
Observer Source of light
Figure 5.4. Relative configuration of observer (O), a source of light (S), and a localized system emitting
of gravitational waves (D). The gravitational waves deflect light rays which are emitted at the moment
t0 at the point S and received at the moment t at the point O. The point E on the line OS corresponds to
the moment of the closest approach of light ray to the system D. Notations for distances are OS = R,
DO = r, DS = r0 , DE = d - the impact parameter, OE = τ = r cos θ, ES = τ0 = τ − R. The
distance R is much smaller than both r and r0 . There is no limitation on the impact parameter d which
can be or may be not small as compared to all other distances.
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6 On the Backreaction Problem in Cosmology
Toshifumi Futamase
Astronomical Institute, Tohoku University, Sendai 980-8578, Japan
6.1 Introduction
The standard Friedman-Lemaitre-Robertson-Walker (FLRW) universe model assuming exact homo-
geneity and isotropy have had tremendous success in describing the early universe. Even at the present
epoch our universe where there exit local inhomogeneities in various scales, recent observations of the
cosmic microwave background radiation [1, 2] and large scale galaxy surveys, such as SDSS [3], shows
that our universe is remarkably isotropic and homogeneous over scales larger than 100 Mpc. Then one
can ask the following question. Let suppose there are two universes. One has a completely smooth
distribution of matter and the other has a very inhomogeneous distribution of matter but with the same
averaged density. Is these two universes expands exactly same way or not? One would think that there
is some kind of gravitational potential energy in the inhomogeneous case and thus there will be some
difference in the expansion from the homogeneous case. This question leads to another question as well.
The distribution of matter is nowhere homogeneous and isotropic in real universe, and thus how can
one make sense of homogeneous and isotropic "background" metric? Physically one can think of global
homogeneity and isotropy as the result of some sort of averaging. However the problem is not simple
because of nonlinear nature of general relativity. The solution of the Einstein equation with averaged
homogeneous matter distribution does not solve the Einstein equation with realistic matter distribution.
Thus it is naturally conjectured that if we average somehow a locally inhomogeneous universe, the ex-
pansion of the averaged spacetime will be affected by the local inhomogeneities. We call this effect as
the backreaction due to local inhomogeneities.
Neglecting nonlinear nature of general relativity in cosmology was questioned by Ellis [4] and the
effect was identified as the backreaction in 1980’s by the present author using the averaged Friedman
equation in the framework of cosmological post-Newtonian approximation realizing that the metric per-
turbation remains small even if the density contrast is highly nonlinear. It was found that the backreaction
acts as a curvature term in the averaged Friedman equation [5, 6]. An attempt to make a rigorous state-
ment for averaging Einstein equation has been developed by Zalaletdinov [7, 8], but no explicit treatment
based on the scheme appears in realistic cosmological situation. The Issacson averaging [9, 10] was em-
ployed for this problem in ref. [11] where the Hubble parameter is defined by the trace of the second fun-
dermental form K and the Friedmann equations with the backreaction term are derived by the averaging
the Hamiltonian constraint and the evolution equation for K within the framework of the cosmological
post-Newtonian approximation. Later an averaging scheme without assuming post-Newtonian type met-
ric is developed in the synchronous comoving coordinate by Buchert [12, 13]. There have been many
studies motivated by the work of Buchert [14–16]. Recently the averaging scheme using one-parameter
family of spacetimes developed by Burnett [17, 18] is applied in the cosmological situation [19].
On the other hand a relativistic treatment has been developed by generalizing Zeldovich approxima-
tion [20, 21]. Although it is not easy to compare these expressions because of different gauge choices
and create some confusion. both approaches coincide that the effect is of order of 10−6 compared with
the critical density and is sufficiently negligible. Recently relativistic higher-order study of Lagrangian
254 Chapter 6 On the Backreaction Problem in Cosmology
approximation is developed by Buchert and his collaborators [22, 23]. There is a series of study on
this problem by using Lagrange approach which claims the effect is not negligible [12]. There have
been renewed interest on the averaging problem in 2000’s by the discovery of accelerated expansion of
the present universe [24, 25]. The hope is to explain the cosmic acceleration by the backreaction and
to avoid the introduction of the cosmological constant or more generally the dark energy [26, 27]. A
possible large effect on the cosmic expansion by the backreaction in some of the works comes from
the contribution from superhorizon perturbation, but such perturbations should be regarded as a part of
background. Thus the redefinition of the background will eliminate such a contribution, and keep the
backreaction small [37]. Perturbation treatment on the backreacion problem are also discussed in detail
[29, 30], and conclude that the backreaction cannot explain the observed cosmic acceleration.
There is another approach to explain the observed acceleration using the voids of matter distribution.
The SDSS commissioning data indicates the existence of a void of scales of the order of 250/h Mpc [31].
It is not totally impossible to assume that we are living inside of such a void. The void expands more
rapidly compared with the surrounding region and thus it is natural to expect that the distance-redshift
relation in this situation will change dramatically compared with that in the homogeneous universe.
There have been some studies of the luminosity distance in such a situation [32, 33]. Moreover there
exits a model using the Lemaitre-Tolman-Bondi(LTB) spherical symmetric dust solution which gives a
good fit to the Type Ia supernova observation [34]. There is an arbitrary function in the LTB solution
which can be chosen to fit the observational data [35, 36]. Although such models are very useful to see
the general nature of the backreaction, but it seems to the present author that they are too special to apply
for realistic universes like ours.
Here we shall investigate the backreaction problem and clarifies some of confusion on the nature of
backreaction based on our works [37, 38]. Because there is no unique choice of the averaged spacetime
in inhomogeneous universe, the averaging should be defined in such a way that it corresponds to the
actual definition of the background geometry from the observation using theory. Namely the background
FLRW spacetime is defined as the geometry generated by the conserved mass density in the comoving
space. However any averaging of the mass distribution does not conserve in the comoving space once
the backreaction is taken into account. Thus it is natural to defined the conserved averaged quantity in
the comoving space for the effective mass term in the Friedman equations. This is what we will do in
this article.
There are obviously many other approaches and references which the author unfortunately cannot
cover here and does not even realize. The reader may consult with the recent review article and different
opinion on this problem by Clarkson et al [39] and by Buchert & Rasanen [40].
This paper is organized as follows. In S2 we present the basic equation in 3+1 formalism in order to
formulate the problem as generally as possible, and the averaging is introduced. In S3, we introduce an
averaging and the scale factor to rewrite Einstein equations using these variables. Then the conserved
effective density for the matter in the comoving space to make a direct comparison with the observation
, and the averaged Friedman equations are write down in S4. There we also discuss the nature of back-
reaction in some detail. Finally we discuss observational effect of the backreaction and give conclusion
in S5.
6.2 Formulation and Averaging

In this section we would like to set the problem as general as possible. In order to do so, we shall
employ the 3+1 formalism in general relativity to write down our basic equations. The line element may
Section 6.2 Formulation and Averaging 255
be written in 3+1 formalism as follows.

ds2 = − (N dt)2 + γij dxi + N i dt dxj + N j dt , (6.2.1)
where N, N i are the lapse function and shift vector, respectively. Without any loss of generality we can
set N i = 0. The unit vector normal to a hypersurface foliated by t = const. is defined by

1
nµ = , 0, 0, 0 . (6.2.2)
N
Then extrinsic curvature is defined as

1 ik
K ij ≡ γ γ̇ kj , (6.2.3)
2N
where the dot denotes differentiation with respect to time coordinate t. Then the basic equations are
obtained by projecting the Einstein equation
Gµν + Λgµν = 8πTµν , (6.2.4)
onto the normal and tangential direction to the hypersurface:

2
(3)
R + K ii − K ij K ji = 16πGE + 2Λ (6.2.5)
K ij|i − K ji|j = 8πGJi (6.2.6)

|i
K̇ ii + N K ij K ji − N |i = −4πGN (E + S) + N Λ , (6.2.7)
where (3) R is a 3-dimensional Ricci scalar curvature, and | denotes the 3-dimensional covariant deriva-
tive. The other symbols are as follows;
1
E = Tµν nµ nν = T00 (6.2.8)
N2
1
Ji = −Tνi nν = T0i (6.2.9)
N
S = Tij γ ij . (6.2.10)
The scale factor may be introduced by the following consideration. Let us define a one-parameter family
γs (t) of timelike geodesic(world line of galaxies), and let η µ be the orthogonal deviation vector from γ0
which is defined to be the worldline of our galaxy. Thus η µ represents a spatial displacement from us to
a neighbouring galaxy. We define the spatial distance by
δ` = (hµν η µ η ν )1/2 , (6.2.11)
where hµν = gµν + nµ nν . Then the rate of change of the distance is calculated to be

d 1
(δ`) = uµ (δ`);µ = K + σij ei ej δ` , (6.2.12)
dτ 3
where ei = η i /δ` with η i = hiµ η µ , and σij is the trace free part of the extrinsic curvature:
1
Kij = σij + Kγij . (6.2.13)
3
The above equation may be interpreted as describing the cosmic expansion where the first and second
terms represent isotropic and anisotropic expansion, respectively. Therefore one would like to introduce
the scale factor proportional to K. Thus introduced scale factor is, however, a function of event. In
the split of 3+1 formalism, we would like to have the scale factor which is independent of position on
t = const. hyprersurfaces. This mean we need to introduce a spatial averaging over sufficiently large
compact domain D. We define the spatial averaging as follows.
√
Z
1
< A >≡ A γ d3 x . (6.2.14)
V D
This is consistent with the definition of the volume of the domain D.

√ 3
Z
VD = γ d x, (6.2.15)
D
where γ = det(γij ). By averaging over the domain D of Eq. (6.2.12) we have

d 1
(δ`) =< N K + σij ei ej > δ` . (6.2.16)
dt 3
Thus we can define the scale factor which depends on the domain D as follows.
ȧD 1
= < NK > . (6.2.17)
aD 3
One can see that this is consistent with the definition of the volume as well. In fact,
1 ij √ 3
Z
ȧ V̇ 1
3 ≡ = γ γ̇ ij γ d x . (6.2.18)
a V V D2
If you remember Eq. (6.2.3), one can see that the last expression is same with Eq. (6.2.13). Then we
can define the deviation from the uniform Hubble flow as
ȧD i
V ji ≡ N K ij − δ j = 3N σ ij , (6.2.19)
aD
and this satisfies < V ii >= 0.

Using the scale factor the averaged Einstein equations are written as follows:
2
ȧ 8πG 1 Λ
= < N 2 E > − < N 2 (3) R > + < N 2 >
a 3 6 3
1 i 2 i j
− < (V i ) − V j V i >, (6.2.20)
6
ä 4πG 1
= − < N 2 (E + S) > + < (V ii )2 − V ji V ij >
a 3 3
1 |i Λ
+ < N N |i + Ṅ K > + < N 2 > . (6.2.21)
3 3
Up to this point the treatment is completely general.
Section 6.3 Calculation in the Newtonian gauge 257
6.3 Calculation in the Newtonian gauge

In this section we give more detailed and practical treatment for the backreaction. This requires explicit
expression for the matter and for the inhomogeneous metric. For the matter we use irrotational dust with
the following stress energy tensor
T µν = ρuµ uν , (6.3.1)
where uµ is the four velocity of the fluid flow. This is reasonable because we are interested in the
backreaction due to the collective effect of clumpy matter distribution much later than decoupling.
For the metric we specify a particular form motivated by the post-Newtonian approximation in the
cosmological circumstance [41–43] since by doing so physical meaning of the backreaction is easy to
interpret and the equations are closed in perturbative sense. This choice is reasonable since the metric
perturbation remains small even when the density contrast is highly nonlinear [5]. As seen below this
assumption allows us to solve the Friedman equations iteratively. To see the validity of this approxi-
mation we shall introduce two smallness parameters and κ. The parameter characterizes the order
of the gravitational potential φ of the material clumps, φ ' 2 . The parameter κ is the ratio between
the horizon scale, 1/H, and the scale, `, of the density fluctuation, κ = `/H. The relative size of
and κ depends on the system we have in mind. The metric perturbation is generated by the density
contrast δ = δρ/δb (where the choice of the background is not relevant in the discussion of the order of
magnitude) via the Poisson equation and may be evaluated as
∆φ 2
δ' ' 2 , (6.3.2)
Gρb κ
where ρb is the averaged density and φ is the Newtonian potential generated by the density contrast. Thus
the linear and nonlinear regions may be characterized by the conditions κ and κ, respectively.
Thus we are interested in the parameter range κ 1 which will be assumed in the below. It has
been shown that the use of the post-Newtonian approximation is guaranteed by the following parameter
range [38].
3 κ . (6.3.3)
This condition is satisfied for almost all practical situations since the metric perturbation is of the order
of 10−3 at most except very special places such as the vicinity of a compact stars.
Neglecting higher order terms the cosmological post-Newtonian, the metric may be written as the
following form
ds2 = − (1 + 2φ) dt2 + a2 (1 − 2φ) δij dxi dxj , (6.3.4)
where δij denotes the Kronecker delta, and we assumed the totally flat universe. Since we restrict our-
selves to inhomogeneities with scales much smaller than the horizon scale κ 1, we can safely ignore
the time dependence of the potential. Actually the potential begins to decay when the cosmological
constant begins to dominate the cosmic expansion. However as the time goes on, small scale structures
drop out from the expansion and larger structures are homogenized. Then the time dependence will
be neglected. Only period we cannot ignore the time dependence would be during nonlinear structure
formation. Although we do not consider here, it would be interesting to see the effect of backreaction in
this period.
One may wonder the relation between the volume dependent scale factor aD and the volume indepen-
dent scale factor a introduced in Eq. (6.3.4). The difference may be calculate as follows. First Einstein
equations give
1 3 ä2 1
2
φ i,i = (2φ + δ) − Λδ (6.3.5)
a 2 a2 2
ȧ −1
δij v j = −2 3ȧ2 − a2 Λ φ,i , (6.3.6)
a
where δ = (ρ − ρb )/ρb , and ρb =< T00 > is the averaged density. Then one can calculate V ji as
follows.
ȧD ȧ ȧD ȧ
V ji = N K ij − = − δ ij − φ2 δ ij + O(φ2 ) . (6.3.7)
aD a aD a
Remembering < V ii >= 0, we find

ȧ ȧD ȧ 2
− = φ + O(φ2 ) . (6.3.8)
a aD a
We also note
√
Z
1
< φ,i,i > = φ,i,i γ d3 x (6.3.9)
V
Z
|i √ √
Z
1
= φ |i γ d3 x + 3 φ,i φ,i γ d3 x (6.3.10)
V
= 3 < φ,i φ,i > , (6.3.11)
where we used periodic boundary condition. Using the above equations we finally get the averaged
Einstein equations as follows.
2
ȧ 8πG 1 Λ
= < T00 > + 2 < φ,i φ i > + , (6.3.12)
a 3 a 3
ä 4πG 1 Λ
=− < T00 + ρb a2 v 2 > − 2 < φ,i φ,i > + , (6.3.13)
a 3 3a 3
where v 2 = δij v i v j . The potential is calculated by Eq. (6.3.5) and thus the above equations are closed
and solved iteratively at least.
Although the effect of local inhomogeneities is described mathematically in the above equations, it is
not straightforward to connect these equations with observation. In order to do so, we need to consider
how we compare the observation and theory which will treated in the next section.
6.4 Definition of the background

Naively one might think that the above equations (33) and (34) indicate that the nonlinear backreaction
give a contribute to positive acceleration. However we should remember how we compare the observa-
tion with theory. The cosmological parameters are determined, at least in principle, by comparing the
observed expansion behaviour against the standard Friedman equation with the conserved matter energy
density in the comoving space.
This suggests to use a newly defined conserved energy density ρ̄ instead of the naive averaged density
ρb . which satisfies the usual conservation law.
ȧ
ρ̄˙ + 3 ρ̄ = 0 . (6.4.1)
a
In order to find ρ̄ we set
ρ̄ =< T00 > +a−3 A(a) . (6.4.2)
Section 6.4 Definition of the background 259
Then one finds

Ȧ(a) = −a3 < T i0 φ,i > +a4 ȧρ < v 2 > . (6.4.3)
From this expression and Eq. (6.4.1), it is easy to find that

5 ΩΛ
ρ̄ =< T00 > + a−2 +a < φ,i φ,j δ ij > , (6.4.4)
12πG 24πGΩm
where we have used Eq. (6.3.7) to rewrite the term containing < v 2 > in terms of < φ,i φ,i >. The
density parameters Ωm , ΩΛ are defined as usual using the critical density ρcr = H02 /8πG, H0 being
the Hubble parameter.
Now we rewrite Eqs. (33) and (34) in terms of the conserved density ρ̄;
2
ȧ 8πG Λ 1 1 ΛΛ
= ρ̄ + − + a < φ,i φ i > (6.4.5)
a 3 3 9 a2 Ωm
ä 4πG Λ ΩΛ
= − ρ̄ + − a < φ,i φ i > . (6.4.6)
a 3 3 6Ωm
These are the Friedmann equations for the averaged FLRW model of a locally inhomogeneous universe,
and they should be used to interpret the effect of the local inhomogeneities on the background expansion
. The third terms on the right hand side of these equations are interpreted as the nonlinear backreaction
due to local inhomogeneities. As seen from the above expressions, there are two types of backreaction.
One is proportional to a−2 , and thus behaves as a positive curvature. The other is proportional to a and
the cosmological constant. Thus if there is no cosmological constant, the backreaction does not appear
in the equation for the cosmic acceleration. This means one cannot explain the cosmic acceleration by
the backreaction. With the cosmological constant, the backreaction does affect the cosmic acceleration,
but it tends to decelerate the cosmic expansion.
It is interesting to see the effective density and pressure giving to backreaction due to the existence of
the cosmological constant only. We find
ΩΛ
ρBR
Λ = −a < φ,i φ i > (6.4.7)
24πGΩm
ΩΛ
PΛBR =a < φ,i φ i > . (6.4.8)
18πGΩm
Thus, the equation of state is PΛbr = − 34 ρbr

Λ , that looks like the equation of state of a phantom energy
but the effective energy density is negative. It satisfies the null energy condition, but violates the weak
energy condition.
We note that the cosmological constant begins to dominate the expansion, the potential starts decay-
ing. Thus the backreaction obtained above also decays. As mentioned above, we have ignored the time
dependence of the potential. This is allowed as long as we consider perturbations with scales much
smaller than the horizon scale. This may be not a bad assumption for our universe because density con-
trast becomes small for larger scales. In fact we can evaluate the backreaction using the power spectrum
P φ (k) of gravitational potential as
d3 k 2 φ
Z
< φ,i φ,i >= k P (k) , (6.4.9)
(2π)3
where
9Ωm H02
P φ (k) = Pδ (k) , (6.4.10)
4a2 k4
Pδ is the matter power spectrum. Employing a realistic Λ CDM model for the power spectrum, we find
that < φ,i φ,i > changes less than 0.3% when the lower bound of the integral region changes from H0
to 0.1H0 .
Finally we mention the averaging issues and gauge issues in the backreaction problem. One may
wonder if the results here would change by employing different averaging and other gauge. Changes
induced by using a different averaging scheme occur simultaneously only inside ρ̄ of Eqs.(32) and (33).
Furthermore, the nonlinear backreaction term is invariant to second order with respect to the choice of
the averaging procedure. Concerning gauge issue. one may wonder the result here would change under
a different gauge condition. So far only the comoving synchronous gauge(CS) except post-Newtonian
gauge is used to investigate the backreaction problem [44, 45], and the result in CS gauge obtained
equations similar to Eqs.(32) and (35) showing no change in the cosmic acceleration in the case of
no cosmological constant [45]. Although there is no general proof available at present, this strongly
suggests that the above conclusion is valid for any choice of the gauge.
6.5 Conclusions
We have studied the backreaction problem in cosmology based on our previous works. We find that
the backreaction on the global cosmic expansion due to local inhomogeneities does exit and acts as a
curvature term in the Friedman equations, but the effect is of the order of 10−6 or so in the critical density
and thus negligibly small at least for the present observations. In fact recent PLANCK observation with
the addition of BAO data sets the constraint of the total density parameter as |Ωtotal − 1| ≤ 10−3 [2].
More explicitly we can calculate the change due to the backreaction obtained above in the luminosity
distance from the present to z = 1,000 is less than 0.001% from the standard distance in the ΛCDM
model with Ωm = 0.3 and ΩΛ = 0.7. I However the progress of the cosmological observation would
reach such a accuracy sometime in future.
Although the effect is small, its nature is very interesting. First of all we find that no cosmic accel-
eration occurs as a result of the backreaction of a Newtonian perturbed FLRW metric in the case of no
cosmological constant. This is clear contrast with other works which show only the smallness of the
backreaction [29, 30]. Second the cosmological constant does induce the backreaction on the cosmic
acceleration, but in a negative direction. Namely the development of the structure tends to decrease the
cosmic acceleration. If we write the Friedman equations as follows
2
ȧ 8πG
= (ρ̄ + ρeff ) (6.5.1)
a 3
ä 4πG
= − (ρ̄ + ρeff + 3Peff ) , (6.5.2)
a 3
and
Peff = ωeff ρeff , (6.5.3)
then, ωeff + 1, changes sign from negative to positive at around z ' 0.25 for ΩΛ = 0.7 flat cosmology
although the deviation from ωΛ = −1 is of the order of 10−6 .
This will set the fundamental limit of the observation to determine the nature of the dark energy if it
is the cosmological constant or a time dependent vacuum energy.
Acknowledgments
I would like to thank Prof. S. Kopeikin for giving me a chance to contribute this article to a volume
celebrating Prof. V. A. Brumberg on his 80. I would also like to thank Prof. M. Kasai and Prof.
Section 6.5 Conclusions 261
H. Asada for fruitful collaborations and useful discussions. I apologize not to cite and mention many
relevant works on the problem of averaging and strongly recommend reader to read the review articles
mentioned in the text. This work is supported by a Grant-in-Aid for Scientific Research from JSPS (Nos.
18072001, 20540245).
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7 Post-Newtonian Approximations in Cosmology
Alexander Petrov1 and Sergei Kopeikin2
1
Sternberg Astronomical Institute, Moscow Lomonosov State University,
Universitetskiy Prospect 13, Moscow 119992, Russia
2
Department of Physics & Astronomy, University of Missouri, 322 Physics Bldg.,
Columbia, Missouri 65211, USA
7.1 Introduction
Post-Newtonian celestial mechanics is a branch of fundamental gravitational physics [18, 19, 66, 104]
that deals with the theoretical concepts and experimental methods of measuring gravitational fields and
testing general relativity both in the solar system and beyond [20, 118]. In particular, the relativistic
celestial mechanics of binary pulsars (see [82], and references therein) was instrumental in providing
conclusive evidences for the existence of gravitational radiation as predicted by Einstein’s theory of
relativity [105, 115].
Over the last few decades, various groups within the International Astronomical Union (IAU) have
been active in exploring the application of general relativity to the modelling and interpretation of high-
accuracy astrometric observations in the solar system. A Working Group on Relativity in Celestial
Mechanics and Astrometry was formed in 1994 to define and implement a relativistic theory of reference
frames and time scales. This task was successfully completed with the adoption of a series of resolutions
on astronomical reference systems, time scales, and Earth rotation models by 24-th General Assembly
of the IAU, held in Manchester, UK, in 2000. The IAU resolutions are based on the first post-Newtonian
approximation of general relativity which is a conceptual basis of the fundamental astronomy in the solar
system [103].
The mathematical formalism of the Post-Newtonian approximations is getting progressively compli-
cated as one goes from the Newtonian to higher orders [31, 99]. For this reason the theory has been
primarily developed for an isolated astronomical systems with a matter distribution having a compact
support and under simplifying assumptions that gravitational field perturbation is weak everywhere, de-
cays rapidly enough at infinity, and the background spacetime is asymptotically flat. Mathematically, it
means that the full spacetime metric, γαβ , is decomposed around the background Minkowskian metric,
ηαβ = diag(−1,1,1,1), into a linear combination
gαβ = ηαβ + hαβ , (7.1.1)
where the perturbation hαβ is represented as the post-Minkowskian1 series decomposition with respect
to the powers of the universal gravitational constant,
hαβ = Ghαβ + G2 hαβ + G3 hαβ + . . . , (7.1.2)

