Einstein Equations-Cacciatori

Tutorials, Schools, and Workshops in the
Mathematical Sciences
Sergio Luigi Cacciatori

Alexander Kamenshchik
Editors
Einstein Equations:
Local Energy,
Self-Force, and
Fields in General
Relativity
Domoschool 2019
Tutorials, Schools, and Workshops
in the Mathematical Sciences
This series will serve as a resource for the publication of results and developments
presented at summer or winter schools, workshops, tutorials, and seminars. Written
in an informal and accessible style, they present important and emerging topics
in scientific research for PhD students and researchers. Filling a gap between
traditional lecture notes, proceedings, and standard textbooks, the titles included
in TSWMS present material from the forefront of research.
Sergio Luigi Cacciatori • Alexander Kamenshchik
Editors
Einstein Equations: Local

Energy, Self-Force, and
Fields in General Relativity
Domoschool 2019
Editors
Sergio Luigi Cacciatori Alexander Kamenshchik
Department of Science and High Department of Physics and Astronomy
Technology University of Bologna
University of Insubria Bologna, Italy
Como, Italy
ISSN 2522-0969 ISSN 2522-0977 (electronic)

Tutorials, Schools, and Workshops in the Mathematical Sciences
ISBN 978-3-031-21844-6 ISBN 978-3-031-21845-3 (eBook)
https://doi.org/10.1007/978-3-031-21845-3
Mathematics Subject Classification: 83C05, 83F05, 83C20, 83C25, 83C35, 83C40, 83C47, 83C57,
83B05
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland
AG 2022
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Preface
This book presents four lecture courses and in addition eight talks, given at the
second edition of the Domoschool, the International Alpine School in Mathematics
and Physics, held in Domodossola in July 2019 with the title “Einstein Equations:
Physical and Mathematical aspects of General Relativity.” Unfortunately, the
COVID-19 pandemic delayed the writing of the present book and prevented the
realization of the 2020 and 2021 editions.
Domoschool is a 5-day summer school aiming to bring together young math-
ematicians and physicists working in the same field of physics. In 2019, general
relativity was chosen as the topic, the same as the previous edition, but with a very
different focus. Four days were devoted to four lecture courses given by prominent
experts, while an additional day of Domoschool was devoted to social activities
(such as a hiking tour and a dinner), aiming to show the participants the town of
Domodossola and its surroundings. The school also included a public meeting with
the local community in order to explain to the citizens of Domodossola the contents
of the summer school and the scientific relevance. Public lectures were given by
Andrea Accomazzo and Marco Giammarchi.
The book is divided into two parts. The first part is devoted to the courses given
at the school by outstanding world experts.
The first course, given by Pengzi Miao, had the primary scope of introducing
graduate students to the mathematical problem of defining local energy in general
relativity. Indeed, while global energy is well defined, the definition of local
energy is still an open problem. Therefore, the notion of quasi-local energy is
a basic subject of study in general relativity. In a four-dimensional spacetime,
there are different ways of defining the quasi-local energy for a spacelike, closed
twosurface. In particular, the “surface Hamiltonian approach” is naturally tied to the
classic problem of isometric embeddings in differential geometry. The lectures by
Pengzi Miao provide a short introduction to the Wang-Yau quasi-local energy of
closed spacelike surfaces in spacetimes. After introducing the subject by discussing
a geometric problem of isometrically embedding a surface into the Minkowski
spacetime, the lecture proceeds with a review of the formula of surface Hamiltonian
and its properties. Then, the definition of the Wang-Yau energy is explained,
v
vi Preface
focusing on its physical and variational features, and relating it to some previously
known quasi-local quantities. In the end, the ideas behind the proof of its positivity
are outlined.
The second course was given by Donato Bini, and concerns the method of
gravitational self-force in curved background. The most important application of
gravitational self-force concerns metric and curvature perturbations in black hole
spacetimes due to moving particles or evolving fields. In these notes, for teaching
purposes, the method is illustrated and discussed at the level of a (massless) scalar
field. Indeed, in this simple case one can look at the various steps implicit in any
self-force computation without facing the additional difficulties of implementing
them in a more involved tensorial background.
The third course was given by Lars Andersson and is devoted to geometry and
analysis in black hole spacetimes. Black holes play a central role in general relativity
and astrophysics. The problem of proving the dynamical stability of the Kerr black
hole spacetime, which describes a rotating black hole in vacuum, is one of the
most important open problems in general relativity. This course presents features
of the geometry of the Kerr spacetime, including its algebraically special nature and
consequences thereof. Then it introduces the analysis of some aspects of the black
hole stability problem and presents the main steps in the recent proof of linearized
stability of the Kerr black hole spacetime.
The fourth course was given by Marco Giammarchi and it provides a quick
introduction to the experimental search for antimatter gravity. Gravitational proper-
ties of antimatter are related to both the validity of the CPT theorem in particle
physics and the weak equivalence Principle of General Relativity. The course
presents the motivation and techniques of the main approaches to this topic.
The school also included a further course, not transposed in the present book,
given by Alexei A. Starobinsky, “Inflation and pre-inflation: The present status and
expected discoveries.”
The second part of the book presents the talks given by participants in a
conference proceeding type manner. In Domoschool 2019, eight of the participants
presented a short talk:
Lennart Brocki et al. (University of Wroclaw, Poland): Quantum ergosphere
and brick wall entropy;
Francesco Cremona (University of Milan, Italy): Geodesic structure and linear
instability of some wormholes
Vittorio De Falco (University in Opava, Czech Republic): New trends in the
general relativistic Poynting-Robertson effect modeling;
Mario L. Gutierrez Abed (Newcastle University, UK): Brief Overview of
numerical relativity
Colin MacLaurin (University of Queensland Brisbane, Australia): Length-
contraction in curved spacetime
Jı̌rí Ryzner et al. (Charles University, Czech Republic): Exact solutions of
Einstein-Maxwell(-dilaton) equations with discrete translational symmetry
Tereza Vardanyan et al. (University of Bologna, Italy): Exact solutions of the
Einstein equations for an infinite slab with constant energy density
Preface vii
Adamantia Zampeli (Charles University in Prague): Emergence of classicality

from an inhomogeneous universe
Domoschool was a project initiated by Sergio Cacciatori, but its realization was
made possible thanks to the openminded people and the enthusiastic support of the
municipality of Domodossola, which supported the school both financially and
concretely in its organization. For this, we are indebted to the mayor, Lucio Pizzi;
the councilor for culture, Daniele Folino, who was the first one welcoming our
proposal and connecting us to the right people; and, especially, the deputy mayor,
Angelo Tandurella, who was fully available to us, assisting step-by-step through
the evolution, the logistics, the public event, and so on.
Additionally, we want to acknowledge the nonprofit association ARS.UNI.VCO
(Associazione per lo Sviluppo della Cultura di Studi Universitari e della Ricerca
nel Verbano Cusio Ossola); its vice president, Giulio Gasparini; and the president,
Stefania Cerutti. We are particularly indebted to the secretary Andrea Cottini and
the communication manager Federica Fili for their continuous support, organizing
the school, assisting all participants, and, in particular, together with the Rosminian
Fathers (especially padre Fausto), making it possible to gain Collegio Rosmini as
the location of the School.
Furthermore, we are particularly grateful to the following speakers of
Domoschool who kindly accepted our invitations and because of their participation
brought high prestige to the school:
Lars Andersson from Albert Einstein Institute (Max-Planck Institute for Gravi-
tational Physics), Potsdam, Germany
Donato Bini from Istituto per le Applicazioni del Calcolo “M. Picone,” CNR,
Rome (Italy)
Marco Giammarchi from Istituto Nazionale di Fisica Nucleare – Sezione di
Milano, Italy
Pengzi Miao from the University of Miami, Miami, Florida
Alexei A. Starobinsky from Landau Institute for Theoretical Physics, Moscow,
Russia
We acknowledge the speakers who accepted our invitation for the public lectures
given at Rovereto Square just outside the city hall of Domodossola. Thanks to them,
the public event was a complete success. They are:
Andrea Accomazzo from ESA, Germany
Marco Giammarchi from Istituto Nazionale di Fisica Nucleare – Sezione di
Milano, Italy
Moreover, we extend our acknowledgments to the participants of the School,
including two high school teachers, who, with their active participation, contributed
to making the atmosphere of the school welcoming and pleasant. And, to the
other members of the scientific board, Francesco Belgiorno, Alessandro Carlotto,
Simone Noja, Batu Güneysu, Stefano Pigola, Riccardo Re, Mauro Giudici,
and Pietro Antonio Grassi, as well as the members of the organizing committee,
Andrea Cottini and Giorgio Mantica, we thank you for your support.
viii Preface
Last but not least, we are grateful to the sponsors of Domoschool: Città
di Domodossola, Università dell’Insubria, INFN, FONDAZIONE CRT, and
CONSIGLIO REGIONALE DEL PIEMONTE.
Como, Italy Sergio Luigi Cacciatori

Bologna, Italy Alexander Kamenshchik
April 13 2022
Contents
Part I Main Lectures

Introduction to the Wang–Yau Quasi-local Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Pengzi Miao
Gravitational Self-force in the Schwarzschild Spacetime. . . . . . . . . . . . . . . . . . . . 25
Donato Bini and Andrea Geralico
Geometry and Analysis in Black Hole Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Lars Andersson
Study of Fundamental Laws with Antimatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Marco Giammarchi
Part II Proceedings
Quantum Ergosphere and Brick Wall Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Lennart Brocki, Michele Arzano, Jerzy Kowalski-Glikman, Marco Letizia,
and Josua Unger
Geodesic Structure and Linear Instability of Some Wormholes . . . . . . . . . . . . 133
Francesco Cremona
New Trends in the General Relativistic Poynting–Robertson
Effect Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Vittorio De Falco
Brief Overview of Numerical Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Mario L. Gutierrez Abed
Length-Contraction in Curved Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Colin MacLaurin
ix
x Contents
Exact Solutions of Einstein–Maxwell(-Dilaton) Equations with

Discrete Translational Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Jiří Ryzner and Martin Žofka
Exact Solutions of the Einstein Equations for an Infinite Slab
with Constant Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Tereza Vardanyan and Alexander Yu. Kamenshchik
Emergence of Classicality from an Inhomogeneous Universe. . . . . . . . . . . . . . . 251
Adamantia Zampeli
Part I
Main Lectures
Introduction to the Wang–Yau
Quasi-local Energy
Pengzi Miao
2010 Mathematics Subject Classification Primary 83C99; Secondary 53A05
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Isometric Embedding of Spheres into R3,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Surface Hamiltonian and Its Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1 Surface Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Graphical Surfaces in R3,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Some Related Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Construction of the Wang–Yau Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.1 The Choice of {T , n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 A Variational Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.3 Expression of the Wang–Yau Quasi-local Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.4 Relation to Other Quasi-local Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5 Positivity of EW Y (, τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.1 A Motivation to Jang’s Equation in R3,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 Jang’s Equation on a General (, g, k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.3 Comparison with a Euclidean Domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.4 A Physical Interpretation of H̃ − Ỹ , ν̃ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.5 Some Comment on Ỹ and ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1 Introduction
In [36, 37], Wang and Yau gave a new definition of quasi-local energy for
closed, spacelike surfaces in a spacetime. The Wang–Yau definition is based on a
Hamiltonian approach relevant to the Einstein equation. Starting from the surface
Hamiltonian, which is the boundary term in the Hamiltonian formulation, one
P. Miao ()
Department of Mathematics, University of Miami, Coral Gables, FL, USA
e-mail: pengzim@math.miami.edu
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 3

S. L. Cacciatori, A. Kamenshchik (eds.), Einstein Equations: Local Energy,
Self-Force, and Fields in General Relativity, Tutorials, Schools, and Workshops
in the Mathematical Sciences, https://doi.org/10.1007/978-3-031-21845-3_1
4 P. Miao
isometrically embeds the physical surface into a reference spacetime, and the quasi-
local energy is taken as the difference between the two surface Hamiltonians.
The Wang–Yau definition involves elegant use of classic results on isometric
embedding from differential geometry, and it makes a prudent analysis of a
boundary value problem of elliptic differential equations and applies fundamental
results on manifolds with nonnegative scalar curvature that include the positive
mass theorem. The construction leading to the Wang–Yau energy demonstrates the
intimate link among geometry, PDEs, and physics.
In this chapter, we want to give a succinct introduction to the Wang–Yau energy
definition. The note is by no means a survey on the topic of quasi-local mass.
It is not a survey on the current development of the Wang–Yau energy or mass
neither. Instead, it aims to quickly introduce audiences whose background is mainly
in a geometric analysis to the subject of the Wang–Yau quasi-local energy. The
interested readers are then encouraged to further explore many of the intriguing and
challenging problems related to this topic.
2 Isometric Embedding of Spheres into R3,1
Let be a closed surface that is topologically a sphere. Let σ denote a smooth

Riemannian metric on . In this section, we discuss some results concerning the
existence of isometric embedding of (, σ ) into the Minkowski spacetime R3,1 .
The following fact is a direct consequence of a theorem of Pogorelov [30] on the
existence of isometric embedding into hyperbolic spaces.
Theorem 2.1 There exist infinitely many isometric embeddings ι : (, σ ) → R3,1
such that ι(), the image of , is spacelike in R3,1 and is a graph over some 2-
surface in R3 = {t = 0}.
Proof Let K denote the Gauss curvature of σ . Choose κ > 0 to be any constant
so that K > −κ 2 . By the result of Pogorelov [30], (, σ ) can be isometrically
embedded into the hyperbolic space H3−κ 2 with constant sectional curvature −κ 2 .
Using the hyperboloid model of the hyperbolic space, one can take
H3−κ 2 = {(x, t) | t 2 − |x|2 = κ −2 , t > 0, x ∈ R3 } ⊂ R3,1 .
As a result, (, σ ) isometrically embeds into R3,1 . Since the hyperboloid H3−κ 2
is spacelike and is a graph over {t = 0} in R3,1 , so is the image of under the
embedding. As κ can be arbitrarily chosen to satisfy K > −κ 2 , such embeddings
are infinitely many.
Remark 2.1 The image ι() above is also convex in H3−κ 2 . It may be useful to
explore the implication of this convexity on the projection of ι() in R3 .
Introduction to the Wang–Yau Quasi-local Energy 5
In [37], Wang–Yau provided an embedding theorem that allows one to prescribe

the time function of the embedding. It makes use of a result of Nirenberg [26] (and
also Pogorelov [29]) on the existence of isometric embeddings into the Euclidean
space.
Theorem 2.2 ([37]) Given a Riemannian metric σ on , suppose τ is a function
on such that σ + dτ ⊗ dτ is a Riemannian metric with positive Gauss curvature.
Then, there exists an isometric embedding ι : → R3,1 , unique up to congruence in
ˆ in R3 = {t = 0}.
R3,1 , such that ι() is the graph of t = τ over a convex surface
Proof Let σ̂ = σ + dτ ⊗ dτ . Since σ̂ is a metric with positive Gauss curvature,
by the result in [26, 29], there exists an isometric embedding ι̂ : (, σ̂ ) → R3 ,
unique up to congruence in R3 , and ι̂() is convex. Define ι : → R3,1 by
ι(q) = (ι̂(q), τ (q)), q ∈ , and then ι is an isometric embedding of (, σ ) in
R3,1 with the required property. The uniqueness of ι follows from that of ι̂.
Remark 2.2 Given a metric σ on , it is interesting to know if there exist functions
τ so that σ + dτ ⊗ dτ is a metric with positive Gauss curvature. By Theorem 2.1,
one does know there exist many functions τ so that (, σ + dτ ⊗ dτ ) isometrically
embeds in R3 .
3 Surface Hamiltonian and Its Properties
Let S denote a four-dimensional spacetime with a Lorentz metric ḡ. For conve-
nience, the metric will also be denoted by ·, ·.
Boundary terms in a Hamilton–Jacobi analysis of the Einstein–Hilbert action
were considered by Brown–York [8, 9], Hawking–Horowitz [18], and Kijowski
[20]. The collection of such terms leads to a surface integral known as the surface
Hamiltonian. In this section, we review its definition and geometric properties.
3.1 Surface Hamiltonian
Let be a closed spacelike 2-surface in a spacetime S. Suppose = ∂ for some

compact spacelike hypersurface . Let n denote the future-directed, timelike unit
normal to . Along , let ν denote the outward unit normal to in .
Given a future-directed, timelike unit vector field T along , one can decompose
T as
T = X + Nn , (1)
where X is tangential to and N is a scalar function on . The surface Hamiltonian

of in S, with respect to T and n , is the integral
6 P. Miao

1
H(T , n ) = − NH − (K − (trK)g)(X, ν ) dσ. (2)
8π
Here,
• H is the mean curvature of in with respect to ν .
• K is the second fundamental form of in S with respect to n , and trK denotes
the trace of K on .
• g is the induced Riemannian metric on , and dσ is the area form on .
(An exposition of the derivation of (2) can be found in [24] and the references
therein.)
To see the role of T and more explicitly in (2), one can further decompose X
along as
X = X, ν ν + T . (3)
Here, T is the projection of X (hence of T ) to the tangent space of . Then,
(K − (trK)g)(X, ν ) = K(T , ν ) − (tr K)X, ν , (4)
where tr K is the trace of K restricted to . The mean curvature vector H of in

S is given by
H = (tr K)n − H ν . (5)
Let J denote the dual of H in (T )⊥ , the normal bundle of , obtained by reflecting
H across the inward future null direction. Then,
J = −(tr K)ν + H n , (6)
and
J , T = −NH − X, ν tr K. (7)
In terms of J , H(T , n ) takes the form of

1
H(T , n ) = J , T + ∇ T n , ν dσ. (8)
8π
The right side of (8) indicates that H(T , n ) depends on only through the 1-form
αν (·) on , given by
αν (·) = ∇ (·) n , ν , (9)
where ∇ denotes the spacetime connection.


Remark 3.1 The term − J , T dσ relates to the total future null expansion of
. Suppose {l, k} ⊂ (T )⊥ is a future-directed null frame along , chosen so that
l is outward-pointing and l, k = −2. It can be checked

− J , T dσ = H , lk, T dσ + H , T dσ. (10)

As k, T < 0, one can scale k so that k, T = −1. Let θ+ = −H , l for the
corresponding future-directed null l, and then one has

− J , T dσ = θ+ dσ + H , T dσ. (11)

In particular, − J , T dσ = θ+ dσ if T is a timelike Killing vector field on
S.
The connection term in (8) transforms suitably under a change of choice of .
Suppose {ν, n} and {ν̃, ñ} are two orthonormal frames in (T )⊥ so that n and ñ are
future-pointing and ν and ν̃ are outward-pointing. Let
ν = cosh φ ν̃ + sinh φ ñ and n = sinh φ ν̃ + cosh φ ñ
for some function φ : → R. Then, ∀ v ∈ T , and it is readily checked
∇ v ν, n = ∇ v ν̃, ñ − v, ∇φ. (12)
In terms of αν (·) = ∇ (·) n, ν, one has
αν (·) = αν̃ (·) + dφ. (13)
As a result, by (8) and (13),

H(T , n) = H(T , ñ) + T , ∇φ dσ. (14)

Here, ∇ denotes the gradient on .

Remark 3.2 The timelike vector field T in H(T , n) serves as a physical observer.
In the case of T , ν = 0, i.e., T = Nn + T , one has J , T = NH , ν and

1
H(T , n) = NH , ν + ∇ T n, ν dσ. (15)
8π
8 P. Miao
3.2 Graphical Surfaces in R3,1
Take S = R3,1 and suppose ⊂ R3,1 . Given any constant, future timelike, unit
vector field T0 , one can decompose
T0 = T0⊥ + T0 along ,
where T0⊥ ∈ (T )⊥ and T0 ∈ T . As T0⊥ = 0, one can normalize T0⊥ to define
n0 = N −1 T0⊥ , (16)

where N = −|T0⊥ |2 .
Suppose is a graph over a closed surface ˆ in a hyperplane ζ that is orthogonal
to T0 . Let ν̂ denote the outward unit normal to ˆ in ζ . Let ν0 denote the lift of ν̂
ˆ where C()
along T0 in R3,1 . Then, ν0 is orthogonal to C(), ˆ is the cylinder over
ˆ in R3,1 . As ⊂ C(),ˆ one has
{ν0 , n0 } ⊂ (T )⊥ and ν0 ⊥ n0 . (17)

Lemma 3.1 ([37]) With the above choice of {ν0 , n0 }, along ⊂ R3,1 ,
0
NH0 , ν0 (q) + ∇ T n0 , ν0 (q) = −N Ĥ (q̂). (18)
0
0
Here, H0 is the mean curvature vector of in R3,1 , ∇ denotes the connection
in R3,1 , q denotes a point in , q̂ is the projection of q on ζ , and Ĥ is the mean
curvature of ˆ in ζ with respect to ν̂.
ˆ is a timelike hypersurface in R3,1 . Let (·,
Proof C() ˜ ·) denote its second
˜ 0
fundamental form with respect to ν0 , i.e., (v, w) = ∇ v ν0 , w for v, w tangential
ˆ The trace of
to C(). ˜ on C()ˆ satisfies
˜
tr(q) ˜ q̂).
= tr( (19)
˜ q̂) = Ĥ (q̂) and

By definition, tr(
0
˜
tr(q) =
0
∇ eα ν0 , eα − ∇ n0 ν0 , n0 , (20)
α
where {eα }α=1,2 is an orthonormal frame in T at q. Plug in T0 = Nn0 + T , and

then one has
Ĥ (q̂) = −H0 , ν0 − N −1 ∇ (T −T ) ν0 , n0 , (21)
0
which proves (18) as ∇ T ν0 = 0.
Next, suppose τ : → R is the time function on with respect to T0 and ζ .

For instance, one may choose standard coordinates (x1 , x2 , x3 , t) on R3,1 so that
T0 = ∂t = −∇t, ζ = {t = 0}
and is the graph of t = τ over . ˆ By abuse of nation, τ can be viewed as a

ˆ
function on both and .
Let σ̂ and σ denote the induced Riemannian metrics on ˆ and , respectively.
Then,
σ̂ = σ + dτ ⊗ dτ.

Let ∇
denote the gradient on (, σ ). Then T = −∇τ , N = 1 + |∇τ |2 , and
d σ̂ = 1 + |∇τ |2 dσ . Thus, by (18),

−N H0 , ν0 − ∇ T n0 , ν0 dσ = N Ĥ dσ = Ĥ d σ̂ . (22)
0 ˆ

It follows from (15) and (22) that

− H(T0 , n0 ) = Ĥ d σ̂ . (23)
ˆ

(See equation (3.4) in [37].)
3.3 Some Related Inequalities
Suppose is a graph of t = τ over a convex surface ˆ ⊂ R3 = {t = 0}. Let ||

ˆ ˆ
and | denote the area of and , respectively. The Minkowski inequality in R3
gives
ˆ
1
1 || 2
Ĥ d σ̂ ≥ .
8π ˆ
4π
Let T0 = ∂t . By (23) and (8),
ˆ
1 1
1 0 || 2
|| 2
− J0 , T0 − ∇ T n0 , ν0 dσ ≥ ≥ , (24)
8π 0 4π 4π

ˆ ≥ || is due to the fact d σ̂ =
where J0 is the dual of H0 , and || 1 + |∇τ |2 dσ .
10 P. Miao
For a properly chosen future-directed null vector field l along in R3,1 , one
knows

− J0 , T0 dσ = θ+ dσ

with θ+ = −H , l (see Remark 3.1). An inequality involving − J0 , T0 dσ was
conjectured by Penrose [28].
Conjecture 3.1 ([28]) For a suitable class of spacelike closed surfaces ⊂ R3,1 ,
1
|| 2
− J0 , T0 dσ ≥ . (25)
4π
Motivation to and results on (25) can be found in [5, 6, 17, 23, 27, 28, 34] and
the references therein. In relation to the setting of isometric embeddings into R3,1
from Sect. 2, we want to reflect on the following theorem of Brendle–Hung–Wang
[6] (more precisely, Theorem 8.1 in [35]).
Theorem 3.1 ([6, 35]) Let be a closed surface in the hyperbolic space H3−1 ⊂
R3,1 . Suppose has positive mean curvature in H3−1 and is star-shaped with respect
to the point (0, 0, 0, 1) ∈ H3−1 . Then,
1
|| 2
− J0 , ∂t dσ ≥ . (26)
4π
Suppose ⊂ H3−1 ⊂ R3,1 . Let be the domain bounded by in H3−1 . By (8),

the surface Hamiltonian of , with respect to ∂t and n in R3,1 , satisfies

1
H(∂t , n ) = J0 , ∂t dσ, (27)
8π
0
where ∇ (∂t ) n , ν = 0 as H−1
3 is totally umbilic in R3,1 . Thus, by Theorem 3.1,
if has positive mean curvature and is star-shaped in H3−1 , then
1
|| 2
− H(∂t , n ) ≥ . (28)
4π
Note that, by (14) and (23),

H(∂t , n ) = − Ĥ d σ̂ + (τ )ψ dσ. (29)
ˆ

Here, τ = t| , the time function on , ˆ is the projection of in R3 , and ψ

is the function determined by cosh ψ = −n , n0 . As n = X, the position
−τ
vector in R3,1 , one has n , n0 = X, N −1 ∂t = . Thus, ψ =
1 + |∇τ |2
τ
cosh−1 .
1 + |∇τ |2
4 Construction of the Wang–Yau Energy
In a Hamiltonian approach toward defining energy for ⊂ S, one considers the

difference between two surface Hamiltonians: one is from the physical surface and
the other one is from a reference surface. In the construction of the Wang–Yau quasi-
local energy [36, 37], the reference is chosen as an isometric embedding of in
R3,1 , with T0 and n0 specified in Sect. 3.2. An intriguing step next is to make a
judicious choice of {T , n} associated with the surface in the physical spacetime.
4.1 The Choice of {T , n}
Suppose ⊂ S, and let σ denote the metric on . Given an isometric embedding

ι : (, σ ) → R3,1 and a constant, future timelike unit vector T0 ∈ R3,1 , let n0 be
given in (16). Along in S, one wants to construct a pair (T , n) so that it matches
(T0 , n0 ) along in R3,1 suitably. To do so, one requires
T = Nn + T , (30)
a relation satisfied by T0 = Nn0 + T0 in R3,1 . Here, T is the pullback of T0 to

via ι. If τ denotes the time function on with respect to T0 in R3,1 , then

T = −∇τ and N = 1 + |∇τ |2 . (31)
Viewing T and T0 as observers of in S and R3,1 , respectively, one may further

require the expansions of , observed in S and R3,1 , to be the same. That is,
H , T = H0 , T0 . (32)
Without losing generality, taking T0 = ∂t , and writing ι = (x1 , x2 , x3 , τ ), then
H0 = (x1 , x2 , x3 , τ ) and H0 , T0 = −τ. (33)

12 P. Miao
In what follows, one assumes the mean curvature vector H of in S is spacelike.

Normalizing {H , J } to get a reference frame in (T )⊥ , one can let
H J
νH = − and nH = .
|H | |H |
Writing
n = sinh φ νH + cosh φ nH , (34)
it follows from (30) and (32) that φ is determined by
τ
sinh φ = . (35)
|H | 1 + |∇τ |2
Thus, a pair {T , n} along ⊂ S, matching {T0 , n0 } along in R3,1 , is determined

by (30), (34), and (35).
4.2 A Variational Property
The choice of T by (32) indeed has a variational characterization. Under the relation
T = N n + T , by (15), one has

8πH(T , n) = 1 + |∇τ |2 ν, H − ∇ ∇τ n, ν dσ. (36)

Consider this as a functional of {ν, n}, an orthonormal frame in (T )⊥ so that n is

future-pointing and ν is outward-pointing (meaning that ν, νH > 0). Representing
ν and n as
ν = cosh φ νH + sinh φ nH
(37)
n = sinh φ νH + cosh φ nH ,
then

− 1 + |∇τ |2 ν, H + ∇ ∇τ n, ν
(38)
= 1 + |∇τ |2 cosh φ|H | + ∇ ∇τ nH , νH + ∇τ, ∇φ.
Integrating on , one has


−8πH(T , n) = 1 + |∇τ |2 |H | cosh φ − (τ )φ dσ + ∇ ∇τ nH , νH dσ.

(39)
The right side of (39) is easily seen to be a convex functional of φ. Thus, one has
the following lemma:
Lemma 4.1 ([37]) Under the relation

T = 1 + |∇τ |2 n − ∇τ,
H(T , n) is uniquely maximized by the choice {ν, n} given by (37) with φ satisfying
τ
sinh φ = . (40)
|H | 1 + |∇τ |2
4.3 Expression of the Wang–Yau Quasi-local Energy
Given a pair (ι, T0 ), let {T , n} be chosen above. Then,

−8πH(T , n) = (1 + |∇τ |2 )|H |2 + (τ )2 − φ τ
(41)
+ ∇ ∇τ nH , νH dσ,
where φ is specified by (35) or (40). In terms of the 1-form αν (·), the connection
H
term satisfies

∇ ∇τ nH , νH dσ = αν (∇τ ) dσ = − τ div[αν (·)] dσ. (42)
H H
Here, div(·) denotes the divergence on (, σ ).

Thinking as a surface in R3,1 , one has an identity

−8πH(T0 , n0 ) = (1 + |∇τ |2 )|H0 |2 + (τ )2 − φ0 τ
(43)
0
+ ∇ ∇τ nH , νH dσ,
0 0
where φ0 is given by
τ
sinh φ0 = . (44)
|H0 | 1 + |∇τ |2
14 P. Miao
Since the dependence of H(T , n), H(T0 , n0 ) on (ι, T0 ) is only via
τ = −T0 , ι, (45)
the Wang–Yau quasi-local energy of ⊂ S, defined by
1
[H(T , n) − H(T0 , n0 )] ,
8π
is also a functional of τ . Denote it by EW Y (, τ ), and it follows from (41)–(43) that
8π EW Y (, τ )

= (1 + |∇τ |2 )|H0 |2 + (τ )2 − (1 + |∇τ |2 )|H |2 + (τ )2 (46)

+ (φ − φ0 )τ + τ div[αν (·) − ᾱν (·)] dσ.