(1) (2) (3)
1 The term “post-Minkowskian” was introduced by Damour and Blanchet [14] to emphasize that the metric tensor
gαβ is built as a perturbative series around the Minkowski metric ηαβ , and it does not assume any limitation on
the velocity of matter generating gravitational field.
266 Chapter 7 Post-Newtonian Approximations in Cosmology
where each term, hαβ , (k = 1,2,3, . . .) of the post-Minkowskian series is decomposed into the post-
(k)
Newtonian series
[2] [3] [4]
hαβ = c−2 hαβ + c−3 hαβ + c−4 hαβ + . . . , (7.1.3)
(k) (k) (k) (k)
with respect to the powers of 1/c, where c is the speed of gravity in general relativity 2 . Post-Minkowskian
series (7.1.2) is analytic with respect to the parameter G while the post-Newtonian series (7.1.3) loses
analyticity at higher-order approximations where the backreaction of gravitational radiation becomes
important [13].
Post-Newtonian approximations suggest that there exists a method to determine hαβ by doing suc-
cessive iterations of Einstein’s field equations with the tensor of energy-momentum of matter field Φ of
the localized astronomical system, Tαβ (Φ,gαβ ), taken as a source of the gravitational field perturbation
hαβ . The iterations start from hαβ = 0 which is inserted to the expression for Tαβ which becomes
a well-defined function of the matter variables Φ. Einstein’s equations are solved at the first iteration
yielding hαβ . This solution is substituted back to the tensor Tαβ which is used to find hαβ , and so on.
(1) (2)
The post-Minkowskian solution for the metric perturbations hαβ naturally depend on the retarded time
(k)
s = t − r/c which accounts for the finite speed of propagation of gravity passing the distance r from the
mass emitting gravitational radiation. The post-Newtonian decomposition (7.1.3) of the metric tensor
perturbation represents an additional expansion of the retarded functions around the time event t. Thus,
the post-Newtonian expansion assumes r/c T or r λ, where λ is a characteristic wavelength of
gravitational radiation. It means that the post-Newtonian series (7.1.3) is valid only in the near zone of
the isolated astronomical system.
The solution of the field equations and the equations of motion of the astronomical bodies are derived
in some coordinates rα = {ct,r} where t is the coordinate time, and r = {x,y,z} are spatial coordi-
nates. The post-Newtonian theory in asymptotically-flat spacetime has a well-defined Newtonian limit
determined by:
[2]
1) solution of Poisson’s equation for the Newtonian potential, U ≡ h00 /2,
(1)
ρ(t,r 0 )d3 r0
Z
U (t,r) = , (7.1.4)
V |r − r 0 |
where ρ = c−2 T00 , is the mass density of matter producing the gravitational field,
2) equation of motion for massive particles
r̈ = ∇U , (7.1.5)
where ∇ = {∂x ,∂y ,∂z } is the operator of gradient, r = r(t) is time-dependent position of a
particle (worldline of the particle), and the dot denotes a total derivative with respect to time t,
3) equation of motion for light (massless particles)
r̈ = 0 . (7.1.6)
These equations are foundational for creation of astronomical ephemerides of celestial bodies in the
2 A common convention is to call c the speed of light irrespectively of the nature of the fundamental interaction
under consideration [118] but it may lead to confusion and misinterpretation of gravitational experiments and
astronomical observations [42, 68].
solar system [19, 66] and in any other localized system of self-gravitating bodies like a binary pulsar
[82]. In all practical cases they have to be extended to take into account the post-Newtonian corrections
sometimes up to the 3-d post-Newtonian order of magnitude [119]. It is important to notice that in the
Newtonian limit the coordinate time t of the gravitational equations of motion (7.1.5), (7.1.6) coincides
with the proper time of observer τ that is practically measured with an atomic clock.
So far, the post-Newtonian theory was mathematically successful and passed through numerous ex-
perimental tests with a flying colour. Nevertheless, it hides several pitfalls. The first one is the problem
of convergence of the post-Newtonian series and regularization of divergent integrals that appear in
the post-Newtonian calculations at higher post-Newtonian orders [99]. The second problem is that the
background manifold is not asymptotically-flat Minkowskian spacetime but the FLRW metric, ḡαβ . We
live in the expanding universe which rate of expansion is determined by the present value H0 of the
Hubble parameter H = H(t) depending on time. Therefore, the right thing would be to replace the
post-Newtonian decomposition (7.1.1) with a more adequate post-Friedmannian series [109]
gαβ = ḡαβ + καβ , (7.1.7)
where
{0} {1} {2}
καβ = καβ + Hκαβ + H 2 καβ + . . . , (7.1.8)
is the metric perturbation around the cosmological background represented as a series with respect to the
Hubble parameter, H. Each term of the series has its own expansion into the post Minkowskian/Newtonian
{0}
series like (7.1.2) and (7.1.3). For example, καβ = hαβ , and there is no asymptotically-flat spacetime
{1} {2}
analogue to καβ , καβ , etc. Generalization of the theory of Post-Newtonian approximations from the
Minkowski spacetime to that of the expanding universe is important for extending the applicability of
the post-Newtonian celestial dynamics to testing cosmological effects, for more deep understanding of
the process of formation of large and small scale structure in the universe and gravitational interaction
between pairs of galaxies and their clusters.
Whether cosmological expansion affects gravitational dynamics of bodies inside a localized astro-
nomical system was a matter of considerable efforts of many researchers [17, 24, 37, 38, 71, 86, 87, 100].
Most of the previous works on celestial dynamics in cosmology assumed spherical symmetry of matter
distribution and gravitational field which allowed to use exact spherically-symmetric solutions of Ein-
stein’s equations approximating the Schwarzschild solution near the body and a cosmological solution
far outside of it. Matching of the two solutions in the intermediate zone was achieved in several different
ways but all of them suggest some kind of a fine tuning of the size of the matching zone to the cosmolog-
ical parameters and the mass of the central body. This fine-tuning is physically unrealistic. Furthermore,
real astronomical systems in cosmology (galaxies, clusters, filaments, etc.) have no spherical symmetry.
McVittie’s solution [87] is perhaps the most successful mathematically among the spherically-symmetric
approaches but yet lacks a clear physical interpretation [24].
Cosmological observations are now performed so accurately that we need a precise mathematical for-
mulation of the post-Newtonian theory for interpretation of these observations. This theory is not to be
limited by the assumption of the spherical symmetry of the isolated astronomical system which must be
coupled to the time-dependent background geometry through the gravitational interaction. Theoretical
description of the post-Newtonian dynamics of a localized astronomical system in expanding universe
should correspond in the limit of vanishing H to the post-Newtonian dynamics in the asymptotically-flat
spacetime. Such a description will allow us to directly compare the equations of the standard post-
Newtonian celestial dynamics with its cosmological counterpart. Therefore, the task is to derive a set
of the post-Newtonian equations in cosmology in some coordinates introduced on the background man-
ifold, and to map them onto the set of the Newtonian equations (7.1.4)–(7.1.6) in asymptotically-flat
spacetime. The post-Newtonian celestial dynamics would be of a paramount importance for extending
the tools of experimental gravitational physics to the field of cosmology, for example, to properly for-
mulate the cosmological extension of the PPN formalism [117]. The present chapter discusses the main
ideas and principal results of such a theoretical approach in the linearized approximation with respect
to the gravitational perturbations of the cosmological background caused by the presence of a localized
astronomical system. The formalism of the present chapter has been employed in [69] to check the the-
oretical consistency of equations (7.1.4)–(7.1.6) on expanding cosmological background and to analyse
the outcome of some experiments like the excessive Doppler effect discovered by J. Anderson et al [3, 4]
in the hyperbolic motion of Pioneer 10 and 11 space probes in the solar system.
The original goal in developing the theory of cosmological perturbations was to relate the physics
of the early universe to CMB anisotropy and to explain the formation and growth of large-scale struc-
ture from a primordial spectrum. The ultimate goal of this theory is to establish a mathematical link
between the fundamental physical laws at the Planck epoch and the output of the gravitational wave
detectors which are the only experimental devices being capable to measure the parameters and the state
of the universe at that time [64]. Originally, two basic approximation schemes for calculating cosmo-
logical perturbations have been invented by E. Lifshitz with his collaborators [78, 79] and, later on, by
J. Bardeen [8]. Lifshitz [78] worked out a coordinate-dependent theory of cosmological perturbations
in a synchronous gauge while Bardeen [8] concentrated on finding the gauge-invariant combinations for
perturbed quantities and derivation of a perturbation technique based on gauge-invariant field equations.
At the same time, Lukash [83] had suggested an original approach for deriving the gauge-invariant scalar
equations based on the thermodynamic theory of the Clebsch potential [102] also known in cosmology
as the scalar velocity potential [73, 102] or the Taub potential [108]. It turns out that the variational
principle with a Lagrangian of cosmological matter formulated in terms of the Clebsch potential, is the
most useful mathematical device for developing the theory of relativistic celestial dynamics of localized
astronomical systems embedded in expanding cosmological manifold [70].
In the years that followed, the gauge-invariant formalism was refined and improved by Durrer and
Straumann [34, 35], Ellis et al. [39–41] and, especially, by Mukhanov et al. [91, 93]. Irrespectively of
the approach a specific gauge must be fixed in order to solve equations for cosmological perturbations.
Any gauge is allowed and its particular choice is simply a matter of mathematical convenience. Impos-
ing a gauge condition eliminates four degrees of freedom in the cosmological metric pertrubations and
brings the differential equations for them to a solvable form. Nonetheless, the residual gauge freedom
associated with the tensor nature of the gravitational field remains. This residual gauge freedom leads
to appearance of spurious perturbations which must be disentangled from the physical modes. Lifshitz’s
theory of cosmological perturbations [78, 79] is worked out in a synchronous gauge and contains the
spurious modes but they are easily isolated from the physical perturbations and suppressed [49]. The
other gauges are described in Bardeen’s article [8] and used in cosmological perturbation theory as well.
Among them, the longitudinal (conformal or Newtonian) gauge is one of the most common. This gauge
is advocated by Mukhanov [91] because it removes spurious coordinate degrees of freedom in scalar
perturbations. Detailed comparison of the cosmological perturbation theory in the synchronous and
conformal gauges was given by Ma and Bertschinger [84].
Unfortunately, none of the previously known cosmological gauges can be applied for analysis of the
cosmological perturbations caused by localized matter distributions like an isolated astronomical system
which can be a single star, a planetary system, a galaxy, or even a cluster of galaxies. The reason is
that the synchronous gauge has no Newtonian limit and is applicable only for freely falling test particles
while the longitudinal gauge separates the scalar, vector and tensor modes in the metric tensor perturba-
tion in the way that is incompatible with the technique of the post-Newtonian approximation schemes
having been worked out in asymptotically flat spacetime [66]. We also notice that standard cosmological
perturbation technique often operates with harmonic (Fourier) decomposition of both the metric tensor
and matter perturbations when one is interested in statistical statements based on the cosmological prin-
ciple. This technique is unsuitable and must be avoided in sub-horizon approximation for working out
the post-Newtonian celestial dynamics of self-gravitating isolated systems.
Current paradigm is that the cosmological generalization of the Newtonian field equations of an iso-
lated gravitating system like the solar system or a galaxy or a cluster of galaxies can be easily obtained
by just making use of the linear principle of superposition with a simple algebraic addition of the local
system to the tensor of energy momentum of the background matter. It is assumed that the superposition
procedure is equivalent to operating with the Newtonian equations of motion derived in asymptotically-
flat spacetime and adding to them (”by hands”) the tidal force due to the presence of the external universe
(see, for example, [86]). Though such a procedure may look pretty obvious it lacks a rigorous math-
ematical analysis of the perturbations induced on the background cosmological manifold by the local
system. This analysis should be done in the way that embeds cosmological variables to the field equa-
tions of standard Post-Newtonian approximations not by “hands” but by precise mathematical technique
which is the goal of the present article. The variational calculus on manifolds is the most convenient
for joining the standard theory of cosmological perturbations with the Post-Newtonian approximations
in asymptotically-flat spacetime. It allows us to track down the rich interplay between the perturbations
of the background manifold with the dynamic variables of the local system which cause these pertur-
bations. The output is the system of the post-Newtonian field equations with the cosmological effects
incorporated to them in a physically-transparent and mathematically-rigorous way. This system can be
used to solve a variety of physical problems starting from celestial dynamics of localised systems in cos-
mology to gravitational wave astronomy in expanding universe that can be useful for deeper exploration
on scientific capability of such missions as LISA and Big Bang Observer (BBO) [30]
In fact, the problem of whether the cosmological expansion affects the long-term evolution of an iso-
lated N-body system (galaxy, solar system, binary system, etc.) had a long controversial history. The
reason is that there was no an adequate mathematical formalism for describing the cosmological pertur-
bations caused by an isolated system so that different authors have arrived to opposite opinions. It seems
that McVittie [87] was first who had considered the influence of the expansion of the universe on the
dynamics of test particles orbiting around a massive point-like body immersed to the cosmological back-
ground. He found an exact solution of the Einstein equations in his model which assumed that the mass
of the central body is not constant but decreases as the universe expands. Einstein and Straus [37, 38]
suggested a different approach to discuss motion of particles in gravitationally self-interacting systems
residing on the expanding background. They showed that a Schwarzschild solution could be smoothly
matched to the Friedman universe on a spherical surface separating the two solutions. Inside the surace
("vacuole") the motion of the test particles is totally unaffected by the expansion. Thus, Einstein and
Straus [37, 38] concluded that the cosmic expansion is irrelevant for the Solar system. Bonnor [17]
generalized the Einstein-Straus vacuole and matched the Schwarzschild region to the inhomogeneous
Lemaître-Tolman-Bondi model thus, making the average energy density inside the vacuole be indepen-
dent of the exterior energy density while in the Einstein-Straus model they must be equal. Bonnor [17]
concluded that the local systems expand but at a rate which is negligible compared with the general
cosmic expansion. Similar conclusion was reached by Mashhoon et al. [86] who analysed the tidal
dynamics of test particles in the Fermi coordinates.
The vacuole solutions are not appropriate for adequate physical solution of the N-body problem in
the expanding universe. There are several reasons for it. First, the vacuole is spherically-symmetric
while majority of real astronomical systems are not. Second, the vacuole solution imposes physically
unrealistic boundary conditions on the matching surface that relates the central mass to the size of the
surface and to the cosmic energy density. Third, the vacuole is unstable against small perturbations. In
order to overcome these difficulties a realistic approach based on the approximate analytic solution of the
Einstein equations for the N-body problem immersed to the cosmological background, is required. In
the case of a flat spacetime there are two the most advanced techniques for finding approximate solution
of the Einstein equations describing gravitational field of an isolated astronomical system – the post-
Newtonian and Post-Minkowskian approximations [31] that have been briefly discussed in introduction.
The post-Newtonian approximation technique is applicable to the systems with weak gravitational field
and slow motion of matter. The Post-Minkowskian approximations also assume that the field is weak but
does not imply any limitation on the speed of matter. The post-Newtonian iterations are based on solving
the elliptic-type Poisson equations while the post-Minkowskain approach operates with the hyperbolic-
type (wave) D’Alembert equations. The Post-Minkowskian approximations naturally include descrip-
tion of the gravitational radiation emitted by the isolated system while the post-Newtonian scheme has
to use additional mathematical methods to describe generation of the gravitational waves [25]. In the
present chapter we concentrate on the development of a generic scheme for calculation of cosmological
perturbations caused by a localized distribution of matter (small-scale structure) which preserves many
advantages of the post-Minkowskian approximation scheme. The cosmological Post-Newtonian approx-
imations are derived from the post-Minkowskian perturbation scheme by making use of the slow-motion
expansion with respect to a small parameter v/c where v is the characteristic velocity of matter in the
N-body system and c is the fundamental speed.
There were several attempts to work out a physically-adequate and mathematically-rigorous approxi-
mation schemes in general relativity in order to construct and to adequately describe dynamics of small-
scale structures in the universe. The most notable work in this direction has been done by Kurskov and
Ozernoy [72], Futamase et al. [12, 45, 46, 107] (see also Chapter 6 of this book), Buchert and Ehlers
[22, 36], Mukhanov et al. [1, 91–93], Zalaletdinov [120]. These approximation schemes have been
designed to track the temporal evolution of the cosmological perturbations from a very large down to
a small scale up to the epoch when the perturbation becomes isolated from the expanding cosmologi-
cal background. These approaches looked hardly connected between each other until recent works by
Clarkson et al [28, 29], Li & Schwarz [76, 77], Räsänen [98], Buchert & Räsänen [23] and Wiegand
& Schwarz [116]. In particular, Wiegand & Schwarz [116] have shown that the idea of cosmic vari-
ance (that is a standard way of thinking) is closely related to the cosmic averages defined by Buchert
and Ehlers [22, 36]. All researchers agree that the Post-Newtonian approximations are important to
understand the backreaction of the cosmological perturbations on the expansion rate of the universe
[1, 45, 56, 57, 93, 120]).
Development of observational cosmology and gravitational wave astronomy demands to extend the
linearized theory of cosmological perturbations to second and higher orders of approximation. A fair
number of works have been devoted to solving this problem. Non-linear perturbations of the metric
tensor and matter affect evolution of the universe and this backreaction of the perturbations should
be taken into account. This requires derivation of the effective stress-energy tensor for cosmological
perturbations formed by freely-propagating gravitational waves and scalar field [1, 91–93]. The laws
of conservation for the effective stress-energy tensor are important for derivation of the post-Newtonian
equations of motion of the isolated astronomical system.
In the present chapter we construct a non-linear theory of successive cosmological perturbations for
isolated systems which generalizes the post-Minkowskian approximation scheme in asymptotically flat
spacetime. We implement the Lagrangian-based theory of dynamical perturbations of gravitational field
on a curved background manifold which has been worked out in [50, 96] (see also [6]). This theory has
a number of specific advantages over other perturbation methods among which the most important are:
• Lagrangian-based approach is covariant and can be implemented for any curved background space-
time that is a solution of the Einstein gravity field equations;
• the system of the partial differential equations describing dynamics of the perturbations is deter-
mined by a dynamic Lagrangian LD which is derived from the total Lagrangian L by making
use of its Taylor expansion with respect to the perturbations and accounting for the background
field equations. The dynamic Lagrangian LD defines the conserved quantities for the perturbations
(energy, angular momentum, etc.) that depend on the symmetries of the background manifold;
• the dynamic Lagrangian LD and the corresponding field equations for the perturbations are gauge-
invariant in any order of the perturbation theory. Gauge transformations map the background man-
ifold onto itself and are associated with arbitrary (analytic) coordinate transformations on the back-
ground spacetime;
• the entire perturbation theory is self-reproductive and is extended to the next perturbative order out
of a previous iteration by making use of the same equations with a corresponding substitution of
quantities from the previous iteration. The linearized approximation is the basic starting point of
the theory.
Perhaps, it would be more appropriate to call the perturbative technique explained in this chapter as the
post-Friedmannian approximations – the term proposed by M. Tegmark [109]. However, we shall con-
tinue to use the conventional name of post-Newtonian approximations to emphasize that it is applicable
not only to large-scale perturbations but also to the discussion of formation and dynamics of small-scale
structures in cosmology – the topic being intimately related to relativistic celestial mechanics.
The chapter is organized as follows. In section 7.2 we describe the variational and Lie derivatives on
manifold. These derivatives are crucial for understanding the mathematical technique of the present arti-
cle. Section 7.3 introduces the Lagrangian of gravitational field and matter of the background cosmolog-
ical model as well as the Lagrangian of an isolated astronomical system which perturbs the background
cosmological manifold. Section 7.4 describes the geometric structure of the background spacetime man-
ifold of the cosmological model and the corresponding equations of motion of the matter and field vari-
ables. Section 7.5 introduces the reader to the theory of the Lagrangian perturbations of the cosmological
manifold and the dynamic variables. Section 7.6 makes use of the preceding sections in order to derive
the field equations in the gauge-invariant form. Beginning from section 7.7 we focus on the spatially-
flat universe in order to derive the post-Newtonian field equations that generalize the post-Newtonian
equations in the asymptotically-flat spacetime. These equations are coupled in the scalar sector of the
proposed theory. Therefore, we consider in section 7.8 a few particular cases when the equations can
be fully decoupled one from another, and solved in terms of the retarded potentials. This section also
provides a proof of the Lorentz-invariance of the retarded potentials for the wave equations describing
propagation of weak gravitational and sound waves on the background cosmological manifold.
We use G to denote the universal gravitational constant and c for the ultimate speed in the Minkowski
spacetime. Every time, when there is no confusion about the system of units, we shall choose a ge-
ometrized system of units such that G = c = 1. We put a bar over any function that belongs to the
background manifold of the FLRW cosmological model. Any function without such a bar belongs to the
perturbed manifold.
The other notations used in the present chapter are as follows:
• Greek indices α,β,γ, . . . run through values 0,1,2,3, and Roman indices i,j,k, . . . take values 1,2,3,
• Einstein summation rule is applied for repeated (dummy) indices, for example, P α Qα ≡ P 0 Q0 +
P 1 Q1 + P 2 Q2 + P 3 Q3 , and P i Qi ≡ P 1 Q1 + P 2 Q2 + P 3 Q3 ,
• gαβ is a full metric on the cosmological spacetime manifold,
√
• gαβ ≡ −gg αβ ,
• ḡαβ is the FLRW metric on the background spacetime manifold,
• fαβ is the metric on the conformal spacetime manifold,

• ηαβ ≡ diag{−1, + 1, + 1, + 1} is the Minkowski metric,
• T and X i ≡ {X,Y,Z} are the coordinate time and isotropic spatial coordinates on the background
manifold,
• X α ≡ {X 0 ,X i } = {cη,X i } are the conformal coordinates with η being a conformal time,
• xα ≡ {x0 ,xi } = {ct,xi } is an arbitrary coordinate chart on the background manifold,
• a bar, F̄ above a geometric object F , denotes the unperturbed value of F on the background mani-
fold,
• a prime F 0 ≡ dF/dη denotes a total derivative with respect to the conformal time η,
• a dot Ḟ ≡ dF/dT denotes a total derivative with respect to the cosmic time T ,
• ∂α ≡ ∂/∂xα is a partial derivative with respect to the coordinate xα ,
• a comma with a following index F,α ≡ ∂α F is another designation of a partial derivative with
respect to a coordinate xα ,
• a vertical bar, F|α denotes a covariant derivative of a geometric object F (a scalar, a vector, a tensor)
with respect to the background metric ḡαβ ,
• a semicolon, F;α denotes a covariant derivative of a geometric object F (a scalar, a vector, a tensor)
with respect to the conformal metric fαβ ,
• the tensor indices of geometric objects on the background manifold are raised and lowered with the
background metric ḡαβ ,
• the tensor indices of geometric objects on the conformal spacetime are raised and lowered with the
conformal metric fαβ ,
• the scale factor of the FLRW metric is denoted as R ≡ R(T ), or as a ≡ a(η) = R[T (η)],
• the Hubble parameter, H ≡ Ṙ/R, and the conformal Hubble parameter, H = a0 /a.
Other notations will be introduced and explained as they appear in text.
7.2 Derivatives on the geometric manifold

7.2.1 Variational derivative
Theory of perturbations of physical fields on manifolds rely upon the principle of the least action of
a functional S called action. Variational derivative arises in the problem of finding solutions of the
gravitational field equation that extremize the action
Z
S = Fd4 x , (7.2.1)
√
where F ≡ f −g, is a scalar density of weight +1. Let the set of F = F (Q,Qα ,Qαβ ) depends
on the field variable Q, its first - Qα ≡ Q,α and second - Qαβ ≡ Q,αβ partial derivatives that play
here a similar role as velocity and acceleration in the Lagrangian mechanics of point-like particles. The
field variable Q can be a tensor field of an arbitrary type with the covariant and/or contravariant indices.
For the time being, we suppress the tensor indices of Q as it may not lead to a confusion. Function
Section 7.2 Derivatives on the geometric manifold 273
F depends on the determinant g of the metric tensor and can also depend on its derivatives. We shall
discuss this case in the sections that follow.
A certain care should be taken in choosing the dynamic variables of the Lagrangian formalism in
case when the variable Q is a tensor field. For example, if we choose a covariant vector field Aµ as an
independent variable, the corresponding “velocity” and “acceleration” variables must be chosen as Aµ,α
and Aµ,αβ respectively. On the other hand, if the independent variable is chosen as a contravariant vector
Aµ , the corresponding “velocity” and “acceleration” variables must be chosen as Aµ ,α and Aµ ,αβ . The
same remark is applied to any other tensor field. The reason behind is that Aµ and Aµ are interrelated
via the metric tensor, Aµ = g µν Aν . Therefore, derivative of Aµ differs from that of Aµ by an additional
term involving the derivative of the metric tensor which, if being improperly introduced, can bring about
spurious terms to the field equations derived from the principle of the least action.
Variational derivative, δF/δQ, taken with respect to the variable Q relates a change, δS, in the
functional S to a change, δF, in the function F that the functional depends on,
Z
δS = δFd4 x , (7.2.2)
where
∂F ∂F ∂F
δF = δQ + δQα + δQαβ . (7.2.3)
∂Q ∂Qα ∂Qαβ
This is a functional increment of F. The variational derivative is obtained after we single out a total
divergence in the right side of (7.2.3) by making use of the commutation relations, δQα = (δQ),α
and δQαβ = (δQ),αβ . The total divergence is reduced to a surface term in the integral (7.2.2) which
vanishes on the boundary of the volume of integration. Thus, the variation of S with respect to Q is
given by Z
δF
δS = δQd4 x , (7.2.4)
δQ
where
δF ∂F ∂ ∂F ∂2 ∂F
≡ − α
+ . (7.2.5)
δQ ∂Q ∂x ∂Qα ∂x ∂xβ ∂Qαβ
α
Similar procedure can be applied to S by varying it with respect to Qα and Qαβ . In such a case we get
the variational derivatives of F with respect to Qα
δF ∂F ∂ ∂F
≡ − , (7.2.6)
δQα ∂Qα ∂xβ ∂Qαβ
and that of F with respect to Qαβ ,
δF ∂F
≡ (7.2.7)
δQαβ ∂Qαβ
Let us assume that there is another geometric object, T (Q,Qα ,Qαβ ), which differs from the original
one F (Q,Qα ,Qαβ ) by a total divergence
T (Q,Qα ,Qαβ ) = F (Q,Qα ,Qαβ ) + ∂β H β (Q,Qα ) . (7.2.8)
It is well-known [53, 90] that taking the variational derivative (7.2.5) from T and F yields the same
result
δT δF
≡ , (7.2.9)
δQ δQ
because the variational derivative from the divergence is zero identically. In fact, it is straightforward to
prove that the variational derivative (7.2.5), after it applies to a partial derivative of an arbitrary smooth
function, vanishes
δ ∂F
≡0. (7.2.10)
δQ ∂xα
However, this property does not hold on for a covariant derivative in the most general case [90].
The variational derivatives are covariant geometric object that is they do not depend on the choice
of a particular coordinates on manifold [66, 90]. In case, when the dynamic variable Q is not a metric
tensor, this statement can be proved by taking the first, Qα ≡ Q;α , and second, Qαβ ≡ Q;αβ , covariant
derivatives of Q as independent dynamic variables instead of its partial derivatives, Qα and Qαβ . In this
case the procedure of derivation of variational derivatives (7.2.5), (7.2.6) remains the same and the result
is

δF ∂F ∂F ∂F
= − + . (7.2.11)
δQ ∂Q ∂Qα ;α ∂Qαβ ;βα
The order, in which the covariant derivatives are taken, is imposed by the procedure of the extracting
the total divergence from the variation of the action in (7.2.2). The order of the derivatives is important
because the covariant derivatives do not commute.
Variational derivative of F with respect to the metric tensor gµν is defined by the same equations
(7.2.5)–(7.2.7) where we identify Q ≡ gµν , Qα ≡ gµν,α , and Qµν ≡ gµν,αβ . It yields
δF ∂F ∂ ∂F ∂2 ∂F
≡ − α
+ (7.2.12)
δgµν ∂gµν ∂x ∂gµν,α ∂x ∂xβ ∂gµν,αβ
α
δF ∂F ∂ ∂F
≡ − , (7.2.13)
δgµν,α ∂gµν,α ∂xβ ∂gµν,αβ
δF ∂F
≡ (7.2.14)
δgµν,αβ ∂gµν,αβ
Covariant generalization of (7.2.12)–(7.2.14) is not quite straightforward because the covariant derivative
of the metric tensor gµν;α = 0, and we cannot use it as a dynamic variable. In this case, we consider the
set of the metric tensor, gµν , the Christoffel symbols Γα µν , and the Riemann tensor Rα βµν as a set of
√
independent dynamic variables. The action is given by (7.2.1) where F ≡ −gf (gµν ,Γα µν ,Rα βµν )
is a scalar density of weight +1. Variation of F is
∂F ∂F ∂F
δF = δgµν + δΓα µν + δRα βµν , (7.2.15)
∂gµν ∂Γα µν ∂Rα βµν
where variations of the Christoffel symbols and the Riemann tensor are tensors that can be expressed in
terms of the variation δgµν of the metric tensor [112]
1 ασ
δΓα µν = g [(δgσµ );ν + (δgσν );µ − (δgµν );σ ] , (7.2.16)
2
α
δR βµν = (δΓα βν );µ − (δΓα βµ );ν . (7.2.17)
Section 7.2 Derivatives on the geometric manifold 275
Now, we replace (7.2.16), (7.2.17) in (7.2.15) and single out a total divergence 3 . It yields
δF
δF = δgµν + Bα ,α , (7.2.18)
δgµν
where the total divergence vanishes on the boundary of integration of the action, and the covariant
variational derivative is

δF ∂F 1 ∂F ∂F ∂F
= − g σµ σ + g σν σ − g σα σ (7.2.19)
δgµν ∂gµν 2 ∂Γ να ∂Γ µα ∂Γ µν ;α

∂F ∂F ∂F
+ g σµ + g σν − g σα
∂Rσ αβν ∂Rσ µβα ∂Rσ µβν ;βα
Variational derivative with respect to the contravariant metric tensor is
δF ∂gαβ δF δF
= = −gαµ gβν . (7.2.20)
δg µν ∂g µν δgαβ δgαβ
The variational derivatives are not linear operators. For example, they do not obey Leibniz’s rule [48,
Section 2.3]. More specifically, for any geometric object, H = FT, that is a corresponding product of
two other geometric objects, F = F (Q,Qα ,Qαβ ) and T = T (Q,Qα ,Qαβ ), the variational derivative
δ (FT) δF δT
6= T+F , (7.2.21)
δQ δQ δQ
in the most general case. The chain rule with regard to the variational derivative is preserved in a limited
sense. More specifically, let us consider a geometric object F = F (Q,Qα ,Qαβ ) where Q is a function
of a variable P , that is Q = Q(P ). Then, the variational derivative
δF δF ∂Q
= , (7.2.22)
δP δQ ∂P
that can be confirmed by inspection [96]. On the other hand, if we have a singled-valued function
H = H(Q), and Q = Q (P,Pα ,Pαβ ), the chain rule
δH ∂F δQ
= , (7.2.23)
δP ∂Q δP
is also valid. The chain rule (7.2.23) will be often used in calculations of the present chapter.
7.2.2 Lie Derivative

Lie derivative on the manifold can be viewed as being induced by a diffeomorphism
x0α = xα + ξ α (x) , (7.2.24)
such that a vector field ξ α has no self-intersections, thus, defining a congruence of curves which provides
a natural mapping of the manifold into itself. Lie derivative of a geometric object F is denoted as £ξ F.
It is defined by a standard rule
£ξ F = F 0 (x) − F(x) , (7.2.25)
3 The fact that F is a scalar density is essential for the transformation of covariant derivatives to the total divergence.
The total divergences can be converted to surface integrals which vanish on the boundary of integration and, hence,
can be dropped off the calculations.
where F 0 is calculated by doing its coordinate transformation induced by the change of the coordinates
(7.2.24) with subsequent pulling back the transformed object from the point x0α to xα along the con-
µ ...µ
gruence ξ α [66]. In particular, for any tensor density F = Fν11...νqp of type (p,q) and weight m one
has
µ ...µ µ ...µ µ ...µ
£ξ Fν11...νqp = ξ α Fν11...νqp ,α + mξ α ,α Fν11...νqp (7.2.26)
µ ...µ µ ...µ α...µ
+ Fα...ν
1
q ξ ,ν1 + . . . + Fν1 ...α ξ ,νq − Fν1 ...νq ξ
p α 1 p α p µ1
,α − . . . − Fν1 ...νq ξ
µ1 ...α µp
,α .
We notice that all partial derivatives in the right side of equation (7.2.26) can be simultaneously replaced
with the covariant derivatives because the terms containing the Christoffel symbols cancel each other.
The Lie derivative commutes with a partial (but not a covariant) derivative
∂α (£ξ F) = £ξ (∂α F) . (7.2.27)
This property allows us to prove that a Lie derivative from a geometric object F (Q,Qα ,Qαβ ) can be
calculated in terms of its variational derivative. Indeed,
∂F ∂F ∂F
£ξ F = £ξ Q + £ξ Qα + £ξ Qαβ . (7.2.28)
∂Q ∂Qα ∂Qαβ
Now, after using the commutation property (7.2.27) and changing the order of partial derivatives in
£ξ Qα and £ξ Qαβ , one can express (7.2.28) as an algebraic sum of the variational derivative and a total
divergence
δF ∂ δF δF
£ξ F = £ξ Q + α
£ξ Q + £ξ Qβ . (7.2.29)
δQ ∂x δQα δQαβ
This property of the Lie derivative indicates its close relation to the variational derivative on the manifold
and will be used in the calculations that follow. It is also worth pointing out that (7.2.29) is used for
derivation of Noether’s theorem of conservation of the canonical stress-energy tensor of the field Q in
case when F = L is the Lagrangian density of the field for which the variational derivative vanishes
on-shell, δF/δQ = δL/δQ = 0, and £ξ L = ∂α (ξ α L).
7.3 Lagrangian and Field Variables

We accept the Einstein’s theory of general relativity and consider a universe filled up with matter con-
sisting of three components. The first two components are: (1) an ideal fluid composed of particles of
one type with transmutations excluded; (2) a scalar field; and (3) a matter of the localized astronomical
system. The ideal fluid consists of baryons and cold dark matter, while the scalar field describes dark
energy [2]. We assume that these two components do not interact with each other directly, and are the
source of the Friedmann-Lemître-Robertson-Walker (FLRW) geometry. There is no dissipation in the
ideal fluid and in the scalar field so that they can only interact through the gravitational field. It means
that the equations of motion for the fluid and the scalar field are decoupled in the main approximation,
and we can calculate their evolution separately. Mathematically, it means that the Lagrangian of the
ideal fluid and that of the scalar field depend only on their own field variables and the metric tensor.
The tensor of energy-momentum of matter of the localized astronomical system is not specified in
agreement with the approach adopted in the post-Newtonian approximation scheme developed in the
asymptotically-flat spacetime [31, 67]. This allows us to generate all possible types of cosmological per-
turbations: scalar, vector and tensor modes. We are the most interested in developing our formalism for
application to the astronomical system of massive bodies bound together by intrinsic gravitational forces
Section 7.3 Lagrangian and Field Variables 277
like the solar system, galaxy, or a cluster of galaxies. It means that our approach admits a large density
contrast between the background matter and the matter of the localized system. The localized system
perturbs the background matter and gravitational field of FLRW metric locally but it is not included
to the matter source of the background geometry, at least, in the approximation being linearized with
respect to the metric tensor perturbation. Our goal is to study how the perturbations of the background
matter and gravitational field are incorporated to the gravitational field perturbations of the standard
post-Newtonian theory of relativistic celestial dynamics .
Let us now consider the action functional and the Lagrangian of each component.
7.3.1 Action Functional

We shall consider a theory with the action functional
Z
S= Ld4 x , (7.3.1)
M
where the integration is performed over the entire spacetime manifold M. The Lagrangian L is com-
prised of four terms
L = Lg + Lm + Lq + Lp , (7.3.2)
where Lg , Lm , Lq are the Lagrangians of gravitational field, the dark matter, the scalar field that governs
the accelerated expansion of the universe [47], and Lp is the Lagrangian describing the source of the
cosmological perturbations. Gravitational field Lagrangian is
1 √
Lg = − −gR , (7.3.3)
16π
where R is the Ricci scalar built of the metric gαβ and its first and second derivatives [89]. Other
Lagrangians depend on the metric and the matter variables. Correct choice of the matter variables is
a key element in the development of the Lagrangian theory of the post-Newtonian perturbations of the
cosmological manifold caused by a localized astronomical system.
7.3.2 Lagrangian of the Ideal Fluid

The ideal fluid is characterized by the following thermodynamic parameters: the rest-mass density ρm ,
the specific internal energy Πm (per unit of mass), pressure pm , and entropy sm where the sub-index ’m’
stands for ’matter’. We shall assume that the entropy of the ideal fluid remains constant, that excludes
it from further consideration. The standard approach to the theory of cosmological perturbations preas-
sumes that the constant entropy excludes rotational (vector) perturbations of the fluid component from
the start, and only scalar (adiabatic) perturbations are generated [2, 91, 113, 114]. However, the present
chapter deals with the cosmological perturbations that are generated by a localized astronomical system
which is described by its own Lagrangian (see section 7.3.4) which is left as general as possible. This
leads to the tensor of energy-momentum of the matter of the localized system that incorporates the rota-
tional motion of matter which is the source of the rotational perturbations of the background ideal fluid.
This extrapolates the concept of the gravitomagnetic field of the post-Newtonian dynamics of localized
systems in the asymptotically-flat spacetime [19, 27, 66] to cosmology. Further details regarding the
vector perturbations are given in section 7.6.5 of the present chapter.
The total energy density of the fluid
m = ρm (1 + Πm ) . (7.3.4)
One more thermodynamic parameter is the specific enthalpy of the fluid defined as
m + pm pm
µm = = 1 + Πm + . (7.3.5)
ρm ρm
In the most general case, the thermodynamic equation of state of the fluid is given by equation pm =
pm (ρm ,Πm ), where the specific internal energy Πm is related to pressure by the first law of thermody-
namics.
Since the entropy has been assumed to be constant, the first law of thermodynamics reads

1
dΠm + pm d =0. (7.3.6)
ρm
It can be used to derive the following thermodynamic relationships
dpm = ρm dµm , (7.3.7)

dm = µm dρm , (7.3.8)
which means that all thermodynamic quantities are solely functions of the specific enthalpy µm , for
example, ρm = ρm (µm ), Πm = Πm (µm ), etc. The equation of state is also a function of the variable µm ,
that is
pm = pm (µm ) . (7.3.9)
Derivatives of the thermodynamic quantities with respect to µm can be calculated by making use of
equations (7.3.7) and (7.3.8), and the definition of the (adiabatic) speed of sound vs of the fluid
∂pm v2
= 2s , (7.3.10)
∂m c
where the partial derivative is taken under a condition that the entropy, sm , of the fluid does not change.
Then, the derivatives of the thermodynamic quantities take on the following form
∂pm ∂m c2 ∂ρm c 2 ρm

= ρm , = 2 ρm , = 2 , (7.3.11)
∂µm ∂µm vs ∂µm vs µm
where all partial derivatives are performed under the same condition of constant entropy.
√
The Lagrangian of the ideal fluid is usually taken in the form of the total energy density, Lm = −gm
[89]. However, this form is less convenient for applying the variational calculus on manifolds. The
above thermodynamic relationships and the integration by parts of the action (7.3.1) allows us to recast
√ √
the Lagrangian Lm = −gm to the form of pressure, Lm = − −gpm , so that the Lagrangian density
becomes (see [66, pp. 334-335 ] for more detail)
√ √
Lm = − −gpm = −g (m − ρm µm ) . (7.3.12)
Theoretical description of the ideal fluid as a dynamic system on spacetime manifold is given the most
conveniently in terms of the Clebsch potential, Φ which is also called the velocity potential [102]. In the
case of a single-component ideal fluid the Clebsch potential is introduced by the following relationship
µm wα = −Φ,α . (7.3.13)
In fact, equation (7.3.13) is a solution of relativistic equations of motion of the ideal fluid [73].
The Clebsch potential is a primary field variable in the Lagrangian description of the isentropic ideal
fluid. The four-velocity is normalized to wα wα = gαβ wα wβ = −1, so that the specific enthalpy can
Section 7.3 Lagrangian and Field Variables 279
be expressed in the following form

p
µm = −g αβ Φ,α Φ,β . (7.3.14)
One may also notice that

µm = wα Φ,α . (7.3.15)
It is important to notice that the Clebsch potential Φ has no direct physical meaning as it can be changed
to another value Φ → Φ0 = Φ + Φ̃ such that the gauge function, Φ̃, is constant along the worldlines of
the fluid: wα Φ̃,α = 0.
In terms of the Clebsch potential the Lagrangian (7.3.12) of the ideal fluid is
√
Lm = −g m − ρm −g αβ Φ,α Φ,β .
p
(7.3.16)
Metrical tensor of energy-momentum of the ideal fluid is obtained by taking a variational derivative of
the Lagrangian (7.3.16) with respect to the metric tensor,
m 2 δLm
Tαβ = √ . (7.3.17)
−g δg αβ
Calculation yields
m
Tαβ = (m + pm ) wα wβ + pm gαβ , (7.3.18)
α α
where w = dx /dτ is the four-velocity of the fluid, and τ is the proper time of the fluid element taken
along its worldline. This is a standard form of the tensor of energy-momentum of the ideal fluid [89].
Because the Lagrangian (7.3.16) is expressed in terms of the dynamical variable Φ, the Noether approach
based on taking the variational derivative of the Lagrangian with respect to the field variable, can be
applied to derive the canonical tensor of the energy-momentum of the ideal fluid. This calculation has
been done in [66, pp. 334-335 ] and it leads to the expression (7.3.18). It could be expected because we
assumed that the ideal fluid consists of bosons. The metrical and canonical tensors of energy-momentum
for the liquid differ, if and only if, the liquid’s particles are fermions (see [66, pp. 331-332] for more
detail). We do not consider the fermionic liquids in the present chapter.
7.3.3 Lagrangian of Scalar Field

The Lagrangian of the scalar field Ψ is given by
√

1 αβ
Lq = −g g ∂α Ψ∂β Ψ + W , (7.3.19)
2
where W ≡ W (Ψ) is a potential of the scalar field. We assume that there is no direct coupling between
the scalar field and the matter of the ideal fluid. They can interact only through the gravitational field.
Many different potentials of the scalar field are used in cosmology [2]. At this step, we do not chose a
specific form of the potential which will be selected later.
Metrical tensor of energy-momentum of the scalar field is obtained by taking a variational derivative
q 2 δLq
Tαβ = √ , (7.3.20)
−g δg αβ
that yields
q 1 µν
Tαβ = ∂α Ψ∂β Ψ − gαβ g ∂µ Ψ∂ν Ψ + W (Ψ) . (7.3.21)
2
The canonical tensor of energy-momentum of the scalar field is obtained by applying the Neother theo-
rem and leads to the same expression (7.3.21).
One can formally reduce the tensor (7.3.21) to the form similar to that of the ideal fluid by making
use of the following procedure. First, we define the analogue of the specific enthalpy of the scalar field
"fluid" p
µq = −g σν ∂σ Ψ∂ν Ψ , (7.3.22)
and the effective four-velocity, v α , of the "fluid"
µq vα = −∂α Ψ . (7.3.23)
The four-velocity v α is normalized to vα v α = −1. Therefore, the scalar field enthalpy µq can be
expressed in terms of the partial derivative from the scalar field
µq = v α ∂α Ψ . (7.3.24)
Then, we introduce the analogue of the rest mass density ρq of the scalar field "fluid" by defining,
p
ρq ≡ µq = v α ∂α Ψ = −g σν ∂σ Ψ∂ν Ψ . (7.3.25)
As a consequence of the above definitions, the energy density, q and pressure pq of the scalar field
"fluid" can be introduced as follows
1 1
q ≡ − g σν ∂σ Ψ∂ν Ψ + W (Ψ) = ρq µq + W (Ψ) , (7.3.26)
2 2
1 σν 1
pq ≡ − g ∂σ Ψ∂ν Ψ − W (Ψ) = ρq µq − W (Ψ) . (7.3.27)
2 2
One notices that a relationship
q + pq
µq = , (7.3.28)
ρq
between the specific enthalpy µq , the density ρq , the pressure pq and the energy density q , of the scalar
field "fluid" formally holds on the same form (7.3.5) as in the case of the barotropic ideal fluid.
After applying the above-given definitions in equation (7.3.21), it is formally reduced to the tensor of
energy-momentum of an ideal fluid
q
Tαβ = (q + pq ) vα vβ + pq gαβ . (7.3.29)
It is worth emphasizing that the analogy between the tensor of energy-momentum (7.3.29) of the scalar
field "fluid" with that of the barotropic ideal fluid (7.3.18) is rather formal since the scalar field, in the
most general case, does not satisfy all required thermodynamic equations because of the presence of
the potential W = W (Ψ) in the energy density q , and pressure pq of the scalar field. For example,
equation of continuity (7.4.66) for scalar field differs from that for the ideal fluid (7.4.58) if the potential
W (Ψ) 6= 0.
7.3.4 Lagrangian of a Localized Astronomical System

The Lagrangian Lp of matter of a localized astronomical system (a small-scale structure inhomogeneity)
which perturbs the geometry of the background manifold of the FLRW metric, can be chosen arbitrary.
We shall call the perturbation of the background manifold that is induced by Lp , the bare perturbation.
We assume that the matter of the bare perturbation is described by a (multi-component) field variable, θ,
which physical meaning depends on a specific problem we want to solve. The Lagrangian density of the
Section 7.4 Background manifold 281
√
bare perturbation is given by Lp = −gLp (θ, gαβ ). tensor of energy-momentum of the matter of the
bare perturbation, Tαβ , is obtained by taking a variational derivative
2 δLp
Tαβ = √ . (7.3.30)
−g δg αβ
Tensor Tαβ is a source of the small-scale gravitational perturbation of the background manifold that is
associated with a particular solution of the linearised Einstein equations which will be derived in next
sections.
7.4 Background manifold

7.4.1 Hubble Flow
We shall consider the background universe as described by the Friedmann-Lemître-Robertson-Walker
(FLRW) metric. The functional form of the metric depends on the coordinates introduced on the man-
ifold. Because the FLRW metric describes homogeneous and isotropic spacetime there is a preferred
class of coordinates which clearly reveal these properties of the background manifold. These coordi-
nates materialize a special set of freely falling observers, called comoving observers. These observers
are following with the flow of the expanding universe and have constant values of spatial coordinates.
The proper distance between the comoving observers increases in proportion to the scale factor R(T ). In
the preferred cosmological coordinates, the time coordinate of the FLRW metric is just the proper time
as measured by the comoving observers. A particle moving relative to the local comoving observers has
a peculiar velocity with respect to the Hubble flow. An observer with a non-zero peculiar velocity does
not see the universe as isotropic.
For example, the peculiar velocity of the solar system implies the dipole anisotropy of cosmic mi-
crowave background (CMBR) radiation corresponding to |v | = 369.0 ± 0.9 km·s−1 , towards a point
with the galactic coordinates (l,b) = (264◦ , 48◦ ) [52, 59]. Such a solar system’s velocity implies a
velocity |vLG | = 627 ± 22 km·s−1 toward (l,b) = (276◦ , 30◦ ) for our Galaxy and the Local Group of
galaxies relative to the CMBR [43, 63]. The existence of the preferred frame in cosmology should not
be understood as a violation of the Einstein principle of relativity. Indeed, any coordinate chart can be
used in order to describe the FLRW metric. A preferred frame exists merely because the FLRW metric
admits only six-parametric group (3 spatial translations and 3 spatial rotations) as contrasted with the
ten-parametric group of Minkowski (or De Sitter) spacetime which includes the time translation and
three Lorentz boosts as well. The metric of FLRW does not remain invariant with respect to the time
translation and the Lorentz transformations because its expansion makes different spacelike hypersur-
faces non-equivalent. It may lead to some interesting observational predictions of cosmological effects
within the solar system [69].
7.4.2 Friedmann-Lemître-Robertson-Walker Metric

In what follows, we shall consider the problem of calculation of the post-Newtonian perturbations in the
expanding universe described by the FLRW class of models. The FLRW metric is an exact solution of
Einstein’s field equations of general relativity that describes a homogeneous, isotropically expanding or
contracting universe. The general form of the metric follows from the geometric properties of homo-
geneity and isotropy of the manifold[113, 114]. Einstein’s equations are only needed to derive the scale
factor of the universe as a function of time.
The most general form of the FLRW metric is given by
dρ2

ds2 = −dT 2 + R2 + ρ 2
d 2
ϑ + sin 2
ϑd2
υ , (7.4.1)
1 − kρ2
where T is the coordinate time, {ρ,ϑ,υ} are spherical coordinates, R = R(T ) is the scale factor de-
pending on time and characterizing the size of the universe compared to the present value of R = 1.
The time T has a physical meaning of the proper time of a comoving observer that is being at rest with
respect to the cosmological frame of reference. The present epoch corresponds to the value of the time
T = T0 . The constant k can take on three different values k = {−1,0, + 1}, where k = −1 corresponds
to the spatial hyperbolic geometry, k = 0 does the spatially flat FLRW model, and k = +1 does the
spatially closed world [89].
The Hubble parameter H characterizes the rate of the temporal evolution of the universe. It is defined
by
Ṙ 1 dR
H≡ = . (7.4.2)
R R dT
For mathematical reasons, it is convenient to introduce a conformal time, η, via differential equation
dT
dη = . (7.4.3)
R(T )
If the time dependence of the scale factor is known, the equation (7.4.3) can be solved, thus, yielding
T = T (η). It allows us to re-express the scale factor R(T ) in terms of the conformal time, R (T (η)) ≡
a(η). The conformal Hubble parameter is, then, defined as
a0 1 da
H≡ = . (7.4.4)
a a dη
The two expressions for the Hubble parameters are related by means of equation
H
H= , (7.4.5)
a
that allows us to link their time derivatives
a2 Ḣ = H0 − H2 , (7.4.6)
00 0
3
a Ḧ = H − 4HH + 2H , 3
(7.4.7)
and so on.
It is also convenient to introduce the isotropic Cartesian coordinates X i = {X,Y,Z}, by transforming
the radial coordinate
r
ρ= , (7.4.8)
k
1 + r2
4
and defining r2 = X 2 + Y 2 + Z 2 = δij X i X j . In the isotropic coordinates the interval (7.4.1) takes on
the following form
ds2 = Gαβ dX α dX β , (7.4.9)
where the coordinates X α = {X 0 ,X 1 ,X 2 ,X 3 } = {η,X,Y,Z}, and the metric has a conformal form
Gαβ = a2 (η)gαβ (7.4.10)

δij
g00 = −1 , g0i = 0 , gij = 2 . (7.4.11)
k 2
1+ r
4
−6
Determinant of the metric Gαβ is G = det(Gαβ ) = a8 g, where g = det(gαβ ) = − 1 + kr2 /4 .
The spacetime interval in the isotropic Cartesian coordinates reads
 
δij dX i dX j 
 
ds2 = a2 (η) −dη 2 + 2  . (7.4.12)

k
1 + r2
 
4
The distinctive property of the isotropic coordinates in the FLRW metric is that the radial coordinate r
is defined in such a way that the three-dimensional space looks exactly Euclidean and null cones appear
in it as round spheres irrespectively of the value of the space curvature k. The isotropic coordinates
do not represent proper distances on the sphere, nor does the radial coordinate r represents a proper
radial distance measured with the help of radar astronomy technique. The proper spatial distance in the
isotropic coordinates is (1 + kr2 /4)−1 ar [113].
The FLRW metric presented in the conformal form by equation (7.4.12) singles out a preferred cos-
mological reference frame defined by the congruence of worldlines of the fiducial test particles being at
rest with respect to the spatial coordinates X i . Four-velocity of a fiducial particle is denoted as Ū α =
dX α /dτ , where dτ = −ds is the proper time on the worldline of the particle. In the isotropic confor-
mal coordinates, Ū α = (1/a,0,0,0). The four-velocity is a unit vector, Ū α Ūα = Gαβ Ū α Ū β = −1.
It implies that the covariant components of the four-velocity are Ūα = (−a,0,0,0). In the preferred
frame the universe looks homogeneous and isotropic. The choice of the isotropic Cartesian coordinates
reflects these fundamental properties explicitly in the symmetric form of the metric (7.4.10). However,
the set of the fiducial particles is a mathematical idealization. In reality, any isolated astronomical sys-
tems (galaxy, binary star, the solar system, etc.) have a peculiar velocity with respect to the preferred
cosmological frame formed by the Hubble flow. We have to introduce a locally-inertial coordinate chart
which is associated with the isolated system and moves along with it. Transformation from the preferred
cosmological frame to the local chart must include the Lorentz boost and a geometric part due to the
expansion and curvature of cosmological spacetime. It can take on multiple forms which originate from
certain geometric and/or experimental requirements [24, 26, 54, 61].
We do not impose specific limitations on the choice of coordinates on the background manifold and
keep the overall formalism of the Post-Newtonian approximations, covariant. The arbitrary coordinates
are denoted as xα = (x0 ,xi ) and they are related to the preferred isotropic coordinates X α = (η,X i )
by the coordinate transformation xα = xα X β . This transformation has inverse X α = X α xβ , at

least in some local domain of the background manifold. In this domain, the matrices of the coordinate
transformations
∂xα ∂X α
Λα β = , Mα β = , (7.4.13)
∂X β ∂xβ
α γ α α γ α
and they satisfy to the apparent equalities Λ γ M β = δβ and M γ Λ β = δβ .
Four-velocity of the Hubble observers written in the arbitrary coordinates has the following form
ūα = Λα β Ū β = a−1 Λα 0 , ūα = Mβ α Uβ = −aM0 α . (7.4.14)
The background FLRW metric written down in the arbitrary coordinates, xα , takes on the following
form
ḡαβ (xα ) = a2 f̄αβ (xα ) . (7.4.15)
Here the scalar function a(xα ) ≡ a [η(xα )], and the conformal metric
f̄αβ (xα ) = Mµ α Mν β gµν (X i ) . (7.4.16)
Any metric admits 1+3 decomposition with respect to a congruence of a timelike vector field [89].
FLRW metric admits a privileged congruence formed by the four-velocity ūα of the Hubble observers
which is a physically privileged vector field. The 1+3 decomposition of the FLRW metric is applied in
arbitrary coordinates and has the following form
ḡαβ = −ūα ūβ + P̄αβ , (7.4.17)
where the tensor

P̄αβ = a2 Mi α Mj β gij , (7.4.18)
describes the metric on the spacelike hypersurface being everywhere orthogonal to the four-velocity
ūα of the Hubble flow. Tensor P̄αβ is the operator of projection on this hypersuface. It can be also
interpreted as a metric on the hypersurace of orthogonality to the Hubble vector flow. Equation (7.4.17)
can be used in order to prove that P̄αβ satisfies the following relationship
P̄ βµ P̄β ν = P̄ µν , (7.4.19)
which can be confirmed by inspection. The trace P̄ α α = ḡ αβ P̄αβ = P̄ αβ P̄αβ = 3.

Now, we consider how to express the partial derivatives of any scalar function F = F (η), which
depends only on the conformal time η = η(xα ), in terms of the four-velocity ūα of the Hubble flow.
Taking into account that η = x0 and applying equation (7.4.14), we obtain
∂F dF ∂η F0
F,α = α
= α
= F 0 M0 α = − ūα = −Ḟ ūα . (7.4.20)
∂x dη ∂x a
In particular, the partial derivative from the scale factor, a,α = −ȧūα = −Hūα , and the partial
derivative from the Hubble parameter H,α = −Ḣūα .
7.4.3 Christoffel Symbols and Covariant Derivatives
In the following sections of the chapter we will need to calculate the covariant derivatives from var-
ious geometric objects on the background cosmological manifold covered by an arbitrary coordinate
chart xα = (x0 ,xi ). The calculation engages the affine connection Γ̄α
βγ of the background manifold
which is decomposed into an algebraic sum of two connections (the Christoffel symbols) because of the
conformal structure of the FLRW metric [111]. By definition,
1 αν
Γ̄α βγ = ḡ (ḡνβ,γ + ḡνγ,β − ḡβγ,ν ) , (7.4.21)
2
where
ḡαβ,γ = −2H ḡαβ ūγ + a2 f̄αβ,γ . (7.4.22)
Separating terms in the right side of (7.4.21) yields
Γ̄α βγ = Āα βγ + B̄ α βγ , (7.4.23)
where
Āα βγ = −H δβα ūγ + δγα ūβ − ūα ḡβγ

, (7.4.24)
and
1 αµ
B̄ α βγ = f̄ (f̄µβ,γ + f̄µγ,β − f̄βγ,µ ) . (7.4.25)
2
The non-vanishing components of the connections are given in the isotropic Cartesian coordinates X α
by
k δpi Xq + δqi Xp − δpq X i
Āα 0β = Hδβα , Ā0 ij = Hgij , B̄ i pq = − , (7.4.26)
2 k
1 + r2
4
where Xq ≡ δqj X j , and all the other components of the connections vanish.
A covariant derivative of a geometric object (scalar, vector, etc.) on the background manifold is
denoted in this chapter with a vertical bar. For example, the covariant derivative of a vector field F α is
F α |β = F α ,β + Γ̄α βγ F γ , (7.4.27)
where a comma in front of sub-index β denotes a partial derivative with respect to coordinate xβ . Equa-
tion (7.4.27) can be brought to yet another form if we denote the covariant derivative of the affine
connection B̄ α βγ with a semicolon. Making use of (7.4.23) in equation (7.4.27) transforms it to the
following form
F α |β = F α ;β + Āα βγ F γ . (7.4.28)
The covariant derivative of a covector Fα is defined in a similar way,
Fα|β = Fα,β − Γ̄γ αβ Fγ (7.4.29)
which is equivalent to
Fα|β = Fα;β − Āγ αβ Fγ , (7.4.30a)

γ
Fα;β = Fα,β − B̄ αβ Fγ (7.4.30b)
Equations for tensors of higher rank can be presented in a similar way. Of course, the covariant derivative
of a scalar field F always coincides with its covariant derivative by definition,
F|α = F;α = F,α . (7.4.31)
We also provide an equation for the covariant derivative of the four-velocity of the Hubble flow. Doing
calculations in the isotropic coordinates X α for the four-velocity Ū α , and applying the tensor law of
transformation to arbitrary coordinates xα , results in
ūα |β = H δβa + ūα ūβ ūα|β = H P̄ αβ ,

ūα|β = H P̄αβ , , (7.4.32)
where the tensor indices are raised and lowered with the metric ḡαβ .
7.4.4 Riemann Tensor
The Riemann tensor is defined by
R̄α βµν = Γ̄α βν,µ − Γ̄α βµ,ν + Γ̄α µγ Γ̄γ βν − Γ̄α νγ Γ̄γ βµ . (7.4.33)
and can be calculated directly from this equation. We prefer a slightly different way by making use of
the algebraic decomposition of the Riemann tensor into the irreducible parts
1 R̄
R̄αβµν = C̄αβµν + S̄αµ ḡβν + S̄βν ḡαµ − S̄αν ḡβµ − S̄βµ ḡαν + (ḡαµ ḡβν − ḡαν ḡβµ ) ,
2 12
(7.4.34)
where C̄αβµν is the Weyl tensor,
1
S̄µν = R̄µν − R̄ḡµν , (7.4.35)
4
R̄µν = ḡ αβ R̄αµβν is the Ricci tensor, and R = ḡ αβ R̄αβ is the Ricci scalar. The Weyl tensor of a
conformally-flat spacetime vanishes identically,
C̄αβµν ≡ 0 . (7.4.36)
Therefore, FLRW cosmological metric (7.4.1) has a remarkable property – it can be always brought up
to the conformally-flat form by applying an appropriate coordinate transformation [55].
Direct evaluation of other tensors entering (7.4.34) by making use of the FLRW metric (7.4.10),
(7.4.11) yields
1 0
H (ḡµν − 2ūµ ūν ) + 2 H2 + k (ḡµν + ūµ ūν ) ,

R̄µν = (7.4.37)
a2
2 1
−H0 + H2 + k ūµ ūν + ḡµν ,

S̄µν = (7.4.38)
a2 4
6 0
H +H +k .
2
R̄ = (7.4.39)
a2
Making use of equations (7.4.36) – (7.4.39) in the decomposition (7.4.34) of the Riemann tensor, yields
the following result
1 0
H (ḡαµ ḡβν − ḡαν ḡβµ ) − H0 − H2 − k P̄αµ P̄βν − P̄αν P̄βµ ,

R̄αβµν = 2
(7.4.40)
a
where
P̄αβ = ḡαβ + ūα ūβ , (7.4.41)
is the operator of projection that was introduced earlier in (7.4.18).
7.4.5 The Friedmann Equations
The Einstein tensor Ēαβ ≡ R̄αβ − ḡαβ R̄/2 of the FLRW cosmological model is derived from equations
(7.4.37) and (7.4.39). It reads
1
Ēαβ = − 2 H0 − H2 − k P̄αβ + 3 H2 + k ḡαβ .

(7.4.42)
a2
Einstein’s field equations on the background spacetime takes on the following form
Ēαβ = 8π T̄αβ , (7.4.43)
where the tensor of energy-momentum of the background spacetime manifold includes the background
matter and the scalar field
m q
T̄αβ = T̄αβ + T̄αβ . (7.4.44)
Here, tensors of energy-momentum in the right side of Einstein’s equations are derived from the La-
grangians (7.3.16) and (7.3.19), and represent an algebraic sum of tensors (7.3.18) and (7.3.22). Each
m q
tensor of energy-momentum, T̄αβ and T̄αβ , is Lie-invariant with respect to the group of symmetry of
the background FLRW metric independently, and each of them has the form of the tensor of energy-
momentum of the perfect (ideal) fluid. Hence, the tensor of energy-momentum T̄αβ in the right side of
(7.4.43) has the form of a perfect fluid as well,
T̄αβ = (¯ + p̄) ūα ūβ + p̄ ḡαβ . (7.4.45)
It imposes a certain restriction on the effective energy density ¯ and pressure p̄ which must obey
Dalton’s law for a partial energy density and pressure of the background matter and the scalar field
components [10]
¯ = ¯m + ¯q , (7.4.46)

p̄ = p̄m + p̄q . (7.4.47)
Here, ¯m and p̄m are the energy density and pressure of the ideal fluid, and ¯q and p̄q are the energy
˙ of the scalar field and
density and pressure of the scalar field which are related to the time derivative Ψ̄
its potential W̄ = W̄ (Ψ̄) by equations (7.3.26), (7.3.27). On the background spacetime these equations
takes on the following form
1
¯q = ρ̄q µ̄q + W̄ , (7.4.48)
2
1
p̄q = ρ̄q µ̄q − W̄ , (7.4.49)
2
where µ̄q is the background specific enthalpy of the scalar field defined by (7.3.22), and ρ̄q = µ̄q is
the background density of the scalar field "fluid". It is worthwhile to remind to the reader that due to
the homogeneity and isotropy of the FLRW metric, all matter variables on the background manifold are
functions of the conformal time η only when being expressed in the isotropic Cartesian coordinates.
Einstein’s equations (7.4.43) can be projected on the direction of the background four-velocity of
matter and on the spatial hypersurface being orthogonal to it. It yields two Friedmann equations for the
evolution of the scale factor a,
8π k
H2 = ¯ − 2 , (7.4.50)
3 a
k
2Ḣ + 3H 2 = −8π p̄ − 2 (7.4.51)
a
where ¯ and p̄ are the effective energy density and pressure of the mixture of matter and scalar field as
defined above.
A consequence of the Friedmann equations (7.4.50), (7.4.51) is an equation
k
Ḣ = −4π (¯ + p̄) + , (7.4.52)
a2
relating the time derivative of the Hubble parameter with the sum of the overall energy density and pres-
sure, which can be expressed in terms of the density and specific enthalpy of the background components
of matter,
¯ + p̄ = ρ̄m µ̄m + ρ̄q µ̄q . (7.4.53)
In order to solve the Friedmann equations (7.4.50), (7.4.51) we have to employ the equation of state
of matter. Customarily, it is assumed that each matter component obeys its own cosmological equation
of state,
p̄m = wm ¯m , p̄q = wq ¯q , (7.4.54)
where wm and wq are parameters lying in the range from −1 to +1. In the most simple cosmological
models, parameters wm and wq are fixed. More realistic models admit that the parameters of the equation
of state may change in the course of the cosmological expansion, that is they may depend on time. The
equation of state does not close the system of the Friedmann equations, which have to be complemented
with the equations of motion of the scalar field and of the ideal fluid in order to make the system of
differential equations for the gravitational and matter field variables complete.
7.4.6 Hydrodynamic Equations of the Ideal Fluid

The background value of the Clebsch potential of the ideal fluid, Φ̄, depends only on the conformal
time η of the FLRW metric. The partial derivative of the potential, taken in arbitrary coordinate chart
on the background manifold, can be expressed following equation (7.3.14) in terms of the background
four-velocity ūα as follows
Φ̄|α = −µ̄m ūα , (7.4.55)
where the background value of the specific enthalpy is
q
µ̄m = −ḡ αβ Φ̄,α Φ̄,β (7.4.56)
in accordance with definition (7.3.14). It allows us to write down the specific enthalpy of the ideal fluid
in terms of a derivative from the Clebsch potential Φ̄. Multiplying both sides of (7.4.55) with ūα , and
accounting for ūα ūα = −1, we obtain
˙ .
µ̄m ≡ ūα Φ̄|α = Φ̄ (7.4.57)
The background equation of continuity for the rest mass density ρ̄m of the ideal fluid is
(ρ̄m ūα )|α = 0 , (7.4.58)
that is equivalent to
ρ̄m|α − 3H ρ̄m ūα = 0 , (7.4.59)
where we have used (7.4.32). The background equation of conservation of energy is
¯m|α − 3H (¯m + p̄m ) ūα = 0 , (7.4.60)
where we have employed definition of the energy (7.3.4), and equation (7.4.59) along with (7.3.6).
7.4.7 Scalar Field Equations

Background equation for the scalar field Ψ̄ is derived from the action (7.3.1) by taking variational deriva-
tives with respect to Ψ̄. It yields
∂ W̄
ḡ αβ Ψ̄|αβ − =0. (7.4.61)
∂ Ψ̄
In terms of the time derivatives with respect to the Hubble time T , equation (7.4.61) reads
Ψ̄ ˙ + ∂ W̄ = 0 .
¨ + 3H Ψ̄ (7.4.62)
∂ Ψ̄
Here, we have taken into account that the background value of the scalar field, Ψ̄, depends only on time
T = T (η), and its derivative with respect to T (denoted with a dot) is proportional to the background
four-velocity
˙ ūα ,
Ψ̄|α = −Ψ̄ (7.4.63)
which follows directly (7.4.20). If we use the definition of the background enthalpy of the scalar field
˙ ,
µ̄q ≡ ūα Ψ̄|α = Ψ̄ (7.4.64)
and account for definition (7.3.26) of the specific energy q of the scalar field, equation (7.4.62) will
become
¯q|α − 3H (¯q + p̄q ) ūα = 0 . (7.4.65)
that looks similar to the hydrodynamic equation (7.4.59) of conservation of energy of the ideal fluid. Be-
cause of this similarity, the second Friedmann equation (7.4.51) can be derived from the first Friedmann
equation (7.4.50) by taking a time derivative and applying the energy conservation equations (7.4.60)
and (7.4.65).
The background density ρ̄q of the scalar filed "fluid" is ρ̄q = µ̄q in accordance with (7.3.25). The
equation of continuity for the density ρ̄q of the ideal fluid is obtained by differentiating definition of ρ̄q ,
and making use of (7.4.62). It yields
∂ W̄
(ρ̄q ūα )|α = − , (7.4.66)
∂ Ψ̄
or, equivalently,
∂ W̄
ρ̄q|α − 3H ρ̄q ūα = ūα , (7.4.67)
∂ Ψ̄
which shows that the "density" ρ̄q of the scalar field "fluid" is not conserved in the most general case of
an arbitrary potential function W̄ (Ψ̄). We emphasize that there is no any violation of physical laws, since
(7.4.67) is simply another way of writing equation (7.4.61), and the scalar field is not thermodynamically
equivalent to the ideal fluid. Equation (7.4.67) is convenient in the calculations that follow in next
sections.
7.4.8 Equations of Motion of Matter of the Localized Astronomical System

Matter of the localized astronomical system is described by the tensor of energy-momentum Tαβ defined
in (7.3.30) in terms of the Lagrangian derivative. It can be given explicitly as a function of field variables
after we chose a specific form of matter, for example, gas, liquid, solid, or something else. We do not
restrict ourselves with a particular form of this tensor, and shall develop a more generic approach that is
applicable to any kind of matter comprising the localized astronomical system..
Background equation of motion of matter of the astronomical system is given by the conservation law
Tαβ |β = 0 . (7.4.68)
It can be also written down in terms of a covariant derivative of the conformal metric
√ √
−ḡTαβ + −ḡ Āα βγ Tβγ = 0 , (7.4.69)
;β
where the connection Āα βγ is defined in (7.4.24). Equation (7.4.68) tells us that the matter of the small-
scale perturbation follows geodesics of the background manifold. This is the starting point for doing the
Post-Newtonian approximations in cosmology. In the geodesic approximation the matter of the isolated
astronomical system has no self-interaction through its own gravitational field. The self-interaction
appears at the next step of the post-Newtonian iteration procedure.
It is natural to write down equation (7.4.68) in 1+3 form by projecting it on the direction of 4-velocity
of the Hubble flow, ūα , and on the hypersurface being orthogonal to it. This is achieved by introducing
the following projections
σ ≡ ūµ ūν Tµν , (7.4.70a)

µν
τ ≡ P̄ Tµν , (7.4.70b)
τα ≡ −P̄α µ ūν Tµν , (7.4.70c)
µ ν
ταβ ≡ P̄α P̄β Tµν , (7.4.70d)
which corresponds to the kinemetric-invariant decomposition 4 of Tµν introduced by A. L. Zelmanov

[123, 124]. Quantity σ is the energy density of matter of the localized system, τα is a density of a linear
momentum of the matter, and ταβ is the stress tensor of the matter.
Equations of motion (7.4.68) of the localized matter can be rewritten in terms of the chronometric
quantities as follows,
(σ ūα + τ α )|α = −Hτ , (7.4.71a)

τ αβ + ūβ τ α = −H (τ α − ūα τ ) , (7.4.71b)
|β
where τ α ≡ ḡ αβ τβ and τ αβ ≡ ḡ αµ ḡ βν τµν . Equation (7.4.71a) is equivalent to the law of conservation

of energy of matter of the localized system. Equation (7.4.71b) is analogues to the Euler equation of
motion of fluid or the equation of the force balance in case of solids.
7.5 Lagrangian Perturbations of FLRW Manifold

7.5.1 The Concept of Perturbations
In the present chapter, FLRW background manifold is defined by the metric ḡαβ which dynamics is
governed by background matter fields - the Clebsch potential Φ̄ of the ideal fluid and the scalar field
Ψ̄. We assume that the background metric and the background values of the fields are perturbed by a
localized astronomical system which is considered as a bare perturbation associated with a field variable
Θ. Perturbations of the metric and the matter fields caused by the bare perturbation are considered to be
small so that the perturbed metric and the matter fields can be split in their backgrounds values and the
corresponding perturbations,
gαβ = ḡαβ + καβ , Φ = Φ̄ + φ , Ψ = Ψ̄ + ψ . (7.5.1)
These equations are exact. We emphasize that all functions entering equation (7.5.1) are taken at one
and the same point of the background manifold. The bare perturbation does not remain the same in
the presence of the perturbations of the metric and the matter fields. Therefore, the field variable Θ
corresponding to the bare perturbation, is also perturbed
Θ = Θ̄ + θ . (7.5.2)
4 This decomposition is also known as a threading approach or 1 + 3 orthonormal frame approach [110].
Section 7.5 Lagrangian Perturbations of FLRW Manifold 291
We consider the perturbations of the metric - καβ , the Clebsch potential - φ, and the scalar field - ψ as
being weak with respect to their corresponding background values ḡαβ , Φ̄, and Ψ̄, which dynamics is
governed by the background equations that have been explained in section 7.4. Because the field variable
Θ is the source of the bare perturbation, we postulate that its background value is equal to zero: Θ̄ = 0.
The perturbations καβ , φ, and ψ have the same order of magnitude as θ.
Perturbation of the contravariant component of the metric is determined from the condition gαγ g γβ =
ḡαγ ḡ γβ = δαβ , and is given by
g αβ = ḡ αβ − κ αβ + κ α γ κ γβ + . . . , (7.5.3)
where the ellipses denote terms of the higher order.