H H0
0
Here, ᾱν (·) = ∇ (·) nH , νH is the connection 1-form on in R3,1 .
H0 0 0
In the case that is embedded in R3,1 as a graph over some ˆ in a hyperplane

orthogonal to T0 , by (23), EW Y (, τ ) also takes the form of

1 1
EW Y (, τ ) = Ĥ d σ̂ + H(T , n). (47)
8π ˆ
8π
4.4 Relation to Other Quasi-local Energy
Suppose the metric σ has positive Gauss curvature. In this case, (, τ ) isometrically
embeds into R3 = {t = 0}. Let ι be such an embedding, and let T0 = ∂t . Then τ = 0
and EW Y (, 0) becomes

EW Y (, 0) = (H0 − |H |) dσ, (48)

where H0 denotes the mean curvature of the isometric embedding of (, σ ) into R3 .
EW Y (, 0) agrees with the quasi-local energy of defined by Liu and Yau [21, 22]
(also see the work of Booth–Mann [7], Epp [16], and Kijowski [20]).
Suppose S satisfies the dominant energy condition. Liu and Yau [21, 22] proved
the positivity of the right side of (48), under assumptions that bounds a compact
spacelike hypersurface and that H is inward-pointing relative to . The latter
means H , ν < 0, where ν is the outward unit normal to in .
In the special case that bounds a compact time-symmetric hypersurface and
H is inward-pointing relative to , EW Y (, 0) further reduces to

EW Y (, 0) = (H0 − H ) dσ, (49)

where H = −H , ν is the mean curvature of in . The right side of (49) is

known as the Brown–York mass [8, 9] of in . Its positivity was guaranteed by
the following theorem of Shi and Tam [33].
Theorem 4.1 ([33]) Let (, g) be a three-dimensional compact Riemannian man-
ifold with nonnegative scalar curvature, with boundary . Suppose the induced
metric σ on has positive Gauss curvature, and the mean curvature H of in
is positive. Then,

H0 dσ ≥ H dσ, (50)

where H0 is the mean curvature of the isometric embedding of in R3 . Moreover,

equality holds if and only if (, g) is isometric to a domain in R3 .
Shi-Tam’s theorem is a fundamental result on compact manifolds with nonnega-
tive scalar curvature, with boundary. Its proof made use of the Riemannian positive
mass theorem of Schoen–Yau [31] and Witten [39]. The positive mass theorem
itself is an assertion of the positivity of the ADM mass [1, 2] on asymptotically
flat manifolds.
The following theorem of Wang–Yau [37] provides a generalization of Theo-
rem 4.1. It plays a key role in the proof of the positivity of the Wang–Yau quasi-local
energy.
Theorem 4.2 ([37]) Let (, g) be a three-dimensional compact Riemannian man-
ifold with boundary . Suppose there exists a vector field Y on such that
R ≥ 2|Y |2 − 2 divY, (51)
where R is the scalar curvature of g, divY is the divergence of Y , and
H > Y, ν , (52)
where H is the mean curvature of in with respect to the outward unit normal
ν . If the induced metric σ on has positive Gauss curvature, then

H0 dσ ≥ H − Y, ν dσ, (53)

where H0 is the mean curvature of the isometric embedding of in R3 .

16 P. Miao
5 Positivity of EW Y (, τ )
If the spacetime S satisfies the dominant energy condition, Wang–Yau [37] proved
that EW Y (, τ ) is always nonnegative for surfaces ⊂ S, under suitable
assumptions. The proof in [37] consists of several key ingredients, which includes a
boundary value problem of Jang’s equation [19, 32], an application of Theorem 4.2
of Wang–Yau, an intriguing physical interpretation of boundary terms from Jang’s
equation, and the variational characterization of {T , n} used in defining EW Y (, τ ).
5.1 A Motivation to Jang’s Equation in R3,1
Suppose ˆ ⊂ R3 = {t = 0} ⊂ R3,1 . Given a function f :

ˆ → R, let be the
graph of f in R3,1 , i.e.,
ˆ ⊂ R3,1 .
= {(x, f (x)) | x ∈ }
Identifying with ˆ via x → (x, f (x)), one can view f as a function on .

Suppose is spacelike. Let g denote the induced Riemannian metric on . The
following are two basic facts about f on (, g):
ˆ ⊂ R3 .
(i) ĝ := g + df ⊗ df is the Euclidean metric on
(ii) If kij denotes the second fundamental form of in R3,1 with respect to the
future-directed timelike normal, then
D2f
k= . (54)
1 + |Df |2
Here, D 2 and D denote the Hessian and the gradient on (, g), respectively.
In particular, f on (, g) satisfies an elliptic equation
D2f
trĝ −k = 0. (55)
1 + |Df |2
In local coordinates, (55) takes the form of
f if j f;ij
g ij − − kij = 0. (56)
1 + |Df |2 1 + |Df |2
Here, f i = g ij f,i and “ ;” denotes the covariant differentiation on (, g).

Suppose ˆ has boundary .ˆ Let τ = f | ˆ and = ∂ be the graph of f over

ˆ
. Then, f on satisfies a boundary condition
f = τ on . (57)
5.2 Jang’s Equation on a General (, g, k)
Let be a compact spacelike hypersurface with boundary in a spacetime S. Let

g and k denote the metric and the second fundamental form of in S, respectively.
Suggested by the preceding fact concerning functions and their graphs in R3,1 , one
can impose the following equations on (, g):
⎧
⎨ g ij − f if j
√ f;ij
− kij = 0 on
1+|Df |2 1+|Df |2 (58)
⎩
f = τ at .
The equation in (58) is known as Jang’s equation, first proposed by Jang [19]. It was
used by Schoen–Yau in their proof of the spacetime positive mass theorem [32]. The
boundary value system (58) was analyzed by Wang–Yau [37].
To focus on the idea behind the positivity of EW Y (, τ ), we suppose f is a
f f i j
smooth solution to (58). The term g ij − 1+|Df |2
, hence the tensor g + df ⊗ df ,
suggests that it can be useful to consider the graph of f over in the Riemannian
product
( × R, g + dt 2 ).
˜ and identify and

Denote this graph by , ˜ in the usual way. The following
holds:
˜
(i) g̃ := g + df ⊗ df is the induced metric on .
(ii) If p̃ denotes the second fundamental form of ˜ in × R with respect to the
downward unit normal ñd , then
D2f
p̃ = .
1 + |Df |2
(In what follows, by abuse of notation, we also use ñd to denote the vector field
on × R, obtained by parallel translating ñd along the R-factor.)
In terms of g̃ and p̃, the PDE in (58) takes the form of
h̃ = trg̃ k̃. (59)

18 P. Miao
˜ in × R with respect to ñd , and k̃ =

Here, h̃ = trg̃ p̃ is the mean curvature of
π ∗ (k), where π : × R → denotes the usual projection map.
As shown by Schoen–Yau [32], a crucial implication of (59) is that the scalar
curvature R̃ of (,˜ g̃) satisfies
R̃ − 2|Ỹ |2 + 2 divỸ ≥ |p̃ − k̃|2 + 2(μ̃ − |J˜|). (60)
Here, Ỹ , μ̃, and J˜ are as follows:

˜ g̃) that is dual to the 1-form
• Ỹ is the vector field on (,
η(·) = D̃ñd ñd , · − k̃(ñd , ·), (61)
where D̃ denotes the connection on ( × R, g + dt 2 ).

• μ̃ = μ ◦ π and J˜ = π ∗ (J ) are the lift of μ and J , where
1
μ= R − |k|2 + (trg k)2 and J = divg (k − (trg k)g)
2
denote the local energy density and local current density on in S.
Remark 5.1 The solvability of Jang’s equation with boundary value τ was analyzed
in [37, Section 4.3].
5.3 Comparison with a Euclidean Domain
Suppose the spacetime S satisfies the dominant energy condition. Then, μ ≥ |J | on

. As a result, (60) implies
R̃ − 2|Ỹ |2 + 2 divỸ ≥ 0. (62)
˜ g̃), provided
This indicates that Theorem 4.2 of Wang–Yau is applicable to (,
(i) (, σ + dτ ⊗ dτ ) has positive Gauss curvature, here σ is the metric on .
˜ = ∂ .
(ii) H̃ − Ỹ , ν̃ > 0 along ˜ Here, ν̃ is the outward unit normal to
˜ in
˜
˜ ˜
and H̃ is the mean curvature of in with respect to ν̃.
Condition (i) also calls for the application of Theorem 2.2 of Wan–Yau, i.e., there
exists an isometric embedding
ι : (, σ ) → R3,1
so that ι() is the graph of t = τ over a convex surface ˆ ⊂ R3 = {t = 0}. Here,

the metric on ˆ is σ̂ = σ + dτ ⊗ dτ . Consequently, ˆ is the image of an isometric
embedding of ˜ = ∂ ˜ in R3 .
By Theorem 4.2, one therefore has

Ĥ d σ̂ ≥ H̃ − Ỹ , ν̃ d σ̂ . (63)
ˆ
˜

5.4 A Physical Interpretation of H̃ − Ỹ , ν̃

˜ let q = π(q̃) ∈ . As d σ̂ =
Given any q̃ ∈ , 1 + |∇τ |2 dσ , one has

− H̃ − Ỹ , ν̃ (q̃) d σ̂ (q̃) = 1 + |∇τ |2 −H̃ + Ỹ , ν̃ (q̃) dσ (q).
˜

In [37], Wang–Yau made an intriguing discovery that recognizes the term

1 + |∇τ |2 −H̃ + Ỹ , ν̃ (q̃) (64)
as a surface Hamiltonian density associated with some frame {ν , n } along ⊂ S.

More precisely, Wang and Yau proved the following:
Theorem 5.1 ([37]) Along ⊂ S, let {ν , n } be an orthonormal frame in (T )⊥
given by

ν = cosh θ ν + sinh θ n
(65)
n = sinh θ ν + cosh θ n ,
where θ is the function on determined by
1 ∂f
sinh θ = − .
1 + |∇τ |2 ∂ν
˜ = ∂
Then, H̃ − Ỹ , ν̃ along ˜ satisfies

1 + |∇τ |2 −H̃ + Ỹ , ν̃ (q̃)
(66)
= 1 + |∇τ |2 H , ν − ∇ ∇τ n , ν (q).
As a result,

− H̃ − Ỹ , ν̃ d σ̂ = 1 + |∇τ |2 H , ν − ∇ ∇τ n , ν dσ. (67)
˜

20 P. Miao
Combined with (63) and the definition of H(T , n ), Theorem 5.1 gives

Ĥ d σ̂ ≥ −H(T , n ). (68)
ˆ

On the other hand, by the maximal property of H(T , n) (Lemma 4.1),
− H(T , n ) ≥ −H(T , n). (69)

Recall that, by (47), 8π EW Y (, τ ) = ˆ
Ĥ d σ̂ +H(T , n). Therefore, one concludes
EW Y (, τ ) ≥ 0, (70)
under the assumptions (i) and (ii) in Sect. 5.3.
5.5 Some Comment on Ỹ and ν
˜ which is the graph of the

The vector field Ỹ , by definition, is a vector field on ,
solution to Jang’s equation. It may be convenient to compute Ỹ via quantities on
directly.
For this purpose, let {vi } be a local frame on . Let wi = vi + fi ∂s , where
fi = vi (f ) and s is the coordinate on R in × R. {wi } forms a local frame on .˜
Using the fact ñd = √ 1 2 (Df − ∂s ), one has
1+|Df |
1
D̃ñd ñd , wi = D̃(Df −∂s ) (Df − ∂s ), wi
1 + |Df |2
(71)
1
= DDf Df, vi .
1 + |Df |2
Similarly,
1
k̃(ñd , wi ) = k̃(Df − ∂s , vi + fi ∂s )
1 + |Df |2
(72)
1
= k(Df, vi ).
1 + |Df |2
˜ given by (61), is determined by

Therefore, the 1-form η on ,

∗ 1 1
(η) = D f (Df, ·) − k(Df, ·) .
2
(73)
1 + |Df |2 1 + |Df |2
Here, : → ˜ is the map with (x) = (x, f (x)).

As a result, Ỹ = 0 on ˜ in the setting of Sect. 5.1, where is the graph of a
function f over ˆ ⊂ R3 in R3,1 . In this case, (,
˜ g̃) is isometric to the Euclidean
domain ,ˆ and thus H̃ = Ĥ . It would be worthy of computing ν too. Let D̂ denote
the gradient on R3 . Using the fact n = √ 1 2 (D̂f + ∂t ), it is easily seen that
1−|D̂f |
if one writes
ν0 = cosh θ ν + sinh θ n , (74)
where ν0 is the vector field given in Sect. 3.2, then
1 ∂
sinh θ = − f.
1 + |∇τ |2 ∂ν
In other words, ν = ν0 in this model case. Consequently, Theorem 5.1 may be

viewed as a generalization of Lemma 3.1.
We would like to end this note with a brief description of the Wang–Yau quasi-
local mass [37]. Assumptions (i) and (ii) in Sect. 5.3, together with the assumption
that (58) admits a solution, correspond to the admissibility definition of the time
function τ in [37]. By minimizing EW Y (, τ ) among all admissible time functions
τ , the Wang–Yau mass of ⊂ S is defined to be

mW Y () = inf EW Y (, τ ) | admissible τ .
The variational feature of mW Y () leads to many interesting questions. For instance,
in the time-symmetric case, whether EW Y (, ·) is minimized by τ = 0, or
equivalently whether mW Y () agrees with the Brown–York mass, seems to be
an intriguing question. Other related questions include understanding a relation
between critical points of EW Y (, ·) and the concept of time-flat surfaces and the
asymptotic behavior of the quasi-local energy. We refer the interested readers to
[3, 4, 10–15, 25, 38] for further exploration of this subject.
Acknowledgments I am deeply grateful to the organizers of the second “Domoschool – Interna-

tional Alpine School of Mathematics and Physics” held in Domodossola, on July 15–19, 2019, for
the kind invitation and warm hospitality.
The author’s research was partially supported by NSF grant DMS-1906423.
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Gravitational Self-force in the
Schwarzschild Spacetime
Donato Bini and Andrea Geralico
Mathematics Subject Classification (2000) Primary 83C25; Secondary 83C57
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 The Various Steps of Gravitational Self-force Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1 Metric Perturbations: The Regge–Wheeler–Zerilli Formalism . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Curvature Perturbations: The Teukolsky Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Scalar Self-force: Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1 Scalar Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Angular Part of the Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Radial Part of the Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Computing the Scalar Field Along the Source World Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5 Solutions of the Radial Homogeneous Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5.1 Heun Confluent (Exact) Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5.2 PN Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5.3 MST or Hypergeometric-Like Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6 Analytical Versus Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 So Far So Good: A List of SF Accomplishments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1 Introduction
When a particle or a field evolves on a given gravitational background, its presence

induces a perturbation on the background, which in turn acts back on the particle
or the field itself, implying modifications to the particle’s world line, or the support
region of the field. The modified background as well as the modified dynamics can
then be fully determined by using the self-force (SF) formalism. To make a long
story short, one can say that SF is the actual terminology to mean a complete (either
D. Bini () · A. Geralico

Istituto per le Applicazioni del Calcolo “M. Picone”, Rome, Italy

26 D. Bini and A. Geralico
analytical or numerical) determination of the gravitational perturbation of a certain

background due to a small mass, charge, or field. Whereas a numerical treatment
of the problem can always be performed, it is only recently that SF techniques have
allowed for analytical results (see, e.g., Ref. [1] for a recent review). Indeed, starting
from 2013 [2], novel analytical SF calculations have been carried on successfully in
black hole spacetimes (Schwarzschild and Kerr). Several gauge-invariant quantities
have been computed within the framework of black hole first-order perturbation
theory through a very high post-Newtonian (PN) accuracy, including the redshift
function, the periastron advance, and the precession angle of a gyroscope for motion
along both circular and eccentric orbits.
The theoretical foundations of the SF program were posed more than a decade
before, thanks to the pioneering works of Quinn and Wald, Mino, Sasaki and
Tanaka, and Detweiler and Whiting (see, e.g., the Living Review article by Poisson
[3] for the fundamental formulation of the SF in curved spacetime, that by Sasaki
and Tagoshi [4] for applications to black hole spacetimes, and the review by Barack
[5] for a historical presentation of developments in SF research). However, all the
various steps which are necessary to accomplish the SF tasks have been elucidated
and standardized only very recently. The problem is the old one of solving
the differential equations for the backreaction induced on a given gravitational
background by a moving body or an evolving field as a consequence of the
(linearized) Einstein’s equation. Mode-sum decomposition and Fourier analysis are
the preliminary tools necessary to obtain a set of (coupled) ordinary differential
equations instead of (coupled) partial differential equations. Then, the symmetries
of the background allow for separated radial and angular equations and the (simple)
motion of the source of the perturbation implies the possibility to fully integrate (in
PN sense) the complete set of the perturbation equations. The final goal is that of
reconstructing the perturbed metric to compute several orbital gauge-invariants.
2 The Various Steps of Gravitational Self-force

Computations
Let us consider a particle with mass μ moving in the background gravitational field
of a much larger mass M described by the Schwarzschild spacetime, with metric
written in standard spherical-like coordinates
(0) dr 2 2M
gαβ dx α dx β = −f (r)dt 2 + + r 2 (dθ 2 + sin2 θ dφ 2 ) , f (r) = 1 −
.
f (r) r
(1)
We will assume geometrical units throughout all this work, with G = 1 = c. The
particle moves along a geodesic orbit with parametric equations xpα = xpα (τ ), τ
dxpα
denoting the proper time parameter, and four velocity U = dτ ∂α (U · U = −1).
The associated energy-momentum tensor is then given by
Gravitational Self-force in the Schwarzschild Spacetime 27
∞ 1
T μν
=μ δ (4) (x α − xpα (τ ))U μ U ν dτ
−∞ −g (0)
U μU ν
=μ δ(r − rp (t))δ(θ − θp (t))δ(φ − φp (t)) , (2)
rp2 sin θp U t
where g (0) is the metric determinant for the background, and δ (4) is the four
dimensional delta function. According to the Detweiler-Whiting formulation [6],
μ
through O( M ) the particle effectively moves along a geodesic of a smooth regularly
perturbed spacetime with metric
(0) μ R
R
gαβ = gαβ + h . (3)
M αβ
The main goal of the SF program is to determine the (regularized) metric pertur-
bation hRαβ . The first-order perturbation can be obtained by using standard tools
of perturbation theory following either the Regge–Wheeler–Zerilli (RWZ) [7, 8] or
the Teukolsky approach [9–11]. The former is usually adopted in the spherically
symmetric case, whereas the latter is most suited for stationary black holes and
general type D spacetimes. In both cases, after separation of variables and for simple
motion of the source, the problem is reduced to solve a single Schrödinger-like
equation for a master function of the radial variable, with a Dirac-delta source term
having support at the particle position. We will shortly review below the main steps
in the simplest case of circular motion, i.e.,
U = U t (∂t + ∂φ ) , (4)
with
−1/2
3M M
U = 1−
t
, = . (5)
r0 r03
2.1 Metric Perturbations: The Regge–Wheeler–Zerilli

Formalism
After decomposing both the perturbed metric and the energy-momentum tensor
associated with the perturbing mass in tensor harmonics, the perturbation equations
can be separated into two different sets, with even parity and odd parity, respectively.
All ten metric perturbation components can then be summarized by a single radial
function, irrespective of their parity and reflection properties about the equatorial
plane. Therefore, the general situation is that of a Regge–Wheeler (–Zerilli, when
including a unified treatment of odd and even waves) equation with different source
terms involving a Dirac-delta function and its derivatives, that is
(r) (even/odd) (even/odd)

L(RW) [Rlmω ] = Slmω (r) . (6)
Here

(r) 2M 2 d 2 2M 2M d
L(RW) = 1 − + 1 − + [ω2 − V(RW) (r)] (7)
r dr 2 r2 r dr
denotes the RW operator with the potential

2M l(l + 1) 6M
V(RW) (r) = 1 − − 3 ,
r r2 r
and the even/odd source terms are of the type
(r0 )δ (r − r0 ) + s2 (r0 )δ (r − r0 ) ,
(even/odd) (e/o) (e/o) (e/o)
Slmω (r) = s0 (r0 )δ(r − r0 ) + s1
with s2 (r0 ) = 0. Recalling the properties of the Green function G(r, r ) of L(RW) ,
(o) (r)
i.e.,
(r )
L(RW) [G(r, r )] = f (r )δ(r − r ) = L(RW) [G(r, r )] ,
(r)
(8)
we have

dr G(r, r )f (r )−1 Slmω (r ) .
(even/odd) (even/odd)
Rlmω (r) = (9)
Here
1
G(r, r ) = Xin (r)Xup (r )H (r − r) + Xin (r )Xup (r)H (r − r ) , (10)
W
where W is the (constant) Wronskian
d d
W = f (r) Xin (r) Xup (r) − Xin (r)Xup (r) = const. (11)
dr dr
and H (x) is the Heaviside step function. The functions X(in) (r) and X(up) (r) are
the solutions of the homogeneous RW equation satisfying the proper regularity
conditions at the horizon and at infinity, obtained by using different methods: PN
approximation (for generic l, in the weak-field and slow motion regime), Wentzel-
Kramers-Brillouin (WKB) approximation (for large l, useful for regularization
purposes), Mano, Suzuki, and Takasugi (MST) [12, 13] technique (specified values
of l, useful to cure the shortcomings of the PN solutions). Finally, both even-parity
and odd-parity solutions are given by

(even/odd) (e/o) G(r, r0 ) (e/o) d G(r, r )
Rlmω (r) = s0 (r0 ) − s1 (r0 ) lim
f (r0 ) r →r0 dr f (r )

(e/o) d2 G(r, r )
+s2 (r0 ) lim . (12)
r →r0 dr 2 f (r )
For instance, for r < r0 we have
(even/odd) (e/o) X(in) (r)X(up) (r0 )

Rlmω,− (r) = s0 (r0 )
Wf (r0 )

(e/o) d X(in) (r)X(up) (r )
−s1 (r0 ) lim
r →r0 dr Wf (r )

(e/o) d 2 X(in) (r)X(up) (r )
+s2 (r0 ) lim
r →r0 dr 2 Wf (r )
(e/o)
= C− (r0 )X(in) (r) , (13)
where

(e/o) (e/o) X(up) (r0 ) (e/o) d X(up) (r )
W C− (r0 ) = s0 (r0 ) − s1 (r0 ) lim
f (r0 ) r →r0 dr f (r )

(e/o) d2 X(up) (r )
+s2 (r0 ) lim . (14)
r →r0 dr 2 f (r )
Once the radial function is known for both parities, the perturbed metric
components are then computed by Fourier anti-transforming, multiplying by the
angular part, and summing over m (between −l and +l), and then over l (between
0 and ∞). The final step consists in building up the gauge-invariant quantity one is
interested in, and evaluating it at the source location, where the perturbed metric is
singular. The sum over m is done by using standard spherical harmonic identities.
The sum over l, instead, requires some care. In fact, the series is in general divergent,
so that a regularization procedure is needed. Furthermore, the contribution of the
non-radiative modes l = 0, 1 must be computed separately, corresponding to a shift
in the black hole mass and spin due to the energy and angular momentum of the
particle.
2.2 Curvature Perturbations: The Teukolsky Formalism
The metric perturbation can be also computed in a radiation gauge by using the
Teukolsky formalism. Einstein’s equations reduce to a single master equation for
the perturbed Weyl scalars ψ0 (spin-weight s = 2) or ψ4 (spin-weight s = −2),
which can be solved by separation of variables. The radiative part of the metric
perturbation (l ≥ 2) can then be reconstructed from the “Hertz potential,” through
the Chrzanowski–Cohen–Kegeles (CCK) procedure [14–16] (see also Refs. [17–
20]).
Let us start with the Teukolsky master equation governing the perturbations on
a Kerr background due to a field of spin-weight s. The radial and angular part can
be separated. The eigenfunctions of the angular part are the spin-weighted spherical
harmonics for vanishing black hole rotation parameter (spheroidal harmonics in the
general Kerr case), so that in the frequency domain
ψs = e−iωt s Rlmω (r)s Ylmω (θ, φ) . (15)

lmω
The radial equation turns out to be
Lrs s Rlmω (r) = −8π Tslmω , (16)
where

−s d d ω2 r 2 − 2is(r − M)ωr 2
Lrs = s+1
+ + 4isωr − λ , (17)
dr dr
with = r 2 − 2Mr and λ = l(l + 1) − s(s + 1), and the source term Tslmω follows
from the harmonic decomposition of the energy-momentum tensor
U μU ν π
T μν = μ 2
δ(r − r0 ) s Ylmω (θ, φ) s Ȳlmω , t , (18)
r0 U t 2
l,m
with ω = m . In the case s = +2 (i.e., for ψs=2 = ψ0 ), we have
Ts=2 = L1 (L2 (T13 ) − L3 (T11 )) + L4 (L5 (T13 ) − L6 (T33 )) , (19)
where T11 = Tll , T13 = Tlm , T33 = Tmm are the frame components of the stress-
energy tensor with respect to the Newman-Penrose principal frame
1 2
l= (r ∂t + ∂r ) ,

1
n = 2 (r 2 ∂t − ∂r ) ,
2r

1 i
m= √ ∂θ + ∂φ , (20)
2r sin θ
with nonvanishing spin coefficients
1 cos θ M
ρ=− , β=− √ = −α , μ=− , γ = ,
r 2 2r sin θ 2r 3 2r 2
(21)
and associated frame derivatives
D = l μ ∂μ , = nμ ∂μ , δ = m μ ∂μ . (22)
The differential operators entering Eq. (19) are then given by
L1 = δ − 2β , L2 = D − 2ρ , L3 = δ ,
L4 = D − 5ρ , L5 = δ − 2β , L6 = D − ρ .
The Teukolsky radial equation (16) can be solved by using the Green’s function
method. For s = 2 the (retarded) Green’s function satisfies the equation
1
Lrs=2 (Glm (r, r )) = δ(r − r ) , (23)

which yields
( )2
Glm (r, r ) = Rin (r)Rup (r )H (r − r) + Rin (r )Rup (r)H (r − r ) ,
Wlm
(24)
where H (x) denotes the Heaviside step function, ≡ (r ), Rin (r) and Rup (r)
are two independent solutions to the homogeneous radial equation that are ingoing
at the outer horizon and outgoing at infinity, respectively, and Wlm is the associated
(constant) Wronskian. One needs –also in this case– different types of solutions:
PN, MST, WKB solutions.
The final solution for the Weyl scalar ψ0 thus has the form

ψ0 = −8π (r )2 T (x , x0 )G(x, x )dr d(cos θ )dφ , (25)
where T (x , x0 ) and G(x, x ) stand for the full source term and the full Green’s
function of the Teukolsky radial equation, respectively, x denoting the spatial point
with coordinates (r, θ, φ). According to the CCK procedure, all the components of
the radiative part (l ≥ 2) of the perturbed metric are then evaluated by applying
a suitable differential operator on a scalar quantity , the Hertz potential, which
also satisfies the homogeneous Teukolsky equation but with opposite spin as the
Weyl scalar from which it is constructed. The (outgoing radiation gauge) metric
perturbation is thus given by
hαβ = −r 4 {nα nβ Dnn + m̄α m̄β Dm̄m̄ − n(α m̄β) Dnm̄ } + c.c. , (26)
where
Dnn = (δ̄ + 2β)(δ̄ + 4β) ,

Dm̄m̄ = ( + 5μ − 2γ )( + μ − 4γ ) ,
Dnm̄ = (δ̄ + 4β)( + μ − 4γ ) + ( + 4μ − 4γ )(δ̄ − 4α) , (27)
and the Hertz potential is related to ψ0 by
1 4
ψ0 = L ¯ + 12M∂t , (28)
8
with
L4 = L1 L0 L−1 L−2 , Ls = −[∂θ − s cot θ + i csc θ ∂φ ] − ia sin θ ∂t . (29)
Finally, the radial part of the harmonic decomposition of
= e−iωt 2 Rlmω (r)2 Ylmω (θ, φ) (30)

lmω
turns out to be
(−1)m D 2 R̄l,−m,−ω + 12iMω 2 Rlmω

2 Rlmω =8 , (31)
D2 + 144M 2 ω2
with D = l(l − 1)(l + 2)(l + 1).

The evaluation of gauge-invariant quantities then proceeds along the lines
sketched before for the RWZ approach, requiring the calculation of the contribution
due to the non-radiative modes [21] and the regularization procedure after summa-
tion over all multipoles.
3 Scalar Self-force: Computational Details
The problem of scalar self-force (SSF) is easier to handle, even if the various steps
needed to compute a gauge-invariant quantity are practically the same as in the
gravitational case. Let the source of the perturbation be a particle endowed with a
scalar charge q, which generates a (massless) scalar field ψ. The latter acts back
on the scalar charge world line, determining modifications to the otherwise free
evolution.
In order to illustrate how SSF computations are explicitly carried out, let us
assume that the scalar charge moves along a timelike circular (equatorial) geodesic
orbit, with parametric equations
π
t = τ , r = r0 , θ= , φ= τ = t, (32)
2
where τ is a proper time parameter along the orbit and
1 M
=√ , M = u3/2 , u= . (33)
1 − 3u r0
The associated 4-velocity of the scalar charge is
U = k , k = ∂t + ∂φ (34)
and is aligned with the Killing vector k (i.e., a linear combination of the temporal
and azimuthal Killing vectors, ∂t and ∂φ ), implying additional simplifications.
3.1 Scalar Wave Equation
The scalar field ψ satisfies a wave equation sourced by the scalar charge density ρ,
i.e.,
ψ = −4πρ , (35)
where

1
ρ=q √ δ (4) (x α − zα (τ ))dτ . (36)
−g
The density ρ determines a perturbation to the scalar field evolution itself. For
circular motion it reads

1 π
ρ=q δ(t − τ )δ(r − r0 )δ θ − δ(φ − t)dτ
r 2 sin θ 2
q π
= 2
δ(r − r0 )δ θ − δ(φ − t) . (37)
r0 2
Having this decomposition for ρ, one can show that a corresponding decomposition
in (scalar) spherical harmonics of the field ψ exists too. We will show this in detail
below.
3.2 Angular Part of the Perturbation
Let us recall the spherical harmonic identity
∞ l
∗
δ (cos θ − cos θ0 ) δ(φ − φ0 ) = Ylm (θ, φ)Ylm (θ0 , φ0 ) , (38)
l=0 m=−l
with

∂ 2 Ylm ∂Ylm m2
+ cot θ + l(l + 1) − Ylm = 0 . (39)
∂θ 2 ∂θ sin2 θ
Replacing θ0 → π/2, φ0 → 0, φ → φ − t, one has
∞ l
∗
δ (cos θ ) δ(φ − t) = Ylm (θ, φ − t)Ylm (π/2, 0) . (40)
l=0 m=−l
Therefore, because of the property δ (cos θ) = δ(θ − π/2),
∞ l
π ∗
δ θ− δ(φ − t) = Ylm (θ, φ − t)Ylm (π/2, 0) , (41)
2
l=0 m=−l
with
Ylm (θ, φ) = Clm eimφ LegendreP(l, m, cos θ ) ,