It turns out [50, 96] that a more convenient field variable of the gravitational field in the theory of
Lagrangian perturbations of curved manifolds, is a contravariant (Gothic) metric
√
gαβ = −gg αβ . (7.5.4)
The convenience of the Gothic metric stems from the fact that it enters the de Donder (harmonic) gauge
conditions which significantly simplifies the Einstein equations [75, 111]. The Gothic metric variable
is also indispensable for concise and elegant formulation of dynamic field theories on curved manifolds
[32]. Making use of the Gothic metric allows us to significantly reduce the amount of algebra in tak-
ing the first and second variational derivatives from the Hilbert Lagrangian and the Lagrangian of the
background matter in FLRW metric as explains in the rest of this section.
The covariant Gothic metric gβγ is defined by means of equation
gαβ gβγ = δγα , (7.5.5)

√
that yields gαβ = gαβ / −g. We accept that gαβ is expanded around its background value, ḡαβ =
√
−ḡ ḡ αβ , as follows
gαβ = ḡαβ + hαβ , (7.5.6)
which is an exact equation.
√
Further calculations prompt that it is more suitable to single out −ḡ from hαβ , and operate with a
variable
hαβ
lαβ ≡ √ . (7.5.7)
−ḡ
This variable splits the dynamic degrees of freedom of the gravitational perturbations from the back-
ground manifold which evolves in according with the unperturbed Friedmann equations. Tensor indices
of lαβ are raised and lowered with the help of the background metric, for example, lαβ ≡ ḡαµ ḡβν lµν .
The field variable lαβ relates to the perturbation καβ of the metric tensor. To establish this relationship,
we start from (7.5.4), substitute equation (7.5.6) to its left side, and expand its right side in the Taylor
series with respect to καβ . It results in
∂ ḡαβ 1 ∂ 2 ḡαβ
hαβ = κµν + κµν κρσ + . . . . (7.5.8)
∂ ḡµν 2 ∂ ḡµν ∂ ḡρσ
where the partial derivatives are calculated by successive application of the following rules
∂ ḡαβ 1 √ αµ βν
= − −ḡ ḡ ḡ + ḡ αν ḡ βµ − ḡ αβ ḡ µν , (7.5.9a)
∂ ḡµν 2
∂ ḡ αβ 1 αµ βν
= − ḡ ḡ + ḡ αν ḡ βµ , (7.5.9b)
∂ ḡµν 2
√
∂ −ḡ 1√
= + −ḡ ḡ µν , (7.5.9c)
∂ ḡµν 2
which can be easily confirmed by inspection. Replacing the partial derivatives in (7.5.8) and making use
of the definition (7.5.7), yields the relationship between lαβ and κ αβ as follows

1 1 1 1
lαβ = −κ αβ + ḡ αβ κ + κ µ(α κ β) µ − κ αβ κ − ḡ αβ κ µν κµν − κ 2 + . . . , (7.5.10)
2 2 4 2
where κ ≡ κ σ σ = ḡ ρσ κρσ , and ellipses denote terms of the cubic and higher order in καβ .
Perturbations of four-velocities, wα and v α , entering definitions of the energy-momentum tensors
(7.3.18), (7.3.29), are fully determined by the perturbations of the metric and the potentials of the mat-
ter fields. Indeed, according to definitions (7.3.13) and (7.3.25) the four-velocities are defined by the
following equations
Φ,α Ψ,α
wα = − , vα = − . (7.5.11)
µm µq
p p
where µm = −g αβ Φ,α Φ,β and µq = −g αβ Ψ,α Ψ,β in accordance with (7.3.14) and (7.3.22)
respectively. We define perturbation of the covariant components of the four-velocities as follows
wα = ūα + δwα , vα = ūα + δvα , (7.5.12)
where the unperturbed values of the four-velocities coincide and are equal to the four-velocity of the
Hubble flow due to the requirement of the homogeneity and isotropy of the background FLRW metric.
Substituting these expansions to the left side of definitions (7.5.11), and expanding its right side by
making use of the expansions (7.5.1) and (7.5.3) of the scalar fields and the metric, yields
1 β 1 1 β 1
δwα = − P̄ α φ|β − qūα , δvα = − P̄ α ψ|β − qūα , (7.5.13)
µ̄m 2 µ̄q 2
where we have introduced a new notation
q ≡ −ūα ūβ καβ , (7.5.14)
for the gravitational perturbation of the metric tensor projected on the background four-velocity of the
Hubble flow. Making use of lαβ , the previous equation can be recast to
l
q ≡ ūα ūβ lαβ + , (7.5.15)
2
where l ≡ lα α = ḡ αβ lαβ . Remembering that ḡ αβ = P̄ αβ − ūα ūβ , we can put equation (7.5.15) yet to
another form
1 α β
q≡ ū ū + P̄ αβ lαβ , (7.5.16)
2
which is useful in the calculations that follow.
7.5.2 The Perturbative Expansion of the Lagrangian

We have introduced the Lagrangian of the theory in section 7.3. The Hilbert Lagrangian of the gravi-
√
tational field is Lg = − −gR/16π, where R is the Ricci scalar. The Lagrangian density of matter
√ √
is Lm = −gLm (Φ, gαβ ), and the Lagrangian density of the scalar field Lq = −gLq (Ψ, gαβ ).
The matter, the scalar field as well as the spacetime manifold are perturbed by a matter of an iso-
lated astronomical system described by a set of field variables Θ with the Lagrangian density Lp =
√
−gLp (Θ, gαβ ).
The action of the unperturbed FLRW metric is a functional
Z
S̄ = d4 xL̄ , (7.5.17)
M
depending on the unperturbed Lagrangian
L̄ = L̄g + L̄m + L̄q , (7.5.18)
taken on the background values of the field variables ḡαβ , Φ̄, and Ψ̄.
The presence of a localized astronomical system perturbs the spacetime manifold and the background
values of the field variables. The perturbed Lagrangian becomes an algebraic sum of four terms
L = Lg + Lm + Lq + Lp , (7.5.19)
where the Lagrangian Lp describes the bare perturbation, and Lg , Lm , Lq are perturbed values of the
Lagrangian of the FLRW metric.
The perturbed Lagrangian can be decomposed in a Taylor series with respect to the perturbed values
of the field variables. It is achieved by substituting expansions (7.5.1) to the Lagrangian (7.5.19) and
expanding it around the background values of the variables in an infinite Taylor series. It yields
∞
X
L = Lp + Ln , (7.5.20)
n=0
where Lp is the Lagrangian of the bare perturbation, L0 ≡ L̄ is the Lagrangian describing the dynamic
properties of the background manifold, and for any n ≤ 1,

1 δLn−1 δLn−1 δLn−1
Ln = hµν + φ + ψ , (7.5.21)
n δ ḡµν δ Φ̄ δ Ψ̄
is the Lagrangian perturbation defined iteratively by taking variational derivatives from the Lagrangian
perturbations of the previous iteration. In particular,
δ L̄ δ L̄ δ L̄
L1 = hµν µν + φ +ψ , (7.5.22a)
δ ḡ δ Φ̄ δ Ψ̄

1 δL1 δL1 δL1
L2 = hµν µν + φ +ψ , (7.5.22b)
2 δ ḡ δ Φ̄ δ Ψ̄
and so on. Here, the variational derivatives from the Lagrangian density, L̄, depending on the field
variables and their derivatives, are defined as follows (see section 7.2.1 for more detail)
δLn ∂Ln ∂ ∂Ln ∂2 ∂Ln

≡ − + , (7.5.23a)
δ ḡµν ∂ ḡµν α µν
∂x ∂ ḡ ,α ∂x ∂xβ ∂ ḡµν ,αβ
α
2
δLn ∂Ln ∂ ∂Ln ∂ ∂Ln
≡ − + , (7.5.23b)
δ Φ̄ ∂ Φ̄ ∂xα ∂ Φ̄,α ∂xα ∂xβ ∂ Φ̄,αβ
δLn ∂Ln ∂ ∂Ln ∂2 ∂Ln
≡ − + , (7.5.23c)
δ Ψ̄ ∂ Ψ̄ ∂xα ∂ Ψ̄,α ∂xα ∂xβ ∂ Ψ̄,αβ
The variational derivative with respect to the metric density ḡµν relates to the derivative with respect to
the metric ḡ µν by an algebraic operator
δ ∂ ḡ αβ δ 1 α β δ
µν
= µν αβ
= √ δµ δν + δνα δµβ − ḡµν ḡ αβ . (7.5.24)
δ ḡ ∂ ḡ δ ḡ 2 −ḡ δ ḡ αβ
One has to notice that the expansion (7.5.20) is defined up to the divergence terms which are rep-
resented as a total covariant derivative from a vector density. The divergences can be important in the
discussion of the boundary conditions but they do not enter equations of motion of fields which represent
a system of the differential equations in partial derivatives for the perturbations of the dynamic (field)
variables. Furthermore, it is straightforward to prove that any of the Lagrangian derivatives (7.5.23a)-
(7.5.23c), applied to a partial derivative of a geometric object F = F (ḡαβ ; Φ̄; Ψ̄; ḡαβ ,γ ; Φ̄,γ ; Ψ̄,γ ; . . .),
vanishes [90] (see (7.2.10))

δ ∂F δ ∂F δ ∂F
= 0 , = 0 , =0. (7.5.25)
δ ḡαβ ∂xα δ Φ̄ ∂xα δ Ψ̄ ∂xα
Equations (7.5.25) does not hold for a covariant derivative [90]. We shall use equation (7.5.25) for
simplification of the Lagrangian derivatives.
The field equations are obtained by taking the variational derivatives from the perturbed action with
respect to various variables subject to the least action principle. In accordance with this principle, the
variational derivatives from the perturbed Lagrangian must vanish,
δLg δLm δLq δLp

+ + = − , (7.5.26a)
δgαβ δgαβ δgαβ δgαβ
δLm
= 0, (7.5.26b)
δΦ
δLq
= 0. (7.5.26c)
δΨ
The Post-Newtonian approximations of these equations are easily derived by the following proce-
dure. We substitute the Taylor decomposition (7.5.20) of the Lagrangian to equations (7.5.26a) and
separate the background value of the derivatives from their perturbed values. We assume that gravita-
tional dynamics of the unperturbed FLRW metric obeys the background field equations shown below
in subsection 7.5.3. Then, the perturbed part of the equations represent a series of the post-Newtonian
equations of the first, second, third, etc. order, which can be solved by successive iterations. In this
chapter we restrict ourselves with the linearized approximation of the first order with respect to the per-
turbations. It generalizes the first post-Newtonian field equations in asymptotically flat spacetime to the
case of the expanding universe.
7.5.3 The Background Field Equations

Dynamics of the background universe is governed exclusively by the background matter. Hence, we
have to take into account only the background values of functions entering variational Euler-Lagrange
equations (7.5.26a)-(7.5.26c) and drop out the Lagrangian Lp of the bare perturbation. We get
δ L̄g δ L̄m δ L̄q

+ + = 0, (7.5.27a)
δ ḡαβ δ ḡαβ δ ḡαβ
δ L̄m
= 0, (7.5.27b)
δ Φ̄
δ L̄q
= 0. (7.5.27c)
δ Ψ̄
After performing the derivatives, equation (7.5.27a) becomes the Einstein equation (7.4.43), equation
(7.5.27b) is reduced to equation of continuity (7.4.58) after taking into account (7.4.55), (7.4.56), and
equation (7.5.27c) is equivalent to (7.4.61). These equations have been thoroughly discussed in section
7.4. Solution of these equations depends on the equation of state of the background matter. We assume
that the solution exists and that the time dependence of the FLRW metric ḡαβ = ḡαβ (η), the Clebsch
potential Φ̄ = Φ̄(η), and the scalar field Ψ̄ = Ψ̄(η) are explicitly known.
7.5.4 The Lagrangian Equations for Gravitational Field Perturbations
We substitute the Taylor decomposition of the Lagrangian in equation (7.5.26a) and account for the
background field equations (7.5.27a)-(7.5.27c). After taking the variational derivatives we find out [70]
that the gravitational field perturbations obey the following (exact) differential equation
Fµν = 8π (Tµν + Tµν ) , (7.5.28)
which generalizes the Einstein field equations in asymptotically flat spacetime to the case of the expand-
ing FLRW metric. Tensor Fµν is an algebraic superposition
g m q
Fµν ≡ Fµν + Fµν + Fµν , (7.5.29)
where the linear operators in the right side are defined through the Lagrangian derivatives as follows,
αβ δ L̄
g

g 16π δ
Fµν ≡ −√ h , (7.5.30a)
−ḡ δ ḡ µν δ ḡαβ
δ L̄m δ L̄m

16π δ
Fµνm
≡ −√ µν
hαβ αβ + φ , (7.5.30b)
−ḡ δ ḡ δ ḡ δ Φ̄
δ L̄ q
δ L̄ q

16π δ
Fµνq
≡ −√ hαβ αβ + ψ . (7.5.30c)
−ḡ δ ḡ µν δ ḡ δ Ψ̄
The right side of equation (7.5.28) contains the tensor of energy-momentum Tµν of the bare gravita-
tional perturbation which is generated by the matter of the localized astronomical system and can be
calculated as the Lagrangian derivative (7.3.30) . The right side of (7.5.28) contains the non-linear
corrections of the second and higher order of magnitude. They are given by

2 δL2 δL3
Tµν = √ + + . . . . (7.5.31)
−ḡ δ ḡ µν δ ḡ µν
Tensor Tµν can be split in two algebraically-independent parts
Tµν = tµν + τµν , (7.5.32)
where tµν is the stress-energy tensor of pure gravitational perturbations hµν while τµν is the stress-
energy tensor characterizing gravitational coupling of the matter field φA with the gravitational pertur-
bations hµν . If we restrict ourselves only with the second order non-linear corrections, the corresponding
stress-energy tensors are given by variational derivatives

1 δ ρσ g 1 ρσ g
tµν = − √ h F ρσ − h ḡ F ρσ , (7.5.33)
16π −ḡ δ ḡ µν 2
√

1 δ 1
τµν = − √ µν
hρσ Fρσ
m
− hg ρσ Fρσm
+ −ḡφFΦm (7.5.34)
16π −ḡ δ ḡ 2
√

1 δ 1 ρσ q
− √ hρσ Fρσ
q
− hg Fρσ + −ḡψFΨq ,
16π −ḡ δ ḡ µν 2
where FΦm and FΨq are defined below in (7.5.47), (7.5.53).

It is worth noticing that the gravitational stress-energy tensor tµν can be derived in exact analytic
form and reads [50]

1 ρ σ 1 ρσ

tµν = δµ δν − ḡµν ḡ Gα ρβ Gβ σα − Gα ρσ Gβ αβ (7.5.35)
8π 2

1 1 1
+ hµν ḡ ρβ Gα αβ − ḡµν hαβ Gρ αβ − hρ (µ Gα ν)α
8π 2 2
i
+ hρ β ḡα(µ Gα ν)β + hβ (µ Gρ ν)β − hβ (µ ḡν)α ḡ ρσ Gα βσ ,
|ρ
α α α
where G βγ ≡ Γ βγ − Γ̄ βγ is the difference between the Christoffel symbols of the perturbed and
unperturbed (background) manifolds
1 αρ
Gα βγ

= g κρβ|γ + κργ|β − κβγ|ρ , (7.5.36)
2
where κµν ≡ gµν − ḡµν . We emphasize that the geometric object Gα βγ is a tensor since it represents
the difference between the two Christoffel symbols defined on one and the same background manifold
possessing two metrics, ḡµν and gµν = ḡµν + κµν . It does not mean, of course, that we employ a
bi-metric theory of gravity being different from general theory of relativity. The background metric ḡµν
is simply the lowest (unperturbed) state of the full metric gµν , and each metric obeys the Einstein field
equations. As the background metric ḡµν , its perturbation κµν , and the symbol Gα βγ are tensors, tµν
is also a tensor. It defines energy density, a linear momentum density, and other physical characteristics
of the gravitational perturbations at each point of the background spacetime [95]. Contribution of Tµν
to the linearised field equations should be neglected as it is of the higher order as compared with other
terms in (7.5.28).
g
The differential operator, Fµν , represents a linearized perturbation of the Ricci tensor, and after cal-
culation of the variational derivative in (7.5.30a), is given by
1 |α
g
Fµν = lµν |α + ḡµν lαβ |αβ − lαµ|ν |α − lαν|µ α , (7.5.37)
2
where each vertical bar denotes a covariant derivative with respect to the background metric ḡµν .
m q
Operators Fµν and Fµν depend essentially on a particular choice of the Lagrangian of matter and
scalar field, and take on different forms depending on the specific analytic dependence of Lm and Lq on
the field variables. In the particular case of the ideal fluid, the term embraced in the round parentheses
in the right side of equation (7.5.30b) is
δ L̄m δ L̄m √

1 1
hαβ αβ + φ = hαβ T̄αβ m
− ḡαβ T̄ m + φ ∂α ρ̄m −ḡ ūα ,

(7.5.38)
δ ḡ δ Φ̄ 2 2
where ūα ≡ −ḡ αβ Φ̄,β /µ̄m , and T̄αβ

m
is given in (7.3.18). We emphasize that though the ideal fluid sat-
isfies the equation of continuity (7.4.58), it should not be immediately implemented in (7.5.38) because
this expression is to be further differentiated with respect to the metric tensor according to (7.5.30b).
For the scalar field, the term enclosed to the round parentheses in the right side of (7.5.30c) is
δ L̄q δ L̄q √ ∂ W̄ √

1 1
hαβ αβ + ψ = hαβ T̄αβ q
− ḡαβ T̄ q + ψ + ∂α ρ̄q −ḡ ūα

−ḡ , (7.5.39)
δ ḡ δ Ψ̄ 2 2 ∂ Ψ̄
q
where ūα ≡ −ḡ αβ Ψ̄,β /µ̄q , ρ̄q = µ̄q , T̄αβ is given in (7.3.29), and the equation of continuity for the
scalar field (7.4.66) should not be implemented until differentiation with respect to the metric tensor
(7.5.30c) is completed.
Taking the variational derivatives with respect to ḡ µν from the expressions (7.5.38) and (7.5.39),
and applying thermodynamic equations (7.3.11), allows us to write down the right sides of equations
(7.5.30b), (7.5.30c) in a more explicit form as follows,
c2

m
Fµν = −4π (p̄m − ¯m )lµν + 1 − 2 (¯m + p̄m )qūµ ūν (7.5.40)
vs
c2

+8π ρ̄m ūµ φ,ν + ūν φ,µ + 1 − 2 ūµ ūν − ḡµν ūα φ,α , ,
vs

∂ W̄
Fµνq
= −4π (pq − q ) lµν − 2ḡµν ψ + 8π ρ̄q (ūµ ψ,ν + ūν ψ,µ − ḡµν ūα ψ,α ) (7.5.41)
,
∂ Ψ̄
where ρ̄q ≡ Ψ̄ ˙ = µ̄q in accordance with definition (7.3.25). The potential energy of the scalar field,
W̄ = W̄ (Ψ̄), is kept arbitrary.
It is important to emphasize that in the most general case the ratio vs2 /c2 of the speed of sound in fluid
to the fundamental speed c, may be not equal to the parameter wm of the equation of state (7.4.54), that
is there are physical equations of state such that wm 6= (vs /c)2 . Indeed, the speed of sound is defined as
a partial derivative of pressure pm with respect to the energy density m taken under the condition of a
constant entropy sm ,
vs2

∂pm
2
= . (7.5.42)
c ∂m sm =const.
This equation is equivalent to the following relation
vs2 (∂pm /∂µm )sm =const.

= , (7.5.43)
c2 (∂m /∂µm )sm =const.
which is a consequence of thermodynamic relations and a definition of the partial derivative. The
ratio of the partial derivatives in (7.5.43) is not reduced to wm in case when wm depends on some
other thermodynamic parameters which are functions of the specific enthalpy. For example, in case
of an ideal gas the equation of state pm = wm m , where wm = kT /mc2 , k is the Boltzmann con-
stant, m - mass of a particle of the ideal fluid, and T is the fluid temperature. The speed of sound
vs2 = c2 (∂pm /∂m )sm =const. = Γwm > wm = pm /m , where Γ > 1 is the ratio of the heat capacities of
the gas taken for the constant pressure and the constant volume respectively [74].
The scalar field with the potential function W (Ψ) 6= 0 does not bear all thermodynamic properties
of an ideal fluid. Nevertheless, we can formally define the speed of "sound" cs propagating in the scalar
field "fluid", by equation being similar to (7.5.43). More specifically,
c2s (∂pq /∂µq )Ψ=const.

= . (7.5.44)
c2 (∂q /∂µq )Ψ=const.
Simple calculation reveals that the speed of "sound" for scalar field is always equal to the fundamental
speed
cs = c , (7.5.45)
irrespectively of the value of the potential function W (Ψ). It explains why the terms being proportional
to the factor 1 − c2 /c2s , do not appear in the expression (7.5.41) as contrasted with (7.5.40).
7.5.5 The Lagrangian Equations for Dark Matter Perturbations

The perturbed field equations for the dark matter which is modelled by the ideal fluid, are obtained by
taking the variational derivatives with respect to the field Φ from the Lagrangian (7.5.19) - it corresponds
to the middle equation in (7.5.26a). Taking into account the background equation (7.5.27b) yields the
equation of sound waves propagating in the fluid as small perturbations of the potential φ,
FΦm = 8πΣm , (7.5.46)
where the linear differential operator
δ L̄m δ L̄m

1 δ
FΦm ≡ − √ hµν µν
+φ , (7.5.47)
−ḡ δ Φ̄ δ ḡ δ Φ̄
and the source term

δLm δLm

1 2
Σm ≡ √ + 3
+ ... . (7.5.48)
8π −ḡ δ Φ̄ δ Φ̄
In the case of a single-component ideal fluid, the Lagrangian (7.3.16) depends merely on the derivative
of the Clebsch potential Φ and on the metric tensor. Therefore, the explicit form of the linear operator
FΦm is reduced to a covariant divergence
FΦm = Y α |α , (7.5.49)
where a vector field

∂ L̄m ∂ L̄m

∂ 1 µν 1
Yα ≡ lµν − lḡ − ḡµν L̄m + φ,β , (7.5.50)
∂ Φ̄,α 2 ∂ ḡ µν 2 ∂ Φ̄,β
and the partial derivatives are taken from the Lagrangian Lm . More specifically, calculations yield
c2

ρ̄m |α ρ̄m α β 1
Yα ≡ φ − ρ̄m lαβ ūβ + 1 − 2 ū ū φ|β − ρ̄m ūα q . (7.5.51)
µ̄m vs µ̄m 2
Similar expression was derived by Lukash [83] who used the variational method to analyze the produc-
tion and quantization of sound waves in the early universe.
7.5.6 The Lagrangian Equations for Dark Energy Perturbations

Equations for the perturbations ψ of dark energy, which is modelled by a scalar field Ψ, are derived by
taking the variational derivative from the Lagrangian (7.5.19) with respect to the field variable Ψ - see
equation (7.5.26c). Subtracting the background equation (7.5.27c) from (7.5.26c) and making use of the
Lagrangian decomposition in the Taylor (post-Newtonian) series leads to
FΨq = 8πΣq , (7.5.52)
where the linear differential operator
δ L̄q δ L̄q

1 δ
FΨq ≡ − √ hµν µν + ψ , (7.5.53)
−ḡ δ Ψ̄ δ ḡ δ Ψ̄
and the source term

δLq2 δLq3

1
Σq ≡ √ + + ... . (7.5.54)
8π −ḡ δ Ψ̄ δ Ψ̄
√
According to equation (7.3.19), the Lagrangian density of the scalar field Lq = −gLq depends on
both the field Ψ and its first derivative, Ψ,α . For this reason, the differential operator F q is not reduced
to the covariant derivative from a vector field as the partial derivative of the Lagrangian with respect to
Ψ does not vanish. We have
l ∂ W̄ ∂ 2 W̄
FΨq ≡ Z α |α − −ψ (7.5.55)
2 ∂ Ψ̄ ∂ Ψ̄2
where l ≡ ḡ αβ lαβ , and vector field
∂ L̄q ∂ L̄q

∂ 1 1
Zα ≡ lµν − lḡ µν − ḡµν L̄q
+ ψ ,β . (7.5.56)
∂ Ψ̄,α 2 ∂ ḡ µν 2 ∂ Ψ̄,β
Performing the partial derivatives in equation (7.5.56), yields a rather simple expression
Z α ≡ ψ |α − ρ̄q lαβ ūβ , (7.5.57)
where we have used equation Ψ̄|α = −ūβ Ψ̄|β ūα = −ρ̄q ūα . The reader is invited to compare equation
(7.5.57) with (7.5.51) to observe the differences between the Lagrangian perturbations of the ideal fluid
and the scalar field. One may observe that (7.5.51) becomes identical with (7.5.57) in the limit vs → c,
and ρ̄m → µ̄m . This corresponds to the case of an extremely stiff equation of state wm = 1 in equation
(7.4.54). According to the discussion following equations (7.5.44), (7.5.45) the speed of ’sound’ cs in
the scalar field ’fluid’ is always equal to c. However, it does not assume that the parameter wq of the
equation of state of the scalar field, p̄q = wq ¯q , in (7.4.54) is equal to unity. This is because the scalar
field is not completely equivalent to the ideal fluid in the sense of thermodynamic [2]. .
7.5.7 Linearised post-Newtonian Equations for Field Variables
Equations for the metric tensor perturbations
Linearized field equations for gravitational field variables, lµν , are obtained from (7.5.28) after neglect-
ing in its right side the non-linear source Tµν , and making a series of transformations to sort out similar
terms. First, let us make use of equations (7.5.40) and (7.5.41) to find out
m q
Fµν + Fµν = 4π (¯ − p̄) lµν (7.5.58)
2

c 1
+ 8π ρ̄m ūµ φ,ν + ūν φ,µ − ḡµν uα φ,α + 1 − 2 ūα φ,α − µ̄m q ūµ ūν
vs 2

∂ W̄ ( Ψ̄) ψ
+ 8π ρ̄q ūµ ψ,ν + ūν ψ,µ − ḡµν uα ψ,α + ḡµν ,
∂ Ψ̄ µ̄q
where we used the superposition ¯ = ¯m + ¯q , p̄ = p̄m + p̄q . Second step is to transform the linear
g
differential operator Fµν in (7.5.37) to a more convenient form that will allow us to single out the gauge-
dependent vector denoted by
Aµ ≡ lµν |ν . (7.5.59)
Changing the order of the covariant derivatives in (7.5.37) and taking into account that the commutator of
the second covariant derivatives is proportional to the Riemann tensor, we recast (7.5.37) to the following
form,
1 |α
g
Fµν ≡ lµν |α + ḡµν Aα |α − Aµ|ν − Aν|µ − R̄α (µ lν)α − R̄µαβν lαβ , (7.5.60)
2
where the round brackets around indices denote symmetrization. The terms with the Ricci and Riemann
tensors can be expressed in terms of the total background energy and pressure of the ideal fluid and
scalar field by making use of equations (7.4.34), (7.4.36) and Einstein’s equations (7.4.43). It yields

5¯ l ¯
Rα (µ lν)α + R̄µαβν lαβ = 4π − p̄ lµν + p̄ − ḡµν (7.5.61)
3 2 3

+ (¯ + p̄) (2ūα ūµ lνα + 2ūα ūν lµα − ūµ ūν l − ḡµν q) .
Finally, substituting equations (7.5.58), (7.5.60) and (7.5.61) to (7.5.28) results in
lµν |α |α + ḡµν Aα |α − Aµ|ν − Aν|µ (7.5.62)

¯ l ¯ 1 α 1 1
−16π lµν + p̄ − ḡµν + (¯ + p̄) ū ū(µ lν)α − ūµ ūν l − ḡµν q
3 4 3 2 2 2
2

c 1
+16π ρ̄m ūµ φ,ν + ūν φ,µ − ḡµν uα φ,α + 1 − 2 ūα φ,α − µ̄m q ūµ ūν
vs 2

∂ W̄ (Ψ̄) ψ
+16π ρ̄q ūµ ψ,ν + ūν ψ,µ − ḡµν uα ψ,α + ḡµν = 16πTµν ,
∂ Ψ̄ µ̄q
where the non-linear term, Tµν , was neglected in the right side of (7.5.62).
The very first term in (7.5.62) is a tensor Laplace-Beltrami operator, lµν |α |α ≡ ḡ αβ lµν|αβ , that is
a rather complicated geometric object. Its explicit expression can be developed by making use of the
Christoffel symbols given in (7.4.23). Tedious but straightforward calculation yields [70]
lµν |α |α = lµν ;α ;α + 2H ūα lµν;α − 2 (H ūα lαµ )|ν − 2 (H ūα lαν )|µ (7.5.63)
α α
+ 2H (ūµ Aν + ūν Aµ ) + 2Ḣ (lµν − ū ūµ lνα − ū ūν lµα )

+ 2H 2 2lµν + 3ūµ ūα lαν + 3ūν ūα lαµ − ḡµν ūα ūβ lαβ − ūµ ūν l ,
where the semicolon denotes a covariant derivative that is calculated with the Christoffel symbols B α µν
like in (7.4.30b), and the spatial Laplace-Beltrami operator, lµν ;α ;α ≡ ḡ αβ lµν;αβ .
Further derivation of the differential post-Newtonian field equation for the linearized metric tensor
perturbations can be significantly simplified if we re-define the gauge function, Aα , in the following
form
Aα = −2Hlαβ ūβ + 16π (ρ̄m φ + ρ̄q ψ) ūα + B α , (7.5.64)
where B α is an arbitrary gauge vector function. This choice of the gauge function Aα allows us to
eliminate two terms in equation (7.5.63) which depend on the first covariant derivatives with respect to
the background metric ḡαβ . Moreover, it allows to eliminate a number of terms depending on the first
derivatives of the fields φ and ψ in equation (7.5.62). Since we keep the gauge function B α arbitrary,
the equation (7.5.64) does not fix any gauge. The choice of the gauge is controlled by the gauge function
Bα.
One substitutes the gauge function (7.5.64) to equations (7.5.63) and (7.5.62) and make use of the
background Friedmann equations (7.4.50), (7.4.51) to replace the background values of the energy den-
sity, ¯, and pressure, p̄, with the Hubble parameter H and its time derivative Ḣ. It brings about equation
(7.5.62) to the following form
lµν ;α ;α + 2H ūα lµν;α + 2 Ḣ + H 2 (lµν + ūµ ūα lαν + ūν ūα lαµ − lūµ ūν )

(7.5.65)

2k α α l
− 2 lµν + 2ūµ ū lαν + 2ūν ū lαµ − lūµ ūν − q + ḡµν
a 2
c2

1 ∂ W̄
+16π ūµ ūν ρ̄m 1 − 2 ūα φ,α − µ̄m q − 2 ψ − 4H (ρ̄m φ + ρ̄q ψ)
vs 2 ∂ Ψ̄
+ḡµν B |α − Bµ|ν − Bν|µ + 2H (ūµ Bν + ūν Bµ − ḡµν ūα Bα )
α
= 16πTµν .
This equation is fully covariant and is valid in any gauge and/or coordinate chart.
Now, let us fix the gauge by selecting a specific gauge function B α in (7.5.64). The task is to
decouple the linearised field equations for l00 , l0i and lij components of the metric tensor perturba-
tions. For this purpose, let us work in the isotropic coordinates associated with the Hubble flow, where
ūα = (1/a,0,0,0) and choose the gauge condition, B α = 0. It brings equation (7.5.65) for different
components of the metric perturbations to the form
c2

q + 2Hq;0 + 4kq − 4π 1 − 2 ρ̄m µ̄m q = 8πa2 (T00 + Tkk ) − (7.5.66a)
vs
c2

8πa3 1 − 2 ρ̄m φ,0 −
vs

∂ W̄
2a ψ − H (ρ̄m φ + ρ̄q ψ) ,
∂ Ψ̄
l0i + 2Hl0i;0 + 2kl0i = 16πa2 T0i , (7.5.66b)
l<ij> + 2Hl<ij>;0 + 2 H0 − k l<ij> 2

= 16πa T<ij> , (7.5.66c)
lkk + 2Hlkk;0 + 2 H0 + 2k lkk 16πa2 Tkk .

= (7.5.66d)
Here, H is a conformal Hubble parameter (7.4.4), a prime means a derivative with respect to η, and we
denoted lµν ≡ f̄αβ lµν;αβ , q ≡ (l00 + lkk ) /2, lkk ≡ l11 +l22 +l33 , l<ij> ≡ lij −(1/3)δij lkk , and the
same index notations are applied to the tensor of energy-momentum Tij of the localized astronomical
system, T<ij> = Tij − (1/3)δij Tkk . These equations are clearly decoupled from one another, thus,
demonstrating the advantage of the gauge condition B α = 0.
Equations (7.5.66b)–(7.5.66d) can be solved independently if the initial and boundary conditions are
known, and the tensor of energy-momentum of the localized astronomical system is well-defined. Equa-
tion (7.5.66a) for a scalar q demands besides knowledge of Tαβ , knowing the scalar field perturbations,
φ and ψ, that contribute to the source of the field equation for q in the right side of (7.5.66a). Equations
for these perturbations are obtained below.
Equations for the dark matter perturbations
The dark matter perturbations, φ, evolve in accordance with the Lagrangian equation (7.5.46). In the
linear approximation we can neglect the non-linear source term Σm in its right side. The covariant
derivative in the definition of the linear operator F m given in the right side of (7.5.49), can be explicitly
performed, thus, yielding equation for the Clebsch potential
c2

1
φ|α |α − 2µ̄m Hq + 16π µ̄m (ρ̄m φ + ρ̄q ψ) + 1 − 2 ūα ūβ φ|αβ − µ̄m ūα q,α = µ̄m ūα Bα ,
vs 2
(7.5.67)
where equation (7.5.64 has been used. The gauge B α remains yet unspecified so that equation (7.5.67) is
covariant and is valid in any coordinate chart. To make it compatible with equations (7.5.66a)-(7.5.66d)
for the metric tensor perturbations, we have to choose B α = 0.
Equations for the dark energy perturbations

Linearized equation for the dark energy perturbations, ψ, is obtained from the Lagrangian equation
(7.5.52) after neglecting the (non-linear) source term Σq . After performing the covariant differentiation
in equation (7.5.55), we conclude that the dark energy perturbation obeys the following equation
∂ 2 W̄

∂ W̄
ψ |α |α − 2µ̄q H + q + 16π µ̄q (ρ̄m φ + ρ̄q ψ) − ψ = µ̄q ūα Bα , (7.5.68)
∂ Ψ̄ ∂ Ψ̄2
where equation (7.5.64) has been used along with the equality ρ̄q = µ̄q . The gauge function B α is kept
unspecified so that equation (7.5.68) is covariant and is valid in any coordinates. To make it compatible
with equations (7.5.66a)-(7.5.66d) for the metric tensor perturbations, we have to choose B α = 0.
7.6 Gauge-invariant Scalars and Field Equations

in 1+3 Threading Formalism
7.6.1 Threading Decomposition of the Metric Perturbations.
We have derived the system of coupled differential equations (7.5.65), (7.5.67), (7.5.68) for the field
variables lαβ , φ and ψ, describing perturbations of the gravitational field, dark matter and dark energy
respectively. These equations are gauge-invariant and written down in arbitrary coordinates on the back-
ground manifold. Nonethelees, they operate with the field variables which are not gauge-invariant in
themselves. Therefore, solutions of equations (7.5.65), (7.5.67), (7.5.68) that are found in a particular
gauge has no direct physical interpretation and must be connected to physical observables to match the-
ory with observations. Another way around is to find out some gauge-invariant geometric objects built
out of lαβ , φ and ψ which will not depend on a particular choice of gauge and coordinates. This pro-
gram was initiated by Bardeen [8] who proposed to split the perturbations of the metric tensor in scalar,
vector, and tensor components by making use of 3+1 spacetime slicing ADM technique [5], and to build
gauge-invariant cosmological variables out of these elements. Gauge-invariant scalars are the most im-
portant quantities in cosmology as they describe the structure formation in the universe. Ellis and Bruni
[39] pointed out that Bardin’s variables are not directly related to the density fluctuations but to it second
derivatives which makes them less useful in relativistic calculations of structure formation. They pro-
posed their own gauge-invariant variables that are build out of gradients of the geometric objects which
vanish on the background manifold so that only their perturbations make physical sense.
In this section we propose even more direct approach to the definition of the gauge-invariant scalars
by making use of the scalar potentials Φ and Ψ for description of the dark matter and dark energy. In
this way we shall find out the gauge-invariant scalars that are equivalent to the matter density fluctuation
itself but not to its gradient or a second order derivative. We shall employ 1+3 threading approach to
split four-dimensional tensors into scalar, vector, and three-dimensional tensors. The original idea was
proposed by A.L. Zelmanov [121] who called the elements of the tensorial decomposition the chrono-
metric invariants. Later on, the theory of chronometric invariants was reinvented by a number of re-
searchers. The central ingredient of the theory is a congruence of worldlines threading spacetime. In
FLRW cosmology, this congruence is naturally associated with the Hubble flow and the Hubble velocity
ūα . Threading (chronometric) decomposition is achieved with the invariant operator of projection P̄αβ
onto a hypersurface being orthogonal to the congruence of world lines of the Hubble flow,
P̄αβ = ḡαβ + ūα ūβ , (7.6.1)

Section 7.6 Gauge-invariant Scalars and Field Equations
in 1+3 Threading Formalism 303
where ḡαβ is FLRW background metric. The operator P̄αβ can be considered as a metric on the spatial
hypersurface of the background FLRW manifold.
The post-Newtonian theory under development admits four, algebraically-independent scalar pertur-
bations. Two of them are the Clebsch potential of the ideal fluid φ and the scalar field ψ. The two other
scalars characterize the scalar perturbations of the gravitational field. They can be chosen, for example,
as a projection of the metric tensor perturbation on the direction of the background four-velocity field,
ūα ūβ lαβ , and the trace of the metric tensor perturbation, l = ḡ αβ lαβ . However, it is more convenient
to work with two other scalars, defined as their linear combinations,
1 α β
q ≡ ū ū + P̄ αβ lαβ , (7.6.2a)
2
p ≡ P̄ αβ lαβ , (7.6.2b)
Notice that the scalar q has been introduced earlier in (7.5.16). The scalar p is, in fact, a projection of lαβ
onto the space-like hypersurface being orthogonal everywhere to the worldlines of Hubble observers.
Vectorial chronometric perturbations are defined by a spacial-temporal projection
pα ≡ −P̄α β ūγ lβγ , (7.6.3)
where minus sign was taken for the sake of mathematical convenience. Due to its definition, vector
pα = ḡ αβ pβ is orthogonal to the four-velocity ūα , that is ūα pα = 0. Hence, it describes a space-like
vector-like gravitational perturbations with three algebraically-independent components.
Tensorial chronometric perturbations are associated with the projection
1
p|αβ ≡ pαβ − P̄αβ p , (7.6.4)
3
where
pαβ ≡ P̄α µ P̄β ν lµν . (7.6.5)
Here, the tensor pαβ is a double projection of lαβ onto space-like hypersurface being orthogonal to the
worldlines of Hubble observers. The trace of this tensor coincides with the scalar p. Indeed,
ḡ αβ pαβ = ḡ αβ P̄α µ P̄β ν lµν = P̄ βµ P̄β ν lµν = P̄ µν lµν = p , (7.6.6)
where the property of the projection tensor P̄ βµ P̄β ν = P̄ µν has been used. Equation (7.6.6) makes it
clear that tensor p|αβ is traceless, that is ḡ αβ p|αβ = 0. Because of this property, and four orthogonality
conditions, ūα p|αβ = 0, the symmetric tensor p|αβ has only five, algebraically-independent components.
Gravitational perturbation lαβ can be decomposed into the algebraically-irreducible scalar, vector and
tensor parts as follows

1
lαβ = p|αβ + ūα pβ + ūβ pα + ūα ūβ + P̄αβ p + 2ūα ūβ (q − p) . (7.6.7)
3
One should not confuse the pure algebraic (threading) decomposition of the metric tensor perturbation
with its functional (slicing) decomposition. The slicing (or kinemetric, according to A.L. Zelmanov
[122]) decomposition was pioneered by Arnowitt, Deser and Misner [5, 89]. It is commonly used in the
research on the relativistic theory of formation of the large-scale structure in the universe. The ADM
decomposition of the metric tensor perturbations is done by foliating spacetime [8, 62] with a set of
spacelike hypersurfaces and making use of three dimensional Helmholtz theorem [7] which singles out
the longitudinal (L), transversal (T) and transverse-traceless (TT) parts of the perturbations. In other
words, the slicing decomposition make vector pα and tensor parts of the gravitational perturbation, p|αβ ,
are further decomposed in the functionally-irreducible components which include two more scalars,
and two transverse spatial vectors each having only two (out of three) independent components. The
remaining part of the tensor perturbations, p|αβ , is transverse-trackless and has only two functionally-
independent components denoted as pTT αβ . The ADM decomposition of the metric tensor is a powerful
technique in the theory of gauge-invariant cosmological perturbations [9, 91]. However, it is not con-
venient in the development of the systematic Post-Newtonian approximations and celestial dynamics of
inhomogeneities in cosmology. Thus, we do not use it in the present chapter.
Our next step is a to find the gauge-invariant scalars directly reproducing the density fluctuation and
to derive the post-Newtonian field equations for the algebraically-irreducible components of matter and
gravitational field. We, first, discuss the gauge transformations of the corresponding field variables.
7.6.2 Gauge Transformation of the Field Variables
Gauge invariance is a cornerstone of the modern theoretical physics with an interesting but somehow
controversial history [58]. The gauge invariance should be distinguished from coordinate (diffeomor-
phism) invariance or general covariance because, by definition, gauge transformation changes merely
the field variables of the theory under consideration but not coordinates on the underlying spacetime
manifold. Discussing gauge-transformation and gauge-invariance requires introduction of a supplemen-
tary gauge field and associated with it geometric structures – an affine connection and a fibre bundle
manifold describing the intrinsic degrees of freedom of corresponding field variables of the gauge field
theory [33, 101].
The present chapter discusses physical perturbations of tensor lαβ , and scalars Φ, Ψ in the framework
of general relativity where the affine connection is represented by the Christoffel symbols of spacetime
manifold while the gauge transformation is generated by a flow of an arbitrary vector (gauge) field ξ α
that maps the manifold into itself. Generic gauge transformation of the fields on a curved manifold
is associated with their Lie transport along the vector flow ξ α [75, 113] while an infinitesimal gauge
transformation is a Lie derivative of the field taken at the value of the parameter on the curves of the
vector flow equal to 1 [66, chapter 3.6].
Let us consider a mapping of spacetime manifold into itself induced by a vector flow, ξ α = ξ α (xβ ).
It means that each point of the manifold with coordinates xα is mapped to another point with coordinates
x̂α = xα − ξ α (x) . (7.6.8)
This mapping of the manifold into itself can be interpreted as a local diffeomorphism which transforms
the field variables in accordance to their tensor properties. The transformed value of the field variable
is pulled back to the point of the manifold having the original coordinates xα , and is compared with the
original value of the field at this point. The difference between the transformed and the original value of
the field, generated by the diffeomorphism (7.6.8) is the gauge transformation of the field that is given
by the Lie derivative taken along the vector flow ξ α at the point of the manifold with coordinates xα .
Let us denote the transformed values of the field variables with a hat. In the linearised perturbation
theory of the cosmological manifold, the gauge transformations of the field variables (the metric tensor
perturbation καβ , the scalar field φ and ψ) are given by equations
κ̂αβ = καβ + ξα|β + ξβ|α , (7.6.9a)

lˆαβ = lαβ − ξα|β − ξβ|α + ḡαβ ξ γ |γ , (7.6.9b)
φ̂ = φ + Φ̄|α ξ α , (7.6.9c)
ψ̂ = ψ + Ψ̄|α ξ α , (7.6.9d)
where the hat above each symbol denotes a new value of the field variable after applying the gauge
transformation (7.6.8), and all functions are calculated at the same value of coordinates xα . The gauge
transformations of the field variables are expressed in terms of the covariant derivatives on the manifold
and are coordinate-independent. Equation (7.6.9b) is derived from the Lie transformation (7.6.9a) of the
metric tensor perturbation, and the relation (7.5.10) connecting καβ and lαβ .
Gauge invariance of the Lagrangian perturbation theory of geometric manifolds means that the gauge
transformations of the field variables can not change the content of the theory. In other words, the
equations for the field variables must be invariant with respect to the gauge transformations (7.6.9a)-
(7.6.9d). However, direct inspection of equations (7.5.65), (7.5.67), (7.5.68) shows that they do depend
on the choice of the gauge in the form of the gauge function B α introduced in equation (7.5.64). To find
out the gauge-invariant content of the theory one should search for the gauge-invariant field variables
and to derive the gauge-invariant equations for them. This program has been completed by Bardeen [9]
who used the functional 3+1 slicing decomposition of the metric tensor perturbations and the vector field
ξ α to build the gauge-invariant variables out of the various projections of the metric tensor components
on space an time. Modifications of Bardeen’s approach can be found in [21, 36, 39, 41, 83, 93] and
in the book by Mukhanov [91]. We use algebraic 1+3 threading decomposition of the metric tensor
perturbations (7.6.7) that allows us to build gauge-invariant scalars. Vector and tensor perturbations
remain gauge-dependent in the threading approach. In order to suppress the gauge degrees of freedom
in these variables we impose a particular gauge condition B α = 0 in equation (7.5.64). This limits the
freedom of the gauge field ξ α by a particular set of differential equations which are discussed in section
(7.6.7).
7.6.3 Gauge-invariant Scalars
The existence of the preferred four-velocity, ūα , of the Hubble flow in the expanding universe provides a
natural way of separating the perturbations of the field variables in scalar, vector, and tensor components.
This section discusses how to build the gauge-invariant scalars. Vector and tensor perturbations are
discussed afterwards.
The gauge-invariant scalar perturbations can be build from the perturbation of the Clebsch potential,
φ, the perturbation of the scalar field ψ, and a scalar q defined in (7.6.2a). To build the first gauge-
invariant scalar, we introduce the scalar perturbations
φ ψ
χm ≡ , χq ≡ , (7.6.10)
µ̄m µ̄q
that normalize perturbations of the Clebsch potential φ and that of the scalar field ψ to the corresponding
background values of the specific enthalpy, µ̄m and µ̄q . The gauge transformations for the three scalars
q, χm , and χq are obtained from (7.6.9b)–(7.6.9d), and read
q̂ = q − 2ūα ūβ ξα|β , (7.6.11a)

χ̂m = χm − ūα ξ α , (7.6.11b)
χ̂q = χq − ūα ξ α , (7.6.11c)
where we have used the definition of the background four-velocity ūα = −Φ̄|α /µ̄m = −Ψ̄|α /µ̄q
in terms of the partial derivatives of the background values of the scalar fields Φ and Ψ. Equations
(7.6.11b), (7.6.11c) immediately reveal that the linear combination
χ ≡ χm − χq , (7.6.12)
is gauge-invariant, χ̂ = χ, that is the diffeomorphism (7.6.8) does not change the value of the scalar
variable χ.
Two other gauge-invariant scalars are defined by the following equations,
q
Vm ≡ ūα χm|α − , (7.6.13a)
2
q
Vq ≡ ūα χq|α − , (7.6.13b)
2
or, more explicitly,
1 α q v2
Vm = ū φ|α − + 3 2s Hχm , (7.6.14a)
µ̄m 2 c
1 α q χq ∂ W̄
Vq = ū ψ|α − + 3Hχq + , (7.6.14b)
µ̄q 2 µq ∂ Ψ̄
where the last terms in the right side of these equations were obtained by making use of thermodynamic
relationships (7.3.11), the equality ρ̄q = µ̄q , and the equations of continuity (7.4.59) and (7.4.67) for the
density of the ideal fluid, ρ̄m , and that of the scalar field, ρ̄q , respectively.
One can easily check that both scalars, Vm and Vq remain unchanged after making the infinitesimal
coordinate transformation (7.6.8). Indeed, the gauge transformation of the derivatives
χ̂m|α = χm|α − H P̄αβ ξ β − ūβ ξ β |α , (7.6.15a)

β β
χ̂q|α = χq|α − H P̄αβ ξ − ūβ ξ |α , (7.6.15b)
where P̄αβ = ḡαβ + ūα ūβ is the operator of projection on the hypersurface being orthogonal to the
Hubble flow of four-velocity ūα . After performing the gauge transformation (7.6.8), and substituting
the gauge transformations of functions q, χm and χq to the definitions of Vm and Vq , we find out
V̂m = Vm , V̂q = Vq , (7.6.16)
that proves the gauge-invariant property of the scalars Vm and Vq .

Physical meaning of the gauge-invariant quantity Vm can be understood as follows. We consider the
perturbation of the specific enthalpy µm defined in equation (7.3.14). Substituting the decomposition
(7.5.1) of the field variables to equation (7.3.14) and expanding, we obtain
µm = µ̄m + δµm , (7.6.17)
where the perturbation δµm of the specific enthalpy is defined (in the linearized order) by
1
δµm = ūα φ|α − µ̄m q . (7.6.18)
2
It helps us to recognize that
δµm v2
Vm = + 3 2s Hχm . (7.6.19)
µ̄m c
Fractional perturbation of the specific enthalpy can be re-written with the help of thermodynamic equa-
tions (7.3.11) in terms of the perturbation δm of the energy density of the ideal fluid,
δµm v 2 δm
= 2s , (7.6.20)
µ̄m c ¯m + p̄m
or, by making use of equation (7.3.8), in terms of the perturbation δρm of the density of the ideal fluid
δµm v 2 δρm
= 2s . (7.6.21)
µ̄m c ρ̄m
This allows us to write down equation (7.6.19) as follows
v 2 δρm

Vm = 2s + 3Hχm , (7.6.22)
c ρ̄m
which elucidates the relationship between the gauge-invariant variable Vm and the perturbation δρm of the
rest mass density of the dark matter. More specifically, Vm is an algebraic sum of two scalar functions,
δρm and χm neither of each is gauge-invariant. The gauge transformation of the dark matter density
perturbation is
δ ρ̂m = δρm − ρ̄m|α ξ α = δρm + 3H ρ̄m ūα ξ α , (7.6.23)
and the gauge transformation of the variable χm is given by (7.6.11b). Their algebraic sum in equation
(7.6.22) does not change under the diffeomorphism (7.6.8) showing that Vm is the gauge-invariant density
fluctuation that does not depend on a particular choice of coordinates on spacetime manifold.
Similar considerations, applied to function Vq reveals that it can be represented as an algebraic sum
of the perturbation, δρq , of the density of the dark energy, and function χq ,
δρq
Vq = + 3Hχq . (7.6.24)
ρ̄q
It is easy to check out that each term in the right side of this equation taken separately, is not gauge-
invariant but their linear combination does. It is worth emphasizing that standard textbooks on cosmo-
logical theory (see, for example, [80, 94, 113, 114]) derive equations for the density perturbations δρ/ρ̄
but those equations are not gauge-invariant and, hence, their solutions have no direct physical meaning
and should be interpreted with care (see discussion in section 7.8.1).
7.6.4 Field Equations for the Scalar Perturbations.
Equation for a scalar q.
Function q was defined in (7.6.2a). In order to derive a differential equation for q, we apply the co-
variant Laplace-Beltrami operator to q, and make use of the covariant equations (7.5.62) and (7.5.64).
Straightforward but fairly long calculation yields
c2 v2

2k
q|α |α − 2 Ḣ + H 2 − 2 q + 8π ρ̄m µ̄m 1 − 2 Vm − 1 + 3 2s Hχm (7.6.25)
a vs c

∂ W̄
−16π ρ̄q + 2H µ̄q χq − 2ūα ūβ Bα|β − 4H ūα Bα = 8π (σ + τ ) ,
∂ Ψ̄
where the source density σ + τ for the field q is

σ + τ = ūα ūβ + P̄ αβ Tαβ , (7.6.26)
in accordance with the definitions introduced in (7.4.70a), (7.4.70b). The reader should notice that
equation (7.6.25) depends on the gauge function B α which remains arbitrary so far.
Equation for a scalar p.

Function p was defined in (7.6.2b). In order to derive equation for p, we apply the covariant Laplace-
Beltrami operator to the definition of p, and make use of the covariant equations (7.5.62) and (7.5.64).
It results in a wave equation
4k
p|α |α + p + B α |α − 2ūα ūβ Bα|β − 6H ūα Bα = 16πτ . (7.6.27)
a2
where the source density τ has been defined in (7.4.70b). Equation (7.6.27) depends on the arbitrary
gauge function B α .
Equation for a scalar χ.

Equation for the gauge-invariant scalar, χ = χm − χq , is derived from the definitions (7.6.10) and the
field equations (7.5.67), (7.5.68). Replacing φ and ψ in those equations with χm and χq , and making use
of equations (7.4.52), (7.4.53) for reshuffling some terms, yields

4k
χ|α
m |α + 2H ū α
χ m|α − Ḣ − χm (7.6.28a)
a2
c2

+4HVm + 1 − 2 ūα Vm|α − 16π ρ̄q µ̄q χ = ūα Bα ,
vs

4k
χ|α α
q |α + 2H ū χq|α − Ḣ − 2 χq (7.6.28b)
a
2 ∂ W̄
+4HVq + Vq + 16π ρ̄m µ̄m χ = ūα Bα .
µ̄q ∂ Ψ̄
Subtracting (7.6.28b) from (7.6.28a) cancels the gauge-dependent term, ūα Bα , and brings about the
field equation for χ,
c2

2 ∂ W̄
χ|α |α + 6H ūα χ|α + 3Ḣχ = Vq − 1 − 2 ūα Vm|α . (7.6.29)
µ̄q ∂ Ψ̄ vs
This equation is apparently gauge-invariant since any dependence on the arbitrary gauge function B α
disappeared. It is also covariant that is valid in any coordinates.
Equation (7.6.29) can be recast to the form of an inhomogeneous wave equation:
c2

ρ̄m ∂ W̄
(ρ̄m χ)|α |α = 2 Vq − 1 − 2 ρ̄m ūα Vm|α . (7.6.30)
ρ̄q ∂ Ψ̄ vs
Yet another form of equation (7.6.29) is obtained in terms of the variable ψ = ρ̄q χ = µ̄q χ. By simple
inspection we can check that equation (7.6.29) is transformed to
c2

∂ W̄
ψ |α |α − m2ψ ψ = 2 Vm − 1 − 2 ρ̄q ūα Vm|α , (7.6.31)
∂ Ψ̄ vs
p
where we introduced notation mψ ≡ ∂ 2 W̄ /∂ Ψ̄2 . This is an inhomogeneous Klein-Gordon equation
for the field ψ governed by Vm . The ’mass’ mψ of the scalar field excitation, ψ, depends on the second
derivative of the potential function W̄ which defines the ’coefficient of elasticity’ of the background
scalar field Ψ̄.
Inhomogeneous equations (7.6.29), (7.6.30), (7.6.31) have the source terms that is determined by
variables Vm and Vq . We derive differential equations for these field variables in the next sections.
Equation for a scalar Vm .

Equation for the field variable Vm is derived from the equations for functions χm and q that enter its
definition (7.6.13a). By applying the Laplace-Beltrami operator to function Vm we get
1 |α
Vm|α |α = ūβ χ|α
m |α + 2Hχ|α
m |α − q |α + ūβ R̄α β χm|α (7.6.32)
|β 2

1 1
+2H ūα Vm + q + 3H 2 Vm + q .
2 |α 2
The Laplace-Beltrami operator for function χm is given in equation (7.6.28a) which is not gauge-
invariant. Taking the covariant derivative from this equation and contracting it with ūα brings about
the first term in the right side of equation (7.6.32),
c2

ūβ χ|α
m |α = − 1 − ūα ūβ Vm|αβ − 6H ūα Vm|α (7.6.33)
|β vs2

4k 1 2k
− 5Ḣ + 2 Vm − H ūα q|α − Ḣ + 2 q
a 2 a
vs2 vs2 k

−3H 1 + 2 Ḣ − 3 + 2 χm
c c a2
∂ W̄
+8π ρ̄q (4χq − 3χm )
∂ Ψ̄
v2

3
+16π ρ̄q µ̄q ūα χ|α − 6Hχ + H 1 − 2s χm + ūα ūβ Bα|β .
4 c
The Laplace-Beltrami operator for function q has been derived in (7.6.25). Now, we make use of equa-
tions (7.6.25), (7.6.28a), (7.6.33) in calculating the right side of (7.6.32). After a significant amount of
algebra, we find out that all terms explicitly depending on q and the gauge functions B α cancel out, so
that equation for Vm becomes
c2 c2

Vm|α |α + 1 − 2 ūα ūβ Vm|αβ + 2 3 − 2 H ūα Vm|α + (7.6.34)
vs vs
c2

2k
2 Ḣ + 3H 2 + 2 − 4π ρ̄m µ̄m 1 − 2 Vm −
a vs

1 ∂ W̄
16π ρ̄q µ̄q ūα χ|α − 3 H + χ = −4π (σ + τ ) .
2µ̄q ∂ Ψ̄
Second-order covariant derivatives in this equation read
c2
2
c
ḡ αβ + 1 − 2 ūα ūβ Vm|αβ ≡ − 2 ūα ūβ + P̄ αβ Vm|αβ , (7.6.35)
vs vs
and they form a hyperbolic-type operator describing propagation of sound waves in the expanding uni-
verse from the source of the sound waves towards the field point with the constant velocity vs2 . Additional
terms in the left side of equation (7.6.34) depend on the Hubble parameter H, and take into account the
expansion of the universe.
Equation (7.6.34) contains only gauge-invariant scalars, Vm and χ. Moreover, it does not depend on
the choice of coordinates on the background manifold. It also becomes clear that the field variables
Vm and χ are coupled through the differential equations (7.6.31) and (7.6.34) which should be solved
simultaneously in order to determine these variables. Solution of the coupled system of differential
equations is a very complicated task which cannot be rendered analytically in the most general case.
Only in some simple cases, the equations can be decoupled. We discuss such cases in section 7.8.
Equation for a scalar Vq .

The field variable Vq is not independent since it relates to Vm and χ by a simple relationship
Vq = Vm − ūα χ|α , (7.6.36)
which is obtained after subtraction of equation (7.6.13a) from (7.6.13b). Equation for Vq is derived
directly from (7.6.36) and equations (7.6.34) and (7.6.29) for Vm and χ respectively. We obtain,

1 ∂ W̄
Vq|α |α + 4 H + ūα Vq|α (7.6.37)
2µ̄q ∂ Ψ̄
c2

2k
+ 2 Ḣ + 3H 2 + 2 − 4π ρ̄m µ̄m 1 − 2
a vs
∂ 2 W̄

2 1 ∂ W̄ ∂ W̄
+ 5H + +2 Vq
µ̄q µ̄q ∂ Ψ̄ ∂ Ψ̄ ∂ Ψ̄2
c2 v2

+4π ρ̄m µ̄m 3 + 2 ūα χ|α − 3 2s Hχ = −4π (σ + τ ) .
vs c
This equation can be also derived by the procedure being similar to that used in the previous subsection
in deriving equation for Vm . We followed this procedure and confirm that it leads to (7.6.37) as expected.
Equation (7.6.37) is clearly gauge-invariant. Besides Vq it depends on variable χ and should be solved
along with equation (7.6.29).
7.6.5 Field Equations for Vector Perturbations

Vector perturbations of the ideal fluid and scalar field are gradients, φ|α and ψ|α . However, they are
insufficient to build a gauge-invariant vector perturbation out of the vector perturbation of the metric
tensor pα . Field equations for vector pα can be derived by applying the covariant Laplace-Beltrami
operator to both sides of definition (7.6.3) and making use of equation (7.5.65). After performing the
covariant differentiation and a significant amount of algebra, we derive the field equation

2k
pα |β |β − 2H ūα pβ |β − 2Ḣ + 3H 2 − 2 pα (7.6.38)
a
+P̄α β ūγ Bβ|γ + Bγ|β + 2H ūγ Bβ

= 16πτα ,
where the matter current τα is defined in (7.4.70c). This equation is apparently gauge-dependent as
shown by the appearance of the gauge function B α . This equation reduces to a much simpler form

2k
pα |β |β − 2H ūα pβ |β − 2Ḣ + 3H 2 − 2 pα = 16πτα , (7.6.39)
a
in a special gauge B α =0 which imposes a restriction on the divergence of the metric tensor perturbation
in equation (7.5.64).
Equation (7.6.38) points out that the vector perturbations are generated by the current of matter τa
existing in the localized astronomical system which physical origin may be a relict of the primordial
perturbations. We do not discuss this interesting scenario in the present chapter as it would require a
non-conservation of entropy and non-isentropic background fluid – the case which we have intentionally
excluded in order to focus on derivation of cosmological generalization of the post-Newtonian equations
of relativistic celestial dynamics [66].
7.6.6 Field Equations for Tensor Perturbations
Field equations for traceless tensor p|αβ can be derived by applying the covariant Laplace-Beltrami
operator to the definition (7.6.4) and making use of equation (7.5.65) along with a tedious algebraic
transformations. This yields the following equation

k
p|αβ |γ |γ − 2H ūα p|βγ |γ + ūβ p|αγ |γ − 2 H 2 + 2 p|αβ (7.6.40)
a
2
−P̄α µ P̄β ν Bµ|ν + Bν|µ + P̄αβ P̄ µν Bµ|ν = 16πταβ |
.
3
Here the transverse and traceless tensor source of the tensor perturbations is
| 1
ταβ ≡ ταβ − P̄αβ τ , (7.6.41)
3
where ταβ has been introduced in (7.4.70d), and τ = P̄ αβ ταβ in accordance with equation (7.4.70b).
|
Tensor ταβ is traceless, that is ḡ αβ ταβ
|
= P̄ αβ ταβ
|
= 0.
Equation (7.6.40) is gauge-dependent. The gauge freedom is significantly reduced by imposing the
gauge condition B α = 0 which brings equation (7.6.40) to the following form,

k
p|αβ |γ |γ − 2H ūα p|βγ |γ + ūβ p|αγ |γ − 2 H 2 + 2 p|αβ = 16πταβ |
. (7.6.42)
a
7.6.7 Residual Gauge Freedom
The gauge freedom of the theory under discussion is associated with the gauge function B α appearing
in equation (7.5.64). The most favourable choice of the gauge condition is
Bα = 0 , (7.6.43)
which drastically simplifies the above equations for vector and tensor gravitational perturbations. The
gauge (7.6.43) is a generalization of the harmonic (de Donder) gauge condition used in the gravitational
wave astronomy and in the post-Newtonian dynamics of extended bodies. This choice of the gauge
establishes differential relationships between the algebraically-independent metric tensor components
introduced in section 7.6.1. Indeed, substituting the algebraic decomposition (7.6.7) of the metric tensor
perturbations to equation (7.5.64) and imposing the condition (7.6.43) yields

1
p|αβ |β + ūα pβ |β + ūβ pα |β − ūα ūβ − P̄αβ p|β + 2Hpα (7.6.44)
3
+2ūα ūβ q|β + 2Hqūα = 16π (ρ̄m µ̄m χm + ρ̄q µ̄q χq ) ūα .
Projecting this relationship on the direction of the background 4-velocity, ūα , and on the hypersurface
being orthogonal to it, we derive two algebraically-independent equations between the perturbations of
metric tensor components and of the matter variables. They are
pβ |β + ūβ (2q − p)|β + 2Hq = 16π (ρ̄m µ̄m χm + ρ̄q µ̄q χq ) , (7.6.45a)
|β 1
|
pαβ + ūβ pα |β + P̄αβ p|β + 2Hpα = 0. (7.6.45b)
3
The gauge (7.6.43) does not fix the gauge function ξ α uniquely. The residual gauge freedom is
described by the gauge transformations that preserve equations (7.6.45a), (7.6.45b). Substituting the
gauge transformation (7.6.9b) of the gravitational field perturbation lαβ to equation (7.5.64) and holding
on the gauge condition (7.6.43), yields the differential equation for the vector function ξ α

ξ α|β |β + ḡ αγ ξ β |γβ − ξ β |βγ + 2H ξ α|β ūβ + ξ β|α ūβ − ξ β |β ūα (7.6.46)
−16π (ρ̄m µ̄m + ρ̄q µ̄q ) ξ β ūβ ūα = 0,
which can be further recast to

ξ α|β |β + 2H ξ α|β ūβ + ξ β|α ūβ − ξ β |β ūα (7.6.47)

k 2k
+2 Ḣ − 2 ξ β ūβ ūα + Ḣ + 3H 2 + 2 ξ α = 0.
a a
The gauge function ξ α can be decomposed in time-like, ξ ≡ −ξ β ūβ , and space-like, ζ α ≡ P̄ α β ξ β ,

components,
ξ α = ζ α + ūα ξ . (7.6.48)
Calculating covariant derivatives from ξ and ζ α and making use of equation (7.6.47), yield equations

4k
ξ |β |β + 2H ūβ ξ|β − Ḣ − 2 ξ = 0, (7.6.49a)
a

2k
ζ α|β |β + 2H ūβ ζ α |β − ūα ζ β |β + Ḣ + H 2 + 2 ζ α = 0 . (7.6.49b)
a
These equations have non-trivial solutions which describe the residual gauge freedom in choosing the
coordinates on the background manifold subject to the gauge condition (7.6.43). It is remarkable that
equations (7.6.49a), (7.6.49b) are decoupled and can be solved separately. It means that the residual
gauge transformations along the worldlines of the Hubble flow are functionally independent of those
performed on the hypersurface being orthogonal to the Hubble flow. Equations (7.6.49a)-(7.6.49b) of
the residual gauge freedom in the cosmological setting given in this subsection generalise equations of
the residual gauge freedom in harmonic coordinates of asymptotically flat spacetime [13, 31].
7.7 Post-Newtonian Field Equations in a Spatially-Flat

Universe
7.7.1 Cosmological Parameters and Scalar Field Potential
Linearised equations of the field perturbations given in the previous section are valid for a wide class
of matter models of the FLRW metric. They neither specify the equation of state of dark matter, nor
that of dark energy. We also keep the parameter of the space curvature k free. By choosing a specific
model of matter and picking up a value of k = −1,0, + 1, we can solve, at least, in principle the field
Section 7.7 Post-Newtonian Field Equations in a Spatially-Flat Universe 313
equations governing the time evolution of the background cosmological manifold. Realistic models of
the cosmological dark matter and dark energy are rather sophisticated and, as a rule, include several
components. It leads to the system of coupled field equations which can be solved only numerically
[2]. However, the large scale structure of the universe is formed at rather late stages of the cosmological
evolution being fairly close to the present epoch. Therefore, the study of the impact of cosmological
expansion on the post-Newtonian dynamics of isolated astronomical systems is based on recent and
present equation of state of matter in the universe.
Precise radiometric observations of the relic CMB radiation and photometry of type Ia supernova
explosions reveal that at the present epoch the space curvature of the universe, k = 0, and the evolution
of the universe is primarily governed by the dark energy and dark matter, which make up to 74% and
24% of the total energy density of the universe respectively, while 4% of the energy density of the
universe belongs to visible matter (baryons), and a tiny fraction of the energy density occupies by the
CMBR radiation [43, 52, 59, 63]. It means that we can neglect the effects of the baryonic matter and
CMB radiation field in consideration of the post-Newtonian dynamics of astronomical systems in the
expanding universe.
We model dark matter by an ideal fluid and dark energy is represented by a scalar field with a potential
function W̄ which structure has not yet been specified. We also follow the discussion given in [2] by
assuming that the spatial curvature k = 0, and the potential, W̄ , of the scalar field relates to its derivative
by a simple equation
∂ W̄ √
= − 8πλW̄ , (7.7.1)
∂ Ψ̄
where the time-dependent parameter, λ = λ(Ψ̄), characterizes the slope of the field potential W̄ . The
time evolution of the background universe can be described in terms of the parameter λ and two other
parameters, x1 = x1 (Ψ̄) and x2 = x2 (Ψ̄), which are functions of the density, ρ̄q = µ̄q = Ψ̄, of the
background scalar field, and the potential, W̄ , scaled to the Hubble parameter, H. These parameters are
defined as follows,
3H 2
ρ̄2q = x1 , (7.7.2)
4π
2
3H
W̄ = x2 . (7.7.3)
8π
The energy density of the scalar field, ¯q , is expressed in terms of the parameters x1 and x2 and the
parameter Ωq ≡ 8π ¯q /3H 2 , by a simple relationship
Ωq = x1 + x2 . (7.7.4)
Time evolution of the parameters x1 and x2 is given by a system of two ordinary differential equations
which are obtained by differentiating definitions (7.7.2), (7.7.3) and making use of equations (7.4.67)
taken along with the Friedmann equation (7.4.52) with k = 0. It yields
dx1 √
= −6x1 + λ 6x1 x2 + 3x1 [(1 − wm ) x1 + (1 + wm ) (1 − x2 )] , (7.7.5a)
dω
dx2 √
= −λ 6x1 x2 + 3x2 [(1 − wm ) x1 + (1 + wm ) (1 − x2 )] , (7.7.5b)
dω
where ω ≡ ln a is the logarithmic scale factor characterizing the number of e-folding of the universe,
wm is the parameter entering the hydrodynamic equation of state (7.4.54), and the parameters x1 and x2
are restricted by the condition imposed by the Friedmann equation (7.4.50), that is Ωq + Ωm = 1, or
x1 + x2 = 1 − Ωm , (7.7.6)
where Ωm ≡ 8π ¯m /3H 2 .