1 m (2l + 1) (l − m)!
Clm = (−1) (42)
2 π (l + m)!
and where (see [22] pag. 959, Eq. 8.7561 )
Ylm (π/2, 0) = Clm LegendreP(l, m, 0)

√
2m π
= Clm , (43)
1 − m2 + 2l 1 − m
2 − l
2
for a later use. Substituting Eq. (41) in Eq. (37) one gets the following expression
for the density ρ
∞ l
q ∗
ρ= δ(r − r0 ) Ylm (θ, φ − t)Ylm (π/2, 0)
r02 l=0 m=−l
q
= δ(r − r0 ) e−iωt Ylm (θ, φ)Ylm
∗
(π/2, 0)
r02 l,m
qlm −iωt
= e δ(r − r0 )Ylm (θ, φ) , (44)
4π r0
l,m
where we have used the relation
Ylm (θ, φ − t) = e−im t Ylm (θ, φ) = e−iωt Ylm (θ, φ) , (45)
with ω = m and
4π q ∗
qlm = Y (π/2, 0) . (46)
r0 lm
Separating then the variables in the scalar field equation, i.e., assuming
ψ= ψlmω (r)e−iωt Ylm (θ, φ) , (47)

l,m
one gets a full decoupling of the temporal and angular variables.
3.3 Radial Part of the Perturbation
The remaining problem is to solve the “radial equation”
L(ψlmω ) = Slm
0
δ(r − r0 ) , (48)
where L is the following second-order ordinary differential operator
d2 2(r − M) d 1 l(l + 1)
L= 2
+ 2
+ 2 ω2 − f , (49)
dr r f dr f r2
and
qlm 4π q 1 ∗
0
Slm =− =− 2 Ylm (π/2, 0) , (50)
r0 − 2M r0 f (r0 )
so that explicitly
d2 2(r − M) d ω2 r 2 l(l + 1)
ψlmω + ψlmω + − ψlmω = Slm
0
δ(r − r0 ) .
dr 2 r(r − 2M) dr (r − 2M)2 r(r − 2M)
(51)
in the radial equation, Eq. (51), there appear three parameters: l , M , ω =

Note that
m =m M
. There is also the hidden symmetry
r03
l → −l − 1 , (52)
which leaves invariant the equation (and hence act correspondingly on the solu-
tions). With the parameters M and ω two competing (dimensionless) quantities are
naturally formed:
M
, ωr , (53)
r
coming but with a different post-Newtonian weight, namely
M GM ωr
→ , ωr → , (54)
r c2 r c
characterized by the fact that their product is a constant
GM ωr GMω
= = constant . (55)
c2 r c c3
As we will see, in the solution for ψlmω (r) these two length-scales “compete among
them,” and it is possible to identify regimes where one is dominant with respect to
the other characterizing so far the corresponding behavior of the solution.
Let us consider two independent solutions of the homogeneous equation,
(l,m,ω) (l,m,ω)
R(in) (r), and R(up) (r), such that
(l,m,ω)
L(R(in,up) (r)) = 0 . (56)
The in-solution is regular at the horizon, while the up-solution is regular at infinity.
Such solutions can be of different types, which we will specify later: exact (Heun
confluent functions), post-Newtonian (PN, expansion in the inverse of the speed of
light), WKB (expansion in large l values), etc. With these two solutions (of whatever
type they are) we can form then their (constant) Wronskian
W ≡ W (l,m,ω)
(l,m,ω) d (l,m,ω) (l,m,ω) d (l,m,ω)
= r 2 f (r) R(in) (r) R (r) − R(up) (r) R(in) (r) , (57)
dr (up) dr
and the associated Green’s function

1 (l,m,ω)
G(l,m) (r, r ) = (l,m,ω)
R(in) (r)R(up) (r )H (r − r)
W

+R(in) (r )R(up) (r)H (r − r ) ,
(l,m,ω) (l,m,ω)
(58)
satisfying the equation
δ(r − r )
L(G(l,m,ω) (r, r )) = . (59)
r 2 f (r)
The solution of the inhomogeneous equation (51) then writes as

ψlmω (r) = dr G(l,m,ω) (r, r )r 2 f (r )Slm
0
δ(r − r0 )
= G(l,m,ω) (r, r0 )r02 f (r0 )Slm

0
, (60)
so that the full reconstructed scalar field is given by
0 −iωt
ψ(t, r, θ, φ) = G(l,m,ω) (r, r0 )r02 f (r0 )Slm e Ylm (θ, φ)|ω=m (61)
l,m
4π q ∗
=− G(l,m,m ) (r, r0 ) Ylm (π/2, 0)e−im t Ylm (θ, φ)

l,m
4π q ∗
=− G(l,m,m ) (r, r0 ) Ylm (π/2, 0)eim(φ− t)
Ylm (θ, 0) .

l,m
3.4 Computing the Scalar Field Along the Source World Line
The dependence on the variables t and φ of the scalar field (61) turns out to be of the
type ψ(r, θ, φ − t). This implies that the self-force modification ψ0 to the field along
the source world line only depends on r0 , i.e., ψ(r, θ, φ − t)|(r=r0 ,θ = π2 ,φ= t) = ψ0 (r0 ),
and is given by
4π q ∗
ψ0 (r0 ) = − G(l,m) (r0 ) Y (π/2, 0)e−im t Ylm (π/2, t)
lm
l,m
4π q ∗
=− G(l,m) (r0 ) Ylm (π/2, 0)Ylm (π/2, 0)

l,m
4π q
=− G(l,m) (r0 ) |Ylm (π/2, 0)|2 , (62)

l,m
where
G(l,m) (r0 ) = G(l,m,m ) (r0 , r0 ) , (63)


M3
and, we recall, M = = u3/2 . As we will see below, the Green’s function along
r03
the world line can be then factorized as
G(l,m) (r0 ) = Gk (l)mk , (64)

k
so that the summation over m can be carried out separately,

∞
4π q
ψ0 (r0 ) = ψ0(l) (r0 ) = − Gk (l; r0 )Ck (l) ,

l=0 l k
l
Ck (l) = mk |Ylm (π/2, 0)|2 . (65)
m=−l
For Ck (l) we have a closed form relation. In fact, using
l
2l + 1 lz 1
l (z) = emz |Ylm (π/2, 0)|2 = e 2 F1 , −l; 1; 1 − e2z
4π 2
m=−l
2l + 1 l(l + 1) z2 l(l + 1)(3l 2 + 3l − 2) z4

= 1+ +
4π 2 2 8 4!
l(l + 1)(5l 4 + 10l 3 − 5l 2 − 10l + 8) z6
+ + O(z8 ) , (66)
16 6!
and expanding the left-hand-side in series of z formally yields
l ∞
zk k z2 z3
l (z) = m |Ylm (π/2, 0)|2 = C0 (l) + C1 (l)z + C2 (l) + C3 (l)
k! 2 3!
m=−l k=0
z4 z5
+ C4 (l) + C5 (l) + O(z6 ) . (67)
4! 5!
All odd powers give a vanishing contribution C2k+1 (l) = 0,
z2 z4
l (z) = C0 (l) + C2 (l) + C4 (l) + O(z6 ) , (68)
2 4!
while the even powers give
L L
C2 (l) = ,
8π

L L 1 3
C4 (l) = − + L ,
4π 4 8

L L 1 5 5
C6 (l) = − L + L2 ,
4π 2 8 16

L L 17 77 35 2 35 3
C8 (l) = − + L− L + L , (69)
4π 8 32 32 128
where we have introduced the notation
L = l(l + 1) , L = 1 + 2l , (70)
allowing for more compact expressions. After summing over m, the scalar field (62)
thus becomes
∞
4π q
ψ0 (r0 ) = − [G0 (l; r0 )C0 (l) + G2 (l; r0 )C2 (l)

l=0
+G4 (l; r0 )C4 (l) + G6 (l; r0 )C6 (l) + . . .]

∞
(l)
= ψ0 (r0 ) , (71)
l=0
with
∞
(l) 4π q
ψ0 (r0 ) = − G2k (l; r0 )C2k (l) . (72)

k=0
An additional complication of the present computation concerns the summation

over l . In fact, the latter diverges, and –in order to be carried out– one needs a
regularizing procedure so to subtract the corresponding singular part. More in detail,
one would have in general

(l) d(u) e(u) 1
ψ0 (u) = . . . + a(u)l 2 + b(u)l + c(u) + + 2 +O , (73)
l l l3

divergent convergent
where we have replaced r0 in terms of u = M/r0 . The divergent terms, representing

the singular part of the (self) field,
∞ ∞
d(u)
Dl (u) = . . . + a(u)l + b(u)l + c(u) +
2
(74)
l
l=0 l=0
should be removed, leading to the regularized field ψ0(reg) (u), namely
(reg)
ψ0 (u) = [ψ0(l) (u) − Dl (u)] . (75)
l
From Ref. [23] we know (a priori) the singular part of the l th multipole piece in
(l)
ψ0 (u) given by c(u) only

1 − 3u 2 u
Dl (u) = c(u) = qu EllipticK(k 2 ) , k2 = , (76)
1 − 2u π 1 − 2u
whose expansion in power series of u reads

1 39 385 4 61559 5 622545 6 25472511 7
Dl (u) = q u − u2 − u3 − u − u − u − u
4 64 256 16384 65536 1048576

263402721 8 176103411255 9
− u − u + O(u10 ) . (77)
4194304 1073741824
Unfortunately, additional difficulties arise. In fact, even if one has subtracted the
singular field, the l summation should be performed carefully, since in the terms
G0 (l; u)C0 (l) + G2 (l; u)C2 (l) + G4 (l; u)C4 (l) + . . . = ck (l)uk (78)
k
there can be coefficients diverging for some value of l , i.e., terms of the type
c̄2 (l) c̄3 (l) c̄4 (l)

c2 (l) = , c3 (l) = , c4 (l) = ,... (79)
l−2 l−3 l−4
The presence of diverging coefficients is associated with the solution of the

homogeneous radial equation considered, i.e., with the PN-type solutions mainly.
To overcome this problem, a convenient use of exact (hypergeometric-like) and PN
(polynomial) solutions should be performed. We will discuss this problem in detail
in next sections.
3.5 Solutions of the Radial Homogeneous Equation
In order apply the procedure outlined above one needs to compute explicitly the
(exact or approximated) solutions of the homogeneous radial equation (51), i.e.,
d2 2(r − M) d ω2 r 2 l(l + 1)
ψlmω + ψlmω + − ψlmω = 0 . (80)
dr 2 r(r − 2M) dr (r − 2M)2 r(r − 2M)
As stated above, these solutions can be of different types.

3.5.1 Heun Confluent (Exact) Solutions
Equation (80) belongs to the confluent Heun class of equations and its solution
(formally) writes as

ψlmω (r) = eiωr C1 (−x)i Hc(1) (, L; x) + C2 (−x)−i Hc(2) (, L; x) , (81)
where
Hc(1) (, L; x) = HeunC −2i, 2i, 0, −2 2 , −L + 2 2 , x ,
Hc(2) (, L; x) = HeunC −2i, −2i, 0, −2 2 , −L + 2 2 , x , (82)
with C1 and C2 integration constants and

r
L = l(l + 1) , = 2Mω , x =1− . (83)
2M
The confluent Heun functions can be Taylor-expanded around r = 2M (only). For

example,
(L − 2i) r
Hc(1) (, L; x) ≈ 1 + −1
(1 + 2i) 2M
(2i − 2L − 8 2 − 8iL + L2 ) r 2
+ −1
4(1 + 2i)(1 + i) 2M
r 3
+O −1 . (84)
2M
3.5.2 PN Solutions
PN solutions follow from a PN expansion of the radial equation. This is accom-

plished by restoring physical units, i.e., introducing the “weight” η = 1/c (with
proper powers) in M and ω, that is
M → η2 M , ω → ηω , (85)
and expanding in η all quantities. The homogeneous radial equation (80) then reads
d2 2(r − η2 M) d η2 ω2 r 2 l(l + 1)
ψlmω + ψlmω + + − ψlmω = 0 ,
dr 2 r(r − 2Mη ) dr
2 (r − 2Mη2 )2 r(r − 2Mη2 )
(86)
and we will look for solutions up to a fixed order in the η-expansion, e.g., η8 .
Equation (86) admits solutions which begin as polynomial in r ,
(l,m,ω)
R(in) (r) ≈ r l , (l,m,ω)
R(up) (r) ≈ r −(l+1) , (87)
but then must include some logarithmic term, even if l is not considered as an integer.
These solutions are regular for r → 0 or r → ∞.1 Moreover, replacing l → −l − 1
one passes from in to up solutions. In the general case polynomial (fundamental)
solutions of Eq. (86) are thus given by
(l,m,ω)
R(in) (r) = r l [1 + η2 A1 + η4 A2 + η6 A3 + η8 A4 + η10 A5 + η12 A6 + . . .]
(l,m,ω) (−l−1,m)
R(up) (r) = R(in) (r) , (88)
where Ak = Ak (M/r, ω2 r 2 ; l), k = 1 . . . 5. The logarithms entering for “generic”

(non-integer) l appear at fractional order η6 (η6 A3 ), both in R(in) and R(up) . In
(l,m,ω)
addition, R(up) (r) contains for generic l “dangerous denominators”: 1/ l in η4 A2 ,
1/(l(l+1)) in η6 A3 , 1/(l(l+1)(l+2)) in η8 A4 , etc. When l is integer these denominators
yield additional logarithms.
We find it convenient to introduce the notation
M
X1 = , X2 = ω2 r 2 , X12 X2 = M 2 ω2 = const . (89)
r
Each Ak = Ak (X1 , X2 ; l) is a k -order polynomial in the variables (X1 , X2 ), with

coefficients depending on l (modulo log terms), as indicated below:
(l,m,ω)
R(in) (r) = r l [1 + η2(1) CA(lm) XA + η4(2) CAB
(lm)
XA XB
(lm) (lm)
+η6 ((3) CABC + (3)ln CABC ln(r/R))XA XB XC
(lm) (lm)
+η8 ((4) CABCD + (4)ln CABCD ln(r/R))XA XB XC XD
(lm) (lm)
+η10 ((5) CABCDE + (5)ln CABCDE ln(r/R))XA XB XC XD XE
+ . . .] , (90)
with R a constant (irrelevant for the present computation) with the dimensions of
a length, later to be identified as Mη2 (when G = 1). The PN-order of (X12 X2 )
is η6 . Actually, starting from η6 (and successively, after every each η6 orders) our
polynomial solution would include a constant term (like X12 X2 = M 2 ω2 =const.): the
latter always appears with an accompanying logarithm. Therefore, the PN solution
shows
1 Note that PN solutions “do not know” the horizon at r = 2M, but they are sensitive to the origin
r = 0 only. The horizon, in fact, will be a regular singular point of the radial equation only when
all PN terms will be summed. Practically, we may say that high-PN orders already start “seeing”
the horizon, marked by the presence of successive l − n diverging coefficients at various PN n
orders.
• ln(r/R1 )) at O(η6 );
• ln2 (r/R2 )) at O(η12 );
• ln3 (r/R3 )) at O(η18 ), etc.
Each log can come with a scale; different scales are allowed but inessential. To be
more explicit, the first coefficients of the expansion are listed below.
1 [ω2 r 3 + 2Ml(3 + 2l)]

A1 = − ,
2 r(3 + 2l)
r 4 ω4 Mω2 r(−10 − 5l + l 2 ) M 2 l(l − 1)2
A2 = + + 2 ,
8(5 + 2l)(3 + 2l) 2(l + 1)(3 + 2l) r (−1 + 2l)
ω6 r 6 ω4 r 3 M(3l 3 − 27l 2 − 142l − 136)
A3 = − −
48(5 + 2l)(3 + 2l)(2l + 7) 24(5 + 2l)(3 + 2l)(l + 2)(l + 1)
M 3 l(l − 1)(l − 2)2 2ω2 M 2 (15l 2 + 15l − 11)
− − ln(r/R) ,
3r 3 (−1 + 2l) (3 + 2l)(2l + 1)(−1 + 2l)
r 8 ω8
A4 =
384(5 + 2l)(3 + 2l)(2l + 9)(2l + 7)
ω6 r 5 M(−60l 3 − 1548l − 1108 + 5l 4 − 625l 2 )
+
240(2l + 7)(5 + 2l)(3 + 2l)(l + 3)(l + 2)(l + 1)
r 2 M 2 ω4
− (24l 8 + 156l 7
24(5 + 2l)(3 + 2l)3 (l + 2)(l + 1)(−1 + 2l)(2l + 1)
−1802l 6 − 14843l 5 − 37099l 4 − 33535l 3 + 381l 2 + 11838l + 1392)
M 3 ω2 (2l 5 − 21l 4 − 13l 3 + 24 − 186l 2 + 44l)
−
6r(−1 + 2l)(2l + 1)
M 4 l(l − 1)(l − 2)2 (−3 + l)2
+
6r 4 (−1 + 2l)(2l − 3)
ω2 M 2 (15l 2 + 15l − 11)(4l 2 M + 6lM + ω2 r 3 )
+ ln(r/R) ,
r(3 + 2l)2 (−1 + 2l)(2l + 1)
A5 = Aln
5 ln(r/R)
ω10 r 10
−
3840(2l + 3)(2l + 7)(5 + 2l)(2l + 9)(2l + 11)
Mω8 r 7 (35l 5 − 490l 4 − 8855l 3 − 40754l 2 − 73032l − 43968)
−
13440(l + 1)(2l + 3)(2l + 7)(2 + l)(5 + 2l)(l + 3)(2l + 9)(l + 4)
(1)
M 2 r 4 ω6 Ā5
+
120(l + 1)(2l + 3) (2l + 7)(2 + l)(5 + 2l)2 (l + 3)(1 + 2l)(−1 + 2l)
3
M 3 rω4 Ā(2)
+ 5
24(l + 1)2 (2l + 3)3 (5 + 2l)l(1 + 2l)(−1 + 2l)
M 4 ω2 Ā(3)
+ 5
12r 2 (2l + 3)(2l − 3)l(−1 + 2l)3 (1 + 2l)
(l − 1)(l − 2)(l − 3)2 (−4 + l)2 M 5 l
− , (91)
30r 5 (2l − 3)(−1 + 2l)
where
(1)
Ā5 = 80l 10 + 1040l 9 − 5280l 8 − 128132l 7 − 738413l 6 − 2020400l 5
−2775100l 4 − 1506838l 3 + 363673l 2 + 586770l + 75240 ,

(2)
Ā5 = 40l 10 − 364l 9 − 2030l 8 + 1007l 7 + 541l 6 − 77809l 5 − 226083l 4
−205386l 3 − 2940l 2 + 61728l + 6480 ,

(3)
Ā5 = 40l 10 − 556l 9 + 850l 8 − 5057l 7 + 11821l 6 + 5167l 5 − 30399l 4
+21098l 3 − 2316l 2 − 1800l + 432 , (92)
and
ω6 r 4 M 2 (15l 2 + 15l − 11)

5 = −
Aln
4(2l + 3)2 (5 + 2l)(−1 + 2l)(1 + 2l)
ω4 M 3 r(−5l − 10 + l 2 )(15l 2 + 15l − 11)
−
(l + 1)(2l + 3)2 (−1 + 2l)(1 + 2l)
2lM 4 ω2 (15l 2 + 15l − 11)(l − 1)2
− . (93)
r 2 (2l + 3)(−1 + 2l)2 (1 + 2l)
Note that A4 contains terms like 1/[(l +1)(l +2)(l +3)]: when l → −l −1 they become
1/[l(l − 1)(l − 2)] and hence the up-solution is singular starting from this order and
it can be used only up to η6 included (i.e., the 3PN order). Moreover, using the
notation X1 = Mr and X2 = ω2 r 2 introduced above, we have, for example,
1 X2 + 2X1 l(3 + 2l)

A1 = − , (94)
2 (3 + 2l)
which is a first order polynomial in the variables (X1 , X2 ), with coefficients

depending on l .
3.5.3 MST or Hypergeometric-Like Solutions
The PN solutions do not correctly include the boundary conditions, since they
provide solutions which are regular at the origin (r = 0) rather than at the horizon
(r = 2M ). Solutions which incorporate the correct boundary conditions are obtained
following Mano, Suzuki, and Takasugi [12, 13] (see also Refs. [24–26]). They are
called MST or hypergeometric-like solutions. Let us introduce the notation
= 2Mωη3 ≡ 0 η3 , (95)
and the following three “radial” variables x , y , and z
r 1 z 2M
x =1− =1− =1− , z = ωr , y= . (96)
2M y r
Restoring physical units, y = (2M/r)η2 and z = ωrη are “small quantities” useful for
series expansions, differently from x = 1−r/(2Mη2 ), which is “large” (and negative,
so that it will appear as −x ). One should discuss separately in and up solutions.
The MST in-solutions are written in the form
(l,m,ω)
ψlm (x) = C(in) (x)R(in) (x) , (97)
where
C(in) = eix (−x)−i = ei[x−ln(−x)]

r
i −η(ωr)+η3 2Mω 1−ln −1
=e 2Mη2

1 r 1 3 3
= 1 − iωrη − ω2 r 2 η2 + 2Mω 1 − ln + ω r iη3
2 2Mη2 6

r ω4 r 4 4
+ 2ω2 Mr 1 − ln 2
+ η
2Mη 24

4M 2 ω r 1 5 5
+ − Mω3 r 2 1 − ln − ω r iη5
r 2Mη2 120
2
r
+ 4M 2 ω2 − 2M 2 ω2 1 − ln
2Mη2

1 r ω6 r 6 6
− ω4 Mr 3 1 − ln − η + ... , (98)
3 2Mη2 720
and
∞
(l,m,ω) (in)
R(in) (x) = an φn+ν (x) , (99)
n=−∞
(in)
where φn+ν (x) = hypergeom([a, b], [c], x) ≡ F (a, b; c; x), with
a = n + ν + 1 − i ,
b = −n − ν − i ,
c = 1 − 2i = a + b . (100)
The hypergeometric function is conveniently rewritten using the identity 15.3.8 of

Ref. [22] (see pag. 559) so that its argument becomes y = 1/(1 − x), namely
(c)(b − a)
F (a, b; c; x) = y a F (a, c − b; a − b + 1; y)
(b)(c − a)
(c)(a − b)
+ yb F (b, c − a; b − a + 1; y) (101)
(a)(c − b)
with x = 1 − 1/y or y = 1/(1 − x) [Note: Passing from the first to the second
expression in Eq. (101) is obtained by the map a → b, b → a ; c = a + b is instead
unchanged]. Explicitly
(in) (1 − 2i)(−2n − 2ν − 1)

φn+ν (y) = y n+ν+1−i H1 (y)
2 (−n − ν − i)
(1 − 2i)(2n + 2ν + 1)
+ y −n−ν−i H2 (y) , (102)
2 (n + ν + 1 − i)
where
H1 (y) = hypergeom([n + ν + 1 − i, n + ν + 1 − i], [2n + 2ν + 2], y)
H2 (y) = hypergeom([−n − ν − i, −n − ν − i], [−2n − 2ν + 2], y) . (103)
Differently from x , which is small as approaching the horizon, the variable y =

(2M/r)η2 is a “PN-small quantity,” and one consider the series representation of the
hypergeometric functions around y = 0. Therefore, the hypergeometric functions in
these expressions are replaced by their associated series expansion around y = 0, up
the order N . In the final expression ψlmω is expanded up to ηN . The coefficients an
are determined by solving the following three-point recurrence relation:
αnν an+1 + βnν an + γnν an−1 = 0 , (104)
where
i(n + ν + 1 + i)2 (n + ν + 1 − i)

αnν = ,
(n + ν + 1)(2n + 2ν + 3)
4
βnν = −l(l + 1) + (n + ν)(n + ν + 1) + 2 2 + ,
(n + ν)(n + ν + 1)
−i(n + ν + i)(n + ν − i)2
γnν = . (105)
(n + ν)(2n + 2ν − 1)
In these coefficients n always enters as n + ν and as i .

An approximated solution of the recursive relation (104) can be found for any
given l . Actually, one fixes “a priori” the largest value of n = nmax to be used
(corresponding the maximum PN-order to be reached with the solutions of the radial
equation) and puts
anmax = 0 = a−nmax , a0 = 1 . (106)
Then solves the recursion equations for nmax , nmax − 1, nmax − 2, . . . 3, 2, 1 and for
−nmax , −nmax +1, −nmax +2, . . .−3, −2, −1. Both series of equations include a relation
for a0 (for n = 1 and n = −1) which is then taken as a compatibility condition
between ν and l . The latter, when l is not explicitly chosen, implies
1 (l + 1)4 l4
ν =l+ −2 + − 2 + O( 4 )
2l + 1 (2l + 1)(2l + 2)(2l + 3) (2l − 1)2l(2l + 1)
1
=l+ [−2 + H (l) − H (l − 1)] 2 + O( 4 ) , (107)
2l + 1
where
(l + 1)4
H (l) = . (108)
(2l + 1)(2l + 2)(2l + 3)
As an example, we list below the relations ν vs l for specific values of l =

0, 1, 2, 3, 4 up to O( 12 ). The structure of the coefficients an is shown in Fig. 1 in
the case l = 4 for nmax = 17. As soon as the values of l increases the corresponding
expressions (both for ν and an ) become more and more involved (i.e., written
in terms of “large fractions”), saturating soon the computational facilities of any
computer available today.
7 9449 4 102270817 6 4988909608861 8

ν0 = − 2 − − −
6 7560 33339600 588110544000
72237319625071987 10 2008359560158182591511 12
− − ,
2593567499040000 21153904986614400000
19 1325203 4 1876733084209 6 29333897359675088897 8
ν1 = 1 − 2 − − −
30 3591000 4083505650000 40863314359098000000
845622982964484596588676709 10
−
678049353372765874500000000
447936092572020422484529225371046957 12
− ,
192953716502507138725465668750000000
79 2 708247 4 423940501889 6
ν2 = 2 − − −
210 9261000 11090886778125
59271881276715766819 8
−
2210181713528914125000
564858584456642264465384132611 10
−
25168170200278032013086000000000
Fig. 1 Assuming a−17 = 0 = a17 (as well as a0 = 1), the various an obtained by solving the three-
point recurrence relation are expressed as a sum of terms corresponding to different powers of .
For example: in a−16 (horizontal line to be drawn corresponding to the value −16 on the ordinate
axis) only two terms enter corresponding to the powers 15 and 16 ; in a−15 only three terms
enter corresponding to the powers 14 , 15 and 16 ; etc. Some irregularities (absence/presence of
expected terms) in the behavior an vs n are shown in the plot. In general, to reach a specified
PN-precision the necessary an terms are not necessarily symmetric with respect to a0
26450326014126037391692269487786665402461 12
− ,
1278949899712679600763473140424400000000000
169 2 74380421 4 1008725842043489 6
ν3 = 3 − − −
630 2750517000 156110268213900000
7518839302383545659213 8
−
3544121463974343252000000
53477783012565497991161254235239 10
−
63506761644931323272606052000000000
3880391139017304179697545363306389370120999
− 12 ,
10090012856082277597217971539061154400000000000
289 2 435454879 4 38349366083366489 6
ν4 = 4 − − −
1386 34612505928 21609407767240893600
31009517594899209604558009 8
−
91740604801957499688876576000
80442119512143591442375139998688729
− 10
1057148247265900978506752691728190720000
9289216445076841677647166314725787704865667257
− 12 .
483209980553594392719686017110200867275574323200000
Similarly, the MST up-solution is written in the form

(l,m,ω)
ψlmω (z) = C(up) (z)R(up) (z) , (109)
where
C(up) ≡ ηC̃(up)
= 2ν e−π e−iπ(ν+1) eiz zν+i (z − )−i

ν −π −iπ(ν+1) iz −i
= (2z) e e e 1− , (110)
z
and is further rescaled (and denoted by a tilde) as

1
C̃(up) = 2ωr + 2iω2 r 2 η − ω3 r 3 η2 − 4πMω + iω3 r 3 ωrη3
3

1 4 4 1
+ ω r − 4iπMω2 r ωrη4 + 2πMω4 r 3 + iω6 r 6 + 8iω2 M 2 η5
12 60
76 3 2 ω7 r 7 2
+ − ω M r(ln(2ωrη) − iπ) − + iπω5 Mr 4
15 360 3

+4(π 2 − 2)ω3 M 2 r η6 + O(η7 ) , (111)
in order to start from η0 . The up-solutions are first written in terms of a Kummer
function U
KummerU(ν + 1 − i, 2ν + 2, −2iz) = (ν + 1 − i, 2ν + 2, −2iz) . (112)
The KummerU function (also denoted by and known as “decaying confluent

hypergeometric function”) is then related to hypergeometric functions by the Eq.
13.1.3, pag. 504 of Ref. [22], that is

π hypergeom([a], [b], x)
(a, b, x) =
sin πb (a − b + 1)(b)

hypergeom([a − b + 1], [2 − b], x)
−x 1−b , (113)
(a)(2 − b)
conveniently rewritten as
(1 − b)
(a, b, x) = hypergeom([a], [b], x)
(a − b + 1)
(b − 1)
+x 1−b hypergeom([a − b + 1], [2 − b], x)) . (114)
(a)
In our case
a = n + ν + 1 − i , b = 2(n + ν + 1) , (115)
so that
a − b + 1 = −n − ν − i , 2 − b = −2n − 2ν . (116)
Up-solutions are finally given by

∞ ∞
(l,m,ω) (ν + 1 − i)n (up)
R(up) (z) = (2iz)n an (a, b, −2iz) ≡ an φn+ν ,
n=−∞
(ν + 1 + i)n n=−∞
(117)
with an the same quantities as in the in-case [whereas a and b in the function
here are different from the similar quantities used in the in-case] and (Q)n is the
Pochhammer symbol
(Q + n)
(Q)n = . (118)
(Q)
Note that, recalling that
ν = l + δν , (119)
we can replace, for example, the term x 1−b in the function by
x 1−b ≈ x −1−2n−2l 1 − 2δνlnx + 2(δν)2 ln2 x (120)
and the hypergeometric functions by their series representation up to a fixed number

nmax of terms around x = 0 since x → −2iz = −2iω0 rη is a “small quantity.”
3.6 Analytical Versus Numerical Results
Let us turn to the calculation of the scalar field (62) at the source position. We
have already discussed how to remove the singular part of the field, leading to the
regularized value (75), i.e.,
(reg) (l)
ψ0 (u) = [ψ0 (u) − Dl (u)] = (l)
ψreg (u) , (121)
l l
which has a gauge-invariant character and can be associated with physical, measur-
able quantities. The subtraction of the Detweiler singular field Dl (r0 ) (see Eq. (76))
is enough to ensure that the series converges. We want now to compute the final sum
of this series over l .
Let us consider the various regularized PN contributions (i.e., obtained by using
as fundamental solutions of the homogeneous radial equation the PN solutions)
∞
SN = (l)
ψreg (u) , (122)
l=N
with
S0 = 0 ,

1 2 61 3
S1 = u − u q,
4 192

1 2 181 3 126253 7 2 4
S2 = u + u + − π u q,
10 672 86400 32

9 2 127 3 9741157 7
S3 = u + u + − π 2 u4
140 1344 3725568 32

4938262487443 29 2 5
+ − + π u q,
2712213504000 512

1 2 661 3 7 709490963
S4 = u + u + − π2 + u4
21 11088 32 302702400

27506934151 29 2 5
+ + π u
93884313600 512

279 2 438350563349183
+ − π + u6 q ,
1024 912931065446400

5 2 25525 3 7 13045986313
S5 = u + u + − π2 + u4
132 576576 32 5708102400

232049019479863 29 2 5
+ − + π u
1434605959987200 512

279 2 6200419652418108697
+ − π + u6
1024 1417390688467353600

65213753441830559307389 76585 4 42084587 2 7
+ + π − π u q.
4400487827043284680704 262144 8847360
(123)
The way to read the above results is the following. If we want the sum of the series
representing the regular field we cannot use only PN solutions: in that case, for
example, one can sum from 1 to infinity but the accuracy reached is only up to u3
terms included, and, in any case, the contribution corresponding to l = 0 should
be added separately (taken from MST solutions). The sum of PN solutions from
l = 2 to infinity will allow to include terms O(u4 ) but the terms corresponding to
l = 0, 1 should be added separately (taken from MST solutions). Similarly, the sum
of PN solutions from l = 3 to infinity will allow to include terms O(u5 ) but the
terms corresponding to l = 0, 1, 2 should be added separately (taken from MST
solutions), etc. The MST contributions for specific values of l = 0, 1, 2 . . . are listed
below.

1 131 3 335 4 196141 5 4203457 6 269861411 7
(0)
ψreg (u) = − u2 − u − u − u − u − u
4 192 256 81920 983040 36700160

1716337981 8 1036093931639 9
− u − u q,
146800640 67645734912

3 2 263 3 309619 2 4 4
(1)
ψreg (u) = u − u + − ln(u) − ln(2) − γ u4
20 448 172800 3 3 3

74 5810627131 37 74
+ γ− + ln(u) + ln(2) u5
15 4257792000 15 15
38 11/2
− πu
45

2789 2789 2789 21001766837951
+ − ln(2) − γ− ln(u) − u6
1050 1050 2100 1549836288000
703 13/2
+ πu
225

304 152 157603582067170153 152
+ γ ln(2) + ln(2)ln(u) + + ln(2)2
45 45 3347646382080000 45
16 1792079 1943279
− ζ (3) − γ− ln(2)
3 56700 56700

38 2 38 152 2 2094479 152
− π + ln(u) + 2
γ − ln(u) + γ ln(u) u7
27 45 45 113400 45
52991 15/2
− πu
31500

5624 2812 6836558017030300481369
+ − γ ln(2) − ln(2)ln(u) −
225 225 87641382282854400000
2812 296 14433220177
− ln(2)2 + ζ (3) + γ
225 15 174636000
12803284177 703 2 703 2812 2
+ ln(2) + π − ln(u)2 − γ
174636000 135 225 225

11173348177 2812
+ ln(u) − γ ln(u) u8
349272000 225
18829 17/2
− πu q,
21000
1 2 235 3 35809397 4
(2)
ψreg (u) = u + u − u
28 1344 31046400

4745938586173 32 128 64
+ − ln(u) − ln(2) − γ u5
542442700800 15 15 15

272 1088 149810782311919 544
+ ln(u) + ln(2) − + γ u6
21 21 6509312409600 21
5056 13/2
− πu
1575

131324 636640539318575085631 262648 525296
+ − ln(u) − − γ− ln(2) u7
6615 10224827932999680000 6615 6615
42976 15/2
+ πu
2205

161792 80896
+ γ ln(2) + ln(2)ln(u)
1575 1575
33754151632103257853333497 161792
+ + ln(2)2
42428945990775472128000 1575
1024 1692896596 3385793192 10112 2 10112
− ζ (3) − γ− ln(2) − π + ln(u)2
15 5457375 5457375 945 1575

40448 2 846448298 40448
+ γ − ln(u) + γ ln(u) u8
1575 5457375 1575
67624 17/2
− πu q,
3675
(124)
1 2 221 3 327912173 4 7370880752321 5

(3)
ψreg (u) = u + u + u − u
60 6336 1210809600 3487131648000

68417846729249363 2734 729 5468 5468
+ − ln(u) − ln(3) − ln(2) − γ u6
2608374472704000 525 70 525 525

209149 67462667376314210330119 418298
+ ln(u) − + γ
4725 582815192180981760000 4725

418298 12393
+ ln(2) + ln(3) u7
4725 140
2772107 15/2
− πu
330750

9273669781088817176800037 7258653
+ − − ln(3)
104906734592576716800000 30800

1102504759 1102504759 1102504759
− ln(u) − γ− ln(2) u8
9355500 4677750 4677750
2772107 17/2
+ πu q,
220500
(125)
3 2 983 3 7172009 4 304439656059107 5

(4)
ψreg (u) = u + u + u + u
308 64064 122943744 669482781327360
2575929576296522669 6
− u
661448987951431680

459008 7315317259754538077097871
+ − γ+
19845 110012195676082117017600

229504 1376768
− ln(u) − ln(2) u7
19845 19845

7901056 3950528
+ γ+ ln(u)
31185 31185

63871255032922971739259116577 677120
− + ln(2) u8
155865278833873143390535680 891
]q . (126)
The final result is obtained as follows
ψreg (u) = ψreg

(0)
(u) + S1 + O(u4 )
ψreg (u) = ψreg

(0)
(u) + ψreg
(1)
(u) + S2 + O(u5 )
ψreg (u) = ψreg

(0)
(u) + ψreg
(1)
(u) + ψreg
(2)
(u) + S3 + O(u6 )
ψreg (u) = ψreg

(0)
(u) + ψreg
(1)
(u) + ψreg
(2)
(u) + ψreg
(3)
(u) + S4 + O(u7 )
ψreg (u) = ψreg

(0)
(u) + ψreg
(1)
(u) + ψreg
(2)
(u) + ψreg
(3)
(u) + ψreg
(4)
(u) + S5 + O(u8 )
(127)
and includes more and more MST solutions if one wants to obtain a high-PN order
final result. Again the way to read the above results is the following. Summing from
1 to infinity the accuracy reached is only up to u3 terms included; summing from
l = 2 to infinity will allow to include terms O(u4 ); summing from l = 3 to infinity
will allow to include terms O(u5 ), etc.
The final result corresponding to contributions up to O(u8 ) (i.e., u7.5 terms

included) is then the following [27]

(reg) 35 7 4 4 2
ψ0 (u) = −u3 + − π 2 − γ − ln(2) − ln(u) u4
18 32 3 3 3

1141 29 2 2 18 1
+ + π + γ − ln(2) + ln(u) u5
360 512 3 5 3
38 11/2
− πu
45

23741 279 2 77 1627 729
+ − − π + γ+ ln(2) − ln(3)
1680 1024 6 42 70

77
+ ln(u) u6
12
3
− πu13/2
35
1515589307 6059603 2 76585 4
+ − − π + π
27216000 983040 262144

5321 152 304 152
+ − + γ+ ln(2) + ln(u) γ
900 45 45 45

1786621 152 152 12393
+ − + ln(2) + ln(u) ln(2) + ln(3)
18900 45 45 140

16 10121 38
− ζ (3) + − + ln(u) ln(u) u7
3 1800 45
35633 15/2
+ πu + O(u8 ) q . (128)
3780
Note that here the inverse radial variable u is also a gauge-invariant variable, since
it is related to the orbital frequency u = (M )2/3 which is a standardly used gauge-
invariant variable. Replacing ordinary logarithms by “eulerlogs,” i.e.,
1
eulerlogm (x) = γ + ln(2) + ln(x) + ln(m) , m = 1, 2, 3, . . . , (129)
2
as was first introduced in Ref. [28], allows to absorb also the Euler γ constant and
helps also to highlight the study of the transcendental structure of the various PN
orders. For example, the lowest order (O(u4 )) contains only eulerlog1 (u), at O(u5 )
a combination of eulerlog1 (u) and eulerlog2 (u) appears, etc. Unfortunately, starting
from O(u7 ) this replacement is not enough to completely remove the Euler γ terms,
meaning that the transcendental structure is more involved.
Scalar self-force effects on a Schwarzschild background have been numerically
studied in Ref. [29]. The comparison between the above analytical results and the
numerical results of Ref. [29] shows a reasonable agreement (see Table 1 and Fig. 2).
It is also interesting to study the behavior of this scalar field at the light-ring u = 1/3
Table 1 Comparison between the analytical prediction (128) for the regularized scalar field and
analytic
the numerical values taken from Table I of Ref. [29]. The difference ψ0 = ψ0num − ψ0 and
analytic
the relative error ψ0 /ψ0 are shown in the 3rd and 4th column, respectively. The superscript
“reg” has been suppressed here to ease notation and we have set q = 1 for simplicity
analytic analytic
u ψ0 ψ0 ψ0 /ψ0
1/4 −0.02304519610 −9.43 × 10−4 0.0409
1/5 −0.01022371010 −1.05 × 10−5 0.00102
1/6 −0.005468782560 1.40 × 10−5 −0.00255
1/7 −0.003282635718 7.29 × 10−6 −0.00222
1/8 −0.002130877461 3.37 × 10−6 −0.00158
1/10 −0.001050586634 7.94 × 10−7 −7.55 × 10−4
1/14 −3.701411742 × 10−4 7.66 × 10−8 −2.07 × 10−4
1/20 −1.246786056 × 10−4 5.81 × 10−9 −4.66 × 10−5
1/30 −3.661740186 × 10−5 3.02 × 10−10 −8.24 × 10−6
1/50 −7.889525256 × 10−6 7.26 × 10−12 −9.20 × 10−7
1/70 −2.877222881 × 10−6 8.81 × 10−13 −3.06 × 10−7
1/100 −9.884245218 × 10−7 2.18 × 10−14 −2.21 × 10−8
1/200 −1.239865750 × 10−7 −2.50 × 10−14 2.02 × 10−7
Fig. 2 The superposition of

the numerical data and our
analytical results (see
Table 1)
(see Ref. [30] where a corresponding study has been done for the case of a massive
particle orbiting a Schwarzschild black hole which is quite different from that of the
scalar field). We provide below a simple numerical fit of the data of Table 1
reg, fit u3
ψ0 =− (1 − 7.84u + 47.36u2 − 8.65u3 + 81.77u3 ln(u)) , (130)
(1 − 3u)2
(with a maximal residual of about 2.4 × 10−4 ), suggesting a blow-up of the form
(1−3u)−2 . However, this should be considered as an indication only, whereas a more
conclusive statement would require strong field numerical data still unavailable.
4 So Far So Good: A List of SF Accomplishments
Gravitational self-force accomplishments concern the redshift variable (z1 ), the

gyroscope precession (ψ ), the periastron advance (k ) and the tidal invariants
2 2 3
(λEi ,B ,E ). Any of these quantities, generically denoted as X, has an expression
of the type
0 ,e0 0 ,e2 0 ,e4
X = Xa + Xa + Xa + ...
a 2 ,e0 a 2 ,e2 a 2 ,e4
+X +X +X + ...
a 4 ,e0 a 4 ,e2 a 4 ,e4
+X +X +X + ...
+... (131)
0 0 0 2
where the term Xa ,e refers to circular orbits in the Schwarzschild spacetime, Xa ,e
to slightly eccentric orbits (including second order corrections in eccentricity) in
0 4
the Schwarzschild spacetime, Xa ,e to slightly eccentric orbits (including fourth
2 0
order corrections in eccentricity) in the Schwarzschild spacetime, Xa ,e to circular
2
orbits in the Kerr spacetime including corrections of order a in the Kerr rotational
parameter, etc. We summarize in Table 2 the current knowledge of the gauge-
2 2 3
invariant quantities z1 , ψ , K and λEi ,B ,E .
Table 2 List of gravitational self-force calculations of orbital invariants for both circular and
eccentric orbits in Schwarzschild and Kerr spacetimes and a sample of references
Schwarzschild a 0 e0 a 0 e2 a 0 e4 a 0 e6 a 0 e8
z1 [2, 30–35] [36] [37] [38]
ψ [39, 40] [41, 42]
k [37, 43]
2 ,B 2 ,E 3
λEi [35, 44–47] [48]
Kerr a 2 e0 a 2 e2 a 2 e4 a 4 e0 a 4 e2
z1 [19, 49–51] [20, 52] [53]
ψ [54] [55] [54]
k [56, 57]
2 ,B 2 ,E 3
λE
i [58] [58]
5 Concluding Remarks
The self-force formalism has been developed to deal with the effects of the
perturbation induced by particles or fields on a given gravitational background
as well as on their own motion. The novelty of the last few years is that the
metric perturbation can be fully reconstructed and several orbital invariants can
be analytically computed (i.e., the multipolar decomposition infinite summations
can be explicitly performed). This information can be then translated into other
approaches (e.g., Hamiltonian formalisms, like the effective one-body [59, 60]),
which are currently used to model the two-body gravitational interaction. The main
applications of these big computational efforts concern the modeling of gravitational
wave signals which are being detected by Earth-based interferometers, like those of
the LIGO/Virgo collaboration.
Acknowledgments We thank Thibault Damour for helpful discussions on analytical self-force

computations.
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Geometry and Analysis in Black Hole
Spacetimes
Lars Andersson
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.1 Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.2 Lorentzian Geometry and Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.3 Conventions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.4 Einstein Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.5 The Cauchy Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.6 Asymptotically Flat Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.7 Komar Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.1 The Schwarzschild Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.1.1 Orbiting Null Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2 Black Hole Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3 The Kerr Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4 Spin Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.1 Spinors on Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Spinors on Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3 Fundamental Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Massless Spin-s Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.5 Killing Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.6 Algebraically Special Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.6.1 Petrov Type D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.7 Spacetimes Admitting a Killing Spinor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.8 GHP Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Based on lectures given at the 2019 Summer School on Einstein Equations at Domodossola, Italy
L. Andersson ()
Albert Einstein Institute, Potsdam, Germany
e-mail: laan@aei.mpg.de

64 L. Andersson
5 The Kerr Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.1 Characterizations of Kerr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6 Monotonicity and Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.1 Monotonicity for Null Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 Dispersive Estimates for Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7 Symmetry Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.1 Symmetry Operators for the Kerr Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.1 Teukolsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.2 Stability for Linearized Gravity on Kerr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.3 Nonlinear Stability for Schwarzschild . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.4 Nonlinear Stability for Kerr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
1 Introduction
A solution to the Einstein vacuum equations is a Lorentzian spacetime (M, gab ),

satisfying Rab = 0, where Rab is the Ricci tensor of gab . The Einstein equation
is the Euler-Lagrange equation of the diffeomorphism invariant Einstein–Hilbert
action functional, given by the integral of the scalar curvature of (M, gab ),

Rdμg .
M
The diffeomorphism invariance, or general covariance, of the action has the conse-
quence that Cauchy data for the Einstein equation must satisfy a set of constraint
equations. After introducing suitable gauge conditions, the Einstein equations
can be reduced to a hyperbolic system of evolution equations. For sufficiently
regular Cauchy data satisfying the constraints, the Cauchy problem for the Einstein
equation has a unique solution which is maximal among all regular, vacuum Cauchy
developments. This general result, however, does not give any detailed information
about the properties of the maximal development.
There are two main conjectures about the maximal development. The strong
cosmic censorship conjecture states that a generic maximal development is inex-
tendible, as a regular vacuum spacetime. There are examples where the maximal
development is extendible and has non-unique extensions, which furthermore may
contain closed timelike curves. In these cases, predictability fails for the Einstein
equations, but if strong cosmic censorship holds, they are non-generic. At present,
this is only known to hold in the context of families of spacetimes with symmetry
restrictions, see [3, 62] and the references therein. The weak cosmic censorship
conjecture, on the other hand, states that a generic maximal development of
asymptotically flat data has no singularities in the domain of outer communication,
i.e., the region of spacetime visible to observers at null infinity, cf. [69]. Results on
black hole stability can thus be viewed as supporting weak cosmic censorship.
The Schwarzschild solution is static, spherically symmetric, asymptotically flat
and has a single free parameter M which represents the mass of the black hole.
Geometry and Analysis in Black Hole Spacetimes 65
By Birkhoff’s theorem it is the unique solution of the vacuum Einstein equations

with these properties. In 1963 Roy Kerr [42] discovered a new, explicit family of
asymptotically flat solutions of the vacuum Einstein equations which are stationary,
axisymmetric, and rotating. The Kerr family of solutions is parametrized by the
mass M, and the azimuthal angular momentum per unit mass a. In the limit a = 0,
the Kerr solution reduces to the spherically symmetric Schwarzschild solution.
If |a| ≤ M, the Kerr spacetime contains a black hole, while if |a| > M,
there is a ringlike singularity which is naked, in the sense that it fails to be hidden
from observers at infinity. This situation would violate the weak cosmic censorship
conjecture, and one, therefore, expects that an overextreme Kerr spacetime is
unstable, and in particular, that it cannot arise through a dynamical process from
regular Cauchy data.
For a geodesic γ a (λ) with velocity γ̇ a = dγ a /dλ, in a stationary axisymmetric
spacetime,1 there are three conserved quantities, the mass μ2 = γ̇ a γ̇b , energy
e = γ̇ a (∂t )a , and angular momentum z = γ̇ a (∂φ )a . In a general axisymmetric
spacetime, geodesic motion is chaotic. However, as was discovered by Brandon
Carter in 1968, there is a fourth conserved quantity for geodesics in the Kerr
spacetime, the Carter constant k, see Sect. 5 for details. By Liouville’s theorem, this
allows one to integrate the geodesic equations by quadrature, and thus geodesics in
the Kerr spacetime do not exhibit a chaotic behavior.
2 Background
2.1 Minkowski Space
Minkowski space M is R4 with metric which in a Cartesian coordinate system

(x a ) = (t, x i ) takes the form2
2
dτM = dt 2 − (dx 1 )2 − (dx 2 )2 − (dx 3 )2 .
A tangent vector ν a is timelike, null, or spacelike when gab ν a ν b > 0, = 0, or

< 0, respectively. Vectors with gab ν a ν b ≥ 0 are called causal.
1 We use signature + − −−, in particular timelike vectors have positive norm.

2 Here and below we shall use line elements, e.g., dτ 2
M = (gM )ab dx a dx b and metrics, e.g., (gM )ab
interchangeably.
66 L. Andersson
Let u, v be given by
u = t − r, v = t + r,
with r 2 = (x 1 )2 + (x 2 )2 + (x 3 )2 . In terms of these coordinates the line element

takes the form
2
dτM = dudv − r 2 d2S 2 . (1)
A complex null tetrad is given by

√ 1
la = 2(∂u )a = √ ∂t )a + (∂r )a , (2a)
2
√ 1
na = 2(∂v )a = √ (∂t )a − (∂r )a , (2b)
2

1 i
ma = √ (∂θ )a + (∂φ )a (2c)
2r sin θ
normalized so that na la = 1 = −ma m̄a , with all other inner products of tetrad
legs zero. Complex null tetrads with this normalization play a central role in the
Newman-Penrose and Geroch-Held-Penrose formalisms, see Sect. 4. In these notes
we will use such tetrads unless otherwise stated.
In terms of a null tetrad, we have
gab = 2(l(a nb) − m(a m̄b) ). (3)
Introduce compactified null coordinates U, V, given by
U = arctan u, V = arctan v.
These take values in {(−π/2, π/2) × (−π/2, π/2)} ∩ {V ≥ U}, and we can
thus present Minkowski space in a causal diagram, see Fig. 1. Here each point
represents an S 2 and we have drawn null vectors at 45◦ angles. A compactification
of Minkowski space is now given by adding the null boundaries3 I± , spatial infinity
i0 and timelike infinity i ± as indicated in the figure. Explicitly,
I+ = {V = π/2}
I− = {U = −π/2}
i0 = {V = π/2, U = −π/2}
3 Here I is pronounced “Scri” for “script I.”

Fig. 1 Causal diagram of

Minkowski space
i± = {(V, U) = ±(π/2, π/2)}.
In Fig. 1, we have also indicated schematically the t-level sets which approach
spatial infinity i0 . Causal diagrams are a useful tool which, if applied with proper
care, can be used to understand the structure of quite general spacetimes. Such
diagrams are often referred to as Penrose, or Carter-Penrose diagrams.
2.2 Lorentzian Geometry and Causality
We now consider a smooth Lorentzian 4-manifold (M, gab ) with signature +−−−.
Each tangent space in a 4-dimensional spacetime is isometric to Minkowski space
M, and we can carry intuitive notions of causality over from M to M. We say that
a smooth curve γ a (λ) is causal if the velocity vector γ̇ a = dγ a /dλ is causal. Two
points in M are causally related if they can be connected by a piecewise smooth
causal curve. The concept of causal curves is most naturally defined for C 0 curves.
A C 0 curve γ a is said to be causal if each pair of points on γ a are causally related.
We may define timelike curve and timelike related points in the analogous manner.
We now assume that M is time oriented, i.e., that there is a globally defined
timelike vector field on M. This allows us to distinguish between future and past
directed causal curves, and to introduce a notion of the causal and timelike future
of a spacetime point. The corresponding past notions are defined analogously. If q
is in the causal future of p, we write p q. This introduces a partial order on M.
The causal future J + (p) of p is defined as J + (p) = {q : p q} while the timelike
future I + (p) is defined in the analogous manner, with timelike replacing causal. A
subset ⊂ M is achronal
68 L. Andersson
I + (p)
S
p
if there is no pair p, q ∈ M such that q ∈ I + (p), i.e., does not intersect its
timelike future or past. The domain of dependence D(S) of S ⊂ M
D(S)
S
is the set of points p such that any inextendible causal curve starting at p must
intersect S.
Definition 2.1 A spacetime M is globally hyperbolic if there is a closed, achronal
⊂ M such that M = D( ). In this case, is called a Cauchy surface.
Due to results of Bernal and Sanchez [19], global hyperbolicity is characterized
by the existence of a smooth, Cauchy time function τ : M → R. A function τ
on M is a time function if ∇ a τ is timelike everywhere, and it is Cauchy if the
level sets t = τ −1 (t) are Cauchy surfaces. If τ is smooth, its level sets are then
smooth and spacelike. It follows that a globally hyperbolic spacetime M is globally
foliated by Cauchy surfaces, and in particular is diffeomorphic to a product × R.
In the following, unless otherwise stated, we shall consider only globally hyperbolic
spacetimes.
If a globally hyperbolic spacetime M is a subset of a spacetime M , then the
boundary ∂M in M is called the Cauchy horizon.
Example 2.2 Let O be the origin in Minkowski space, and let M = I + (O) = {t >
r} be√its timelike future. Then M is globally hyperbolic with Cauchy time function
τ = t 2 − r 2 . Further, M is a subset of Minkowski space M, which is a globally
hyperbolic space with Cauchy time function t. Minkowski space is geodesically
complete and hence inextendible. The boundary {t = r} is the Cauchy horizon ∂M
of M. Past inextendible causal geodesics (i.e., past causal rays) in M end on ∂M. In
particular, M is incomplete. However, M is extendible, as a smooth flat spacetime,
with many inequivalent extensions.
We remark that for a globally hyperbolic spacetime, which is extendible, the
extension is in general non-unique. In the particular case considered in Example 2.2,
M is an extension of M, which is also happens to be maximal and globally
hyperbolic. In the vacuum case, there is a unique maximal globally hyperbolic
extension, cf. Sect. 2.5 below. However, a maximal extension is in general non-
unique, and may fail to be globally hyperbolic.
2.3 Conventions and Notation
We shall use mostly abstract indices, but will sometimes work with coordinate
indices, and unless confusion arises will not be too specific about this. We raise
and lower indices with gab , for example ,ξ a = g ab ξb , with g ab gbc = δ a c , where δ a c
is the Kronecker delta, i.e., the tensor with the property that δ a c ξ c = ξ a for any ξ a .
Let a···d be the Levi-Civita symbol, i.e., the skew symmetric expression which
√ system has the property that 1···n = 1. The volume form of gab is
in any coordinate
(μg )abcd = |g|abcd . Given (M, gab ) we have the canonically defined Levi-Civita
covariant derivative ∇a . For a vector ν a , this is of the form
∇a ν b = ∂a ν b + ac
b c
ν ,
b = 1 g bd (∂ g + ∂ g
where ac 2 a dc c db − ∂d gac ) is the Christoffel symbol. In order to
fix the conventions used here, we recall that the Riemann curvature tensor is defined
by
(∇a ∇b − ∇b ∇a )ξc = Rabc d ξd .
The Riemann tensor Rabcd is skew symmetric in the pairs of indices ab, cd, Rabcd =
R[ab]cd = Rab[cd] , is pairwise symmetric Rabcd = Rcdab , and satisfies the first
Bianchi identity R[abc]d = 0. Here square brackets [· · · ] denote antisymmetrization.
We shall similarly use round brackets (· · · ) to denote symmetrization. Further, we
have ∇[a Rbc]de = 0, the second Bianchi identity. A contraction gives ∇ a Rabcd = 0.
The Ricci tensor is Rab = R c acb and the scalar curvature R = R a a . We further let
Sab = Rab − 14 Rgab denote the trace free part of the Ricci tensor. The Riemann
tensor can be decomposed as follows,
Rabcd = − 1
12 gad gbc R + 1
12 gac gbd R + 12 gbd Sac − 12 gbc Sad
− 12 gad Sbc + 12 gac Sbd + Cabcd . (4)
This defines the Weyl tensor Cabcd which is a tensor with the symmetries of
the Riemann tensor, and vanishing traces, C c acb = 0. Recall that (M, gab ) is
locally conformally flat if and only if Cabcd = 0. It follows from the contracted
second Bianchi identity that the Einstein tensor Gab = Rab − 12 Rgab is conserved,
∇ a Gab = 0.
2.4 Einstein Equation
The Einstein equation in geometrized units with G = c = 1, where G, c denote

Newtons constant and the speed of light, respectively, cf. [68, Appendix F], is the
system
70 L. Andersson
Gab = 8π Tab . (5)
This equation relates geometry, expressed in the Einstein tensor Gab on the left-hand
side, to matter, expressed via the energy-momentum tensor Tab on the right-hand
side. For example, for a self-gravitating Maxwell field Fab , Fab = F[ab] , we have
1 1
Tab = (Fac Fbc − Fcd F cd gab ).
4π 4
The source-free Maxwell field equations
∇ a Fab = 0, ∇[a Fbc] = 0
imply that Tab is conserved, ∇ a Tab = 0. The contracted second Bianchi identity
implies that ∇ a Gab = 0, and hence the conservation property of Tab is implied by
the coupling of the Maxwell field to gravity. These facts can be seen to follow from
the variational formulation of Einstein gravity, given by the action

R
I= dμg − Lmatter dμg ,
M 16π M
where Lmatter is the Lagrangian describing the matter content in the spacetime. In
the case of Maxwell theory, this is given by
1
LMaxwell = Fcd F cd .
4π
Recall that in order to derive the Maxwell field equation, as an Euler-Lagrange
equation, from this action, it is necessary to introduce a vector potential for Fab ,
by setting Fab = 2∇[a Ab] , and carrying out the variation with respect to Aa . It is a
general fact that for generally covariant (i.e., diffeomorphism invariant) Lagrangian
field theories which depend on the spacetime location only via the metric and its
derivatives, the symmetric energy-momentum tensor
1 ∂Lmatter
Tab = √
g ∂g ab
is conserved when evaluated on solutions of the Euler-Lagrange equations.

As a further example of a matter field, we consider the scalar field, with action
Lscalar = 12 ∇ c ψ∇c ψ,
where ψ is a function on M. The corresponding energy-momentum tensor is
Tab = ∇a ψ∇b ψ − 12 ∇ c ψ∇c ψgab

and the Euler-Lagrange equation is the free scalar wave equation
∇ a ∇a ψ = 0. (6)
As (6) is another example of a field equation derived from a covariant action which
depends on the spacetime location only via the metric gab or its derivatives, the
symmetric energy-momentum tensor is conserved for solutions of the field equation.
In both of the just mentioned cases, the energy-momentum tensor satisfies the
dominant energy condition, Tab ν a ζ b ≥ 0 for future directed causal vectors ν a , ζ a .
This implies the null energy condition
Rab ν a ν b ≥ 0 if νa ν a = 0. (7)
These energy conditions hold for most classical matter.