The parameter λ obeys the following equation
dλ √
= − 6x1 λ2 (Γq − 1) , (7.7.7)
dω
where
∂ 2 W̄ /∂ Ψ̄2
Γq = 2 W̄ , (7.7.8)
∂ W̄ /∂ Ψ̄
If Γq = 1, the parameter λ is constant, and equation (7.7.1) can be integrated yielding an exponential
potential √
W̄ (Ψ̄) = W̄0 exp(− 8πλΨ̄) . (7.7.9)
In this case, and under assumption that, wm = const., the system of two differential equations (7.7.5a),
(7.7.5b) is closed. If Γq 6= 1, three equations (7.7.5a), (7.7.5b), (7.7.7) must be solved together in order
to describe temporal evolution of the background cosmological manifold.
In the general case, derivatives of the potential W̄ are expressed in terms of the parameters under
discussion. Namely,
∂ W̄ 3λ ∂ 2 W̄
= − √ H 2 x2 , = 3Γq λ2 H 2 x2 . (7.7.10)
∂ Ψ̄ 8π ∂ Ψ̄2
It is also useful to express the products ρ̄q µ̄q and ρ̄m µ̄m in terms of the parameters x1 and x2 . For
µ̄q = ρ̄q , one can use definition (7.7.2) to obtain
3H 2
ρ̄q µ̄q = x1 . (7.7.11)
4π
The product ρ̄m µ̄m = ¯m + p̄m , so that making use of the matter equation of state, p̄m = wm ¯m , and
equation (7.7.6), we derive
3H 2
ρ̄m µ̄m = (1 + wm )Ωm , (7.7.12)
8π
where Ωm = 1 − x1 − x2 . These equations allow us to recast equation (7.4.52) for the time derivative of
the Hubble parameter to the following form
3
Ḣ = − (1 + weff ) H 2 , (7.7.13)
2
where
weff ≡ wm + (1 − wm )x1 − (1 + wm )x2 , (7.7.14)
is the (time-dependent) parameter of the effective equation of state of the mixture of the ideal fluid and
the scalar field.
7.7.2 Conformal Cosmological Perturbations

The FLRW metric (7.4.15) is a product of the scale factor a and a conformal metric f̄αβ . The conformal
spacetime is comoving with the Hubble flow and is not globally expanding. In case of the flat spatial
curvature, k = 0, the conformal spacetime becomes equivalent to the Minkowski spacetime which is
used as a starting point in the standard theory of the Post-Newtonian approximations [31]. Therefore,
it is mathematically instructive to formulate the field equations for cosmological perturbations in the
conformal spacetime. It also allows us to simplify the differential operators in the left side of the equa-
tions for perturbations (see section (7.7.3) below). Nonetheless, the reader must keep in mind that the
conformal spacetime is unphysical and additional scale transformations of coordinates are required to
convert mathematical results from the conformal spacetime to a real physical world [69].
Let us associate the cosmological perturbation, hαβ , of gravitational field in the conformal spacetime
with the background metric f̄αβ with physical perturbation καβ of the metric as follows [65, 70]
καβ = a2 (η)hαβ , (7.7.15)
where perturbation καβ has been defined in (7.5.1) and a(η) is the scale factor of the FLRW metric.
Gravitational perturbation lαβ relates to καβ by equation (7.5.10), and can be also represented in the
conformal form
lαβ = a2 (η)γαβ , (7.7.16)
where
1
γαβ = −hαβ + f̄αβ h , (7.7.17)
2
with h ≡ f̄αβ hαβ . In what follows, tensor indices of geometric objects in the conformal spacetime are
raised and lowered with the help of the conformal metric f̄αβ .
We assume that the scale factor a of the universe remains unperturbed. This assumption is justified
since we can always include the perturbation of the scale factor to the perturbation hαβ of the conformal
metric. Thus, the perturbed physical spacetime interval, ds, of the FLRW metric relates to the perturbed
conformal spacetime interval, ds̃, by the conformal transformation
ds2 = a2 (η)ds̃2 . (7.7.18)
Here, the perturbed conformal spacetime interval reads
ds̃2 = fαβ dxα dxβ , (7.7.19)
where
fαβ = f̄αβ + hαβ , (7.7.20)
is the perturbed conformal metric. Here, f̄αβ is the unperturbed conformal metric defined in (7.4.16),
hαβ is the perturbation of the conformal metric, and xα = (x0 ,xi ) are arbitrary coordinates which are
the same as in the physical spacetime manifold in correspondence with the definition of the conformal
metric transformation [88].
It is worth emphasizing that in case of the space curvature k = 0, the background conformal metric,
gαβ (η,X i ), expressed in the isotropic Cartesian coordinates (η,X i ), is the diagonal Minkowski metric,
gαβ (η,X i ) = ηαβ = diag(−1,1,1,1). Therefore, in this case the background metric f̄αβ remains the
Minkowski metric with the components expressed in arbitrary coordinates by means of tensor transfor-
mation
f̄αβ = Mµ α Mν β ηµν , (7.7.21)
where the matrix of transformation has been defined in (7.4.13). If the matrix of transformation, Mµ α ,
is the Lorentz boost, the conformal metric, f̄αβ , remains flat, f̄αβ = ηαβ . It is worth noticing that, in
general, the unperturbed conformal metric can be chosen flat even in case of k = −1, + 1 [55]. Hence,
all equations given above will remain intact which means that, in fact, our formalism is applicable to
FLRW metric with any space curvature. The only change will be in the conformal factor which, in
the case of k = ±1, is not merely the scale factor a(η) of the FLRW metric but a more complicated
function, a(η,xa ), of time and spatial coordinates [55]. Though it is not difficult to handle all three cases
of k = −1,0, + 1 on the same footing but it burdens equations for the field perturbations with a number
of terms being proportional to k. Morover, consideration of the dark energy equations with k = ±1
given in the preceding section gets complicated [2]. For this reason, we restrict ourselves with the case
of the spatially-flat universe with k = 0 which is an excellent approximation in treating cosmological
observations [60].
Similarly to (7.6.7) the conformal metric perturbation, γαβ , can be split in 1+3 algebraically-irreducible
components

1
γαβ = p|αβ + v̄α pβ + v̄β pα + v̄α v̄β + π̄αβ p + 2v̄α v̄β (q − p) , (7.7.22)
3
where the four-velocity v̄α = aūα , v̄α = f̄αβ v̄β = a−1 ḡαβ ūβ = a−1 ūα , and
π̄αβ = f̄αβ + v̄α v̄β , (7.7.23)
is the operator of projection on the conformal space which represents a hypersurface being everywhere
orthogonal to the congruence of worldlines of four-velocity v̄α . Four-velocity v̄α is an analogue of the
Hubble flow in the conformal spacetime. We also notice that P̄αβ = a2 π̄αβ .
Different pieces of the conformal metric perturbation, γαβ , are related to those of the physical metric
perturbation, lαβ , by the powers of the scale factor,
p|αβ = a2 p|αβ , pα = apα , p=p, q=q. (7.7.24)
More specifically,
1 µ ν
q = (v̄ v̄ + π̄ µν ) γµν , (7.7.25a)
2
p = π̄ µν γµν , (7.7.25b)
β γ
pα = −π̄α v̄ γβγ , (7.7.25c)
| 1
pαβ = pαβ − π̄αβ p , (7.7.25d)
3
where
pαβ = π̄α µ π̄β ν γµν . (7.7.26)
αβ
The trace of the gravitational perturbation, γ ≡ f̄ γαβ = 2(p − q). The components hαβ = −γαβ +
f̄αβ γ/2 are used in calculating dynamical behavior of particles and light in the conformal spacetime as
well as for matching theory with observables. The components of hαβ are
2
hαβ = −p|αβ − v̄α pβ − v̄β pα + π̄αβ p − (v̄α v̄β + π̄αβ ) q , (7.7.27)
3
and h ≡ f̄αβ hαβ = 2(p − q) = γ.

It turns out that the conformal Hubble parameter, H = a0 /a is more convenient in the conformal
spacetime than the “canonical” Hubble parameter, H = Ṙ/R = R−1 dR/dT , where T is the cos-
mological time (see section 7.4.2). Relations between H and H, and their derivatives are shown in
equations (7.4.5)–(7.4.7). These relations are employed along with equations (7.4.6) and (7.7.13) in
order to express the time derivative, H0 , of the conformal Hubble parameter in terms of H2 and the
parameter weff of the effective equation of state

1
H0 = − (1 + 3weff )H2 . (7.7.28)
2
We shall use this expression in the calculations that follow.
7.7.3 Post-Newtonian Field Equations in Conformal Spacetime

The set of the post-Newtonian field equations in cosmology consists of equations for the perturbations
of the background dark matter, dark energy and gravitational field. Perturbations of dark matter and
dark energy are described by four scalars, Vm , Vq , χm and χq but only three of them are functionally-
independent because of equality (7.6.36), that is
Vm − Vq = ūα (χm − χq )|α . (7.7.29)
Depending on a particular situation, any of the three scalars can be taken as independent variables in
description of scalar perturbations.
The gravitational field perturbations are q, p, pα , p|αβ but among them the scalar q is not independent
and can be expressed in terms of χm and Vm in accordance with (7.6.13a),
q = −2 Vm − ūα χm,α ,

(7.7.30)
where we have also used the equality q = q as follows from (7.7.24). The scalar q can be also expressed
in terms of χq and Vq in accordance with (7.6.13b). Hence, as soon as the pairs, Vm and χm or Vq and
χq are known, the scalar gravitational perturbation q can be easily calculated from (7.7.30). Functions
p, pα , p|αβ are independent and decouple both from each other and from the other perturbations. Thus,
the most difficult part of the perturbation theory is to find out solutions of the scalar perturbations which
are coupled one to another.
The post-Newtonian field equations in the conformal spacetime for variables χm , χq , Vm and for
p, pα , p|αβ are derived from the equations of the previous section by transforming all functions and
operators from physical to conformal spacetime. The important part of the transformation technique is
based on formulas converting the covariant Laplace-Beltrami wave operators, defined on the background
spacetime manifold, to their conformal spacetime counterparts.
Laplace-Beltrami Operator in Conformal Spacetime

Let F be an arbitrary scalar, Fα - an arbitrary covector, and Fαβ - an arbitrary covariant tensor of the
second rank. We have three different types of the Laplace-Beltrami operators on the curved background
manifold: a scalar - F |µ |µ , a vector - Fα |µ |µ , and a tensor - Fαβ |µ |µ where the covariant derivatives
are taken with the help of the affine connection Γ̄α βγ being compatible with the metric ḡαβ as shown in
(7.4.21). Covariant derivatives gives the invariant description of differential equations of mathematical
physics on curved manifolds. However, for handling a more pragmatic purpose of finding solution of
a differential equation, the covariant operators must be expressed in terms of partial derivatives with
respect to the coordinates chosen for solving the equation.
Transformation of the covariant Laplace-Beltrami operators to the partial derivatives is achieved after
writing down the covariant derivatives for a scalar F , a vector fα , and a tensor Fαβ in explicit form by
making use of the Christoffel symbols given in (7.4.23)–(7.4.25). Tedious but straightforward calcula-
tions of the covariant derivatives yield the scalar, vector and tensor Laplace-Beltrami operators in the
following form [70]

1h i
F |µ |µ = F − 2H v̄ µ
F ;µ , (7.7.31a)
a2
1 h
Fα |µ |µ = Fα − 2Hv̄µ Fµ;α + 2Hv̄α f̄µν Fµ;ν (7.7.31b)
a2 i
+ H0 + 2H2 Fα − 2H2 v̄α v̄µ Fµ ,

1h
Fαβ |µ |µ = Fαβ + 2Hv̄µ Fαβ;µ − 2Hv̄µ Fµα;β − 2Hv̄µ Fµβ;α (7.7.31c)
a2
0
+2Hf̄ (v̄α Fβµ;ν + v̄β Fαµ;ν ) + 2 H + H Fαβ
µν 2
1 1 i
−4H2 v̄µ v̄α Fβµ + v̄µ v̄β Fαµ − v̄α v̄β f̄µν Fµν − f̄αβ v̄µ v̄ν Fµν ,
2 2
where we have introduced notations
F ≡ f̄µν F;µν , Fα ≡ f̄µν Fα;µν , Fαβ ≡ f̄µν Fαβ;µν , (7.7.32)
of the wave operators for the scalar, vector and tensor fields in the conformal spacetime and in arbi-
trary coordinates. Notice that although the conformal spacetime coincides, in case of k = 0, with the
Minkowski spacetime, the metric f̄αβ is not the diagonal Minkowski metric ηαβ unless the coordinates
are Cartesian. Of course, the covariant derivative from a scalar must be understood as a partial derivative,
that is F;α = F,α .
We will need several other equations to complete the transformation of the Laplace-Beltrami operators
to the conformal spacetime since the wave operator acts on functions like those shown in (7.7.24),
which are made of a product of the scale factor, a = a(η) in some power n (may be not an integer), with
a geometric object, z = z(xα ), which can be a scalar, a vector or a tensor of the second rank (we have
suppressed the tensor indices of z since they do not interfere with the derivation of the equations which
follow). These equations are
(an z);µ = an (z;µ − nHv̄µ z) , (7.7.33a)

0
(an z);µν a z;µν − nH (v̄µ z;ν + v̄ν z;µ ) + n H + nH v̄µ v̄ν , (7.7.33b)
n 2
=
and they allow us to write down the wave operator from the product of an and z in the following form
h i
(an z) = an z − 2nHv̄µ z;µ − n H0 + nH2 z , (7.7.34)
It is easy to confirm that contraction of (7.7.33b) with the conformal four-velocity, v̄α , brings about
another differential operator
h i
v̄µ v̄ν (an z);µν = an v̄µ v̄ν z;µν + 2nHv̄µ z;µ + n H0 + nH2 z . (7.7.35)
We remind that if the object z is a scalar, the covariant derivative z;α = z,α is reduced to a partial
derivative. In case, when z is either a vector or a tensor, the covariant derivative must be calculated with
taking into account the affine connection B̄ α βγ defined in (7.4.25).
It is also interesting to notice that in the expanding universe the conformal Laplace operator, ∆z ≡
π̄ µν z;µν is the scale invariant in the sense that
∆ (an z) = an ∆z , (7.7.36)
where z is a tensor of an arbitrary rank. Equation (7.7.36) can be proven by adding up (7.7.34) and
(7.7.35), and accounting for definition (7.7.23) of the projection operator on the hypersurface being
orthogonal to v̄α .
Now, we are ready to formulate the field equations for cosmological perturbations in the conformal
spacetime.
Equations for perturbations of dark matter and dark energy

Dark matter and dark energy are described by scalar fields Φ and Ψ. The fields themselves are not gauge-
invariant. Therefore, physical meaning have only the equations for the gauge-invariant perturbations
of these fields which are Vm , Vq , and χ. We consider, first, equation (7.6.34) for the gauge-invariant
scalar Vm . We convert the covariant derivatives taken with respect to the background metric, ḡαβ , to
the partial derivatives of the conformally-flat metric, f̄αβ and use equation (7.7.31a) for the Laplace-
Beltrami operator along with the expressions for various cosmological parameters given in section 7.7.2.
After arranging terms with respect to the powers of the Hubble parameter H, we obtain the scalar
equation for function Vm describing the perturbations of dark matter,
c2 c2

Vm + 1 − 2 v̄α v̄β Vm;αβ + 3 − 2 Hv̄α Vm,α (7.7.37)
vs vs
c2

1
+3 1 − weff − (1 + wm ) 1 − 2 Ωm H2 Vm
2 vs
" r ! #
3 x1
2 α
+12H v̄ χ,α − 3 1 − λx2 Hχ = −4πa2 (σ + τ ) .
8x1 a
This is a wave equation with the speed of sound vs which determines the speed of propagation of the
scalar perturbations in the dark matter considered as an ideal fluid. These perturbations can be interpreted
as acoustic or sound waves of different wavelengths propagating in spacetime. Solution of homogeneous
equation (7.7.37) describes the propagation of primordial scalar perturbations of dark matter. A particu-
lar solution of the inhomogeneous equation (7.7.37) tells us how the perturbation of dark matter caused
by the isolated astronomical system propagate.
Similar procedure is applied to equation (7.6.37) and leads to a wave equation for function Vq de-
scribing propagation of perturbations of dark energy considered as a scalar field,
r !
3
Vq + 2 1 − λx2 Hv̄µ Vq,µ (7.7.38)
2x1
c2

1
+3 1 − weff − (1 + wm ) 1 − 2 Ωm H2 Vq
2 vs
" r #
x2 6
+λx2 3λ 2Γq + −5 H2 Vq
x1 x1
c2 v2

3 Ωm
+ H2 (1 + wm ) 3 + 2 v̄µ χ,µ − 3 2s Hχ = −4πa2 (σ + τ ) .
2 vs c a
The speed of propagation of dark energy is naturally equal to the fundamental speed c as contrasted with
dark matter. Dark matter has an intrinsic elasticity associated with the bulk modulus K = (dp/d)
that ispproportional to pressure p, and where is the energy density of the fluid. The speed of sound
vs = K/ < c for a fluid because in this case K < . However, in case of the scalar field K = ||,
and vs = c.
Equations (7.7.37) and (7.7.38) depend on the scalar function χ which obeys equation (7.6.29). Mak-
ing use of the same transformations as applied to derivation of (7.7.37) and (7.7.38), we can recast
(7.6.29) to a wave equation for χ,
r ! "r #
c2

3 9 6
χ+4H 1 − λx2 v̄α χ,α − (1 + weff ) H2 χ = −a λx2 HVm + 1 − 2 v̄α Vm,α .
8x1 2 x1 vs
(7.7.39)
We can observe that the speed of propagation of the field χ is equal to the fundamental speed c. More-
over, (7.7.39) depends on Vm and should be solved simultaneously with equation (7.7.37) for Vm after
imposing certain boundary conditions. As soon as the gauge-invariant scalar χ is known, the potential,
Vq , can be determined either as a particular solution of the inhomogeneous equation (7.7.38) or, more
simple, from algebraic relation (7.6.36).
We also need equations for the normalized Clebsch and scalar potentials, χm and χq . These potentials
are required to determine the gravitational perturbation, q, with the help of (7.7.30) and/or to check
on self-consistency of the solutions of the field equations in the matter sector of perturbation theory.
Conformal-spacetime equations for χm and χq are derived from their definition (7.6.10) and the field
equations (7.5.67) and (7.5.68). They are
c2

3
χm + (1 + weff ) H2 χm = 12H2 x1 χ − a 4HVm + 1 − 2 v̄α Vm,α , (7.7.40)
2 vs
r !
3 6
χq + (1 + weff ) H χq = −6H (1 + wm )Ωm χ − a 4 −
2 2
λx2 HVq .(7.7.41)
2 x1
By subtracting one of these equations from another, we get back to equation (7.7.39). Notice that χm
and χq are not gauge-invariant perturbations and, hence, the solutions of (7.7.40), (7.7.41) should be
interpreted with care.
Equations for the metric perturbations

Post-Newtonian equations for gravitational perturbations in physical spacetime are (7.6.25), (7.6.27),
(7.6.38) and (7.6.40). These equations are gauge-dependent. In order to fix the gauge we imposed the
gauge conditions (7.5.64), (7.6.43). In this gauge, equations for the conformal metric tensor perturba-
tions become
q − 2Hv̄α q,α + (1 + 3weff ) H2 q = 8πa2 (σ + τ ) (7.7.42a)

"r #
3x1 χq
−24H 2
λx2 − Hx1
8 a
−3 (1 + weff ) H2 Ωm
c2 v 2 χm

× 1 − 2 Vm − H 1 + 3 2s ,
vs c a
p − 2Hv̄α p,α = 16πa2 τ , (7.7.42b)
pα − 2Hv̄ pα;β + (1 + 3weff ) H pα
β 2
= 16πaτα , (7.7.42c)
| γ |
pαβ − 2Hv̄ pαβ;γ = 16πταβ . (7.7.42d)
The reader can observe that equations (7.7.42a)-(7.7.42d) for linearized metric perturbations are decou-
pled from each other. Moreover, equations (7.7.42b-(7.7.42d) are decoupled from the matter perturba-
tions Vm , χm , etc. Only equation (7.7.42a) for q is coupled with the matter perturbations governed by
equations (7.7.37), (7.7.40), (7.7.41) so that these equations should be solved together. As we have men-
tioned above, function q is a linear combination of Vm and χm according to (7.7.30). Hence, in order to
determine q it is, in fact, sufficient to solve (7.7.37), (7.7.38) and (7.7.40). Nevertheless, it is convenient
to present the differential equation (7.7.42a) for q explicitly for the sake of mathematical completeness
and rigour. It can be used for independent validation of the solution of the system of equations (7.7.37),
(7.7.40) and (7.7.38). Unfortunately, these equations are strongly coupled and cannot be solved analyti-
cally in the most general situation of a multi-component background universe governed by dark energy
and dark matter. Solution of (7.7.37)-(7.7.41) would require a numerical integration.
It would be instrumental to get better insight to the post-Newtonian theory of cosmological perturba-
tions by making some simplifying assumptions about the background model of the expanding universe
in order to decouple the system of the post-Newtonian equations and to find their analytic solution ex-
plicitly. We discuss these assumptions and the corresponding system of the decoupled post-Newtonian
equations in section 7.8 below.
7.7.4 Residual Gauge Freedom in the Conformal Spacetime
The gauge conditions (7.5.64), (7.6.43) in physical space are given by equations (7.6.45a), (7.6.45b).
After transforming to the conformal spacetime the equations for the gauge condition reads
pβ ;β + v̄β (2q − p),β + 2Hq = 16πa (ρ̄m µ̄m χm + ρ̄q µ̄q χq ) , (7.7.43a)
|αβ β α 1
p ;β + v̄ p ;β + π̄ αβ p,β + 2Hpα = 0. (7.7.43b)
3
The residual gauge freedom in the conformal spacetime is described by two gauge functions, ζ ≡ ξ/a
and ζ α , where ξ and ζ α have been defined in section 7.6.7. Differential equations for ζ and ζ α are
obtained by making transformation of equations (7.6.49a), (7.6.49b) to the conformal spacetime. The
calculation is straightforward and results in
ζ − 2Hv̄β ζ,β + (1 + 3weff ) H2 ζ = 0, (7.7.44a)

α β α
ζ − 2Hv̄ ζ ;β = 0. (7.7.44b)
Solutions of equations (7.7.42a)–(7.7.42d) are determined up to the gauge transformations
q̂ = q + 2v̄α ζ,α + 2Hζ , (7.7.45a)

α α
p̂ = p+ζ ;α + 3v̄ ζ,α + 6Hζ , (7.7.45b)

p̂α = pα + π̄αβ v̄γ ζ β ;γ − ζ ,β + 2Hζ β , (7.7.45c)
ν ν µ α α
p̂αβ = pαβ − (π̄µα π̄β + π̄µβ π̄α ) ζ ;ν + π̄αβ (ζ ;α + v̄ ζ,α + 2Hζ) , (7.7.45d)
where the gauge functions ζ, ζ α are solutions of the differential equations (7.7.44a), (7.7.44b).
7.8 Decoupled System of the Post-Newtonian Field

Equations
7.8.1 The Universe Governed by Dark Matter and Cosmological Constant
Case 1: Arbitrary equation of state of dark matter
Let us consider a special case of the background value of dark energy represented by cosmological
constant Λ = 8π W̄ . In this case, the equation of state of the scalar field is wq = −1, and we have
ρ̄q µ̄q = ¯q + p̄q = 0. The parameter x1 = 0, and x2 = Λ/(3H 2 ). It yields the parameter Ωq = x2 , and
Ωm = 1 − x2 . Since the cosmological constant corresponds to a constant potential W̄ of the scalar field,
we get for its derivative ∂ W̄ /∂ Ψ̄ = 0, and equation (7.7.1) points out that the parameter λ = 0.
In the universe governed by dark matter and cosmological constant the parameter of the effective
equation of state of the dark matter is
Λ
weff = wm − (1 + wm ) . (7.8.1)
3H 2
Hence, the time derivative of the Hubble parameter defined in (7.7.13), is reduced to a more simple
expression,
1
Ḣ = (1 + wm ) Λ − 3H 2 .

(7.8.2)
2
On the other hand, equation (7.4.52) tells us that in this model of the universe the time derivative of the
Hubble parameter is
Ḣ = −4π ρ̄m µ̄m . (7.8.3)
The field equation (7.7.37) for scalar Vm is reduced to that describing the time evolution of the perturba-
tion of the ideal fluid density, δρm . Indeed, the scalar Vm defined by equation (7.6.13a), can be recast to
the form given by equation (7.6.22), that is
vs2
Vm = δm , (7.8.4)
c2
where the gauge-invariant scalar perturbation
δρm
δm ≡ + 3Hχm , (7.8.5)
ρ̄m
is a linear combination of the perturbation of the mass density of the dark matter and the normalized
Clebsch potential χm . Replacing expression (7.8.4) in equation (7.7.37), yields an exact equation for δm
that is
v2 v2 v2

1 − 2s v̄α v̄β δm;αβ − 2s δm + 1 − 3 2s Hv̄α δm,α (7.8.6)
c c c
v2

3
− (1 − 3wm ) 2s + (1 + wm ) H2 δm
2 c
v2

1
+ (1 + wm ) 1 − 3 2s a2 Λδm = 4πa2 (σ + τ ) .
2 c
This equation describes propagation of the density perturbation of dark matter, δm , in the form of sound
waves with velocity vs . Equation (7.8.6) is decoupled from any other perturbation and can be solved
separately after the boundary conditions are specified. For this reason we call (7.8.6) master equation.
Section 7.8 Decoupled System of the Post-Newtonian Field Equations 323
Equation (7.7.39) for potential χ makes no sense since the normalized perturbation χq = ψ/µ̄q of
dark energy in the form of cosmological constant diverges due to the condition µq = ρq = 0. Equation
for the perturbation of dark energy, ψ, itself is obtained from (7.5.68) and is reduced to a homogeneous
wave equation
ψ − 2Hv̄µ ψ,µ = 0 . (7.8.7)
Equation for the normalized Clebsch potential, χm , is derived from equation (7.7.40) and, in the case
of the universe under consideration, reads
v2 v2

1
χm + (1 + wm ) 3H2 − a2 Λ χm = 1 − 2s av̄µ δm,µ − 4aH 2s δm .