There are many interesting matter systems which are worthy of consideration,
such as fluids, elasticity, kinetic matter models including Vlasov, as well as
fundamental fields such as Yang-Mills, to name just a few. We consider only
spacetimes which satisfy the null energy condition, and for the most part we shall
in these notes be concerned with the vacuum Einstein equations,
Rab = 0. (8)
2.5 The Cauchy Problem
Given a spacelike hypersurface4 in M with timelike normal T a , induced metric

hab and second fundamental form kab , defined by kab Xa Y b = ∇a Tb Xa Y b for
Xa , Y b tangent to , the Gauss, and Gauss-Codazzi equations imply the constraint
equations
R[h] + (kab hab )2 − kab k ab =16π Tab T a T b (9a)

∇[h]a (kbc hbc ) − ∇[h]b kab =Tab T b . (9b)
A 3-manifold together with tensor fields hab , kab on solving the constraint
equations is called a Cauchy data set. The constraint equations for general relativity
are analogs of the constraint equations in Maxwell and Yang-Mills theory, in that
they lead to Hamiltonians which generate gauge transformations.
Consider a 3 + 1 split of M, i.e., a 1-parameter family of Cauchy surfaces t ,
with a coordinate system (x a ) = (t, x i ), and let
4 If there is no room for confusion, we shall denote abstract indices for objects on by a, b, c, . . . .
72 L. Andersson
(∂t )a = NT a + Xa
be the split of (∂t )a into a normal and tangential piece. The fields (N, Xa ) are called
lapse and shift. The definition of the second fundamental form implies the equation
L∂t hab = −2Nkab + LX hab .
In the vacuum case, the Hamiltonian for gravity can be written in the form

NH + Xa Ja + boundary terms,
where H and J are the densitized left-hand sides of (9). If we consider only
compactly supported perturbations in deriving the Hamiltonian evolution equation,
the boundary terms mentioned above can be ignored. However, for (N, Xa ) not
tending to zero at infinity, and considering perturbations compatible with asymptotic
flatness, the boundary term becomes significant, cf. Sect. 2.6.
The resulting Hamiltonian√evolution equations, written in terms of hab and its
canonical conjugate π ab = h(k ab − (hcd kcd hab )) are usually called the ADM
evolution equations.
Let ⊂ M be a Cauchy surface. Given functions φ0 , φ1 on and F on M, the
Cauchy problem is the problem of finding solutions to the wave equation

∇ a ∇a ψ = F, ψ = φ0 , L∂t ψ = φ1 .
Assuming suitable regularity conditions, the solution is unique and stable with
respect to initial data. This fact extends to a wide class of non-linear hyperbolic
PDE’s including quasilinear wave equations, i.e., equations of the form
Aab [ψ]∂a ∂b ψ + B[ψ, ∂ψ] = 0
with Aab a Lorentzian metric depending on the field ψ.

Given a vacuum Cauchy data set, ( , hab , kab ), a solution of the Cauchy problem
for the Einstein vacuum equations is a spacetime metric gab with Rab = 0, such
that (hab , kab ) coincides with the metric and second fundamental form induced on
from gab . Such a solution is called a vacuum extension of ( , hab , kab ).
Due to the fact that Rab is covariant, the symbol of Rab is degenerate. In order to
get a well-posed Cauchy problem, it is necessary to either impose gauge conditions
or introduce new variables. A standard choice of gauge condition is the harmonic
coordinate condition. Let gab be a given metric on M. The identity map i : M → M
is harmonic if and only if the vector field
V a = g bc (bc
a
−a
bc )
vanishes. Here bc a , a are the Christoffel symbols of the metrics g , a

bc ab gab . Then V
is the tension field of the identity map i : (M, gab ) → (M, gab ). This is harmonic if
and only if
V a = 0. (10)
Since harmonic maps with a Lorentzian domain are often called wave maps, the
gauge condition (10) is sometimes called wave map gauge. A particular case of this
construction, which can be carried out if M admits a global coordinate system (x a ),
is given by letting
gab be the Minkowski metric defined with respect to (x a ). Then

bc = 0 and (10) is simply
a
∇ b ∇b x a = 0, (11)
which is usually called the wave coordinate gauge condition.

Going back to the general case, let ∇ be the Levi-Civita covariant derivative
defined with respect to
gab . We have the identity
1 √ ab + ∇(a Vb) ,
Rab = − 12 √ ∇ a gg ∇b gab + Sab [g, ∇g] (12)
g
a gcd . Setting
where Sab is an expression which is quadratic in first derivatives ∇
V = 0 in (12) yields Rab , and (8) becomes a quasilinear wave equation
a harm
harm
Rab = 0. (13)
By standard results, the Eq. (13) has a locally well-posed Cauchy problem in
Sobolev spaces H s for s > 5/2. Using more sophisticated techniques, well-
posedness can be shown to hold for any s > 2 [44]. Recently a local existence
has been proved under the assumption of curvature bounded in L2 [46]. Given a
Cauchy data set ( , hab , kab ), together with initial values for lapse and shift N, Xa
on , it is possible to find Lt N, Lt Xa on such that the V a are zero on . A
calculation now shows that due to the constraint equations, L∂t V a is zero on .
Given a solution to the reduced Einstein vacuum equation (13), one finds that V a
solves a wave equation. This follows from ∇ a Gab = 0, due to the Bianchi identity.
Hence, due to the fact that the Cauchy data for V a is trivial, it holds that V a = 0
on the domain of the solution. Thus, in fact the solution to (13) is a solution to the
full vacuum Einstein equation (8). This proves local well-posedness for the Cauchy
problem for the Einstein vacuum equation. This fact was first proved by Yvonne
Choquet-Bruhat [32], see [63] for background and history.
Global uniqueness for the Einstein vacuum equations was proved by Choquet-
Bruhat and Geroch [22]. The proof relies on the local existence theorem sketched
above, patching together local solutions. A partial order is defined on the collection
of vacuum extensions, making use of the notion of common domain. The common
74 L. Andersson
Fig. 2 Partial Cauchy

surface touching ∂U
∂U
domain U of two extensions M, M is the maximal subset in M which is isometric

to a subset in M . We can then define a partial order by saying that M ≤ M
if the maximal common domain is M. One sees from the construction that each
totally ordered subset has an upper bound. Thus, a maximal element exists by Zorn’s
lemma. This is proven to be unique by an application of the local well-posedness
theorem for the Cauchy problem sketched above. For a contradiction, let M, M be
two inequivalent extensions, and let U be the maximal common domain. Due to the
Hausdorff property of spacetimes, this leads to a contradiction. By finding a partial
Cauchy surface which touches the boundary of U , see Fig. 2 and making use of local
uniqueness, one finds a contradiction to the maximality of U . It should be noted that
here, uniqueness holds up to isometry, in keeping with the general covariance of the
Einstein vacuum equations. These facts extend to the Einstein equations coupled
to hyperbolic matter equations. See [64] for a construction of the maximal globally
hyperbolic extension which does not rely on Zorn’s lemma, see also [73]. The global
uniqueness result can be generalized to Einstein-matter systems, provided the matter
field equation is hyperbolic and that its solutions do not break down. General results
on this topic are lacking, see, however, [58] and the references therein. The minimal
regularity needed for global uniqueness is a subtle issue, which has not been fully
addressed. In particular, results on local well-posedness are known, see, e.g., [45]
and the references therein, which require less regularity than the best results on
global uniqueness.
2.6 Asymptotically Flat Data
The Kerr black hole represents an isolated system, and the appropriate data for the
black hole stability problem should, therefore, be asymptotically flat. To make this
precise we suppose there is a compact set K in M and a map : M \ K →
R3 \ B(R, 0), where B(R, 0) is a Euclidean ball. This defines a Cartesian coordinate
system on the end M \ K so that hab − δab falls off to zero at infinity, at a suitable
rate. Here δab is the Euclidean metric in the Cartesian coordinate system constructed
above. Similarly, we require that kab falls off to zero.
Let x a be the chosen Euclidean coordinate system and let r be the Euclidean
radius r = (δab x a x b )1/2 . Following Regge and Teitelboim [61], see also [18], we
assume that gab = δab + hab with
hab =O(1/r), ∂a hbc = O(1/r 2 ),

kab =O(1/r 2 ).
Further, we impose the parity conditions
hab (x) = hab (−x), kab (x) = −kab (−x). (14)
These falloff and parity conditions guarantee that the ADM 4-momentum and
angular momentum are well defined. It was shown in [39] that data satisfying the
parity condition conditions (14) are dense among data which satisfy an asymptotic
flatness condition in terms of weighted Sobolev spaces.
Let ξ a be an element of the Poincare Lie algebra and assume that NT a + Xa
tends in a suitable sense to ξ a at infinity. Then the action for Einstein gravity can be
written in the form

Rdμg = Pa ξ a + π ij ḣij − N H + Xi Ji .
M
Here we may view Pa as a map to the dual of the Poincare Lie algebra, i.e.,
a momentum map. Evaluating Pa ξ a on a particular element of the Poincare Lie
algebra gives the corresponding momentum. These can also be viewed as charges at
infinity. We have

1
P0 = lim (∂i gj i − ∂j gii )dσ i (15a)
16π r→∞ Sr

1
Pi = lim πij dσ j , (15b)
8π r→∞ Sr
where dσ i denotes the hypersurface area element of a family of spheres (which

can be taken to be coordinate spheres) Sr foliating a neighborhood of infinity. See
[54] and the references therein for a recent discussion of the conditions under which
these expressions are well defined.
The energy and linear momentum (P 0 , P i ) provide the components of a 4-
vector P a , the ADM 4-momentum. Assuming the dominant energy condition, then
under the above asymptotic conditions, P a is future causal, and timelike unless
the maximal development (M, gab ) is isometric to Minkowski space. Further, a
√ aP
transforms as a Minkowski 4-vector, and the ADM mass is given by M = P Pa .
The boost theorem [23] implies, given an asymptotically flat Cauchy data set, that
one may find in a boosted slice in its development such that the data is in the rest
frame, i.e., P a = M(∂t )a .
Since the constraint quantities H, Ji vanish for solutions of the Einstein equa-
tions, the gravitational Hamiltonian takes the value Pa ξ a , and hence the ADM mass
and momenta defined by (15) are conserved for an evolution with lapse and shift
(N, Xi ) → (1, 0) at infinity.
76 L. Andersson
2.7 Komar Integrals
Assume that ν a is a Killing vector field. Then we have ∇a νb = ∇[a νb] . A calculation
shows
∇ a (∇a ξb − ∇b ξa ) = −2Rbc ξ c .
Hence, in vacuum,

eabcd ∇ c ξ d
S
depends only on the homology class of the two-surface S. The analogous fact for
the source-free Maxwell equation, were we have ∇ a Fab = 0, ∇[a Fbc] = 0, is the
conservation of the charge integrals S Fab , S abcd F cd , which again depend only
on the homology class of S. These statements are immediate consequences of Stokes
theorem.
If we consider asymptotically flat spacetimes, we have in the stationary case,
with ξ a = (∂t )a ,

1
P a ξa = − abcd ∇ c ξ d ,
8π S
where on the left-hand side we have the ADM 4-momentum evaluated at infinity.
Similarly, in the axially symmetric case, with ηa = (∂φ )a ,

1
J =− abcd ∇ c ηd .
16π S
These integrals again depend only on the homology class of S. See [41, §6] for
background to these facts. For a non-symmetric, but asymptotically flat spacetime,
letting S tend to infinity through a sequence of suitably round spheres yields the
linkage integrals, which again reproduce the ADM momenta [72].
3 Black Holes
3.1 The Schwarzschild Solution
In Schwarzschild coordinates (t, r, θ, φ), the Schwarzschild metric takes the form
gab dx a dx b = f dt 2 − f −1 dr 2 − r 2 d2S 2 (16)

with f = 1 − 2M/r. Here d2S 2 = dθ 2 + sin2 θ dφ 2 is the line element on the unit
2-sphere. The coordinate r is the area radius, defined by 4π r 2 = A(S(r, t)), where
S(r, t) is the 2-sphere with constant t, r. The Schwarzschild metric is asymptotically
flat and the parameter M coincides with the ADM mass.
In order to get a better understanding of the Schwarzschild spacetime, it is
instructive to consider its maximal extension. In order to do this, we first introduce
the tortoise coordinate r∗ ,
r
r∗ = r + 2Mlog( − 1). (17)
2M
r∗
r
r+
This solves dr∗ = f −1 dr, r∗ (4M) = 4M. As r 2M, r∗ diverges

logarithmically to −∞, and for large r, r∗ ∼ r. Inverting (17) yields
r∗
r = 2MW e 2M −1 + 2M, (18)
where W is the principal branch of the Lambert W function.5 We can now introduce
null coordinates
u = t − r∗ , v = t + r∗ .
A null tetrad is given by
2 a
la = ∂ ,
f v
2 a
na = ∂ ,
f u
5 The Lambert W function, or product logarithm, is defined as the solution of W (x)eW (x) = x for
x > 0. It satisfies W (x) = W (x)/((W (x) + 1)x). The principal branch is analytic at x = 0 and is
real valued in the range (−e−1 , ∞) with values in (−1, ∞). In particular, W (0) = 0. See [25].
78 L. Andersson
1 i a
ma = √ (∂θa + ∂ ).
2r sin θ φ
On the exterior region in Schwarzschild, (u, v) take values in the range (−∞, ∞) ×
(−∞, ∞). Let U, V be a pair of coordinates taking values in (−π/2, π/2), and
related to u, v by
u = − 4Mlog(− tan U), U ∈ (−π/2, 0)

v = 4Mlog(tan V), V ∈ (0, π/2).
We have
t = 12 (v + u) = 4Mlog (− tan V tan U)

tan V
r∗ = 12 (v − u) = 4Mlog − .
tan U
In terms of U, V we have
r = 2MW(−e−1 tan U tan V) + 2M (19)
and r > 0 thus corresponds to tan U tan V < 1. The line element now takes the form
dUdV 32M 3 − r
gab dx a dx b = e 2M − r 2 d2S 2 . (20)
cos2 U cos2 V r
The form (20) of the Schwarzschild line element is non-degenerate in the range
(U, V) ∈ (−π/2, π/2) × (−π/2, π/2) ∩ {−π/2 < U + V < π/2}. (21)
In particular, the location r = 2M of the coordinate singularity in the line element

(16) corresponds to UV = 0. The line element (20) has a coordinate singularity,
which is also a curvature singularity, at r = 0 (corresponding to tan U tan V = 1),
and at U = ±π/2, V = ±π/2 (corresponding to u, v taking unbounded values).
Figure 3 shows the region given in (21), with lines of constant t, r indicated. Using
the causal diagram for the extended Schwarzschild solution, one can easily find
the null infinities I± , spatial infinity i0 , timelike infinities i± , the horizons H± at
r = 2M, which are indicated. Region I is the domain of outer communication,
i.e., I − (I+ ) ∩ I + (I− ), while region I I is the future trapped (or black hole) region,
MSchw \ I − (I+ ).
The level sets of t hit the bifurcation sphere B located at U = V = 0, where
∂t = 0. In particular, we see that the Schwarzschild coordinates are degenerate,
since the level sets of t do not foliate the extended Schwarzschild spacetime. On
the other hand, a global Cauchy foliation of the maximally extended Schwarzschild
spacetime is given by the level sets of the Kruskal time function T = 12 (V + U).
Fig. 3 Causal diagram of the maximally extended Schwarzschild solution
Although the region covered by the null coordinates U, V is compact, the line
element (20) is of course isometric to the form given in (16). A conformal factor
= cos U cos V may now be introduced, which brings I± to a finite distance.
Letting g̃ab = 2 gab , and adding these boundary pieces to (M, g̃ab ) provides a
conformal compactification6 of the maximally extended Schwarzschild spacetime.
3.1.1 Orbiting Null Geodesics
Consider a null geodesic γ a in the Schwarzschild spacetime. Due to the spherical

symmetry of the Schwarzschild spacetime, we may assume without loss of gener-
ality that θ̇ = 0 and set θ = π/2, so that γ a moves in the equatorial plane. We
have that the geodesic energy and azimuthal angular momentum e = −ξ a γ̇a and
z = ηa γ̇a are conserved. We have
z = ηa γ̇ b gab = r 2 φ̇.
In fact the same is true for the momenta corresponding to each of the three rotational
Killing fields. Thus, we may consider the total squared angular momentum L2 given
by
L2 = 2r 2 m(a m̄b) γ̇ a γ̇ b = r 4 (gS 2 )ab γ̇ a γ̇ b . (22)
6 There are subtleties concerning the regularity of the conformal boundary of Schwarzschild, and
the naive choice of conformal factor mentioned above does not lead to an analytic compactification.
See [38] for recent developments.
80 L. Andersson
For geodesics moving in the equatorial plane, we have L2 = z 2 . Rewriting

gab γ̇ a γ̇ b = 0 using (3) and these definitions gives
ṙ 2 + V = e2 , (23)
where
f 2
V = L .
r2
Equation (23) can be viewed as the equation for a particle moving in a potential V .
V
r
r+ 3M
An analysis shows that V has a unique critical point at r = 3M, and hence a null
geodesic with ṙ = 0 in the Schwarzschild spacetime must orbit at r = 3M. We call
such null geodesics trapped. The critical point r = 3M is a local maximum for V
and hence the orbiting null geodesics are unstable. The sphere r = 3M is called the
photon sphere. A similar analysis can be performed for massive particles orbiting
the Schwarzschild black hole, see [68, Chapter 6] for further details.
The geometric optics correspondence between waves packets and null geodesics
indicates that the phenomenon of trapped null geodesics is an obstacle to dispersion,
i.e., the tendency for waves to leave every stationary region. For waves of finite
energy, the fact that the trapped orbits are unstable can be used to show that such
waves in fact disperse. This is a manifestation of the uncertainty principle.
3.2 Black Hole Stability
The black hole stability conjecture states that Cauchy data sufficiently close, in a
suitable sense, to Kerr Cauchy data7 have a maximal development which is future
asymptotic to a Kerr spacetime, see Fig. 4. It is important to note that the parameters
of the “limiting” Kerr spacetime cannot be determined in any effective manner from
the initial data.
As discussed above, cf. Sect. 2.7, if we restrict to axial symmetry, then angular
momentum is quasi-locally conserved. This means that if we further restrict to
zero angular momentum, the end state of the evolution must be a Schwarzschild
7 See [5], see also, e.g., [14, 53] for discussions of the problem of characterizing Cauchy data as
Kerr data.
Fig. 4 Causal diagram of a

spacetime that is future
asymptotic to the Kerr
solution
black hole. Thus, the black hole stability conjecture for the axially symmetric
case is that the maximal developments of sufficiently small (in a suitable sense),
axially symmetric, deformations of Schwarzschild Cauchy data with zero angular
momentum, are asymptotic to the future to a Schwarzschild spacetime. In this case,
due to the loss of energy through I+ , the mass of the “limiting” Schwarzschild black
hole cannot be determined directly from the Cauchy data. See Sect. 8.3 below.
3.3 The Kerr Metric
In this section we shall discuss the Kerr metric, which is the main object of our
considerations. Although many features of the geometry and analysis on black hole
spacetimes are seen in the Schwarzschild case, there are many new and fundamental
phenomena present in the Kerr case. Among those are complicated trapping, i.e.,
the fact that trapped null geodesics fill up an open spacetime region, the fact that
the Kerr metric admits only two Killing fields, but a hidden symmetry manifested
in the Carter constant, and the fact that the stationary Killing vector field ξ a fails to
be timelike in the whole domain of outer communications, which leads to a lack of
a positive conserved energy for waves in the Kerr spacetimes. This fact is the origin
of superradiance and the Penrose process. See [67] for a recent survey.
The Kerr metric describes a family of stationary, axisymmetric, asymptotically
flat vacuum spacetimes, parametrized by ADM mass M and angular momentum
per unit mass a. The expressions for mass and angular momentum introduced in
Sect. 2.6 when applied in Kerr geometry yield M and J = aM. In Boyer-Lindquist
coordinates (t, r, θ, φ), the Kerr metric takes the form
( − a 2 sin2 θ )dta dtb 2a sin2 θ (a 2 + r 2 − )dt(a dφb)

gab = + (24)
82 L. Andersson

dra drb sin2 θ (a 2 + r 2 )2 − a 2 sin2 θ dφa dφb
− − dθa dθb − ,

where = a 2 − 2Mr + r 2 and = a 2 cos2 θ + r 2 . The volume element is
| det gab | = sin θ. (25)
There is a ring-shaped singularity at r = 0, θ = π/2. For |a| ≤ M, √ the Kerr

spacetime contains a black hole, with event horizon at r = r+ ≡ M + M 2 − a 2 ,
while for |a| > M, the singularity is naked in the sense that it is causally connected
to observers at infinity. The area of the horizon is AHor = 4π(r+ 2 + a 2 ). This
achieves its maximum of 16π M 2 when a = 0, providing one of the ingredients

in the heuristic argument for the Penrose inequality, see Sect. 3.2. The case |a| = M
is called extreme. We shall here be interested only in the subextreme case, |a| < M,
as this is the only case where we expect black hole stability to hold.
The Boyer-Lindquist coordinates are analogous to the Schwarzschild coordinates
Sect. 3.1 and upon setting a = 0, (24) reduces to (16). The line element takes a
simple form in Boyer-Lindquist coordinates, but similarly for the Schwarzschild
coordinates, the Boyer-Lindquist coordinates have the drawback that they are not
regular at the horizon.
The Kerr metric admits two Killing vector fields ξ a = (∂t )a (stationary) and
η = (∂φ )a (axial). Although the stationary Killing field ξ a is timelike near infinity,
a
since gab ξ a ξ b → 1 as r → ∞, ξ a becomes spacelike for r sufficiently small,

when 1 − 2M/ < 0. In the Schwarzschild case a = 0, this occurs at the event
horizon r = 2M. However, for a rotating Kerr black hole with 0 < |a| ≤ M, there
is a region, called the ergoregion, outside the event a
√ horizon where ξ is spacelike.
The ergoregion is bounded by the surface M + M − a cos θ which touches
2 2 2
the horizon at the poles θ = 0, π, see Fig. 5. In the ergoregion, null and timelike
geodesics can have negative energy with respect to ξ a . The fact that there is no
globally timelike vector field in the Kerr exterior is the origin of superradiance, i.e.,
the fact that waves which scatter off the black hole can leave the ergoregion with
larger energy (as measured by a stationary observer at infinity) than was sent in.
This effect was originally found by an analysis based on separation of variables, but
can be demonstrated rigorously, see [31]. However, it is a subtle effect and not easy
to demonstrate numerically, see [49].
Fig. 5 The ergoregion

Let ωH = a/(r+ 2 + a 2 ) be the rotation speed of the black hole. The Killing
field χ = ξ + ωH ηa is null on the event horizon in Kerr, which is, therefore,

a a
a Killing horizon. For |a| < M, there is a neighborhood of the horizon in the
black hole exterior where χ a is timelike. The surface gravity κ, defined by κ 2 =
− 12 (∇ a χ b )(∇a χb ) takes the value κ = (r+ −M)/(r+ 2 +a 2 ), and is in the subextreme
case |a| < M nonzero. By general results, a Killing horizon with non-vanishing
surface gravity is bifurcate, i.e., there is a cross-section where the null generator
vanishes. In the Schwarzschild case, this is the 2-sphere U = V = 0. See [57, 60]
for background on the geometry of the Kerr spacetime, see also [56].
4 Spin Geometry
The 2-spinor formalism, and the closely related GHP formalism, are important
tools in Lorentzian geometry and the analysis of black hole spacetimes, and we
will introduce them here. A detailed of this material is given by Penrose and
Rindler [59]. Following the conventions there, we use the abstract index notation
with lower case Latin letters a, b, c, . . . for tensor indices, and unprimed and
primed upper-case Latin letters A, B, C, . . . , A , B , C , . . . for spinor indices.
Tetrad and dyad indices are boldface Latin letters following the same scheme,
a, b, c, . . . , A, B, C, . . . , A , B , C , . . . . For coordinate indices we use Greek let-
ters α, β, γ , . . . .
4.1 Spinors on Minkowski Space
Consider Minkowski space M, i.e., R4 with coordinates (x α ) = (t, x, y, z) and

metric
gαβ dx α dx β = dt 2 − dx 2 − dy 2 − dz2 .
Define a complex null tetrad (i.e., frame) (ga a )a=0,··· ,3 = (l a , na , ma , m̄a ), as in

(2) above, normalized so that l a na = 1, ma m̄a = −1, so that
gab = 2(l(a nb) − m(a m̄b) ). (26)
Similarly, let A A be a dyad (i.e., frame) in C2 , with dual frame A A . The complex
conjugates will be denoted ¯A A , ¯A A and again form a basis in another 2-
dimensional complex space denoted C̄2 , and its dual. We can identify the space
of complex 2 × 2 matrices with C2 ⊗ C̄2 . By construction, the tensor products
A A ¯A A and A A ¯A A forms a basis in C2 ⊗ C̄2 and its dual.
84 L. Andersson
Now, with x a = x a ga a , writing

0 2
a AA x x
x ga ≡ (27)
x3 x1
defines the soldering forms, also known as Infeld-van der Waerden symbols ga AA ,
(and analogously gAA a ). By a slight abuse of notation we may write x AA = x a
instead of x AA = x a ga AA or, dropping reference to the tetrad, x AA = x a ga AA .
In particular, we have that x a ∈ M corresponds to a 2 × 2 complex Hermitian matrix
x AA ∈ C2 ⊗ C̄2 . Taking the complex conjugate of both sides of (27) gives
x̄ a = x̄ A A = (x AA )∗ ,
where ∗ denotes Hermitian conjugation. This extends to a correspondence C4 ↔

C2 ⊗ C̄2 with complex conjugation corresponding to Hermitian conjugation.
Note that
det(x AA ) = x 0 x 1 − x 2 x 3 = x a xa /2. (28)
We see from the above that the group

ab
SL(2, C) = A = , a, b, c, d ∈ C, ad − bc = 1
cd
acts on X ∈ C2 ⊗ C̄2 by
X → AXA∗ .
In view of (28) this exhibits SL(2, C) as a double cover of the identity component
of the Lorentz group SO0 (1, 3), the group of linear isometries of M. In particular,
SL(2, C) is the spin group of M. The canonical action
(A, v) ∈ SL(2, C) × C2 → Av ∈ C2
of SL(2, C) on C2 is the spinor representation. Elements of C2 are called (Weyl)

spinors. The conjugate representation given by
(A, v) ∈ SL(2, C) × C2 → Āv ∈ C2
is denoted C̄2 .
Spinors8 of the form x AA = α A β A correspond to matrices of rank one, and

hence to complex null vectors. Denoting oA = 0 A , ιA = 1 A , we have from the
above that
l a = oA oA , na = ιA ιA , ma = oA ιA , m̄a = ιA oA . (29)
This gives a correspondence between a null frame in M and a dyad in C2 .

The action of SL(2, C) on C2 leaves invariant a complex area element, a skew
symmetric bispinor. A unique such spinor AB is determined by the normalization
gab = AB ¯A B .
The inverse AB of AB is defined by AB CB = δA C , AB AC = δC B . As with

gab and its inverse g ab , the spin-metric AB and its inverse AB is used to lower and
raise spinor indices,
λB = λA AB , λA = AB λB .
We have
AB = oA ιB − ιA oB .
In particular,
oA ιA = 1. (30)

An element φA···DA ···D of k C2 l C̄2 is called a spinor of valence (k, l). The
space of totally symmetric9 spinors φA···DA ···D = φ(A···D)(A ···D ) is denoted Sk,l .
The spaces Sk,l for k, l non-negative integers yield all irreducible representations of
SL(2, C). In fact, one can decompose any spinor into “irreducible pieces,” i.e., as
a linear combination of totally symmetric spinors in Sk,l with factors of AB . The
above mentioned correspondence between vectors and spinors extends to tensors of
any type, and hence the just mentioned decomposition of spinors into irreducible
pieces carries over to tensors as well. Examples are given by Fab = φAB A B , a
complex anti-self-dual 2-form, and − Cabcd = ABCD A B C D , a complex anti-
self-dual tensor with the symmetries of the Weyl tensor. Here, φAB and ABCD are
symmetric.
8 It is conventional to refer to spin-tensors, e.g., of the form x AA or ψABA simply as spinors.

9 The ordering between primed and unprimed indices is irrelevant.
86 L. Andersson
4.2 Spinors on Spacetime
Let now (M, gab ) be a Lorentzian 3 + 1 dimensional spin manifold with metric of
signature + − −−. The spacetimes we are interested in here are spin, in particular
any orientable, globally hyperbolic 3 + 1 dimensional spacetime is spin, cf. [35,
page 346]. If M is spin, then the orthonormal frame bundle SO(M) admits a lift
to Spin(M), a principal SL(2, C)-bundle. The associated bundle construction now
gives vector bundles over M corresponding to the representations of SL(2, C), in
particular we have bundles of valence (k, l) spinors with sections φA···DA ···D . The
Levi-Civita connection lifts to act on sections of the spinor bundles,
∇AA : ϕB···DB ···D → ∇AA ϕB···DB ···D , (31)
where we have used the tensor-spinor correspondence to replace the index a by

AA . We shall denote the totally symmetric spinor bundles by Sk,l and their spaces
of sections by Sk,l .
The above mentioned correspondence between spinors and tensors, and the
decomposition into irreducible pieces, can be applied to the Riemann curvature ten-
sor. In this case, the irreducible pieces correspond to the scalar curvature, traceless
Ricci tensor, and the Weyl tensor, denoted by R, Sab , and Cabcd , respectively. The
Riemann tensor then takes the form
Rabcd = − 1
12 gad gbc R + 1
12 gac gbd R + 12 gbd Sac − 12 gbc Sad
− 12 gad Sbc + 12 gac Sbd + Cabcd . (32)
The spinor equivalents of these tensors are
¯ A B C D AB CD ,
Cabcd = ABCD ¯A B ¯C D + (33a)
Sab = − 2ABA B , (33b)
R = 24. (33c)
4.3 Fundamental Operators
Projecting (31) on its irreducible pieces gives the following four fundamental
operators, introduced in [4].
Definition 4.1 The differential operators
Dk,l : Sk,l → Sk−1,l−1 , Ck,l : Sk,l → Sk+1,l−1 ,
†
Ck,l : Sk,l → Sk−1,l+1 , Tk,l : Sk,l → Sk+1,l+1
are defined as
(Dk,l ϕ)A1 ...Ak−1 A1 ...Al−1 ≡ ∇ BB ϕA1 ...Ak−1 B A1 ...Al−1 B , (34a)
(Ck,l ϕ)A1 ...Ak+1 A1 ...Al−1 ≡ ∇(A1 B ϕA2 ...Ak+1 ) A1 ...Al−1 B , (34b)

†
(Ck,l ϕ)A1 ...Ak−1 A1 ...Al+1 ≡ ∇ B(A1 ϕA1 ...Ak−1 B A2 ...Al+1 ) , (34c)
(Tk,l ϕ)A1 ...Ak+1 A1 ...Al+1 ≡ ∇(A1 (A1 ϕA2 ...Ak+1 ) A2 ...Al+1 ) . (34d)
The operators are called, respectively, the divergence, curl, curl-dagger, and twistor
operators.
†
As we will see in Sect. 4.4, the kernels of C2s,0 and C0,2s are the massless spin-
s fields. The kernels of Tk,l , are the valence (k, l) Killing spinors, which we will
discuss further in Sects. 4.5 and 4.7. A complete set of commutator properties of
these operators can be found in [4].
4.4 Massless Spin-s Fields
†
For s ∈ 12 N, ϕA···D ∈ ker C2s,0 is a totally symmetric spinor ϕA···D = ϕ(A···D) of
valence (2s, 0) which solves the massless spin-s equation
†
(C2s,0 ϕ)A···BD = 0.
For s = 1/2, this is the Dirac-Weyl equation ∇A A ϕA = 0, for s = 1, we have

the left and right Maxwell equation ∇A B φAB = 0 and ∇A B ϕA B = 0, i.e.,
†
(C2,0 φ)AA = 0, (C0,2 ϕ)AA = 0.
An important example is the Coulomb Maxwell field on Kerr,
2
φAB = − o(A ιB) . (35)
(r − ia cos θ )2
This is a non-trivial sourceless solution of the Maxwell equation on the Kerr

background. We note that the scalars components, see Sect. 4.8 below, of the
Coulomb field φ1 = (r − ia cos θ )−2 while φ0 = φ2 = 0.
For s > 1, the existence of a non-trivial solution to the spin-s equation implies
curvature conditions, a fact known as the Buchdahl constraint [20],
0 = (A DEF φB...C)DEF . (36)
This is easily obtained by commuting the operators in

†
0 = (D2s−1,1 C2s,0 φ)A...C . (37)
88 L. Andersson
For the case s = 2, the equation ∇A D ABCD = 0 is the Bianchi equation, which
holds for the Weyl spinor in any vacuum spacetime. Due to the Buchdahl constraint,
it holds that in any sufficiently general spacetime, a solution of the spin-2 equation
is proportional to the Weyl spinor of the spacetime.
4.5 Killing Spinors
Spinors A1 ···Ak A1 ···Al ∈ Sk,l satisfying
(Tk,l )A1 ···Ak+1 A1 ···Al+1 = 0
are called Killing spinors of valence (k, l). We denote the space of Killing spinors of
valence (k, l) by KSk,l . The Killing spinor equation is an over-determined system.
The space of Killing spinors is a finite dimensional space, and the existence of
Killing spinors imposes strong restrictions on M, see Sect. 4.7 below. Killing spinors
νAA ∈ KS1,1 are simply conformal Killing vector fields, satisfying ∇(a νb) −
2 ∇ νc gab . A Killing spinor κAB ∈ KS2,0 corresponds to a complex anti-selfdual
1 c
conformal Killing-Yano 2-form YABA B = κAB A B satisfying the equation
∇(a Yb)c − 2ζc gab + ζ(a gb)c = 0, (38)
where in the 4-dimensional case, ζa = 13 ∇b Yb a .
4.6 Algebraically Special Spacetimes
Let ϕA···D ∈ Sk,0 . A spinor αA is a principal spinor of ϕA···D if
ϕA···D α A · · · α D = 0.
An application of the fundamental theorem of algebra shows that any ϕA···D ∈ Sk,0
has exactly k principal spinors αA , . . . , δA , and hence is of the form
ϕA···D = α(A · · · δD) .
(i)
If ϕA···D ∈ Sk,0 has n distinct principal spinors αA , repeated mi times, then ϕA···D
is said to have algebraic type {m1 , . . . , mn }. Applying this to the Weyl tensor leads
to the Petrov classification, see Table 1. We have the following list of algebraic, or
Petrov, types.10
10 The Petrov classification is exclusive, so a spacetime belongs at each point to exactly one Petrov
class.
Table 1 The Petrov I {1, 1, 1, 1} ABCD = α(A βB γC δD)

classification
II {2, 1, 1} ABCD = α(A αB γC δD)
D {2, 2} ABCD = α(A αB βC βD)
III {3, 1} ABCD = α(A αB αC βD)
N {4} ABCD = αA αB αC αD
O {−} ABCD =0
II
D III
A principal spinor oA determines a principal null direction la = oA ōA .

The Goldberg-Sachs theorem states that in a vacuum spacetime, the congruence
generated by a null field l a is geodetic and shear free11 if and only if la is a repeated
principal null direction of the Weyl tensor Cabcd (or equivalently oA is a repeated
principal spinor of the Weyl spinor ABCD ).
4.6.1 Petrov Type D
The Kerr metric is of Petrov type D, and many of its important properties follows
from this fact. The vacuum type D spacetimes have been classified by Kinnersley
[43], see also Edgar et al. [29]. The family of Petrov type D spacetimes includes the
Kerr-NUT family and the boost-rotation symmetric C-metrics. The only Petrov type
D vacuum spacetime which is asymptotically flat and has positive mass is the Kerr
metric, see Theorem 5.1 below.
A Petrov type D spacetime has two repeated principal spinors oA , ιA , and
correspondingly there are two repeated principal null directions l a , na , for the Weyl
tensor. We can without loss of generality assume that l a na = 1, and define a null
tetrad by adding complex null vectors ma , m̄a normalized such that ma m̄a = −1.
By the Goldberg-Sachs theorem both l a , na are geodetic and shear free, and only
one of the 5 independent complex Weyl scalars is non-zero, namely
2 = − l a mb m̄d nc Cabcd . (39)
11 If l a is geodetic and shear then the spin coefficients σ, κ, cf. (50) below, satisfy σ = κ = 0.
90 L. Andersson
In this case, the Weyl spinor takes the form
1
ABCD = 2 o(A oB ιC ιD) .
6
See (53) below for the explicit form of 2 in the Kerr spacetime.
The following result is a consequence of the Bianchi identity.
Theorem 4.2 ([70]) Assume (M, gab ) is a vacuum spacetime of Petrov type D.
Then (M, gab ) admits a one-dimensional space of Killing spinors κAB of the form
κAB = −2κ1 o(A ιB) , (40)
−1/3
where oA , ιA are the principal spinors of ABCD and κ1 ∝ 2 .
Remark 4.3 Since the Petrov classes are exclusive, we have that 2 = 0 for a Petrov
type D space.
4.7 Spacetimes Admitting a Killing Spinor
Differentiating the Killing spinor equation (Tk,l φ)A···DA ···D = 0, and commuting
derivatives yields an algebraic relation between the curvature, Killing spinor, and
their covariant derivatives which restrict the curvature spinor, see [4, §2.3], see also
[5, §3.2]. In particular, for a Killing spinor κA···D of valence (k, 0), k ≥ 1, the
condition
(ABC F κD···E)F = 0 (41)
must hold, which restricts the algebraic type of the Weyl spinor. For a valence (2, 0)
Killing spinor κAB , the condition takes the form
(ABC E κD)E = 0. (42)
It follows from (42) that a spacetime admitting a valence (2, 0) Killing spinor is
of type D, N, or O. The space of Killing spinors of valence (2, 0) on Minkowski
space (or any space of Petrov type O) has complex dimension 10. The explicit form
in Cartesian coordinates x AA is
κ AB = U AB + 2x A (A V B) A + x AA x BB WA B ,
where U AB , V B A , W A B are arbitrary constant symmetric spinors, see [1, Eq.

(4.5)]. One of these corresponds to the spinor in (40), in spheroidal coordinates
it takes the form given in (54) below.
A further application of the commutation properties of the fundamental operators

yields that the 1-form
†
ξAA = (C2,0 κ)AA (43)
is a Killing field, ∇(a ξb) = 0, provided M is vacuum. Clearly the real and imaginary
parts of ξa are also Killing fields. If ξa is proportional to a real Killing field,12 we
can without loss of generality assume that ξa is real. In this case, the 2-form
Yab = 32 i(κAB ¯A B − κ̄A B AB ) (44)
is a Killing-Yano tensor, ∇(a Yb)c = 0, and the symmetric 2-tensor
Kab = Ya c Ycb (45)
is a Killing tensor,
∇(a Kbc) = 0. (46)
Further, in this case,
ζa = ξ b Kab (47)
is a Killing field, see [24, 40]. Recall that the quantity Lab γ̇ a γ̇ b is conserved
along null geodesics if Lab is a conformal Killing tensor. For Killing tensors, this
fact extends to all geodesics, so that if Kab is a Killing tensor, then Kab γ̇ a γ̇ b is
conserved along a geodesic γ a . See [5] for further details and references.
4.8 GHP Formalism
Taking the point of view that the null tetrad components of tensors are sections of
complex line bundles with action of the non-vanishing complex scalars correspond-
ing to the rescalings of the tetrad, respecting the normalization, leads to the GHP
formalism [36].
Given a null tetrad l a , na , ma , m̄a we have a spin dyad oA , ιA as discussed above.
For a spinor ϕA···D ∈ Sk,0 , it is convenient to introduce the Newman-Penrose scalars
ϕi = ϕA1 ···Ai Ai+1 ···Ak ιA1 · · · ιAi oAi+1 · · · oAk . (48)
12 We say that such spacetimes are of the generalized Kerr-NUT class, see [12] and the references
therein.
92 L. Andersson
In particular, ABCD corresponds to the five complex Weyl scalars i , i = 0, . . . 4.

The definition ϕi extends in a natural way to the scalar components of spinors of
valence (k, l).
The normalization (30) is left invariant under rescalings oA → λoA , ιA → λ−1 ιA
where λ is a non-vanishing complex scalar field on M. Under such rescalings, the
scalars defined by projecting on the dyad, such as ϕi given by (48) transform as
sections of complex line bundles. A scalar ϕ is said to have type {p, q} if ϕ →
λp λ̄q ϕ under such a rescaling. Such fields are called properly weighted. The lift
of the Levi-Civita connection ∇AA to these bundles gives a covariant derivative
denoted a . Projecting on the null tetrad l a , na , ma , m̄a gives the GHP operators
þ = la a, þ = na a, ð = ma a, ð = m̄a a.
The GHP operators are properly weighted, in the sense that they take properly
weighted fields to properly weighted fields, for example, if ϕ has type {p, q}, then
þϕ has type {p + 1, q + 1}. This can be seen from the fact that l a = oA ōA has
type {1, 1}. There are 12 connection coefficients in a null frame, up to complex
conjugation. Of these, 8 are properly weighted, the GHP spin coefficients. The other
connection coefficients enter in the connection 1-form for the connection a .
The following formal operations take weighted quantities to weighted quantities,
−
(bar) : l a → l a , na → na , ma → m̄a , m̄a → ma , {p, q} → {q, p},
(prime) : l → n , n → l , m → m̄ , m̄ → m ,
a a a a a a a a
{p, q} → {−p, −q},
∗
(star) : l a → ma , na → −m̄a , ma → −l a , m̄a → na , {p, q} → {p, −q}.
(49)
The properly weighted spin coefficients can be represented as
κ = m b l a ∇a l b , σ = m b m a ∇a l b , ρ = mb m̄a ∇a lb , τ = mb na ∇a lb , (50)
together with their primes κ , σ , ρ , τ .

A systematic application of the above formalism allows one to write the tetrad
projection of the geometric field equations in a compact form. For example, the
Maxwell equation corresponds to the four scalar equations given by
(þ − 2ρ)φ1 − (ð − τ )φ0 = −κφ2 , (51)
with its primed and starred versions.

Working in a spacetime of Petrov type D gives drastic simplifications, in view
of the fact that choosing the null tetrad so that l a , na are aligned with principal null
directions of the Weyl tensor (or equivalently choosing the spin dyad so that oA , ιA
are principal spinors of the Weyl spinor), as has already been mentioned, the Weyl
scalars are zero with the exception of 2 , and the only non-zero spin coefficients
are ρ, τ and their primed versions.
5 The Kerr Spacetime
Taking into account the background material given in Sect. 4, we can now state
some further properties of the Kerr spacetime. As mentioned above, the Kerr
metric is algebraically special, of Petrov type D. An explicit principal null tetrad
(l a , na , ma , m̄a ) is given by the Carter tetrad [74]
a(∂φ )a (a 2 + r 2 )(∂t )a 1/2 (∂r )a

la = √ + √ + √ , (52a)
21/2 1/2 21/2 1/2 2 1/2
a(∂φ )a (a 2 + r 2 )(∂t )a 1/2 (∂r )a
na = √ + √ − √ , (52b)
21/2 1/2 21/2 1/2 2 1/2
(∂θ )a i csc θ (∂φ )a ia sin θ (∂t )a
ma = √ + √ + √ . (52c)
2 1/2 2 1/2 2 1/2
In view of the normalization of the tetrad, the metric takes the form gab = 2(l(a nb) −
m(a m̄b) ). We remark that the choice of l a , na to be aligned with the principal null
directions of the Weyl tensor, together with the normalization of the tetrad fixes the
tetrad up to rescalings.
We have
M
2 = − , (53)
(r − ia cos θ )3
κAB = 23 (r − ia cos θ )o(A ιB) . (54)
With κAB as in (54), Eq. (43) yields
ξ a = (∂t )a , (55)
and from (44) we get
Yab = a cos θ l[a nb] − irm[a m̄b] . (56)
With the normalizations above, the Killing tensor (45) takes the form
Kab = 14 (2 l(a nb) − r 2 gab ) (57)
and (47) gives
ζ a = a 2 (∂t )a + a(∂φ )a . (58)
Recall that for a geodesic γ , the quantity k = 4Kab γ̇ a γ̇ b , known as Carter’s

constant, is conserved. Explicitly,
94 L. Andersson
k = γ̇θ2 + a 2 sin2 θe2 + 2aez + a 2 cos2 θμ2 , (59)
where γ̇θ = γ̇ a (∂θ )a . For a = 0, the tensor Kab cannot be expressed as a tensor
product of Killing fields [70], and similarly Carter’s constant k cannot be expressed
in terms of the constants of motion associated to Killing fields. In this sense Kab and
k manifest a hidden symmetry of the Kerr spacetime. As we shall see in Sect. 7, these
structures are also related to symmetry operators and separability properties, as well
as conservation laws, for field equations on Kerr, and more generally in spacetimes
admitting Killing spinors satisfying certain auxiliary conditions.
5.1 Characterizations of Kerr
Consider a vacuum Cauchy data set ( , hij , kij ). We say that ( , hij , kij ) is
asymptotically flat if has an end R3 \ B(0, R) with a coordinate system (x i )
such that
hij = δij + O∞ (r α ), kij = O∞ (r α−1 ) (60)
for some α < −1/2. The Cauchy data set ( , hij , kij ) is asymptotically
Schwarzschildean if

2A α 2xi xj −3/2
hij = − 1 + δij − − δij + o∞ (r ), (61a)
r r r2

β 2xi xj
kij = 2 − δij + o∞ (r −5/2 ), (61b)
r r2
where A is a constant, and α, β are functions on S 2 , see [13, §6.5] for details. Here,
the symbols o∞ (r α ) are defined in terms of weighted Sobolev spaces, see [13, §6.2]
for details.
If (M, gab ) is vacuum and contains a Cauchy surface ( , hij , kij ) satisfying
(60) or (61), then (M, gab ) is asymptotically flat, respectively, asymptotically
Schwarzschildean, at spatial infinity. In this case there is a spacetime coordinate
system (x α ) such that gαβ is asymptotic to the Minkowski line element with
asymptotic conditions compatible with (61). For such spacetimes, the ADM 4-
momentum P μ is well defined. The positive mass theorem states that P μ is future
directed causal P μ Pμ ≥ 0 (where the contraction is in the asymptotic Minkowski
line element), P 0 ≥ 0, and gives conditions under which P μ is strictly timelike.
This holds in particular if contains an apparent horizon.
Mars [52] has given a characterization of the Kerr spacetime as an asymptotically
flat vacuum spacetime with a Killing field ξ a asymptotic to a time translation,
positive mass, and an additional condition on the Killing form FAB = (C1,1 ξ )AB ,
ABCD F CD ∝ FAB .
A characterization in terms of algebraic invariants of the Weyl tensor has been given
by Ferrando and Saez [30]. The just mentioned characterizations are in terms of
spacetime quantities. It can be shown that Killing spinor initial data propagates,
which can be used to formulate a characterization of Kerr in terms of Cauchy data.
See [12–15].
We here give a characterization in terms spacetimes admitting a Killing spinor of
valence (2, 0).
Theorem 5.1 Assume that (M, gab ) is vacuum, asymptotically Schwarzschildean
at spacelike infinity, and contains a Cauchy slice bounded by an apparent horizon.
Assume further (M, gab ) admits a non-vanishing Killing spinor κAB of valence
(2, 0). Then (M, gab ) is locally isometric to the Kerr spacetime.
Proof Let P μ be the ADM 4-momentum vector for M. By the positive mass
theorem, P μ Pμ ≥ 0. In the case where M contains a Cauchy surface bounded
by an apparent horizon, then P μ Pμ > 0 by [16, Remark 11.5].13
Recall that a spacetime with a Killing spinor of valence (2, 0) is of Petrov type
D, N, or O. From asymptotic flatness and the positive mass theorem, we have
Cabcd C abcd = O(1/r 6 ), and hence there is a neighborhood of spatial infinity where
M is Petrov type D. It follows that near spatial infinity, κAB = −2κ1 o(A ιB) , with
−1/3
κ1 ∝ 2 = O(r). It follows from our asymptotic conditions that the Killing
†
field ξAA = (C2,0 κ)AB is O(1) and hence asymptotic to a translation, ξ μ → Aμ
as r → ∞, for some constant vector Aμ . It follows from the discussion in [2, §4]
that Aμ is non-vanishing. Now, by [17, §III], it follows that in the case P μ Pμ > 0,
then Aμ is proportional to P μ , see also [18]. We are now in the situation considered
in the work by Bäckdahl and Valiente-Kroon, see [14, Theorem B.3], and hence we
can conclude that (M, gab ) is locally isometric to the Kerr spacetime.
Remark 5.1
1. This result can be turned into a characterization in terms of Cauchy data along
the lines in [13].
2. Theorem 5.1 can be viewed as a variation on the Kerr characterization given in
[14, Theorem B.3]. In the version given here, the asymptotic conditions on the
Killing spinor have been removed.
13 Section 11 appears only in the ArXiv version of [16].

96 L. Andersson
6 Monotonicity and Dispersion
The dispersive properties of fields, i.e., the tendency of the energy density contained
within any stationary region to decrease asymptotically to the future is a crucial
property for solutions of field equations on spacetimes, and any proof of stability
must exploit this phenomenon. In view of the geometric optics approximation, the
dispersive property of fields can be seen in an analogous dispersive property of null
geodesics, i.e., the fact that null geodesics in the Kerr spacetime which do not orbit
the black hole at a fixed radius must leave any stationary region in at least one of
the past or future directions. In Sect. 6.1 we give an explanation for this fact using
tools which can readily be adapted to the case of field equations, while in Sect. 6.2
we outline sketch how these ideas apply to fields.
We begin by a discussion of conservation laws. For a null geodesic γ a , we define
the energy associated with a vector field X and evaluated on a Cauchy hypersurface
to be
eX [γ ]( ) = gab Xa γ̇ b | .
Since γ̇ b ∇b γ̇ a = 0 for a geodesic, integrating the derivative of the energy gives

λ2
eX [γ ]( 2 ) − eX [γ ]( 1) = (γ̇a γ̇b )∇ (a Xb) dλ, (62)
λ1
where λi is the unique value of λ such that γ (λ) is the intersection of γ with i.
Formula (62) is particularly easy to work with, if one recalls that
1
∇ (a Xb) = − LX g ab .
2
The tensor ∇ (a Xb) is commonly called the “deformation tensor.” In the following,
unless there is room for confusion, we will drop reference to γ and in referring
to eX .
Conserved quantities play a crucial role in understanding the behavior of
geodesics as well as fields. By (62), the energy eX is conserved if X a is a Killing
field. In the Kerr spacetime we have the Killing fields ξ a = (∂t )a , ηa = (∂φ )a with
the corresponding conserved quantities energy e = (∂t )a γ̇a and azimuthal angular
momentum z = (∂φ )a γ̇a . In addition, the squared particle mass μ = gab γ̇ a γ̇ b , and
the Carter constant k = Kab γ̇ a γ̇ b are conserved along any geodesic γ a in the Kerr
spacetime. The presence of the extra conserved quantity allows one to integrate the
equations of geodesic motion.14
For a covariant field equation derived from an action principle which depends
on the background geometry only via the metric and its derivatives, the symmetric
stress-energy tensor Tab is conserved. As an example, we consider the wave
equation
∇ a ∇a ψ = 0 (63)
which has stress-energy tensor
Tab = ∇(a ψ∇b) ψ̄ − 12 ∇ c ψ∇c ψ̄gab . (64)
Let ψ be a solution to (63). Then Tab is conserved, ∇ a Tab = 0. For a vector field
Xa we have that ∇ a (Tab Xb ) is given in terms of the deformation tensor,
∇ a (Tab Xb ) = Tab ∇ (a Xb) .
Let (JX )a = Tab Xb be the current corresponding to X a . By the above, we have

conserved currents Jξ and Jη corresponding to the Killing fields ξ a , ηa .
An application of Gauss’ law gives the analog of (62),

(JX )a dσ a − (JX )a dσ a = Tab ∇ (a Xb) ,
2 1
where is a spacetime region bounded by 1, 2.
6.1 Monotonicity for Null Geodesics
We shall consider only null geodesics, i.e., μ = 0. In this case we have
k = Kab γ̇ a γ̇ b
= 2 l(a nb) γ̇ a γ̇ b
= 2 m(a m̄b) γ̇ a γ̇ b . (65)
14 In general, the geodesic equation in a 4-dimensional stationary and axisymmetric spacetime

cannot be integrated, and the dynamics of particles may in fact be chaotic, see [34, 50] and the
references therein. Note, however, that the geodesic equations are not separable in the Boyer-
Lindquist coordinates. On the other hand, the Darboux coordinates have this property, cf. [33].
98 L. Andersson
We note that the tensors 2 l(a nb) and 2 m(a m̄b) are conformal Killing tensors, see
Sect. 4.5. From (65) it is clear that k is non-negative. A calculation using (52) gives
1 2
2 l (a nb) ∂a ∂b = [(r + a 2 )∂t + a∂φ ]2 − ∂r2

1
2 m(a m̄b) ∂a ∂b = ∂θ2 + ∂φ2 + a 2 sin2 θ ∂t2 + 2a∂t ∂φ .
sin2 θ
Let Z = (r 2 + a 2 )e + az . Recall that ṙ = γ̇ r = g rr γ̇r where g rr = −/ . Now

we can write 0 = gab γ̇ a γ̇ b in the form
2 2
ṙ + R(r; e, z , k) = 0, (66)
where
R = −Z 2 + k. (67)
Equation (66) allows one to make a qualitative analysis of the motion of null
geodesics in the Kerr spacetime. In particular, we find that the location of orbiting
null geodesics is determined by R = 0, ∂r R = 0. Due to the form of R, the location
of orbiting null geodesics depends only on the ratios k/z 2 , e/z . One finds that
orbiting null geodesics exist for a range of radii r1 ≤ r ≤ r2 , with r+ < r1 <
3M < r2 . Here r1 , r2 depend on a, M and as |a| M, r1 r+ , and r2 4M.
The orbits at r1 , r2 are restricted to the equatorial plane, those at r1 are corotating,
while those at r2 are counterrotating. For r1 < r < r2 , the range of θ depends on r.
There is r3 = r3 (a, M), r1 < r3 < r2 such that the orbits at r3 reach the poles, i.e.,
θ = 0, θ = π, see Fig. 6. For such geodesics, it holds that z = 0. Examples of null
geodesics with non zero z are shown in Fig. 7.
For the following discussion, it is convenient to introduce
(a) (b)
Fig. 6 The Kerr photon region. In (a), |a| M and the ergoregion, see Sect. 3.3, is well separated
from the photon region (bordered in black). The radius r3 where geodesics reach the poles is
indicated by a grey, dashed line. In (b), |a| is close to M and the ergoregion overlaps the photon
region
Fig. 7 Examples of orbiting

null geodesics in Kerr with
a = M/2. In (a), the k/z 2 is
small, while in (b), this
constant is larger
(a) (b)
q = k − 2aez − z 2 = Qab γ̇a γ̇b ,
where
cos2 θ
Qab = (∂θ )a (∂θ )b + (∂φ )a (∂φ )b + a 2 sin2 θ (∂t )a (∂t )b . (68)
sin2 θ
By construction, q is a sum of conserved quantities and is, therefore, conserved.
Further, it is non-negative, since it is a sum of non-negative terms. In the following
we use (e, z , q) as parameters for null geodesics. Since we are considering only
null geodesics, there is no loss of generality compared to using (e, z , k) as
parameters.
For a null geodesic with given parameters (e, z , q), a simple turning point
analysis shows that there is a number ro ∈ (r+ , ∞) so that the quantity (r − ro )γ̇ r
increases overall. This quantity corresponds to the energy eA for the vector field
A = −(r − ro )∂r . Following this idea, we may now look for a function F which
will play the role of −(r − ro ), so that for A = F∂r , the energy eA is non-decreasing
for all λ and not merely non-decreasing overall. For a = 0, both ro and F will
necessarily depend on both the Kerr parameters (M, a) and the constants of motion
(e, z , q); the function F will also depend on r, but no other variables.
We define Aa = F(∂r )a with
F = F(r; M, a, e, z , q).
It is important to note that this is a map from the tangent bundle to the tangent
bundle, and hence Aa = F(∂r )a cannot be viewed as a standard vector field, which
is a map from the manifold to the tangent bundle.
To derive a monotonicity formula, we wish to choose F so that eA has a non-
negative derivative. We define the covariant derivative of A by holding the values of
(e, z , q) fixed and computing the covariant derivative as if A were a regular vector
field. Similarly, we define LA g ab by fixing the values of the constants of geodesic
motion. Since the constants of motion have zero derivative along null geodesics,
Eq. (62) remains valid.
Recall that null geodesics are conformally invariant up to reparameterization.
Hence, it is sufficient to work with the conformally rescaled metric g ab . Fur-
100 L. Andersson
thermore, since γ is a null geodesic, for any function qreduced , we may subtract
qreduced g ab γ̇a γ̇a wherever it is convenient. Thus, the change in eA is given as the
integral of

1
γ̇a γ̇b ∇ (a Ab) = − LA ( g ab ) − qreduced g ab γ̇a γ̇b .
2
The Kerr metric can be written as

1 ab
g ab = −(∂r )a (∂r )b − R , (69)

where the tensorial form of Rab can be read off from the earlier definitions. We now
calculate −LA g ab γ̇a γ̇b using (69). Ignoring distracting factors of , , the most
important terms are
−2(∂r F)γ̇r γ̇r + F(∂r Rab )γ̇a γ̇b = −2(∂r F)γ̇r γ̇r + F(∂r R).
The second term in this sum will be non-negative if F = ∂r R(r; M, a; e, z , q).

Recall that the vanishing of ∂r R(r; M, a; e, z , q) is one of the two conditions for
orbiting null geodesics. With this choice of F, the instability of the null geodesic
orbits ensures that, for these null geodesics, the coefficient in the first term, −2(∂r F),
will be positive. These observations motivate the form of F which yields non-
negativity for all null geodesics.
It remains to make explicit choices of F and qreduced . Once these choices are
made, the necessary calculations are straight-forward but rather lengthy. Let z
and w be smooth functions of r and the Kerr parameters (M, a). Let R̃ denote
∂r ( z R(r; M, a; e, z , q)) and choose F = zw R̃ and qreduced = (1/2)(∂r z)w R̃ . In
terms of these functions,

z1/2
γ̇a γ̇b ∇ A(a b)
= 2 w(R̃ )
1 2
−z 1/2
3/2
∂r w 1/2 R̃ γ̇r2 . (70)

If z and w are chosen to be positive, then the first term on the right-hand side of (70)
which contains a square (R̃ )2 is non-negative. If we now take z = z1 = (r 2 +
a 2 )−2 and w = w1 = (r 2 + a 2 )4 /(3r 2 − a 2 ), then15

z1/2 3r 4 + a 4 3r 4 − 6a 2 r 2 − a 4
−∂r w 1/2 R̃ = 2 2 z 2 + 2 q. (71)
(3r − a )2 2 (3r 2 − a 2 )2
The coefficient of q is positive for r > r+ when |a| < 31/4 2−1/2 M ∼
= 0.93M. Since
q is non-negative, the right-hand side of (71) is non-negative, and hence also the
15 Equation (71) corrects a misprint in [8, Eq. (1.15b)].

right-hand side of Eq. (70) is non-negative, for this range of a. Since Eq. (70) gives
the rate of change, the energy eA is monotone.
These calculations reveal useful information about the geodesic motion. The
positivity of the term on the right-hand side of (71) shows that R̃ can have at most
one root, which must be simple. In turn, this shows that R can have at most two
roots. For orbiting null geodesics R must have a double root, which must coincide
with the root of R̃ . It is convenient to think of the corresponding value of r as
being ro .
The first term in (70) vanishes at the root of R̃ , as it must so that eA can be
constantly zero on the orbiting null geodesics. When a = 0, the quantity R̃ reduces
to −2(r − 3M)r −4 (z 2 + q), so that the orbits occur at r = 3M. The continuity in
a of R̃ guarantees that its root converges to 3M as a → 0 for fixed (e, z , q).
From the geometrics optics approximation, it is natural to imagine that the
monotone quantity constructed in this section for null geodesics might imply the
existence of monotone quantities for fields, which would imply some form of
dispersion. For the wave equation, this is true. In fact, the above discussion, when
carried over to the case of the wave equation, closely parallels the proof of the
Morawetz estimate for the wave equation given in [8], see Sect. 6.2 below. The
quantity (γ̇α γ̇β )(∇ (α Xβ) ) corresponds to the Morawetz density, i.e., the divergence
of the momentum corresponding to the Morawetz vector field. The role of the
conserved quantities (e, z , q) for geodesics is played, in the case of fields, by
the energy fluxes defined via second order symmetry operators corresponding to
these conserved quantities. The fact that the quantity R vanishes quadratically on
the trapped orbits is reflected in the Morawetz estimate for fields, by a quadratic
degeneracy of the Morawetz density at the trapped orbits.
6.2 Dispersive Estimates for Fields
As discussed in Sect. 6.1, one may construct a suitable function of the conserved
quantities for null geodesics in the Kerr spacetime which is monotone along the
geodesic flow. This function may be viewed as arising from a generalized vector
field on phase space. The monotonicity property implies, as discussed there, that
non-trapped null geodesics disperse, in the sense that they leave any stationary
region in the Kerr space time. As mentioned in Sect. 6.1, in view of the geometric
optics approximation for the wave equation, such a monotonicity property for null
geodesics reflects the tendency for waves in the Kerr spacetime to disperse.
At the level of the wave equation, the analog of the just mentioned monotonicity
estimate is called the Morawetz estimate. For the wave equation ∇ a ∇a ψ = 0, a
Morawetz estimate provides a current Ja defined in terms of ψ and some of its
derivatives, with the property that ∇ a Ja has suitable positivity properties, and that
the flux of Ja can be controlled by a suitable energy defined in terms of the field.
Let ψ be a solution of the wave equation ∇ a ∇a ψ = 0. Define the current Ja by
102 L. Andersson
Ja = Tab Ab + 12 q(ψ̄∇a ψ + ψ∇a ψ̄) − 12 (∇a q)ψ ψ̄,
where Tab is the stress-energy tensor given by (64). We have
∇ a Ja = Tab ∇ (a Ab) + q∇ c ψ∇c ψ̄ − 12 (∇ c ∇c q)ψ ψ̄. (72)
We now specialize to Minkowski space, with the line element gab dx a dx b = dt 2 −

dr 2 − dθ 2 − r 2 sin2 θ dφ 2 . Let

E(τ ) = Ttt d 3 x
{t=τ }
be the energy of the field at time τ , where Ttt is the energy density. The energy is
conserved, so that E(t) is independent of t.
Setting Aa = r(∂r )a , we have
∇ (a Ab) = g ab − (∂t )a (∂t )b . (73)
With q = 1, we get
∇ a Ja = −Ttt .
With the above choices, the bulk term ∇ a Ja has a sign. This method can be used
to prove dispersion for solutions of the wave equation. In particular, by introducing
suitable cutoffs, one finds that for any R0 > 0, there is a constant C, so that
t1
Ttt d 3 xdt ≤ C(E(t0 ) + E(t1 )) ≤ 2CE(t0 ), (74)
t0 |r|≤R0
see [55]. The local energy, |r|≤R0 Ttt d 3 x, is a function of time. By (74) it is
integrable in t, and hence it must decay to zero as t → ∞, at least sequentially.
This shows that the field disperses. Estimates of this type are called Morawetz or
integrated local energy decay estimates.
†
For a solution φAB of the Maxwell equation (C2,0 φ)AA = 0, the stress-energy
tensor Tab given by
Tab = φAB φ̄A B
is conserved, ∇ a Tab = 0. Further, Tab has trace zero, with T a a = 0.