(7.8.8)
2 c c
This is an inhomogeneous equation that can be solved as soon as one knows δm from the master equation
(7.8.6). The potential χm is necessary to determine the perturbation of the four velocity of dark matter.
We also need it to find out the metric perturbation q.
Gravitational potential, q, can be determined directly from equation (7.7.30) after solving equations
(7.8.6) and (7.8.8) or by solving equation (7.7.42a) which (in the dark matter+cosmological constant
universe) takes on the following form,
q − 2Hv̄µ q,µ + (1 + 3wm ) H2 − (1 + wm ) a2 Λ q

= (7.8.9a)
v2 v2

8πa2 σ + τ + ρ̄m µ̄m 1 − 2s δm + H 1 + 3 2s χm .
c c
Equations for other components of the metric tensor perturbations are found from (7.7.42b)-(7.7.42d).
In dark matter+cosmological constant universe they read
p − 2Hv̄µ p,µ = 16πa2 τ , (7.8.9b)

pα − 2Hv̄ pα;µ + (1 + 3wm ) H − (1 + wm ) a2 Λ pα
µ 2

= 16πaτα , (7.8.9c)
|
pαβ − 2Hv̄µ p|αβ;µ = |
16πταβ . (7.8.9d)
Equations given in this section are valid for arbitrary cosmological equation of state of dark matter,
p̄m = wm ¯m , that is physically reasonable and makes sense. The parameter wm of the equation of state
should not be replaced with the ratio of vs2 /c2 which characterizes the derivative of pressure, p̄m , with
respect to the energy density, ¯, of dark matter. This is because the parameter wm can depend in the
most general case on the other thermodynamic quantities (like enthropy, temperature, etc.) which may
implicitly depend on ¯. Equations (7.8.6)–(7.8.9d) are decoupled in the sense that all of them can be
solved one after another starting from solving the master equation (7.8.6) for δm .
Case 2: Cold dark matter
Equations of the previous section can be further simplified for some particular equations of state of dark
matter. For example, in the case of cold dark matter (CDM) we can think about it as being made out of
collisionless dust. Background pressure of dust drops down to zero making the parameter of the cold
dark matter equation of state wm = 0. Sound waves do not propagate in dust. Hence, the speed of sound
vs = 0. For this reason all terms being proportional to vs2 and wm vanish in equation (7.8.6). Moreover,
dust has the specific enthalpy, µm = 1 making the energy density of dust equal to its rest mass density
¯m = ρ̄m , and the normalized perturbation χm of the Clebsch potential of dust equal to the perturbation
φ of the Clebsch potential itself, χm = φ. The Friedmann equation (7.4.50) (for k = 0) tells us that
a2
H2 = (8π ρ̄m + Λ) . (7.8.10)
3
Accounting for this result in the master equation (7.8.6), and neglecting all terms being proportional to
the speed of sound, vs and wm , we obtain
v̄α v̄β δm;αβ + Hv̄α δm,α − 4πa2 ρ̄m δm = 4πa2 (σ + τ ) , (7.8.11)
where the terms depending on the cosmological constant, Λ, have cancelled out. This equation is more
familiar when is written down in the preferred FLRW frame, where v̄α = (1,0,0,0). Equation (7.8.11)
assumes the following form
δm00 + Hδm0 − 4πa2 ρ̄m δm = 4πa2 (σ + τ ) , (7.8.12)
where the time derivatives (denoted with a prime) are taken with respect to the conformal time η. Con-
verting the time derivatives in (7.8.12) from the conformal time η to the cosmic time T reduces it to a
canonical form
δ̈m + 2H δ̇m − 4πδm = 4π (σ + τ ) (7.8.13)
which can be found in many textbooks on cosmology [80, 91, 94, 113, 114]. All textbooks always
dropped off the source of the bare perturbation in the right side of (7.8.13) as they are concerned with
the description of the formation of the large scale structure in the universe out of the primordial pertur-
bations. However, omitting the bare perturbation in the right side of (7.8.13) is equivalent to neglecting
the contribution of the small-scale density fluctuations in the early universe to the formation of the large
scale structures – the process which can be physically significant in the cold dark matter scenario of
galaxy formation [15, 16].
Equation (7.8.13) has been derived by previous researchers without resorting to the concept of the
Clebsch potential of the ideal fluid. For this reason, the density contrast, δm , was interpreted as the ratio
of the perturbation of the dust density to its background value, δ = δρm /ρ̄m , without taking into account
the perturbation, φ, of the Clebsch potential. However, the quantity δ is not gauge-invariant which was
considered as a drawback. The scrutiny analysis of the underlying principles of hydrodynamics in the
expanding universe given in the present chapter, reveals that equation (7.8.12) is, in fact, valid for the
gauge-invariant density perturbation δm defined above in (7.8.5). Another distinctive feature of equation
(7.8.12) is the presence of the source of a bare perturbation in its right side. The bare perturbation is
caused by the effective density σ + τ of the matter which comprises the isolated astronomical system
and initiates the growth of instability in the cosmological matter that, in its own turn, induces formation
of the large scale structure of the universe [94, 114]. Standard approach to cosmological perturbation
theory always set σ + τ = 0 and operates with the spectrum of the primordial perturbation of the density
δρm /ρm (but not with the spectrum for δm ).
Equation (7.8.8) in case of dust reads,
1
3H2 − a2 Λ χm = av̄α δm,α ,

χm + (7.8.14)
2
where χm = φ is reduced to the perturbation φ of the Clebsch potential Φ for the reason that in case of
dust µm = 1. If equations (7.8.12) and (7.8.14) are solved, the gravitational perturbations can be found
from equations (7.8.9a)–(7.8.9d), which take on the following form
q − 2Hv̄α q,α + H2 − a2 Λ q 8πa2 [σ + τ + ρ̄m (δm + Hχm )] , (7.8.15a)

=
p − 2Hv̄α p,α = 16πa2 τ , (7.8.15b)

β
+ H2 − a2 Λ pα

pα − 2Hv̄ pα;β = 16πaτα , (7.8.15c)
p|αβ − 2Hv̄γ p|αβ;γ = |
16πταβ . (7.8.15d)
It is interesting to notice that besides the bare density perturbation, σ + τ , caused by an isolated as-
tronomical system, the source for the scalar gravitational perturbation, q, contains in the right side of
equation (7.8.15a) also the induced density perturbation ρ̄m (δm + Hχm ) = δρm + H ρ̄m φ of the back-
ground dark matter . This induced density perturbation is depends on time and leads to a temporal change
of the initial (bare) mass of the isolated astronomical system in the course of the Hubble expansion of the
universe. Thus, our post-Newtonian approach to cosmology explains the origin of the time-dependence
of the central, point-like mass in the cosmological solution found by McVittie [87] (see also discussion
in [24]).
Case 3: Hot dark matter
Hot dark matter (HDM) is a hypothetical form of dark matter which consists of ultrarelativistic particles
that travel with velocities being very close to the fundamental speed c. A plausible candidate for the
hot dark matter is neutrino. Hot dark matter taken alone, cannot explain how individual galaxies were
formed from the primordial perturbations. Therefore, hot dark matter is discussed only as part of a mixed
dark matter theory [11]. Nonetheless, the case of the hot dark matter is interesting from mathematical
point of view. Equation of state of the hot dark matter is approximated by the radiative equation of state,
pm = (1/3)m which yields the parameter
p wm = 1/3. We assume that this parameter is constant and,
hence, the speed of sound vs = 1/3c. This value of vs is comparable with the fundamental speed
c which means that we have to keep the terms with the speed of sound in the master equation (7.8.6).
The values of wm and vs for the hot dark matter equation of state reduce the master equation for the
gauge-invariant HDM density perturbation δm to the following form
s δm + 6H2 δm = −12πa2 (σ + τ ) , (7.8.16)
where
c2 α β

s ≡ − v̄ v̄ + π̄ αβ ∂αβ , (7.8.17)
vs2
√
is the D’Alembert wave operator with the speed of propagation of sound waves vs = c/ 3. Equation
for the perturbation of the Clebsch potential of the hot dark matter is derived from (7.8.8) and reads

1 2a µ
χm + 2 H2 − a2 Λ χm = (v̄ δm,µ − 2Hδm ) . (7.8.18)
3 3
Equations for the gravitational perturbations are

2 2
q − 2Hv̄µ q,µ + 2 H2 − a2 Λ q = 8πa2 σ + τ + ρ̄m µ̄m (δm + 3Hχm ) . (7.8.19a)
3 3
Equations for other components of the metric tensor perturbations are found from (7.7.42b)-(7.7.42d).
In dark matter+cosmological constant universe they read
p − 2Hv̄µ p,µ = 16πa2 τ , (7.8.19b)

2
pα − 2Hv̄µ pα;µ + 2 H2 − a2 Λ pα = 16πaτα , (7.8.19c)
3
p|αβ − 2Hv̄µ p|αβ;µ = |

16πταβ . (7.8.19d)
7.8.2 The Universe Governed by Dark Energy
In this section we explore the case of the universe governed primarily by a dark energy (scalar field
Ψ) with dark matter constituent being unimportant. In this case, the time evolution of the background
universe is defined exceptionally by equations (7.7.5a), (7.7.5b). The most general solution of (7.7.5a),
(7.7.5b) is complicated and can not be achieved analytically. Numerical analysis shows that the solution
evolves in the phase space of the two variables {x1 ,x2 } from an unstable to a stable fixed point by
passing through a saddle point [2]. The cosmic acceleration is realized by the stable point with the
values of x1 = λ2 /6 and x2 = 1 − λ2 /6, which is equivalent to the equations of state (7.4.54) with
the values of the parameters, wm = 0, and, wq = −1 + λ2 /3. It also requires the energy density of the
background matter ¯m = 0, that is Ωm = 0. In such a universe the derivatives of the potential of the
scalar field are
1 ∂ W̄ 3 ∂ 2 W̄ 9
= H 2 1 − wq2 .

= − H (1 − wq ) , (7.8.20)
µ̄q ∂ Ψ̄ 2 ∂ Ψ̄2 2
Moreover, because ρ̄m µ̄m = ¯m + p̄m = 0, the time derivative of the Hubble parameter is
3
Ḣ = −4π ρ̄q µ̄q = − H 2 (1 + wq ) . (7.8.21)
2
In the point of the attractor of the scalar field, perturbations of the dark matter are fully suppressed
that is the normalized value of the perturbed Clebsch potential of the dark matter, χm = 0. It makes the
function Vm = q/2, that is reduced to the perturbation of the scalar component of the gravitational field
only. Perturbations of the scalar field are described by the scalar field variable, χq . In particular, after
substituting the derivatives (7.8.20) of the scalar field potential along with the derivative (7.8.21) of the
Hubble parameter, to (7.6.37), we obtain the post-Newtonian equation for function Vq ,
3 2
Vq − (1 − 3wq ) Hv̄µ Vq,µ + H (1 − wq ) (1 + 3wq ) Vq = −4πa2 (σ + τ ) . (7.8.22)
2
Field equation for the perturbation of the scalar field, χq , is reduced to
1
χq + (1 + 3wq ) H2 χq = − (1 + 3wq ) aHVq . (7.8.23)
2
Post-Newtonian equations for gravitational perturbations are (7.7.42a)–(7.7.42d). After substituting the
values of the parameters x1 ,x2 , weff , etc., corresponding to the model of the universe governed by the
dark energy alone, the post-Newtonian equations for the metric perturbations become
q − 2Hv̄µ q,µ + (1 + 3wq ) H2 q = 8πa2 (σ + τ ) (7.8.24a)

3
+ (1 + wq ) (1 + 3wq ) H3 χq ,
a
p − 2Hv̄µ p,µ = 16πa2 τ , (7.8.24b)
pα − 2Hv̄ pα;µ + (1 + 3wq ) H2 pα
µ
= 16πaτα , (7.8.24c)
|
pαβ − 2Hv̄µ p|αβ;µ = |
16πταβ . (7.8.24d)
One can see that the field equations for the perturbations of dark energy and gravitational field are
decoupled, and can be solved separately starting from the master equation (7.8.22).
7.8.3 Post-Newtonian Potentials in the Linearized Hubble Approximation
The metric tensor perturbations

The post-Newtonian equations for cosmological perturbations of gravitational and matter field variables
crucially depend on the equation of state of the matter fields in the background universe. It determines
the time evolution of the scale factor a = a(η) and the Hubble parameter H = H(η) which are de-
scribed by the wide range of elementary and special functions of mathematical physics (see, for example,
textbooks [2, 85, 106] and references therein). It is not the goal of the present chapter to provide the
reader with an exhaustive list of the mathematical solutions of the perturbed equations which requires
theoretical development of cosmological Green’s function (see, for example, [51, 78, 79, 97]). We notice
that solving the field equations of the Post-Newtonian approximations in cosmology is more complicated
than in case of the post-Newtonian theory in asymptotically flat spacetime. The reason is twofold: (1)
the system of the post-Newtonian equations on cosmological background involves, besides equations
for the metric tensor perturbations, also the equations for the perturbations of the matter that curves the
background manifold and governs its temporal evolution; and (2) the post-Newtonian field equations in
cosmology depend on the time dependent Hubble parameter that makes finding the Green functions of
the field equations pretty difficult task. If we are interested in finding the far zone (radiative) solution for
the gravitational field of an isolated astronomical system, we have to fulfil this task exactly. This problem
has not yet been solved though it is very important for doing precise cosmology with gravitational wave
astronomy. On the other hand, we can employ the post-Friedmannian approximations by looking for the
solution of the cosmological post-Newtonian equations as a series with respect to the Hubble parameter.
In this section we shall limit ourselves with the linearized Friedmann approximation. In other words,
we shall take into account only the terms which are proportional to the Hubble parameter H, and shall
systematically neglect all terms which are quadratic, cubic, and higher-orders with respect to H.
As we shall see, in the linearized Friedmann approximation the post-Newtonian equations for the
field perturbations have identical mathematical structure so that they are not only decoupled from one
another, but their generic solution can be found irrespectively of the equation of state governing the
background universe. Indeed, if we neglect all quadratic with respect to H terms, the field equations for
the conformal metric perturbations are reduced to the following set,
q − 2Hv̄α q,α = 8πa2 (σ + τ ) , (7.8.25a)

α 2
p − 2Hv̄ p,α = 16πa τ , (7.8.25b)
pα − 2Hv̄β pα;β = 16πaτα , (7.8.25c)
|
pαβ − 2Hv̄γ p|αβ;γ = 16πταβ ,|
(7.8.25d)
where the wave operator has been defined in (7.7.32), and the source of the bare perturbation is the
tensor of energy-momentum of a localized astronomical system having a bounded matter support in
space – see section (7.4.8). The differential structure of the left side of equations (7.8.25a)–(7.8.25d) is
the same for all functions. The equations differ from each other only in terms of the order of H2 which
have been omitted.
In order to bring equations (7.8.25a)-(7.8.25d) to a solvable form, we resort to relation (7.7.34) which
reveals that in the linearized Friedmann approximation, the post-Newtonian equations for metric pertur-
bations can be reduced to the form of a wave equation
(aq) = 8πa3 (σ + τ ) , (7.8.26a)

3
(ap) = 16πa τ , (7.8.26b)
(apα ) = 16πa2 τα , (7.8.26c)
ap|αβ |

= 16πaταβ . (7.8.26d)
So far, we did not impose any limitations on the curvature of space that can take three values: k =
{−1,0, + 1}. Solution of wave equations (7.8.26a)-(7.8.26d) can be given in terms of special functions
in case of the Riemann (k = +1) or the Lobachevsky (k = −1) geometry [78, 79]. The case of the
spatial Euclidean geometry (k = 0) is more manageable, and will be discussed below.
If the FLRW metric is spatially-flat universe, k = 0, and we chose the Cartesian coordinates xα
related to the isotropic coordinates X α of the FLRW metric by a Lorentz transformation, X α = Lα β xβ ,
where Lα β is the matrix of the Lorentz boost. In these coordinates the operator becomes a wave
operator in the Minkowski spacetime,
= η µν ∂µν . (7.8.27)
and equations (7.8.26a)-(7.8.26d) are reduced to the inhomogeneous wave equations which solution
depends essentially on the boundary conditions imposed on the metric tensor perturbations at conformal
past-null infinity J− of the cosmological manifold [89]. We shall assume a no-incoming radiation
condition also known as Fock-Sommerfeld’s condition [31, 44]
lim
r→+∞
nγ ∂γ [a(η)rlαβ (xγ )] = 0 , (7.8.28)
t+r=const.
where xγ = (x0 ,xi ), η ≡ X 0 is the conformal time in isotropic coordinates connected to the coordinates
xα by a Lorentz boost η = η(xγ ) = L0 β xβ , the null vector nα = {1,xi /r}, and r = δij xi xj is the
radial distance. This condition ensures that there is no infalling gravitational radiation arriving to the
localized astronomical system from the future null infinity J+ . Effectively, it singles out the retarded
solution of the wave equation. Whether the boundary condition (7.8.28) is valid or not, we do not
know for sure because our knowledge of the universe is limited by the existence of the cosmological
(also known as light or particle) horizon [81] that represents the boundary between the observable and
the unobservable regions of the universe. Nonetheless, in case of spatially flat (k = 0) universe, the
condition (7.8.28) seems to be highly plausible.
A particular solution of the wave equations satisfying condition (7.8.28), is the retarded integral [75]
a3 [η (s,x0 )] [σ (s,x0 ) + τ (s,x0 )] d3 x0

Z
2
q(t,x) = − , (7.8.29a)
a [η(t,x)] V |x − x0 |
a3 [η(s,x0 )] τ (s,x0 ) d3 x0
Z
4
p(t,x) = − , (7.8.29b)
a [η(t,x)] V |x − x0 |
Z 2 0 0 3 0
4 a [η(s,x )] τα (s,x ) d x
pα (t,x) = − , (7.8.29c)
a [η(t,x)] V |x − x0 |
a [η(s,x0 )] ταβ
|
(s,x0 ) d3 x0
Z
4
p|αβ (t,x) = − , (7.8.29d)
a [η(t,x)] V |x − x0 |
where the scale factor a in front of the integrals depends on the coordinates of the field point a ≡
a [η(t,x)], and the functions in the integrand depend on the retarded time
s = t − |x − x0 | , (7.8.30)
because gravity propagates with finite speed. Equation (7.8.30) describes characteristic of the null cone
in the conformal Minkowski spacetime that determines the causal nature of the gravitational field in
the expanding universe with k = 0 []. Solutions (7.8.29a)–(7.8.29d) are Lorentz-invariant as shown by
calculations in section 7.8.4.
Integration in (7.8.29a)–(7.8.29d) is performed over the finite volume, V, occupied by the matter of
the localized astronomical system. In case of the system comprised of N massive bodies which are
separated by distances being much larger than their characteristic size, the matter occupies the volumes
of the bodies. In this case the integration in equations (7.8.29a)–(7.8.29d) is practically performed
over the volumes of the bodies. It means that each post-Newtonian potential q,p,pα ,p|αβ is split in the
algebraic sum of N pieces
N
X N
X N
X N
X
q= qA , p= pA , pα = pAα , p|αβ = p|Aαβ , (7.8.31)
A=1 A=1 A=1 A=1
where each function with sub-index A has the same form as one of the corresponding equations (7.8.29a)–
(7.8.29d) with the integration performed over the volume, VA , of the body A. This confirms the principle
of superposition in the linearised Friedmann approximation.
The gauge functions

The residual gauge freedom describes arbitrariness in adding a solution of homogeneous wave equations
(7.8.29a)–(7.8.29d). It is described by two functions, ζ ≡ ξ/a and ζ α as discussed in section 7.7.4.
Since we neglected the terms being quadratic with respect to the Hubble parameter, equations (7.7.44a),
(7.7.44b) gets simpler, and read
ζ − 2Hv̄β ζ,β = 0, (7.8.32a)

α β α
ζ − 2Hv̄ ζ ;β = 0. (7.8.32b)
They are equivalent to the homogeneous wave equations in the conformal flat spacetime
(aζ) = 0 , (aζ α ) = 0 , (7.8.33)
which point out that (in the approximation under consideration) the products, aζ and aζ α , are the har-
monic functions.
Potentials q,p,pα ,p|αβ must satisfy the gauge conditions (7.7.43a), (7.7.43b). Neglecting terms being
quadratic with respect to the Hubble parameter, the gauge conditions (7.7.43a), (7.7.43b) can be written
down as follows
(apα ),α + v̄α (2aq − ap),α + Hap = 0, (7.8.34a)

1 αβ
ap|αβ ,β + v̄β (apα ) ,β + π̄ (ap),β + Hapα = 0, (7.8.34b)
3
where we have taken into account a,α = −aHv̄α , and pα v̄α = 0, p|αβ v̄β = 0. The potentials pα
and p|αβ are obtained from pα and p|αβ by rising the indices with the Minkowski metric and taking
|
into account that the indices of τα and ταβ in the integrands of (7.8.29c) and (7.8.29d) should be raised
with the full background metric ḡ αβ
= a−2 η αβ taken at the point of integration. This is because by
convention having been adopted in section 7.4.8, the notations τ α ≡ ḡ αβ τβ and τ αβ ≡ ḡ αµ ḡ βν τµν . It
yields
a4 [η(s,x0 )] τ α (s,x0 ) d3 x0
Z
4
pα (t,x) = − , (7.8.35a)
a [η(t,x)] V |x − x0 |
a5 [η(s,x0 )] τ |αβ (s,x0 ) d3 x0
Z
4
p|αβ (t,x) = − . (7.8.35b)
a [η(t,x)] V |x − x0 |
It is instrumental to write down solutions for the products of the potentials p and pα = η αβ pβ with the
Hubble parameter. Multiplying both sides of equations (7.8.26b), (7.8.26c) with the Hubble parameter
H, and neglecting the quadratic with respect to H terms, we obtain
(aHp) = 16πa3 Hτ , (aHpα ) = 16πa4 Hτ α , (7.8.36)
which solutions are the retarded potential
a3 [η(s,x0 )] H [η(s,x0 )] τ (s,x0 ) d3 x0

Z
aHp(t,x) = −4 , (7.8.37a)
V |x − x0 |
a4 [η(s,x0 )] H [η(s,x0 )] τ α (s,x0 ) d3 x0
Z
aHpα (t,x) = −4 . (7.8.37b)
V |x − x0 |
Substituting functions q,p,pα ,p|αβ and aHp, aHpα to the gauge equations (7.8.34a), (7.8.34b),
bring about the following integral equations
Z h i d3 x0
a4 τ α + v̄α a3 σ ,α + a3 Hτ

= 0, (7.8.38a)
V |x − x0 |
" #
d3 x0
Z
1
a5 τ |αβ + a4 v̄β τ α + π̄ αβ a3 τ + a4 Hτ α = 0, (7.8.38b)
V 3 ,β |x − x0 |
where all functions in the integrands are taken at the retarded time s and at the point x0 , for example,
a ≡ a[η(s,x0 )], H ≡ H[η(s,x0 )], σ ≡ σ[(s,x0 )], and so on. These equations are satisfied by the
equations of motion (7.4.71a), (7.4.71b) of the localized matter distribution. Indeed, divergences of any
vector F α and a symmetric tensor F αβ obey the following equalities
1 √
F α |α −ḡF α ,α ,

= √ (7.8.39)
−ḡ
1 √
F αβ |β = √ −ḡF αβ + Γ̄α
βγ F
βγ
. (7.8.40)
−ḡ ,β
Moreover, the root square of the determinant of the background metric tensor is expressed in terms of
√
the scale factor, −ḡ = a4 , while the four-velocity ūα = v̄α /a. Employing these expressions along
with equations (7.8.39), (7.8.40) in equations of motion (7.4.71a), (7.4.71b), transform them to
a4 τ α + v̄α a3 σ ,α + a3 Hτ

= 0, (7.8.41a)

a4 τ αβ + a3 v̄β τ α + 2a3 Hτ α = 0. (7.8.41b)
,β
Equation (7.8.41a) proves that the integral equation (7.8.38a) is valid. In order to prove the second
integral equation (7.8.38b), we multiply equation (7.8.41b) with the scale factor a, and reshuffle its
terms. It brings (7.8.41b) to the following form

a5 τ αβ + a4 v̄β τ α + a4 Hτ α = 0 . (7.8.42)
,β
Substituting, τ αβ = τ |αβ + (1/3a2 )π̄ αβ τ , to (7.8.42) and comparing with the integrand in (7.8.38b)
makes it clear that (7.8.38b) is valid. We conclude that the retarded integrals (7.8.29a)–(7.8.29d) yield
the complete solution of the linearised wave equations (7.8.26a)–(7.8.26d) in the sense that there is no
residual gauge freedom since the gauge functions ζ = ζ α = 0.
Perturbations of dark matter and dark energy
What remains is to find out solutions for the scalar functions Vm and Vq and χm and χq . In the linearised
Friedmann approximation equation for Vm is obtained from (7.7.37) by discarding all terms of the order
of H2 . It yields
c2 c2

Vm + 1 − 2 v̄α v̄β Vm,αβ + 3 − 2 Hv̄α Vm,α = −4πa2 (σ + τ ) . (7.8.43)
vs vs
Applying relations (7.7.34), (7.7.35) in equation (7.8.43) allows us to recast it to
c2

1 n α β n
(a Vm ) + 1 − v̄ v̄ (a V m ),αβ (7.8.44)
an vs2
2

c
+ 3 + (2n − 1) 2 Hv̄α Vm,α = −4πa2 (σ + τ ) ,
vs
where n is yet undetermined real number. Now, we postulate that the speed of sound vs is constant.
Then, choosing, n ≡ ns , with
v2

1
ns = 1 − 3 2s , (7.8.45)
2 c
annihilates the term being proportional to H in the left side of (7.8.44) and reduces it to
c2

(ans Vm ) + 1 − 2 v̄α v̄β (ans Vm ),αβ = −4πa2+ns (σ + τ ) . (7.8.46)
vs
This equation describes propagation of perturbation Vm with the speed of sound vs . Indeed, let us
introduce the sound-wave Laplace-Beltrami operator (7.8.17). Then, equation (7.8.46) reads
s (ans Vm ) = −4πa2+ns (σ + τ ) . (7.8.47)
This equation has a well-defined Green function with characteristics propagating with the speed of sound
vs . We discard the advanced Green function because we assume that at infinity the function Vm and its
first derivatives vanish. Solution of (7.8.47) is explained below in section 7.8.5, and has the following
form
a2+ns (ς,x0 ) [σ(ς,x0 ) + τ (ς,x0 )] d3 x0
Z
1
Vm (t,x) = n , (7.8.48)
a (t,x) V |x − x0 |
s
s
c2

1 + γ 1 − 2 (β × n)
2 2
vs
p
where the retarded time ς is given by equation (7.8.96), β = β i = v̄i /c, γ = 1/ 1 − β 2 is the Lorentz
factor, and the unit vector n = (x − x0 )/|x − x0 |. The retardation in the solution (7.8.48) is due to the
finite speed of propagation of acoustic (sound) waves in the ideal fluid that represents the dark matter.
Equation for Vq is obtained in the lineazised Friedmann approximation from (7.7.38) after discarding
all terms being proportional to H2 . It yields
r !
3
Vq + 2 1 − λx2 Hv̄µ Vq,µ = −4πa2 (σ + τ ) . (7.8.49)
2x1
Applying relation (7.7.34) in (7.8.49) allows us to recast it to

r !
1 3
n
(a Vq ) + 2 n + 1 − λx2 Hv̄α Vm,α = −4πa2 (σ + τ ) . (7.8.50)
an 2x1
√ p
If, and only if, the ratio λx2 / x1 is constant, we can choose, n ≡ nq = −1+ 3/(2x1 )λx2 , in order to
eradicate the second term in the left side of (7.8.50). In those models of the universe where this condition
is satisfied, the resulting equation for Vq is simplified and reads
(anq Vq ) = −4πa2+nq (σ + τ ) . (7.8.51)
This is the wave equation in flat spacetime. We pick up the retarded solution as the most physical one,
a2+nq (s,x0 ) [σ (s,x0 ) + τ (s,x0 )] d3 x0

Z
1
Vq = nq , (7.8.52)
a (t,x) V |x − x0 |
where the retarded time s has been defined in (7.8.30).

Perturbations χm and χq can be found by integrating equations (7.6.13a) and (7.6.13b) that can be
written as q q
v̄α χm,α = a Vm + , v̄α χq,α = a Vq + . (7.8.53)
2 2
These are the ordinary differential equations of the first order. Their solutions are
Z t
1
χm = a[t,x(t)]{Vm [t,x(t)] + q[t,x(t)]}dt , (7.8.54a)
t0 2
Z t
1
χq = a[t,x(t)]{Vq [t,x(t)] + q[t,x(t)]}dt , . (7.8.54b)
t0 2
where t0 is an initial epoch of integration, and the integration is performed along the Hubble flow of the
background universe
dxi
= v̄i (t,x) . (7.8.55)
dt
Therefore, the most simple way to integrate equations (7.8.53) would be to work in the preferred coor-
dinate frame X α = (η,X i ) where the velocity v̄i = 0, and the spatial coordinates X i = const. After
the calculation in the rest frame of the Hubble flow is finished, the transformation to a moving frame of
observer can be done with the help of the coordinate transformation between the two frames.
7.8.4 Lorentz Invariance of Retarded Potentials
We use a prime in the appendices exclusively as a notation for time and spatial coordinates which are
used as variables of integration in volume integrals (see, for example, equations (7.8.57), (7.8.58), and
so on). It should not be confused with the time derivative with respect to the conformal time used in the
main text of the present chapter.
Let us consider an inhomogeneous wave equation for a scalar field, V = V (η,X), written down in a
coordinate chart X α = (X 0 ,X i ) = (η,X),
V = −4πσX , (7.8.56)
where ≡ η αβ ∂αβ , ∂α = ∂/∂X α , and σX = σX (η,X) is the source (a scalar function) of the field
V with a compact support (bounded by a finite volume in space). Equation (7.8.56) has a solution given
as a linear combination of advanced and retarded potentials. Let us focus only on the retarded potential
which is more common in physical applications. Advanced potential can be treated similarly.
We assume the field, V , and its first derivatives vanish at past null infinity. Then, the retarded solution
(retarded potential) of (7.8.56) is given by an integral,
σX (ζ,X 0 ) d3 X 0
Z
V (η,X) = , (7.8.57)
V |X − X 0 |
where
ζ = η − |X − X 0 | , (7.8.58)
is the retarded time, and we assume the fundamental speed c = 1. Physical meaning of the retardation
is that the field V propagates in spacetime with the fundamental speed c from the source σX , to the
point with coordinates X α = (η,X) where the field V is measured in correspondence with equation
(7.8.57). Left side of equation (7.8.56) is Lorentz-invariant. Hence, we expect that solution (7.8.58)
must be Lorentz-invariant as well. As a rule, textbooks prove this statement for a particular case of the
retarded (Liénard-Wiechert) potential of a moving point-like source but not for the retarded potential
given in the form of the integral (7.8.57). This appendix fulfils this gap.
Lorentz transformation to coordinates, xα = (t,x) linearly transforms the isotropic coordinates
X α = (η,X) of the FLRW metric as follows
x α = Λα β X β , (7.8.59)
where the matrix of the Lorentz boost [89]

γ−1 i j
Λ0 0 = γ , Λi 0 = Λ0 i = −γβ i , Λi j = δ ij + β β , (7.8.60)
β2
the boost four-velocity uα = {u0 , ui } = u0 {1,β i } is constant, and
1
γ = u0 = p , (7.8.61)
1 − β2
is the constant Lorentz-factor.