Restricting to Minkowski space and setting Ja = Tab Ab , with Aa = r(∂r )a we
have
∇ a Ja = −Ttt
which again gives local energy decay for the Maxwell field on Minkowski space.
For the wave equation on Schwarzschild we can choose
(r − 3M)(r − 2M)
Aa = (∂r )a , (75a)
3r 2
6M 2 − 7Mr + 2r 2
q= . (75b)
6r 3
This gives
Mg ab (r − 3M) M(r − 2M)2 (∂r )a (∂r )b

−∇ (a Ab) = − +
3r 3 r4
(r − 3M)2 ((∂θ )a (∂θ )b + csc2 θ (∂φ )a (∂φ )b )
+ , (76a)
3r 5

M|∂r ψ|2 (r − 2M)2 |∂θ ψ|2 + |∂φ ψ|2 csc2 θ (r − 3M)2
−∇a J =
a
+
r4 3r 5
M|ψ|2 (54M 2 − 46Mr + 9r 2 )
+ . (76b)
6r 6
Here, Aa was chosen so that the last two terms (76a) have good signs. The form of
q given here was chosen to eliminate the |∂t ψ|2 term in (76b). The first terms in
(76b) are clearly non-negative, while the last is of lower-order and can be estimated
using a Hardy estimate [8]. The effect of trapping in Schwarzschild at r = 3M is
manifested in the fact that the angular derivative term vanishes at r = 3M.
In the case of the wave equation on Kerr, the above argument using a classical
vector field cannot work due to the complicated structure of the trapping. However,
making use of higher-order currents constructed using second order symmetry
operators for the wave equation, and a generalized Morawetz vector field analogous
to the vector field Aa as discussed in Sect. 6.1. This approach has been carried out
in detail in [8].
If we apply the same idea for the Maxwell field on Schwarzschild, there is no
reason to expect that local energy decay should hold, in view of the fact that the
Coulomb solution is a time-independent solution of the Maxwell equation which
does not disperse. In fact, with
2M
Aa = F(r) 1 − (∂r )a , (77)
r
we have
−Tab ∇ (a Ab) = − φ AB φ̄ A B (T1,1 A)ABA B (78)

(r − 2M)
= |φ0 |2 + |φ2 |2 F (r)
2r
104 L. Andersson

|φ1 |2 r(r − 2M)F (r) − 2F(r)(r − 3M)
− . (79)
r2
If F is chosen to be positive, then the coefficient of the extreme components in

(79) is positive. However, at r = 3M, the coefficient of the middle component is
necessarily of the opposite sign. It is possible to show that no choice of F will give
positive coefficients for all components in (79).
The dominant energy condition that Tab V a W b ≥ 0 for all causal vectors V a , W a
is a common and important condition on stress-energy tensors. In Riemannian
geometry, a natural condition on a symmetric 2-tensor Tab would be non-negativity,
i.e., the condition that for all X a , one has Tab Xa Xb ≥ 0.
However, in order to prove dispersive estimates for null geodesics and the wave
equation, the dominant energy condition on its own is not sufficient and non-
negativity cannot be expected for stress-energy tensors. Instead, a useful condition
to consider is non-negativity modulo trace terms, i.e., the condition that for every
Xa there is a q such that Tab Xa Xb + qT a a ≥ 0. For null geodesics and the wave
equation, the tensors γ̇a γ̇b and ∇a u∇b u = Tab + T γ γ gab are both non-negative, so
γ̇a γ̇b and Tab are non-negative modulo trace terms.
From Eq. (76a), we see that −∇ (a Ab) is of the form f1 g ab + f2 ∂ra ∂rb + f3 ∂θa ∂θb +
f4 ∂φa ∂φb where f2 , f3 and f4 are non-negative functions. That is −∇ (a Ab) is a sum of
a multiple of the metric plus a sum of terms of the form of a non-negative coefficient
times a vector tensored with itself. Thus, from the non-negativity modulo trace
terms, for null geodesics and the wave equation, respectively, there are functions
q such that γ̇a γ̇b ∇ a Ab = γ̇a γ̇b ∇ a Ab + qg ab γ̇a γ̇b ≤ 0 and Tab ∇ a Ab + qT a a ≤ 0.
For null geodesics, since g ab γ̇a γ̇b = 0, the q term can be ignored. For the wave
equation, one can use the terms involving q in Eqs. (72), to cancel the T a a term in
∇ a Ja . For the wave equation, this gives non-negativity for the first-order terms in
−∇ a Ja , and one can then hope to use a Hardy estimate to control the zeroth order
terms.
If we now consider the Maxwell equation, we have the fact that the Maxwell
stress-energy tensor is traceless, T a a = 0 and does not satisfy the non-negativity
condition. Therefore, it also does not satisfy the condition of non-negativity modulo
trace. This appears to be the fundamental underlying obstruction to proving a
Morawetz estimate using Tab . This can be seen as a manifestation of the fact that
the Coulomb solution does not disperse.
In fact, it is immediately clear that the Maxwell stress energy cannot be used
directly to prove dispersive estimates since it does not vanish for the Coulomb
field (35) on the Kerr spacetime. We remark that the existence of the Coulomb
solution on the Kerr spacetime is a consequence of the fact that the exterior of the
black hole contains non-trivial 2-spheres, and the existence of two conserved charge
integrals S Fab dσ ab , S (∗F )ab dσ ab . Hence this is valid also for dynamical black
hole spacetimes.
7 Symmetry Operators
A symmetry operator for a field equation is an operator which takes solutions

to solutions. In order to analyze higher spin fields on the Kerr spacetime, it is
important to gain an understanding of the symmetry operators for this case. In the
paper [4] we have given a complete characterization of those spacetimes admitting
symmetry operators of second order for the field equations of spins 0, 1/2, 1,
i.e., the conformal wave equation, the Dirac-Weyl equation and the Maxwell
equation, respectively, and given the general form of the symmetry operators, up
to equivalence. In order to simplify the presentation here, we shall discuss only the
spin-1 case, and restrict to spacetimes admitting a valence (2, 0) Killing spinor κAB .
We first give some background on the wave equation.
7.1 Symmetry Operators for the Kerr Wave Equation
As shown by Carter [21], if Kab is a Killing tensor in a Ricci flat spacetime, the
operator
K = ∇a K ab ∇b (80)
is a commuting symmetry operator for the d’Alembertian,
[∇ a ∇a , K] = 0.
In particular there is a second order symmetry operator for the wave equation, i.e.,
an operator which maps solutions to solutions,
∇ a ∇a ψ = 0 ⇒ ∇ a ∇a Kψ = 0.
Due to the form of the Carter Killing tensor, Kab , cf. (57), the operator K defined
by (80) contains derivatives with respect to all coordinates.
√
Recall that ∇ a ∇a = μ1g ∂a μg g ab ∂b , where μg = det(gab ) is the volume
element. For Kerr in Boyer-Lindquist coordinates, we have from (25) that μg = µ,
with µ = sin θ . After rescaling the d’Alembertian by , and using the just
mentioned facts, one finds
106 L. Andersson
R(r; ∂t , ∂φ , Q)
∇ a ∇a = − ∂r ∂r + , (81)

where
1
Q= ∂a µQab ∂b . (82)
µ
In view of the form of Qab given in (68), we see that Q contains derivatives only
with respect to θ, φ, t, but not with respect to r. Thus, it is clear from (81) that Q is
a commuting symmetry operator for the rescaled d’Alembertian ∇ a ∇a ,
[ ∇ a ∇a , Q] = 0.
In addition to the symmetry operator Q related to the Carter constant, we have

the second order symmetry operators generated by the Killing fields ξ a ∇a = ∂t ,
ηa ∇a = ∂φ . The operator Q can be termed a hidden symmetry, since it cannot be
represented in terms of operators generated by the Killing fields.
The above shows that we can write
∇ a ∇a = R + S,
where the operators R, S commute, [R, S] = 0, and R contains derivatives with

respect to the non-symmetry coordinate r, and the two symmetry coordinates t, φ,
while S contains derivatives with respect to the non-symmetry coordinate θ, and
with respect to t, φ.
By making a separated ansatz
ψω,",m (t, r, θ, φ) = e−iωt eimφ Rω,",m (r)Sω,",m (θ )
the equation ∇ a ∇a ψ = 0 becomes a pair of scalar ordinary differential equations
RR + λR = 0 (83a)
SS = λS, (83b)
where λ = λω,",m . Here it should be noted that Eq. (83b) is to be considered as

a boundary value problem on [0, π ] with boundary conditions determined by the
requirement that φ be smooth. In the Schwarzschild case a = 0, we can take
S =, the angular Laplacian. The eigenfunctions of are the spherical harmonics
Y",m (θ, φ) = eimφ Y" (θ ). The eigenvalues of
are λ",m = −"(" + 1).
The solutions to the eigenproblem SS = λS are the spheroidal harmonics, the
eigenvalues in this case are not known in closed form, depend on the time frequency
ω, and are indexed by ", m. For real ω, it is known that the eigensystem is complete,
but for general ω this is not known.
One may now apply a Fourier transform and represent a typical solution ψ to the
wave equation in the form

ψ= dω e−iωt eimφ Rω,",m Sω,",m ,
",m
analyze the behavior of the separated modes ψω,",m , and recover estimates for ψ
after inverting the Fourier transform. In order to do this, one must show a priori
that the Fourier transform can be applied. This can be done by applying cutoffs,
and removing these after estimates have been proved using Fourier techniques.
This approach has been followed in, e.g., [9, 10, 27]. In recent work by Dafermos,
Rodnianski and Shlapentokh-Rothman, see [28], proving boundedness and decay
for the wave equation on Kerr for the whole range |a| < M, makes use of the
technical condition of time integrability, i.e., that the solution to the wave equation
and its derivatives to a sufficiently high order is bounded in L2 on time lines,
∞
dt|∂ α ψ(t, r, θ, φ)|.
−∞
This condition is consistent with integrated local energy decay and is removed at
the end of the argument.
However, by working directly with currents defined in terms of second order
symmetry operators, one may prove a Morawetz estimate directly for the wave
equation on the Kerr spacetime. This was carried out for the case |a| M in
[8]. This involves introducing a generalization of the vector field method to allow
for currents defined in terms of generalized, operator valued, vector fields. These
are operator analogs of the generalized vector field Aa introduced in Sect. 6.1.
Fundamental for either of the above mentioned approaches is that the analysis
of the wave equation on the Kerr spacetime is based on the hidden symmetry
manifested in the existence of the Carter constant, or the conserved quantity q, and
its corresponding symmetry operator Q.
8 Outlook
I will end by briefly discuss some further developments and give some references.
Analysis on black hole spacetimes, and the black hole stability problem is the sub-
ject of intense work. Here I will give some brief, and partial remarks and references
on the current (Sept 2021) state of the problem. Recall that the subextreme range
of Kerr is |a|/M < 1, while slowly rotating Kerr black holes are characterized by
|a|/M 1.
108 L. Andersson
8.1 Teukolsky
Linearized gravity on Petrov type D spacetimes, and Kerr in particular, is governed

by the Teukolsky equation, a spin- and boost-weighted wave equation for suitably
rescaled linearized extreme Weyl scalars. The Teukolsky Master Equation (TME) is
admits a symmetry operator generalizing the symmetry operator Q discussed above,
and as a consequence is separable. A generalized Morawetz vector field for the
Teukolsky equation on slowly rotating Kerr spacetimes can be constructed following
the approach of [8], cf. [51].
Similar results can be achieved using Fourier techniques, cf. [28] for the spin-
0 case for subextreme Kerr. For the case of non-zero spin, there is currently16 no
complete proof of a Morawetz estimate or decay for the TME for the full subextreme
range, see, however, [66].
For the Fourier based approaches to estimates for Teukolsky, mode stability, i.e.,
the absence of mode solutions is an essential step. By mode solutions is meant
solutions of the radial Teukolsky equation with boundary conditions at the horizon
and infinity that imply that no radiation is entering the black hole exterior, either
across the horizon of from infinity [11, 65, 71].
8.2 Stability for Linearized Gravity on Kerr
Stability for linearized gravity on slowly rotating Kerr backgrounds is known [6, 37].
The approach in [6] relies on the outgoing radiation gauge (ORG) condition, adapted
to one of the principal null directions of the Kerr geometry. In addition to the gauge
invariant Teukolsky equation, the linearized Einstein equations in ORG imply a
transport system. Provided a Morawetz estimate for the TME is given, as is provided
by [51], improved decay estimates are shown using a hierarchy of equations derived
by commuting the TME with suitable vector fields. The Teukolsky-Starobinsky
Identities are used to get improved control near null infinity. Transport estimates
are then used to close the estimates. The approach in [37], on the other hand, relies
on treating the linearized Einstein equation on the Kerr spacetime as a perturbation
of the Schwarzschild case.
8.3 Nonlinear Stability for Schwarzschild
As discussed in Sect. 2.7, angular momentum is quasilocally conserved on an

axisymmetric spacetime. In particular, the black hole stability conjecture for this
16 Sept. 2021.
case states that Cauchy data that are axially symmetric with vanishing angu-
lar momentum and close to Schwarzschild must evolve to a member of the
Schwarzschild family. This has been proved for the polarized case [47]. See also
[26] for related results.
8.4 Nonlinear Stability for Kerr
The black hole stability conjecture is open. An approach to this problem has been
presented by Klainerman and Szeftel, cf. [48] and the references therein. A complete
proof is not available as of this writing.17
The nonlinear version of the outgoing radiation gauge for deformations of Kerr
has been introduced [7]. The reduced Einstein equation can be written in first-order
symmetric hyperbolic form and is hence locally well-posed. The system implies a
nonlinear Teukolsky equation as well as a transport system, which is analogous to
the one used in [6].
Acknowledgments I am grateful for the organizers for the invitation to lecture at the 2019
Domodossola summer school and for their hospitality during this enjoyable event. I thank Steffen
Aksteiner, Thomas Bäckdahl, Pieter Blue, Siyuan Ma, Marc Mars, and Claudio Paganini for helpful
remarks.
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Study of Fundamental Laws
with Antimatter
Marco Giammarchi
Mathematics Subject Classification (2000) Primary 81V99; Secondary 81V72
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
2 Testing CPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3 Testing the Weak Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4 Anti-hydrogen at CERN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.1 Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2 Anti-hydrogen Beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.3 The Beginning of Anti-hydrogen Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5 Positrons and Positronium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6 The Mu-Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
1 Introduction
Antimatter entered the scene of Modern Physics with the Dirac equation [1] and
the discovery of the positron in 1932 [2], offering a new view of particles (and the
Universe) as well as an interesting ground to test physical laws at a fundamental
level. The question naturally arose whether particles and antiparticles behaved in
identical ways with respect to the laws of Physics.
The theoretical basis for testing particle–antiparticle asymmetries was set by
the CPT theorem, devised in 1957 in the frame of Lagrangian quantum field
theory in a flat spacetime [3]. This theorem relates (at the quantum level) the
properties of particle and antiparticles, by explicitly constructing the relevant CPT
operators in the Lagrangian formalism: for any spin 0, 1/2, and 1 field a particle
M. Giammarchi ()
Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Milano, Italy
e-mail: marco.giammarchi@mi.infn.it
114 M. Giammarchi
and its antiparticle would have the same mass and lifetimes and opposite electric
charges (and magnetic moments). In addition, the Weak Equivalence Principle
(WEP) of General Relativity (given its validity for all forms of energies) relates
the (gravitational) mass of particles and antiparticles at the classical (non-quantum)
level. In its most general form, including all forms of self-energies, this is called the
Einstein Equivalence Principle.
From the observational viewpoint, it is important to note that the effect of a
CPT violation and a violation of the WEP can concur in creating an experimental
evidence of different behavior of a particle and its antiparticle. Let us show this with
an elementary example, assuming that CPT could be applied to the macroscopic
level and using Newton’s second law. In this case, CPT invariance would lead to the
equality of inertial masses:
mi a = mi a (1)
between a particle and its antiparticle. While the separate validity of the WEP for a
particle and an antiparticle would imply
mi = m g (2)
mi = mg (3)
Supposing then that a particle/antiparticle different behavior is observed, say, in the

Earth’s gravitational field, this could mean either a breakdown of CPT invariance,
through the chain:
mg = mi = mi = mg (4)
or a violation of the WEP for antimatter because of
mg = mi = mi = mg (5)
It is important to note that any level of CPT violation would necessarily imply
Lorentz violation (see [4]). However, in this chapter, I will only focus on CPT and
WEP measurements through the behavior of low-energy (T ≤ MeV) antimatter. To
this goal, a variety of fundamental tests can be done or are being considered by
making use of the following neutral systems:
• Anti-hydrogen, the bound state of an antiproton and a positron
• Positronium, the (unstable) bound state of electron and positron
• Mu-atom, the (unstable) bound state of a muon and an electron
In this work, I will shortly summarize the developments in the test for fundamen-
tal laws, using these systems.
Study of Fundamental Laws with Antimatter 115
2 Testing CPT
Tests of CPT can be done in several ways: they are generally based on the fact that
the Standard Model satisfies CPT symmetry and contains the C operator relating a
particle to its own antiparticle. Therefore, in any quantum relativistic system, some
properties of an antiparticle can be deduced by the ones of the particle by means of
the CPT symmetry [5].
As already mentioned above, under mild general assumptions, CPT symmetry
implies the equality of charges, masses, and lifetimes of a particle and its antipar-
ticle. Moreover, in any antimatter bound system (like anti-hydrogen), the transition
frequencies should be the same as for the matter system, the hydrogen atom.
Tests of CPT have been made in various configurations and are summarized in
Fig. 1. Many of those measurements refer to the charge and mass of the particle and
the antiparticle as measured in Penning trap systems: the electron [6, 7], the proton
[8], and the antiproton [9].
For what concern the electron–positron mass difference, the most stringent limit
comes from cosmology and makes use of the null mass of the photon [10, 11], while
Fig. 1 Several CPT tests are compared. Bar’s right-hand side: measured quantity. Length of bar:
relative precision of the test. Let-hand side of the bar: sensitivity on an absolute energy scale.
Blue: performed measurements. Orange: predicted sensitivity if existing precision on hydrogen is
reached for anti-hydrogen. In the case of the Hyperfine Splitting (HFS) and the Lamb shift, the
color refers to different future stages of experimental development
116 M. Giammarchi
in laboratory the obtained relative limit is at the 10−9 level [12]. On the other hand,
the most stringent limit on the antiproton/proton mass ratio comes from antiprotonic
helium [13] and the antiproton charge-to-mass ratio [14, 19].
3 Testing the Weak Equivalence Principle
The weak equivalence principle is a cornerstone of General Relativity and has been
tested for matter in a variety of configurations, reaching a level of about a part
on 1015 for matter systems [15]. This is in striking contrast with the information
we have for antimatter, with only a limit of about 100 on the ratio of inertial to
gravitational mass for anti-hydrogen [16]. This is one of the most fundamental
aspects of the research currently being done on antimatter.
4 Anti-hydrogen at CERN
Anti-hydrogen has a long tradition of being studied for testing fundamental laws.
First produced at high energies in laboratories at CERN [17] and at Fermilab [18],
anti-hydrogen was made available at low energies starting from 2002 [20, 21].
The development of the CERN Antiproton Decelerator machine was of paramount
importance to this achievement, making available p beams with energies at the MeV
scale. Subsequent deceleration has led to antiprotons confined at energies of a few
kiloelectronvolts, to be mixed with positrons, obtained with radioactive sources.
The results of this experimentation have recently brought amazing results to
the field. They were possible thanks to the development of magnetic and electric
confinement and cooling techniques (Penning traps). Energies down to the Kelvin
temperature range have in fact been reached for anti-hydrogen.
These results can be summarized along the following main lines.
4.1 Confinement
The number of anti-hydrogen atoms actually available has always been a matter of
concern to experiments, typically ranging in the thousands at best per bunch of the
Antiproton Decelerator. This fundamental problem has been successfully addressed
by the ALPHA collaboration, making use of a special configuration for confinement.
An octupolar magnetic system, featuring the capability of confining both charged
particles (antiprotons and positrons) and the neutral anti-hydrogen atom, has been
developed. Using this technique, the ALPHA collaboration has been able to confine
anti-atoms for as long as 1000 s [22]. A decisive proof of the confinement
mechanism was provided by the ability to release anti-atoms from the trap by means
of microwave-induced quantum transitions between trappable and untrappable
states [23].
4.2 Anti-hydrogen Beams
The capability of obtaining antiparticle beams is also of decisive importance for

experimentation on the WEP, since in principle a tiny fall during propagation can
be measured by interferometric methods. As a reference, a neutral particle flying at
103 m/s in the Earth gravitational field will fall by about 4 µm along a meter length.
The Asacusa collaboration has been the first experiment capable of producing
anti-hydrogen atoms to be detected at almost 3 m distance from the production point
[24].
4.3 The Beginning of Anti-hydrogen Spectroscopy
The capability of confining anti-atoms has been instrumental to a variety of studies

made by the ALPHA collaboration, actually opening up the field of spectroscopy of
anti-atoms.
Basically, the goal is to study the configuration and radiative transitions analo-
gous to that of hydrogen, see Fig. 2.
The first step in this direction has been the observation of the 1S-2S two-photon
transition [25], soon followed by the observation of the hyperfine transition [26]
and the first observation of the dipole-allowed 1S-2P transition, by means of a
laser system tuned on the Lyman-α frequency [27]. These remarkable series of
observation have de facto opened the field of anti-atoms spectroscopy.
Finally, one has to note the remarkable improvement constituted by the first
extension of the laser cooling technique to antimatter [28], which is of fundamental
importance for the next steps.
5 Positrons and Positronium
Positrons and positronium are also used to test fundamental laws in a variety of
ways, from tests of masses and charges (see, e.g., [12]) to the recent observation of
positron interferometry by the QUPLAS group [29].
In addition to the CPT tests made on positrons in traps of various kinds (see
contributions already shown in Fig. 1), positronium is also being used to test
fundamental laws.
118 M. Giammarchi
Fig. 2 The energy levels of the hydrogen atom are now being studied for the case of the antimatter
system
Positronium (Ps) comes in the two varieties of para-Ps (anti-parallel spins, 124
ps lifetime) and ortho-Ps (parallel spins, 142 ns lifetime). Those states have been
extensively used to test QED radiative corrections and are now being considered
(mainly ortho-Ps) for the study of gravitation.
A neutral and well-collimated Ps beam is in fact being developed by several
groups. This is especially challenging because of the small lifetime of ortho-Ps
and involves the recently demonstrated excitation of Ps to Rydberg states (see,
e.g., [30]). Once this goal is achieved, it will be possible to apply interferometric
techniques to a Ps beam, similar to what was done for positrons.
6 The Mu-Atom
The muonium atom is a bound system composed of a μ+ and an electron.1 Its

interest in fundamental physical studies stems from being relatively (∼2 µs) long-
lived and from being the leptonic system with the smallest possible Bohr radius. The
research underway to prepare Mu-atom experiments is based first on high-energy
beams of positive muons at the PSI Institute in Switzerland and at the J-PARC
complex in Japan.
Mu-atom spectroscopy is an already well-studied problem for its interest in
QED studies [31], and Mu-atom gravitation is being now considered by the current
research programs. See [32] for a review of the activities at PSI.
1 This system is often called—somehow improperly—muonium.

7 Conclusion
Antimatter is a very active research field, motivated by the study of fundamen-

tal physical symmetries like CPT and the Weak Equivalence Principle. While
traditionally it was studied only for a few charged systems like antiproton or
the positron, it is now becoming available in neutral systems like anti-hydrogen
and positronium. These systems hold the tantalizing promise of the possibility of
antimatter gravitational studies.
Acknowledgments I would like to thank Luca Venturelli for several useful discussions.
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Part II
Proceedings
Quantum Ergosphere and Brick Wall
Entropy
Lennart Brocki, Michele Arzano, Jerzy Kowalski-Glikman, Marco Letizia,

and Josua Unger
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
2 Geometry of an Evaporating Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3 Mode Counting and Calculation of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4 Calculation of Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
1 Introduction
The discovery that black holes carry an entropy proportional to their horizon area
A divided by the Planck length squared 2 according to the celebrated Bekenstein–
Hawking formula
A
S= (1)
42
L. Brocki () · J. Kowalski-Glikman · J. Unger

Institute for Theoretical Physics, University of Wrocław, Wroclaw, Poland
e-mail: lennart.brocki@uwr.edu.pl; jerzy.kowalski-glikman@uwr.edu.pl; unger.josua@uwr.edu.pl
M. Arzano
Dipartimento di Fisica “E. Pancini” and INFN, Università degli studi di Napoli “Federico II”,
Napoli, Italy
e-mail: michele.arzano@na.infn.it
M. Letizia
Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
e-mail: mletizia@uwaterloo.ca
124 L. Brocki et al.
is now more than 40 years old [1, 2]. Despite the numerous derivations of the
entropy-area relation (1) existing in a variety of approaches to quantum gravity
(see [3] for a comprehensive listing), the fundamental question concerning the
nature of the degrees of freedom responsible for such entropy has not found yet
a conclusive answer. Since due to quantum effects black holes radiate thermally [2],
one of the earliest attempts at addressing this question focused on the quanta of a
field in thermal equilibrium at the Hawking temperature near the horizon [4, 5] as
possible candidates for the origin the Bekenstein–Hawking entropy. As it turns out
the counting of modes needed for deriving the thermodynamic partition function of
the field yields a divergent result due to an infinite contribution coming from the
black hole horizon. ’t Hooft noticed that introducing a crude regulator by requiring
the vanishing of the field at a small radial distance from the horizon one can obtain
a finite horizon contribution to the entropy proportional to the area. Appropriately
tuning the distance of such “brick wall” from the horizon one can exactly reproduce
the Bekenstein–Hawking formula (1). This result, albeit suggestive, replaces the
question about the origin of the Bekenstein–Hawking entropy with a question about
the nature of the brick wall boundary. In [6] the authors suggested that backreaction
of the Hawking radiation can excite the quasinormal modes of the black hole thus
effectively creating a “wall” of oscillations in the geometry close to the horizon.
The Bekenstein–Hawking entropy can be then seen as emerging from an interplay
between the degrees of freedom of the geometry and those of the field.
In the present letter we adopt a philosophy similar to [6] and study the effect of
backreaction on the field propagating in the vicinity of the black hole horizon. We
do this by replacing the usual Schwarzschild metric by a dynamic, “evaporating”
metric first proposed in [7], in which the effects of backreaction are parametrized by
the luminosity of the radiating black hole. After solving the field equations in such
metric we proceed to the usual mode counting for the field. The key feature of our
model is that the small luminosity creates a “quantum ergosphere,” a region between
the apparent horizon and the event horizon which effectively acts as a brick wall
providing a finite horizon contribution to the entropy. As we show below, within the
small luminosity and quasi-static approximations we use, we are able to reproduce
the Bekenstein–Hawking result within very good accuracy.
2 Geometry of an Evaporating Black Hole
Our starting point is the result by Bardeen [7] (see also [8] and [9]) that the metric
of a spherically symmetric black hole slowly emitting Hawking radiation has the
following form

2m
ds 2 = −e2ψ 1 − dv 2 + 2eψ dvdr + r 2 d, (2)
r
Quantum Ergosphere and Brick Wall Entropy 125
where ψ and m are functions of the advanced time v and the radial coordinate r. For
m constant and ψ = 0 this metric reduces to the Schwarzschild one, while for ψ = 0
and m = m(v) it becomes the Vaidya metric. Following [8] we define the mass of
the black hole at a given time to be M(v) = m(v, r = 2m) and its luminosity to be
L = − dM dv . In this letter we work in the regime of small luminosity L 1 and up
1
to first order in perturbation theory so that we can write
m(v, r) = M(v) M0 − Lv . (3)
In what follows we will focus on the near-horizon features of the metric (2).
To this end we introduce a new “comoving” radial coordinate ρ = r − 2M =
r − 2M0 + 2Lv and assume that ρ is small, of the same order as Lv, so that in our
computations we will only keep terms which are at most linear in ρ and L. Further,
we use the residual coordinate invariance to set ψ(r = 2M) = 0, which in our
approximation conveniently makes the function ψ disappear from all the linearized
expressions. Indeed

∂ψ
ψ(r) = ψ(r = 2M) + ρ (4)
∂r r=2M
and it follows from Einstein equations that ∂ψ/∂r ∼ L at r = 2M, so that the first
term in (4) vanishes, while the second is of higher order and can be neglected. In
terms of the comoving radial coordinate the metric near r = 2M takes the form