The inverse Lorentz transformation is given explicitly as follows
η = γ(t + β · x) , (7.8.62)
γ2
X = r+ (β · r)b , (7.8.63)
1+γ
where
r = x + βt , (7.8.64)
i i 0
and the boost three-velocity, β = {β } = {u /u }.
Let us reiterate (7.8.57) by introducing a one-dimensional Dirac’s delta function and integration with
respect to time η, Z ∞ Z
σX (η 0 ,X 0 )δ(η 0 − ζ) dη 0 d3 X 0
V (η,X) = , (7.8.65)
−∞ V |X − X 0 |
where ζ is the retarded time given by (7.8.58). Then, we transform coordinates X 0α = (η 0 ,X 0 ) to
x0α = (t0 ,x0 ) with the Lorentz boost (7.8.59). The Lorentz transformation changes functions entering
the integrand of (7.8.65) as follows,
σ(η 0 ,X 0 ) = σx (t0 ,x0 ) , (7.8.66)

0
p
|X − X | = |r − r 0 |2 + γ 2 [β · (r − r 0 )]2 , (7.8.67)
where the coordinate difference
r − r 0 = x − x0 + β(t − t0 ) . (7.8.68)
The coordinate volume of integration remains Lorentz-invariant
dη 0 d3 X 0 = dt0 d3 x0 . (7.8.69)
Let us denote Fη (η 0 ) ≡ η 0 − ζ where ζ is given by (7.8.58). After making the Lorentz transformation
this function changes to
Fη (η 0 ) = Ft (t0 ) γ t 0 − t − β · x − x0

= (7.8.70)
p
0 0 0 0
+ |x − x | − (t − t) + γ [β · (x − x ) − (t − t)] ,
2 2 2 2
where we have used equations (7.8.62), (7.8.63) and (7.8.67) and relationship γ 2 β 2 = γ 2 −1, to perform
the transformation. Integral (7.8.65) in coordinates xα becomes
Z ∞ Z
σx (t0 ,x0 )δ(Ft (t0 )) dt0 d3 x0
V (t,x) = p , (7.8.71)
−∞ V |r − r 0 |2 + γ 2 [β · (r − r 0 )]2
The delta function has a complicated argument Ft (t0 ) in coordinates xα . It can be simplified with a
well-known formula
δ(t0 − s)
δ Ft (t0 ) =

, (7.8.72)
Ḟt (s)
where Ḟt (s) ≡ [dFt (t0 )/dt0 ]t0 =s , and s is one of the roots of equation Ft (t0 ) = 0 that is associated with
the retarded interaction. It is straightforward to confirm by inspection that the root is given by formula,
s = t − |x − x0 | . (7.8.73)
The time derivative of function Ft (t0 ) is
β 2 (t0 − t) − β · (x − x0 )
Ḟt (t0 ) = γ + γ 2 p . (7.8.74)
|x − x0 |2 − (t0 − t)2 + γ 2 [β · (x − x0 ) − (t0 − t)]2
After substituting t0 = s, with s taken from equation (7.8.73), we obtain,
1 |x − x0 |
Ḟt (s) = . (7.8.75)
γ |x − x | + β · (x − x0 )
0
Performing now integration with respect to t0 in equation (7.8.71) with the help of the delta-function, we
arrive to
σx (s,x0 ) d3 x0
Z
V (t,x) = 0
, (7.8.76)
V Ḟt (s)|X − X |t0 =s
where |X − X 0 |t0 =s must be calculated from (7.8.67) with t0 = s where s is taken from (7.8.73). It
yields
Ḟt (s)|X − X 0 |t0 =s = |x − x0 | , (7.8.77)
and proves that the retarded potential (7.8.57) is Lorentz-invariant
σX (ζ,X 0 ) d3 X 0 σx (s,x0 ) d3 x0
Z Z
0
= . (7.8.78)
V |X − X | V |x − x0 |
We have verified the Lorentz invariance for the scalar retarded potential. However, it is not difficult to
check that it is valid in case of a source σα1 α2 ...αl that is a tensor field of rank l. Indeed, the Lorentz
transformation of the source leads to Λβ1 α1 Λβ2 α2 ...Λβl αl σβ1 β2 ...βl but the matrix Λα β is constant,
and can be taken out of the sign of the retarded integral. Because of this property, all mathematical
operations given in this appendix for a scalar retarded potential, remain the same for the tensor of any
rank. Hence, the Lorentz invariance of the retarded integral is a general property of the wave operator in
Minkowski spacetime.
7.8.5 Retarded Solution of the Sound-wave Equation

Let us consider an inhomogeneous sound-wave equation for a scalar function U = U(η,X) describing
a perturbation propagation in a medium. This equation written down in the isotropic coordinates X α =
(η,X), reads
s U = −4πτX , (7.8.79)
where τX = τX (η,X) is the source of U having a compact support, and the sound-wave differential
operator s was defined in (7.8.17). It is Lorentz-invariant and reads
c2

s = + 1 − 2 v̄α v̄β ∂αβ , (7.8.80)
cs
where v̄α is four-velocity of motion of the medium with respect to the coordinate chart, vs is the constant
speed of sound in the medium, and we keep the fundamental speed c in the definition of the operator for
dimensional purposes. We assume that vs < c. The case of vs = c is treated in section 7.8.4, and the
case of vs ≥ c makes a formal mathematical sense in discussion of the speed of propagation of gravity
in alternative theories of gravity since the equation describing propagation of gravitational potential U
has the same structure as (7.8.79) after formal replacement of vs with the speed of gravity cg [68, 118].
In particular, in the Newtonian theory the speed of gravity cg = ∞, and the operator (7.8.80) is reduced
to the Laplace operator
∆ = + vα vβ ∂α ∂β = π̄ αβ ∂αβ , (7.8.81)
where the constant projection operator, π̄ αβ , has been defined in (7.7.23).
We are looking for the solution of (7.8.79) in the Cartesian coordinates xα = (t,x) moving with
respect to the isotropic coordinates X α with constant velocity β i . Transformation from X α to xα is
given by the Lorentz transformation (7.8.59). In coordinates X α the four-velocity v̄α = (1,0,0,0).
Therefore, in these coordinates, equation (7.8.79) is just a wave equation for the field U propagating
with speed vs . It has a well-known retarded solution,
τX (ηs ,X 0 ) d3 X 0
Z
U(η,X) = , (7.8.82)
V |X − X 0 |
where
c
ηs = η − |X − X 0 | , (7.8.83)
vs
is the retarded time.
Equation (7.8.79) is Lorentz-invariant. Hence, its solution must be Lorentz-invariant as well. Our
goal is to prove this statement. To this end, we take solution (7.8.82) and perform the Lorentz trans-
formation (7.8.62), (7.8.63). We recast the retarded integral (7.8.82) to another form with the help of
one-dimensional delta-function
Z ∞Z
τX (η 0 ,X 0 )δ(η 0 − ηs ) dη 0 d3 X 0
U(η,X) = . (7.8.84)
∞ V |X − X 0 |
It looks similar to (7.8.57) but one has to remember that the retarded time ηs differs from ζ that was
defined in (7.8.58) on the characteristics of the null cone defined by the fundamental speed c. Transfor-
mation of functions entering integrand in (7.8.84) is similar to what we did in section 7.8.4 but, because
vs 6= c, calculations become more involved. It turns out more preferable to handle the calculations
in tensor notations, making transition to the coordinate language only at the end of the transformation
procedure.
Let us consider two events with the isotropic coordinates X α = (η,X) and X 0α = (η 0 ,X 0 ). We
postulate that in the coordinate chart, xα , these two events have coordinates, xα = (t,x), and, x0α =
(t0 ,x0 ), respectively. We define the components of a four-vector, rα = (t0 − t,x − x0 ) which is
convenient for doing mathematical manipulations with the Lorentz transformations. For instance, the
Lorentz transformation of the Euclidean distance between the spatial coordinates of the two events, is
given by a
|X − X 0 | = π̄αβ rα rβ ,
p
(7.8.85)
where π̄ αβ is the operator of projection on the hyperplane being orthogonal to v̄α (the same operator
as in (7.8.81)). Equation (7.8.85) is a Lorentz-invariant analogue of expression (7.8.67) and matches
it exactly. Transformation of the source, τX (X α ) = τx (xα ) is fully equivalent to that of σX as given
by equation (7.8.66). Coordinate volume of integration transforms in accordance with (7.8.69). We
need to transform the argument, η 0 − ηs , of delta-function which we shall denote in coordinates X α as
fη (η 0 ) ≡ η 0 − ηs . The argument is a scalar function which is transformed as fη (η 0 ) = ft (t0 ) where,
cp
ft (t0 ) = −v̄α rα + π̄αβ rα rβ . (7.8.86)
vs
Transformation of the delta-function in the integrand of integral (7.8.84) is
δ(t0 − ς)
δ ft (t0 ) =

, (7.8.87)
f˙t (ς)
where f˙t (ς) ≡ [dft (t0 )/dt0 ]t0 =ς , and ς is one of the roots of equation ft (t0 ) = 0 that is associated with
the retarded interaction. Eventually, after accounting for transformation of all functions and performing
integration with respect to time, integral (7.8.84) assumes the following form
τx (ς,x0 )d3 x0
Z
U(t,x) = , (7.8.88)
˙ 0
V ft (ς)|X − X |t0 =ς
where |X − X 0 |t0 =ς denotes the expression (7.8.85) taken at the value of t0 = ς. What remains is
to calculate the instant of time, ς, and the value of functions entering denominator of the integrand in
(7.8.88).
Calculation of ς is performed by solving equation ft (ς) = 0, that defines the characteristic cone of
the sound waves, and has the following explicit form,
v2

ηαβ + 1 − 2s v̄α v̄β rα rβ = 0 , (7.8.89)
c
which is derived from (7.8.86). This is a quadratic algebraic equation with respect to the time variable
r0 = ς − t. It reads
A(ς − t)2 + 2B(ς − t) + C = 0 , (7.8.90)
where the coefficients A,B,C of the quadratic form are,
v2

A = −1 + 1 − 2s γ 2 , (7.8.91)
c
v2

B = − 1 − 2s γ 2 β · (x − x0 ) , (7.8.92)
c
v2

2
C = |x − x0 |2 + 1 − 2s γ 2 β · (x − x0 ) ,

(7.8.93)
c
p
and γ = 1/ 1 − β 2 is the Lorentz factor. Equation (7.8.90) has two roots corresponding to the ad-
vanced and retarded times. The root corresponding to the retarded-time solution of (7.8.90) is
1 p
ς =t− B − B 2 − AC , (7.8.94)
A
or, more explicitly,
! v !
vs2 vs2
u
2
u
1− 2 γ (β · n) + t1 − 1− γ 2 [1 − (β · n)2 ]
c c2
0
ς = t − |x − x | ! , (7.8.95)
vs2
1− 1− 2 γ2
c
where the unit vector n = (x − x0 )/|x − x0 |. After some algebra equation (7.8.95) can be simplified
to
αs
ς = t − |x − x0 | , (7.8.96)
vs
where
"s #
1 − β2 c2

c2 2
αs = 1 + 1 − γ 2 (β × n)2 − 1 − γ (β · n) . (7.8.97)
β2 vs2 vs2
1− 2
vs
Coefficient αs defines the speed of propagation of the sound waves, vs ≡ vs /αs , as measured by observer
moving with speed β i with respect to the Hubble flow. Thus, the value of the speed of sound, vs , depends
crucially on the motion of observer.
Derivative of the function, f˙t (ς), is given by
∂ft ∂rα
f˙t (ς) = , (7.8.98)
∂rα ∂ς
where the partial derivative ∂rα /∂ς = δ0α = (1,0,0,0). Making use of (7.8.86), the partial derivative
∂fx c π̄αβ rβ
α
= −v̄α + p , (7.8.99)
∂r vs π̄αβ rα rβ
which has to be calculated at the instant of time, t0 = ς, where ς is given by (7.8.96).

In order to calculate the denominator in the integrand in (7.8.88), we account for (7.8.85), (7.8.89)
and combine (7.8.98), (7.8.99) together. We get
v2

c
|X − X 0 |f˙x (ς) = rα + 1 − 2s v̄α v̄β rβ δ0α . (7.8.100)
vs c
It is straightforward to check that after using (7.8.94) the above equation is reduced to |X −X 0 |f˙x (ς) =
√
(c/vs ) B 2 − AC, or more explicitly,
s
0 ˙ 0 c2
|X − X |fx (ς) = |x − x | 1 + 1 − 2 γ 2 (β × n)2 , (7.8.101)
vs
Finally, the retarded Lorentz-invariant solution of (7.8.79) is
τx (ς,x0 ) d3 x0
Z
U(t,x) = , (7.8.102)
|x − x0 |
v
V u
u 2
!
t1 + γ 2 1 − c
(β × n)2
vs2
with the retarded time ς calculated in accordance with (7.8.96). This solution reduces to the retarded
potential (7.8.78) in the limit of vs → c.
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Index
2l -pole moment, 65 Arminjon, 112

N -body, 1, 3 astrometry, 179
N -body system, 63 astronomical ephemerides, 266
isolated, 69 astronomical system
3+1 formalism, 255 isolated, 177, 182
isolatedi, 265–268, 270, 271, 283, 290, 292,
manifold 313, 319, 324, 325, 327
background, 71 astronomical unit, 209
asymptotic matching, 142
action, 68
active coordinate transformation, 201 backreaction, 253, 254, 257, 259
active galatic nuclei, 179 nonlinear, 258, 260
active internal multipoles, 89 Bardeen, 268, 302, 305
active mass, 99, 103 barycenter, 179, 235
relation to conformal mass, 99 Barycentric Celestial Reference System , 155
adiabatic invariant, 63 bending of light, 220
ADM, 63 bi-tensor, 64
ADM formalism, 38 propagator, 64
ADM-Hamiltonian, 40 Bianchi identity, 69
ADM-harmonic coordinates, 188 binary
advanced solution, 147 coalescing, 63
affine connection, 72, 91 binary pulsar, 63, 112
affine parameter, 191 binary pulsars, 1
AGN, 179 binary system, 179
anti-symmetric, 95 coalescing, 175
anti-symmetrization, 65 main sequence stars, 179
approximation white dwarf, 179
geometric optics, 190 black hole, 68, 79, 141
post-Minkowskian, 184 supermassive, 179
post-Newtonian, 186 black holes, 1, 2, 49
slow-motion, 68 Blanchet, 77
weak-field, 68 Blanchet-Damour theorem, 150
approximations bootstrap effect, 95
post-Friedmannian, 271 boundar, 80
post-Minkowskian, 72, 266, 270 boundary conditions, 72, 85, 86
post-Newtonian, 72, 265–267, 269, 270, no-incoming radiation, 80
283, 289, 294, 304, 315, 327 Brans, 68
348 Index
Brumberg, 112 Cartesian, 181

Buchet, 270 conformal harmonic, 78
harmonic, 63, 73, 78, 186
Caporali, 112 harmonic conformal, 78
Cartesian tensor, 87, 93 inettial, 189
CDM, 260 non-rotating
celestial dynamics, 304, 311 dynamically, 80
celestial ephemeris, 79 kinematically, 80
celestial mechanics, 265, 267–269, 277 post-Newtonian
post-Newtonian, 70 boundary conditions, 79
relativistic, 265 global, 79
celestial mechnaics local, 82
external problem, 79 parametrized, 78
internal problem, 79 spherical, 181
celestial pole, 218 TT, 189
center of mass, 64, 72, 82, 83 Coriolis force, 80, 84
center-of-energy vector, 43, 46, 49 cosmic acceleration, 254, 259
central world-line, 156 cosmic expansion, 260
centrifugal force, 84 cosmological constant, 254
centripetal force, 80 cosmological perturbations, 268, 277
Christoffel symbol, 64 backreaction, 270
Christoffel symbols, 83, 91, 92, 191 field equations, 315, 319
background, 117 gauge-invariant, 304
Christoffel-symbols, 144, 151, 169 Lifshitz’s theory, 268
Clarkson, 270 linearized theory, 270
CMBR, 176 non-linear theory, 270
coalescing binary, 63 post-Newtonian theory, 321, 327
coalescing binary systems, 2 scalar, 276
comsic acceleration, 260 source, 277
conformal compactification, 176 stress-energy tensor, 270
conformal harmonic coordinates, 78 synchronous gauge, 268
conformal infinity, 176 temporal evolution, 270
conformal internal multipoles, 89 covariant derivative, 70, 117
conformal mass, 99, 103 horizontal, 64
congruence, 191 vertical, 64
constraint equations, 38, 39, 42 cross product, 184
convective derivative, 98 curvature, 65
coordinate chart, 177 extrinsic, 255
coordinate conditions, 40, 42 curvature scalar, 145
coordinate time, 177
coordinate transformation D’Alembert, 77
post-Newtonian, 91 d’Alembert criterion, 168
transition functions, 91 D’Eath, 90
coordinates Dallas, 112
ADM, 63 Damour, 77
ADM-harmonic, 188 dark energy, 254, 276, 312, 321, 326, 331
Arnowitt-Deser-Misner, 188 background value, 322
astronomical, 219 cosmological constant, 323
Index 349
density, 307 Effective One Body, 2

model, 313 Ehlers, 124, 270
perturbation, 323 Ehresmann’s connection, 64
perturbations, 317 EIH, 98, 101
speed of propagation, 319 equations of motion, 112
scalar field, 298 Einstein, 63, 97, 98
dark matter, 277, 302, 307, 312, 317, 319, 321, length contraction, 91
323, 326, 331 equations, 256
cold, 276 time dilation, 91
equation of state, 325 Einstein frame metric, 78
field equation, 298, 302 Einstein’s summation rule, 87
hot, 325 Einstein-Infeld-Hoffmann (EIH) equations of
perturbation, 301 motion, 169
de Rham’s operator, 190 electromagnetic wave, 178
de Sitter effect, 57 energy-momentum tensor, 142
degree of polarization, 204, 205 EOP, 82
density of matter, 69 equation
derivative D’Alembert, 270
convective, 98 EIH, 112
covariant, 70, 117 Einstein, 256
partial, 64, 75, 80, 94 Friedmann, 287–289, 291, 300, 313, 314,
variational, 69 324
Dicke, 68, 83, 100, 103, 125
gravitational field, 66
Dicke-Nordtvedt effect, 103
inhomogeneous, 308, 319, 320, 323
differential operator, 296, 298, 299, 318
Laplace, 83, 88, 93
Laplace-Beltrami, 310
Navier-Stokes, 98
linear, 298, 299
Poisson, 82, 93, 266, 270
sound-wave, 335
sound-wave, 335
dimensional regularization, 41
wave, 271, 308, 319, 320, 323, 327–330,
dipole moment
332, 335
active, 100
equation of continuity, 70, 98
conformal, 100
equation of state, 327
external, 84
background matter, 295
internal, 83
cosmological, 288, 323
distribution, 65
dark matter, 312, 322
Dixon, 64, 92
Dixon’s supplementary condition, 65 cold, 323
Doppel tracking, 179 effective, 314, 317
dot product, 184 hydrodynamic, 313
dynamic system, 278 ideal gas, 297
dynamical invariants, 50 matter, 287, 314
dynamical velocity, 64, 124 parameters, 288
radiative, 325
Earth Orientation Parameters, 82 scalar field, 299
eccentricity, 154 stiff, 299
EEP, 83 thermodynamic, 278
effacement of internal structure, 4 equations of motion, 1, 2
effacing principle, 73, 89 rotational, 70
350 Index
translational, 70 gauge condition, 69, 146

equivalence principle, 83 gauge freedom, 76
evolution equations, 42 residual, 76
extended Blanchet-Damour theorem, 158 gauge functions, 232
external multipole, 67 gauge invariance, 164
external multipole moments, 66 general relativity, 64, 139, 140, 176
generalized function, 65
Fermi coordinates, 141 Geocentric Celestial Reference System , 155
Fichtengoltz, 95 geodesic, 151
field equations, 143 geodetic precession, 57, 84, 87, 170
Fierz, 68 geometric optics, 190
Flanagan, 112 Gigaparsec, 209
FLRW, 176, 259, 267, 271, 272, 276, 277, 280– Global Positioning System, 82
284, 286–288, 290–295, 302, 303, GPS, 82
312, 314–316, 324, 328, 333, 352, gravitational energy, 39
355 gravitational field
fluid external, 83, 100
ideal, 276–280, 287–290, 296–300, 303, intrinsic, 83
306, 307, 310, 313, 314, 319, 322, multipolar expansion, 73
324, 331 Schwarzschild, 82
perfect, 287 sperically-symmetric, 73
Fock, 98 stationary, 177
Fokker effect, 57 strong, 68
frame-dragging effect, 57 tidal, 86, 87, 96
Friedmann weak, 72, 79, 82
equation, 287–289, 291, 300, 313, 314, 324 gravitational lens, 234
Friedmann equations, 259, 260 gravitational physics, 175
Futamase, 270 gravitational radiation damping, 53
gravitational radiation reaction, 47
Galilean translation, 91 Gravitational self-force, 2
gamma-ray burst, 222 gravitational wave, 2, 63, 80, 175
gauge plane, 176
conformal, 268 tail, 186
Coulomb, 188 gravitational wave astronomy, 63, 269, 270,
de Donder, 291 311, 327
degrees of freedom, 305 gravitational wave detector, 177
field theory, 304 gravitational wave observatory, 175
freedom, 268, 311 gravitational waves, 80
residual, 312, 321 plane, 179
functions, 329 spectrum, 179
harmonic, 187, 291, 311 stochastic, 179
Newtonian, 257 ultra-long, 179
supplementary field, 304 gravitomagnetic force, 112
synchronous, 268 gravitomagnetic precession, 84
transformation, 304, 305 GRB, 222
dark matter, 307
vector field, 304 Hamiltonian approach, 37
vector function, 300 harmonic coordinates
Index 351
conformal, 78 Issacson averaging, 253

harmonic function, 78
harmonic gauge, 76, 78, 142, 147, 187 Jordan, 68
conformal, 77 Jordan-Fierz frame, 68, 78
harmonic polynomial, 82, 93
Hartle, 72, 98, 101 Kerr geometry, 53
HDM, 325 Kiloparsec, 209
Heviside step function, 183 kinematic rotation, 87
Higgs boson, 68 kinematic velocity, 64, 124
Hoffmann, 98 Kroneker symbol
horizon scale, 257, 259 orthonormal basis, 231
Hubble flow, 256 two-dimensional, 182
Hubble parameter
conformal, 272 Lagrangian, 68
Hublle Lagrangian mechanics, 272
approximation, 327 Landau, 63, 72
expansion, 325 Laplace equation, 83, 88, 93
flow, 281, 283–285, 301, 306, 316, 332, 337 fundamental solution, 93
4-velocity, 290, 292 kernel, 105
velocity, 302 Laplace-Beltrami operator, 69, 77
observers, 283, 303 Laplacian, 146, 188
parameter, 267, 282, 313, 327, 329 lapse function, 255
conformal, 316 length contraction
time derivative, 300, 322 Einstein, 91
time derivative, 287 Lorentz, 91
Lense-Thirring, 112
inertial force, 83 Lense-Thirring effect, 57
Infeld, 98 Lense-Thirring precession, 170
infinity Levi-Civita, 123
null, 178 Levi-Civita symbol, 181
null past, 80 LHC, 68
spatial, 80 Lie transport, 105
timelike, 178 Lifshitz, 63, 72, 268
innermost stable circular orbit, 49 light
intensity direction of propagation, 177
electromagnetic radiation, 204 frequancy, 177
internal multipole, 66, 104 intensity, 177
internal multipole moments, 66 polarization, 177, 179
internal multipoles speed, 177
active, 89 light geodesics, 178
conformal, 89 light ray
mass-type, 89 impact parameter, 179
scalar, 90 line of sight, 179
spin-type, 90 linear connection, 64
internal structure, 76 linear momentum, 64, 123
inverse matrix, 119 LLR, 83, 100
isolated astronomical system, 71, 80 local evolution equations, 166
isolated gravitating system, 89 Lorantz transformation, 222
352 Index
Lorentz metric tensor, 68, 74, 81, 84, 143, 181

length contraction, 91 external, 88
time dilation, 91 external solution, 86
Lorentz group, 201 internal solution, 85
Lorentz-Poincare transformation, 181 perturbation, 86
LTB solution, 254 Minkowski metric, 71, 79, 181, 184
luminosity distance, 260 mixed solution, 147
Lunar Laser Ranging, 83 moment of inertia, 99, 112
momentum-velocity relation, 66, 124
Mach’s principle, 100 monochromatic flux, 206
manifold, 76, 79, 91, 181, 267–269, 274, 275, Mukhanov, 270
277, 280, 284, 287, 290, 296, 302– Multi-chart approach, 3
304, 312–315, 327, 328 multi-chart approach, 4
atlas, 67, 71 multipolar tensor coefficients, 113
background, 117, 118, 123, 267, 269–272, multipolar waveform, 13
280, 281, 283–285, 287–291, 293, multipole
296, 302, 309, 312, 317, 327 external, 83
effective, 117 internal, 83
cosmological, 269 mass-type, 182
curved, 291, 317 multipole moment
differential structure, 71 mass-type, 185
fibre bundle, 304 spin-type, 185
geometric, 272, 305 symmetric trace-free, 185
homegeneity, 281 multipole moments, 140, 148, 158
isotropy, 281 active, 99
perturbed, 271 external, 66, 121
slow, 95 internal, 66, 83
spacetime, 70, 72, 79, 82, 90, 271, 292, 317 multipoles
topological structure, 71 Blanchet-Damour, 177
torsionless, 72 external, 84, 87
mass current-type, 97
active, 99, 103 mass-type, 97
conformal, 99, 103 gravitational, 73, 177
inertial, 108 Hansen, 177
Tolman, 99 internal, 73, 89
mass density spin-type, 182
active, 89 Thorne, 177
conformal, 90 time dependent, 177
mass dipole moment, 64
mass multipole moments, 142 Navier-Stokes equation, 98
Mathisson, 64 near zone, 73
matrix of transformation, 118, 121 neutron star, 68, 141
Maxwell’s theory, 143 neutron stars, 1, 49
merger, 16 Newman-Penrose formalism, 200
metric Newtonian limit, 143
FLRW, 272 Newtonian potential, 85
FLRW, 267, 271, 281–284, 286–288, 292, Nordtvedt, 83, 100, 103
295, 314 effect, 83, 100
Index 353
parameter, 100 post-Newtonian expansion

null geodesic, 191 axioms, 73
null infinity, 218 small parameters, 72
null past infinity, 80 post-Newtonian theory, 267
null rotation, 200 potential
null tetrad, 200, 202 Clebsch, 268, 278, 279, 288, 290, 291, 295,
null vector, 191 298, 301, 303, 305, 322–326
Numerical Relativity, 2 gravitational, 323, 335
nutation, 82 Taub, 268
Nutku, 76 PPN, 70
gauge condition, 77 formalism, 67
theory, 67
operator of projection, 182 PPN formalism, 78, 176
orbit solutions, 51 PPN parameters, 112
Ozernoy, 270 precession, 82
de-Sitter, 84, 95
Papapetrou, 64, 98 geodetic, 95
partial derivative, 64, 75, 80, 94 gravitomagnetic, 95
passive coordinate transformation, 201 Lense-Thirring, 84, 95
past null infinity, 192 post-Newtonian, 91
Penrose diagarm, 178 relativistic, 118
periastron-advance, 21 Thomas, 84, 95
perturbation principle of equivalence, 71, 97
density strong, 109, 125
gauge-invariant, 324 violation, 125
perturbations principle of relativity, 72
gauge-invariant, 320 problem of motion, 1
photon, 177, 192 external, 71
PLANCK, 260 internal, 71
Planck constant, 206 proper motion, 179
plane gravitational wave, 239 proper time, 69
plane of the sky, 219, 220
PMA, 72 quadrupole formula
PNA, 72 Landau-Lifshitz, 63
Poincaré algebra, 43, 44 quantum gravity, 176
Poincaré sphere, 205 quasi-normal-mode, 16
Poisson, 77
equation, 82, 93, 266 Racine, 112
polarization, 202 radiation
polarization tensor, 203 electromagnetic, 179
polarization vector, 205 radiation reaction, 12
polazrization, 201 radiation-reaction, 5
polynomial coefficients, 183 Rendall, 73
post-Minkowskian approximation, 184 residual gauge freedom, 78
post-Minkowskian approximations, 72 resummation, 5, 13
post-Newtonain approximations, 72 retarded solution, 147
post-Newtonian, 1, 139, 142 retarded time, 208
post-Newtonian approximations, 72 Ricci
354 Index
scalar, 255 solar-system ephemerides, 160

Ricci tensor, 190 sound wave, 271
Ricci-tensor, 145 spacetime, 177
Riemann tensor, 65, 144, 190 asymptotically flat, 177
ringdown, 16 asymptotically-flat, 69, 176, 266–269, 271,
rotational matrix, 95 276, 277
Routhian, 42 Minkowski, 71, 267, 271, 314, 318, 328,
Rudolph, 124 335
Runge-Lenz-Laplace-Lagrange vector, 56 Minkowskian, 267
spatial infinity, 80
satellite motion, 151 Special Relativity, 143
scalar specific enthalpy, 278, 280, 287, 288, 297, 305,
gauge-invariant, 304, 305 306, 323
scalar field, 68, 69, 73, 85, 91, 270, 276, 277, specific internal energy, 69
279, 280, 285–292, 295–301, 303– speed
306, 308, 310, 312–314, 319, 322,
fundamental, 270, 297, 319, 320, 325, 333,
326, 332, 354
335, 336
background value, 73
gravity, 266, 335
boundary conditions, 74
light, 266
charge, 125
sound, 278, 297, 319, 323–325, 331, 335,
dark energy, 276, 298, 313, 319, 326
337
dark matter, 319
ultimate, 271
dipole moment, 125
speed of gravity, 73
equation, 74
speed of light, 73, 177
external multipoles, 96, 122
spherical harmonics, 148
external solution, 85
spin, 64
internal solution, 85
spin multipole moments, 142
perturbation, 77
spin-type multipoles, 90
potential, 68
solution of the field equation, 84 spinning bodies, 23
source, 69 spinning particles, 41
speed of sound, 299 Spyrou, 112
scalar field STF, 77, 83, 85, 108, 112
speed of sound, 297 Cartesian tensor, 91, 93
scalar internal multipoles, 90 multipole moments
scalar-tensor theory of gravity, 68 external, 87
Schäfer, 112 multipoles, 112
Schiff effect, 57 external, 86
second law of theormodynamics, 70 tensor, 77
self-force, 79, 95 STF derivative, 212
semi-major axis, 154 STF tensor, 149
SEP, 83, 125 Stokes parameters, 204
shift vector, 255 stress-energy tensor, 186
simultaneity, 91 stress-energy-momentum tensor, 64
skeletonized harmonic gauge, 150 post-Newtonian expansion, 75
Skrotskii effect, 232, 233 skeleton, 65, 79
slow manifold, 95 strong principle of equivalence, 83
slow-motion approximation, 68 superhorizon perturbation, 254
Index 355
symmetrization, 65, 87, 127 time delay of light, 220

Synge, 64 time derivative
Synge’s function, 64 total, 75, 96
system of units–geometrized, 181 Tolman mass, 99
torsion, 72
tail effects, 13 transition functions
tail factor, 14 of coordinate transformation, 91
tangent bundle, 64 Tulczyjew, 64
tensor two-body, 2
Cartesian, 85, 87, 181
transverse, 182 universe
transverse-traceless, 182 FLRW, 277, 280, 281, 283, 284, 287, 291,
energy-momentum, 266, 276, 277, 279–281, 293–295, 312, 315, 316, 328, 333
286, 287, 289, 295, 301, 327
variational calculus, 278
metric, 79, 81, 82, 84, 266, 268–270, 273–
variational derivative, 69
277, 279, 291, 292, 296–305, 310–
Very-Long Baseline Interferometry, 179
312, 320, 323, 325, 327, 328, 330
Vincent, 112
Ricci, 190
Vines, 112
Riemann, 190
virial theorem, 73
STF, 182
VLBI, 179
stress-energy-momentum, 64, 69
skelelton, 79
wave operator, 77
skeleton, 72
waveform, 2, 12
symmetric trace-free, 182
weak effacement condition, 160
tetrad, 159
weak-field approximation, 68
Thomas precession, 84, 170 wedge product, 184
Thorne, 72, 77, 98, 101 world function, 64
tidal expansion, 163 world line, 91
tidal force, 82, 87 world tube, 64
tidal moments, 164 wormhole, 79
tidal multipole, 83 Wu, 112
tidal polarizability, 23
tilt angle, 207 Xu, 112
time
retarded, 266, 328, 330–333, 335–338 Zeldovich approximation, 253

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Frontiers of Relativistic Celestial Mechanics.

Editor: Sergei M. Kopeikin

February 15, 2014

Prof. Dr. Toshifumi Futamase

Prof. Dr. Sergei Kopeikin

Prof. Dr. Gerhard Schäfer

Prof. Dr. Michael Soffel

Dr. Pavel Korobkov

Dr. Alexander Petrov

2.5 Binary motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.5 Gravitational Perturbations of the Light Ray . . . . . . . . . . . . . . . . . . . . . . . . .212

7.4.5 The Friedmann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

5.1 The Penrose Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Editor: Sergei Kopeikin University of Missouri, USA

1.2 Multi-chart approach to the N -body problem

One expands (say in powers of Newton’s constant) the metric,

1.3 EOB description of the conservative dynamics of two

b 1PN (q, p) = 1 (3ν − 1)(p2 )2 − 1 [(3 + ν)p2 + ν(n · p)2 ] 1 + 1 ,

b 2PN (q, p) = 1 (1 − 5ν + 5ν 2 )(p2 )3

b 3PN (q, p) = 1 (−5 + 35ν − 70ν 2 + 35ν 3 )(p2 )4

where the dimensionless coefficients e an , e

A(R; ν = 0) = 1 − 2GM/c2 R = B −1 (R; ν = 0) .

Seff = −Eeff t + Jeff ϕ + Seff (R) , (1.3.13)

where the dimensionless coefficients ceff an , e

Finally, identifying Eeff (n,`)/µc2 to 1 + f (Ereal

4 total = M c2 + E relative = M c2 + Newtonian terms + 1PN/c2 + · · · , while E 2

a4 = a4 ν , de3 = 2(3ν − 26)ν , α3 = 0 , z3 = 2(4 − 3ν)ν .

Here, a4 denotes the number

D(R) ≡ A(R) B(R) .

A3PN (R) = 1 − 2u + 2 ν u3 + a4 ν u4 , (1.3.18)

Eeff relative relative

1.4 EOB description of radiation reaction and of the

which explicitly read

() GM ν () (`+)/2 `−,−m π

ρ`m (x; ν) = [f`m (x; ν)]1/` . (1.4.15)

See Fig. 1 in [65].

where the (instantaneous, circular) GW flux F (`max ) is defined as

1.5 EOB description of the merger of binary black holes

A14 (u; a5 ,ν) = P41 A3PN (u) + νa5 u5 .

1.6.1 EOB[NR] waveforms vs NR ones

A-flexibility parameters ea5 (ν), e

where we defined a5 and a6 so that e a5 (ν) = a5 ν, e

500 1000 1500 2000 2500 3000 3500 4000

precisely, the phase difference, ∆φ = φEOB NR

Here, having in view GW observations from ground-based interferometric detectors we focussed on

−0.3 Merger time

1.6.2 EOB[3PN] dynamics vs NR one

1.7 Other developments

1.7.2 EOB with tidally deformed bodies

1.7.3 EOB and GSF

A(u; ν) = 1 − 2u + ν a(u) + O(ν 2 ) ,

D(u; ν) = (AB)−1 = 1 + ν d(u)

[1] J. Droste, Versl. K. Akad. Wet. Amsterdam, 19, 447-455 (1916).

[3] J. Chazy, La théorie de la Relativité et la Mécanique Céleste, vols. 1 and 2 (Gauthier-Villars,

[8] V.A. Fock, Zh. Eksp. i. Teor. Fiz. 9, 375 (1939).

[14] S.M. Kopeikin, Astron. Zh. 62, 889 (1985).

[34] A. H. Mroue, M. A. Scheel, B. Szilagyi, H. P. Pfeiffer, M. Boyle, D. A. Hemberger, L. E. Kid-

[35] D. Radice, L. Rezzolla and F. Galeazzi, “Beyond second-order convergence in simulations of

[44] I. Hinder, A. Buonanno, M. Boyle, Z. B. Etienne, J. Healy, N. K. Johnson-McDaniel, A. Nagar

[47] F. K. Manasse: J. Math. Phys. 4, 746 (1963).

() GM ν () (`+)/2 `−,−m π

1/2 pak γ kj Saji

{xia , paj } = δij , {Sa(i) , Sa(j) } = ijk Sa(k) , (2.2.35)