ρ
ds = −
2
+ 4L dv 2 + 2dvdρ + (ρ + 2M)2 d. (5)
ρ + 2M
The metric (2) has several horizon-like structures. We first consider the apparent
horizon (AH), defined as the outermost trapped surface, i.e., the surface from which
no light ray can move outwards. One characterizes this feature with the help of
the expansion of a congruence of null geodesics, which describes the fractional
change of the congruence’s area. The apparent horizon is defined as a surface for
which = 0.
We define (see, e.g., [10]) null geodesics by their tangent vectors l μ , with lμ l μ =
0, and introduce an auxiliary vector β μ , an affine tangent vector for ingoing radial
null geodesics, with normalization lμ β μ = −1. This auxiliary vector is needed
because l μ , parametrized by advanced time v, is not an affine tangent vector and
the expansion is defined as the divergence of an affine tangent vector. The quantity
κ = −β μ l ν lμ;ν measures to which extent l μ is non-affine and the expansion is then
given by
a black hole with solar mass the value would be L ∼ 10−38 in Planck units where we set the
1 For
Newton constant G to 1.
= l μ ;μ − κ . (6)
Choosing the null vectors

1 2M
l = (l , l ) = 1,
μ v r
1− , β μ = (0, −1), (7)
2 r
we obtain the following expressions for the expansion in Bardeen coordinates (r, v)
and comoving coordinates (ρ, v)
M 1 M 1
(r) = − , (ρ) = − , (8)
r 2 4M (ρ + 2M)2 4M
showing that the apparent horizon is located in the two coordinate systems at rAH =
2M and at ρAH = 0.
In order to capture another horizon-like structure present in the problem, York [8]
gives a working definition of what we will call York event horizon (YEH), which
lacks the teleological property of the event horizon and is instead based on the local
condition
d 2r
= 0, (9)
dv 2
i.e., it characterizes the YEH as the surface imprisoning photons for times long
compared to the dynamical scale 4M of the black hole. According to this definition
the YEH in the Bardeen and comoving coordinates lies at rY EH = 2M − 8ML and
ρY EH = −8ML.
The region between York event horizon and apparent horizon was dubbed by
York quantum ergosphere [8], and he argues that its presence is an irreducible
property of an evaporating black hole.
We will use this observation to shed new light on the brick wall calculation of ’t
Hooft [5] by including the contributions due to the backreaction, here modelled by
a small luminosity L.
The original result of ’t Hooft is that the free energy of a scalar field living outside
the Schwarzschild black hole has a horizon contribution given by
4
2π 3 2M
F =− + ... (10)
45h β
where β is the inverse Bekenstein–Hawking temperature and h is a small cut-off

parameter with dimensions of length. From (10), using standard manipulations, one
can calculate the thermodynamic entropy associated with the field and the resulting
contribution from the horizon term above is proportional to the area of the black
hole thus qualitatively reproducing the Bekenstein–Hawking entropy-area relation.
As we will see, it is a consequence of finite luminosity that the brick wall thickness
h, which is arbitrary in the original ’t Hooft calculation, can be now naturally

identified with the distance between the event and apparent horizon. Therefore the
quantum ergosphere [8], the region between the AH and the YEH, plays the role of
a physically motivated brick wall.
3 Mode Counting and Calculation of Entropy
In order to proceed with the counting of modes of the field we start by solving the
field equation in the vicinity of the black hole horizon in the comoving coordinates
introduced earlier. A massless scalar field φ in this geometry with metric (5) obeys
the Klein-Gordon equation

ρ ∂ρ φ M
4L + ∂ρ2 φ + 2 1− + 2L + 2∂ρ ∂v φ+
ρ + 2M ρ + 2M ρ + 2M
2 l(l + 1)
∂v φ − φ = 0. (11)
ρ + 2M (ρ + 2M)2
Since we are interested only in the contribution coming from the vicinity of the
horizon in the case of small luminosity assuming ρ/M0 1, L 1 and using the
quasi-static approximation 2MLv
0
1, we can write

2 1 ρ Lv
≈ 1− + (12)
ρ + 2M M0 2M0 M0

2 M 1 Lv
1− + 2L ≈ 1 + 4L + (13)
ρ + 2M ρ + 2M 2M0 M0
and the Klein-Gordon equation (11) becomes

ρ 1 Lv
4L + ∂ρ2 φ + 1 + 4L + ∂ρ φ + 2∂ρ ∂v φ
2M0 2M0 M0

1 ρ Lv l(l + 1)
+ 1− + ∂v φ − φ = 0. (14)
M0 2M0 M0 (2M0 )2
We now make use of the standard WKB ansatz

ρ
k(ρ )dρ
φ(ρ, v) = U (ρ)e−iωv ei , (15)
and find that the real part of (14) takes the form

ρ U l(l + 1)
4L + (U − k 2 U ) + + 2ωkU − U = 0, (16)
2M0 2M0 (2M0 )2
which, as a consequence of our approximation scheme, is v-independent. This part

is sufficient for obtaining the wavenumber k. The imaginary part could be used
to compute the amplitude but since we are only interested in counting the modes
of the field with the help of the wavenumber we can ignore it. Moreover the v-
independence of (16) shows that for the purpose of the entropy computation, to be
presented below, the geometry is static. In the WKB approximation one assumes
that the amplitude U (ρ) varies slowly compared to the wave number
U U
k, k2, (17)
U U
and therefore (16) becomes

ρ l(l + 1)
− 4L + k 2 + 2ωk − = 0, (18)
2M0 (2M0 )2
which can be solved for k giving

ρ l(l+1)
ω± ω2 − 4L + 2M0 (2M0 )2
±
k ≈ ρ , (19)
4L + 2M0
where again we neglected the terms which are of higher order in our approximation
scheme. These two solutions correspond to incoming and outgoing modes, respec-
tively, and can be used to calculate the thermodynamic entropy associated with the
field via a count of its number of modes and the derivation of the statistical partition
function.
By approximating the sum over those l that render the square root real with an
integral, the number of modes with frequency up to ω is given by
lmax
g(ω) = ν(l, ω)(2l + 1)dl, (20)
0
where ν(l, ω) is the number of nodes in the mode with (l, ω) [11]. Such quantity
can be explicitly calculated by considering the modes (19) in the box of the radial
length , which acts as an infrared regulator
λ π 2π
=ν = ν → π ν = k, k= , (21)
2 k λ
where λ is the wavelength of the mode.
In the original brick wall calculation [5] it is assumed that the scalar field, whose
entropy we are going to compute, vanishes beyond the brick wall, situated at a small
distance h from the Schwarzschild black hole horizon at rSch , so that all the relevant
integrals have the lower limit at rSch + h. In the case of the Schwarzschild black
hole considered in [5] the apparent and event horizon coincide, rSch = rEH = rAH ;
however, in our case they are different and we must decide at which of the two we
impose the scalar field boundary conditions. Our argument relies on the observation
that in the brick wall picture the scalar field is to be in thermal equilibrium at
temperature T which is identified with the temperature of Hawking radiation.
However, by invoking the so-called tunneling picture, it can be argued that the
Hawking radiation originates at the vicinity of the apparent, not the event horizon
(see [12] and the references therein). Thus, remembering that the apparent horizon
corresponds to ρ = 0, we choose the integration range in the formula above to go
from 0 to , where is the infrared cut-off introduced before, whose explicit value
will not interest us here, since the expression for the area contribution to the entropy
does not depend on it. The number of nodes is thus given by the integral

ρ l(l+1)
ω+ ω2 − 4L + 2M0 (2M0 )2
2π ν(l, ω) = k + dρ = ρ dρ, (22)
0 0 4L + 2M0
where we used Eq. (19) and only considered the contribution from the outgoing
modes. We notice that the ingoing solutions close to the apparent horizon are moving
towards the singularity and one can argue that they cannot contribute to the entropy.
As we will find below, this choice is also justified a posteriori, by the remarkable
agreement of our final result with the Bekenstein–Hawking entropy relation.
Let us notice that the equation for the number of nodes above differs from the
one obtained previously in the literature in [11] in two aspects. First, due to the
approximations we made there is no dependence on the advanced time v and second,
we do not have to introduce a cut-off close to the horizon, since the finite luminosity
prevents the integrand from diverging at ρ = 0. The integration with respect to l in
Eq. (20) is taken over those values for which the square root is real and yields
5(2GM0 + ρ)4 ω3
g(ω) = dρ. (23)
0 6π(8M0 L + ρ(1 + 4L))2
The leading contributions in the integral in (23) are thus given by
5ω3 M03 5ω3 3

g(ω) = + , (24)
3π L 18π(1 + 8L)
where the second term is the usual volume contribution and has no relevance for our
discussion. The thermodynamic partition function of the field is given by
Z = e−βF , (25)
where F is the free energy

πβF = dg(ω) ln 1 − e−βω . (26)
Using (24) and neglecting the volume contribution to g(ω) we have
1 ∞ dg(ω) M 3π 3
F = ln (1 − e−βω ) dω = − 0 4 , (27)
β 0 dω 9Lβ
from which we can calculate the entropy of the field associated with the horizon
boundary
∂ 4M03 π 3
S = β2 F = . (28)
∂β 9Lβ 3
Comparing our result for the free energy (27) with the standard result obtained
from the brick wall calculation (10) we see that the brick wall width parameter
h introduced by ’t Hooft can be expressed in terms of the luminosity of the black
hole as
32
h= LM0 , (29)
5
and thus the backreaction of the quantum radiance on the horizon structures of the
black hole naturally provides the regulator needed for a finite horizon contribution
to the field entropy. It should be noticed that if we had counted also the ingoing
modes in (22) the resulting brick wall size would have coincided with the quantum
ergosphere, i.e., h = 8LM0 .
4 Calculation of Luminosity
In order to have an expression for the entropy (28) to be compared to the

Bekenstein–Hawking relation (1) we now have to spell out the explicit form of the
luminosity L in terms of the black hole mass M0 . In the first order approximation
used in our calculation the luminosity L is a small quantity so that we can identify it
with the luminosity of Hawking radiation in the case of a Schwarzschild black hole.
To find it, one considers [13] a flux X of radiation with energy ωk
(ωk )
X(ωk ) = , (30)
8π
2π(e M0 ωk − 1)
where the factor models the backscattering. Integrating the flux times the energy
ω we find the luminosity that escapes to infinity
∞
rL = dω ωX(ω). (31)
0
The factor can be approximated by DeWitt [14]
≈ 27π M02 ω2 (32)
and integration over ω yields
1.69
L≈ . (33)
7680π M02
Plugging the expression (33) in (28) we finally obtain
S = 0.987 · 4π M02 = 0.987SBH , (34)
where SBH is the Bekenstein–Hawking entropy. We thus see that our model
reproduces the exact result of Bekenstein–Hawking with an accuracy close to 99%,
which is a remarkable result given the rather crude approximations that we used.
There is an important comment to be made at this point. In our calculation of
the entropy we assumed that the luminosity of black hole results from a single
massless scalar mode, the same that we used to compute entropy. Since we do not
know any massless scalar field it might be argued that one should use instead in our
computations the massless fields that we know about, namely photons and gravitons,
i.e., four massless degrees of freedom. Each degree of freedom will contribute the
amount (28) to the entropy. As for the luminosity one can use the numerical results
of Page [15], to see that the contribution to luminosity of photons and gravitons is
of order of order of 3 × 10−5 1/M02 as compared to 7 × 10−5 1/M02 given by (33).
This means that the final entropy will be by factor 8 larger in the case of photons and
gravitons that it is in the case of a single scalar. On the other hand it is believed that
the Bekenstein–Hawking entropy is fundamental, capturing some essential features
of space-time and from that perspective it is hard to imagine that it could depend
on the number of massless degrees of freedom in nature, which seems to be rather
contingent. The fact that employing a single massless degree of freedom reproduces
in our reasoning reproduces the correct value, with a small error, indicates that there
might be something special about the single massless scalar field model.
5 Conclusion
In this letter we showed how small backreaction effects can be introduced in

the derivation of the thermodynamic entropy of a field in thermal equilibrium
in the proximity of a black hole horizon. The resulting changes due to a small
but non-vanishing luminosity on the horizon structure of the black hole provide
a natural brick wall regulator for the near-horizon modes of the field. Using the
small luminosity and quasi-static approximations we were able to solve the field
equations in the evaporating metric to find an explicit expression for the field modes,
the degrees of freedom contributing to the thermodynamic partition function of
the field. We showed that once the width of the quantum ergosphere is set by the
Hawking luminosity the horizon contribution to the entropy of the field is in very
good agreement with the Bekenstein–Hawking relation for the black hole entropy.
In the original brick wall calculation the width of the brick wall had to be adjusted
by hand in order to have the correct proportionality factor between entropy and the
black hole area. From this point of view we find our result particularly suggestive:
the non-trivial horizon geometry determined by the backreaction of the Hawking
flux creates a “covariant” brick wall and leaves no arbitrary parameter to be tuned
to obtain the desired result.
Acknowledgments This work is based on [16], which is available under the terms of the
Creative Commons Attribution License (CC BY) https://creativecommons.org/licenses/by/4.0/.
For LB, JKG, and JU this work is supported by funds provided by the National Science Center,
projects number 2017/27/B/ST2/01902. ML acknowledges the Fondazione Angelo della Riccia
and the Foundation BLANCEFLOR Boncompagni-Ludovisi, née Bildt for financial support. MA
acknowledges support from the COST Action MP1405 “QSpace” for a Short Term Scientific
Mission Grant supporting a visit to the University of Wroclaw where part of this work was carried
out.
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134887
Geodesic Structure and Linear Instability
of Some Wormholes
Francesco Cremona
Mathematics Subject Classification (2000) 83C10, 83C15, 83C20, 83C25
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2 Modelling the Metric of a Wormhole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
2.1 An Embedding for the EBMT Wormhole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3 Deriving Wormholes from Einstein’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4 An Embedding for the AdS Wormhole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5 The Timelike and Null Geodesics Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.1 General Spherically Symmetric Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.2 The Case of the AdS Wormhole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6 Linear Instability of the EBMT Wormhole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.1 Radial Perturbations of the EBMT Wormhole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.2 A Master Equation for R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.3 Linear Instability of the EBMT Wormhole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.4 A Comparison with [2, 4, 8, 14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
1 Introduction
We consider the metric
1
− 1 + k 2 (x 2 + b2 ) dt 2 + dx 2 +(x 2 +b2 )d2 (−∞ < t < x < +∞).
1 + k 2 (x 2 + b2 )
(1)
where d2 = dθ 2 +sin2 θ dϕ 2 ( 0 < θ < π, 0 < ϕ < 2π ) is the usual line element
of the spherical surface S 2 and b and k are two positive constants, respectively,
F. Cremona ()
Dipartimento di Matematica, Universitá di Milano, Milano, Italy
e-mail: francesco.cremona92@gmail.com
134 F. Cremona
with the dimension of a length and dimensionless. In [3] (and, more recently, in the
survey [2]), Bronnikov finds this metric as a solution to Einstein’s equations for a
gravitational field minimally coupled to a self-interacting scalar field whose action
functional has the kinetic part artificially multiplied by −1; the scalar field and its
self-interacting potential found by Bronnikov have the following expressions:

2 x
φ(x) = arctan , (2)
κ b
2
k κ
V (φ) = − 3 − 2 cos2 φ . (3)
κ 2
In the limit case k = 0, the potential V (φ) vanishes, and the metric (1) reduces to
the well-known metric
− dt 2 + dx 2 + (x 2 + b2 )d2 ; (4)
this was first introduced almost simultaneously by Ellis [7] and Bronnikov [1]
and only at a later time by Morris and Thorne in the famous paper [12] and
describes the configuration of a static spherically symmetric wormhole connecting
two asymptotically flat universes. In order to take into account all the previously
mentioned authors, using the initials of their names, we will refer to the spacetime
structure (4) as “EBMT wormhole” and to Eqs. (4), (2) as “EBMT solution”.
In the first part of this chapter, we focus on the general metric (1) and build an
embedding diagram in a suitable three-dimensional ambient space, justifying the
interpretation of this spacetime as a four-dimensional, static spherically symmetric
wormhole with a throat of size b connecting two separate universes with asymptotic
constant negative scalar curvature −12k 2 (Anti-de Sitter universes); for these
reasons, throughout the chapter, we will refer to the metric (1) as “AdS-AdS
wormhole”, or more simply, “AdS wormhole”, and to Eqs. (1), (2), and (3) as
“AdS-AdS wormhole solution”, or more simply, “AdS wormhole solution”. From
this general construction, in the limit case k = 0, we recover exactly the same
embedding diagram for the EBMT wormhole (1) which was proposed by Hartle in
his textbook [9].
In the second part of the chapter, we recall a general strategy to studying the
geodesic motion in a four-dimensional, static spherically symmetric spacetime and
specialize it to the case of the AdS wormhole (1); the motion of some timelike and
null geodesics is computed numerically and plotted in the embedding diagram of
the wormhole.
The last part of the chapter is dedicated to the linear stability analysis of the
EMBT wormhole (4). Although the linear instability of this configuration has
already been stated in other previous works [2, 4, 8, 14], we will summarize the new
derivation proposed in [5]; in this latter paper, the linearized Einstein equations for
a time-dependent spherically symmetric perturbation of the EBMT solution (4) and
(2) is reduced to a wave-type equation for the perturbed wormhole radius without
Geodesic Structure and Linear Instability of Some Wormholes 135
encountering any singularities in its derivation; we will make a brief comparison

between this approach and the one of [2, 4, 8, 14] (for more details about this
comparison, see the Introduction of [5]); some words will also be spent on a very
recent work [6] in which the linear instability result of the EBMT wormhole is
extended to the AdS wormhole solution (1), (2), and (3) in a general gauge-invariant
setting.
The chapter is organized as follows: in Sect. 2, we build the metric of a
static spherically symmetric spacetime wormhole, starting from its “tunnel-shaped”
embedding in an ambient space with constant negative curvature; in Sect. 3, we
introduce the EBMT wormhole and the AdS wormhole as solutions to the phantom
scalar field Einstein’s equation; in Sect. 4, taking advantage of the considerations
of Sect. 2, we plot the embedding diagram for the AdS wormhole; in Sect. 5, the
qualitative features of the timelike and null geodesics of the AdS wormhole are
studied; finally, Sect. 6 contains a brief overview on the linear stability analysis of
the EBMT wormhole performed in [5] and a comparison with other previous works
on the same subject [2, 4, 8, 14].
Throughout this chapter, we use the signature convention (−, +, +, +, ), and we
stipulate
c=1, h̄ = 1 , κ := 8π G , (5)
where c is the speed of light, h̄ is the reduced Planck constant, and G is the
gravitational constant.
2 Modelling the Metric of a Wormhole
In the coordinate system
(t, x, θ, ϕ) ∈ R × R × (0, π ) × (0, 2π ) , (6)
we consider a four-dimensional spacetime (M, g) with metric
g = −h(x)2 dt 2 + q(x)2 dx 2 + r(x)2 d2 , (7)
where d2 = dθ 2 + sin2 θ dϕ 2 is (again) the usual line element of the spherical
surface S 2 and h, q, r : R → (0, +∞) are smooth functions. Let us start with few
elementary considerations about the metric (7):
(i) Since the coefficients of the metric g do not depend on the angular coordinates
θ and ϕ, the spacetime M has the property of being spherically symmetric.
(ii) Since the coefficients of the metric g do not depend on the temporal coordinate
t, the spacetime M is static: the coordinates (x, θ, ϕ) define a coordinate
136 F. Cremona
system for the “space” of a static observer in this spacetime, while the temporal
coordinate t represents its “time”.
(iii) The first term −h(x)2 dt 2 of g is the proper time (physical time) measured by
someone which is at rest according to the static observer; note that the proper
time depends on the spatial variable x and, therefore, the function h can be
used to quantify the “gap” between the time lengths signed by two clocks at
rest for the static observer depending on their relative positions.
(iv) The second term
q(x)2 dx 2 + r(x)2 d2 (8)
is the metric of each of the three-dimensional spherically symmetric manifolds

t := {A ∈ M | t (A) = const} ⊂ M, which represent the space seen by the
static observer at a fixed time; for this reason, we can say that the functions q
and r determine what we reasonably would call the “shape” of the spacetime
(M, g).
A very useful tool for visualizing the geometrical properties of the spatial slice t
is the embedding diagram. In general, a smooth map ι : M → N between two
differential manifolds M and N is an embedding if both ι and its differential dι are
everywhere injective and ι : M → ι(M) is a homeomorphism. We will see that it
might be impossible to embed the three-dimensional slice t in a four-dimensional
flat space. However, if it is the case, profiting from the spherical symmetry of (8),
one can build up a picture of the embedded slice ι(t) fixing the value of an angle.
In fact, the embedding ι transforms the spacetime slices tθ0 := {A ∈ M | t (A) =
const, θ (A) = θ0 } ⊂ M into two-dimensional surfaces in the three-dimensional
Euclidean space: these representations of the spatial part of a spacetime are called
embedding diagrams. Note that, as the value of θ0 is immaterial, from now on, we
fix θ0 = π/2.
After these preliminary considerations, we can propose a naive definition of a
static spherically symmetric wormhole spacetime: the metric (7) represents a static
spherically symmetric wormhole if each of its t π2 slices (defined by the metric
(8) with θ = π/2), once embedded in a three-dimensional flat space, looks as a
“tunnel-shaped” hypersurface, a form familiar from popular accounts of wormholes.
However, this statement is too restrictive since, as we will see later, there are
spacetimes whose t π2 slices cannot be embedded in a flat space, but nevertheless, in
some sense these slices have still the “shape of a tunnel” if embedded in a different
suitable ambient space.
In the remaining part of this section, we will give a more precise mathematical
setting to the consideration of the previous paragraph with an example: we will build
up the metric of a spacetime whose t π2 slice is embeddable as a “tunnel-shaped”
surface in a three-dimensional space with constant negative curvature.
Let us start considering a four-dimensional Riemannian manifold Mk with
constant scalar negative curvature RMk := −12k 2 , k > 0; this is the ambient space
in which the slice t will be embedded. The reasons why we are interested in Mk
(instead of a four-dimensional flat space) will be clear a little bit later. It is widely
known that introducing the system of coordinates (z, ρ, θ̂ , ϕ̂) ∈ R × (0, +∞) ×
(0, π) × (0, 2π ), the line element of the space Mk reads
1
dσk2 = α(ρ)2 dz2 + ˆ2,
dρ 2 + ρ 2 d α(ρ) = 1 + k2ρ 2 (9)
α(ρ)2
ˆ 2 = d θ̂ 2 + sin2 θ̂ d ϕ̂ 2 ). A very easy way to define a “tunnel-shaped”

(here, again, d
hypersurface S in Mk is
S := {(z, ρ, θ̂ , ϕ̂) : ρ = F (z)} ⊂ Mk ,
where the function F : dom(F ) ⊆ R → (0, +∞) is smooth and possesses a

positive minimum of size b > 0 at a certain point of its domain, let us say, without
loss of generality, at z0 = 0 ∈ dom(F ); moreover, we prescribe that z0 = 0 is the
only minimum point of F . In short, we require that the function F , which can be
effectively regarded as the “profile” of the tunnel, has the following properties:
F ∈ C∞ , F (0) = b > 0 , F (0) = 0 , F (z)z > 0 for all z ∈ dom(F )/{0} .

(10)
With this position, it is clear that the minimum b of the function F is the size of
the tunnel throat, while the large z limits represent the far ends of two separate
hypersurfaces of Mk linked by the tunnel throat; these hypersurfaces are defined,
respectively, by {(z, F (z), θ̂ , ϕ̂) | z > 0}} and {(z, F (z), θ̂, ϕ̂) | z < 0}}. Moreover,
as ρ = F (z) and dρ = F (z)dz on S, the metric of the whole hypersurface S reads

F (z)2
dS = α(F (z)) +
2 2 ˆ2.
dz2 + F (z)2 d (11)
α(F (z))2
Therefore, we are looking for a spacetime M with metric (7) such that, given a
“profile function” F , it exists an embedding ι : M → Mk such that the embedded
slice S := ι(t) ⊂ Mk has the metric (11). Working in coordinates, the embedding ι
is specified by four smooth bijections:
z = z(x, θ, ϕ) , ρ = ρ(x, θ, ϕ) , θ̂ = θ̂(x, θ, ϕ) , ϕ̂ = ϕ̂(x, θ, ϕ) ;
note that, as we are in a radially symmetric configuration, we can take θ̂ = θ , ϕ̂ = ϕ

and the functions z(·) and ρ(·) to be angles-independent, i.e. we can set
z = z(x) , ρ = ρ(x) , θ̂ = θ , ϕ̂ = ϕ . (12)
Provided Im(z) ⊆ dom(F ), one can insert the embedding functions (12) and their
differentials into (11), obtaining the original form of the t slice metric:
138 F. Cremona

F (z(x))2
2 2
dt = α(F (z(x))) z (x) 2
1+ ˆ2.
dx 2 + F (z(x))2 d (13)
α(F (z(x)))4
Without loss of generality, from now on, we also stipulate that z(0) = 0 and z (x) >
0 for all x ∈ R.
By comparing the two expressions (8) and (13) for the metric of the slice t, we
get the expressions of the functions q and r in dependence of F and z = z(x):
r(x) = F (z(x)) , (14)
F (z(x))2
q(x) = α(F (z(x)))z (x) 1 + . (15)
α(F (z(x)))4
Summing up, in this section, we have proved that, for all smooth functions h : R →
(0, +∞), z : R → Im(z), F : dom(F ) ⊆ R → [b, +∞) such that
(i) z is a bijection with z (x) > 0 for all x ∈ R.
(ii) Im(z) ⊆ dom(F ).
(iii) z(0) = 0 ∈ dom(F ).
(iv) F satisfies the requirements in (10).
The metric (7) with r and q as in (14), (15) and α as in (9) represents a static
spherically symmetric wormhole spacetime; indeed, the embedded hypersurface
S = ι(t) ⊂ Mk has a “tunnel-shaped” structure with a throat of size b located
at z = 0. Thanks to the injectivity of the embedding function z, one can easily see
that the slice t of the wormhole spacetime has the throat of size b at x = 0 and links
together the two separate universes defined, respectively, by {A ∈ M | x(A) > 0}
and {A ∈ M | x(A) < 0}.
2.1 An Embedding for the EBMT Wormhole
To our knowledge, there is at least one example in the literature of the previous
construction in the limit case k = 0, namely in the case of a static spherically
symmetric spacetime whose spatial part is embeddable as a “tunnel-shaped”
hypersurface in a flat ambient space (indeed, for k = 0, the space Mk has zero
curvature): in his textbook [9], Hartle describes how to build the embedding of the
t π2 slice of the EBMT wormhole (4) (therein referred generically as “a Wormhole
Spacetime”) in the three-dimensional Euclidean space R3 . The construction made
by Hartle leads to an embedding diagram for this wormhole, described (in our
notation) by the profile and the embedding functions
FEBMT (z; b) := b cosh(z/b) , zEBMT (x; b) := b arcsinh(x/b). (16)

In the following section, we show that these functions can be generalized in the case
k = 0 in order to describe the embedding of the AdS wormhole.
3 Deriving Wormholes from Einstein’s Equation
In a four-dimensional spacetime (M, g), we consider a gravitational field minimally

coupled to a real scalar field φ with a non-null self-interacting potential V (φ). This
system is described by the action functional

R σ
S[gμν , φ] := − ∂ μ φ ∂μ φ dv, (17)
2κ 2
where κ, R and dv are, respectively, the usual coupling constant defined in

Eq. (5), the scalar curvature of the Lorentzian manifold (M, g) and the volume
element corresponding to the metric g (dv = |det(gμν )| λ dx λ in any spacetime
coordinate system (x λ )λ ); σ is a constant which is normally set to 1. Both the metric
and the scalar field are assumed to be smooth.
The first stationarity condition δS/δgμν = 0 is equivalent to Einstein’s equations
1
Rμν − gμν R = κTμν , (18)
2
where stress–energy tensor is given by

1
Tμν := σ ∂μ φ ∂ν φ − gμν ∂ λ φ ∂λ φ − gμν V (φ) , (19)
2
The second stationarity condition δS/δφ = 0 gives the Klein–Gordon equation for
the scalar field φ
φ = σ V (φ) , (20)
where := ∇μ ∇ μ and ∇μ is the covariant derivative induced by the metric g.

As we have already mentioned, if φ is an ordinary field, the constant σ equals
1; however, a different and interesting class of scalar filed is defined by the position
σ = −1: these are called phantom scalar fields. The latter anomalous choice of the
value of σ corresponds to an artificial change of the sign of (the kinetic part of) the
action functional (17) and reproduces, in the background of General Relativity, a
surprising and well-known feature of quantum fields in their vacuum states [10, 15].
In their almost simultaneous papers, Ellis [7] and Bronnikov [1] introduced actually
a phantom scalar field in order to prove that the EBMT wormhole
− dt 2 + dx 2 + (x 2 + b2 )d2 (21)
140 F. Cremona
could be seen as a solution to Einstein’s field equations (18), (19), and (20); the
phantom scalar field φ which supports the metric (21) was found to be non-self-
interactive and to depend only on the spatial coordinate x:

2 x
φ(x) = arctan . (22)
κ b
More recently, in 2006, Bronnikov [3] proposed a generalization of the EBMT

wormhole solution: with some reparameterization of the constants involved therein,
this new general spacetime was found to have a metric of the form (7) with
coefficients

−1
x
h(x) = q (x) = 1+ x 2 +b2 M arctan −K + bMx , r(x) = x 2 + b2
b
(23)
and to be supported by a phantom scalar field with a self-interacting potential given
by

2 x
φ= arctan , (24)
κ b

K κ
V (φ) = 3 − 2 cos2 φ
κ 2

M κ κ κ κ
− 3 sin φ cos φ + φ 3 − 2 cos2 φ ,
κ 2 2 2 2
(25)
where K, M and b > 0 are three constants. Obviously, the EBMT wormhole
solution is recovered in the limit case M, K = 0.
Let us now focus on the particular choices M = 0 and K = −k 2 < 0 (with
k > 0) so that the general solution (23), (24), and (25) reduces to
h(x) = q −1 (x) = 1 + k 2 (x 2 + b2 ) , r(x) = x 2 + b2 , (26)

2 x k2 κ
φ= arctan , V (φ) = − 3 − 2 cos2 φ . (27)
κ b κ 2
In the limit case b = 0, the metric (26) is non-singular for x > 0 and x < 0 and
describes in this two regions as many Anti-de Sitter (AdS) universes with constant
negative scalar curvature RAdS = −12k 2 , while, in the general case b > 0, the
functions h, q, r are smooth for all x ∈ R; since r(x) ∼ |x| and h(x) = q −1 (x) ∼
√
1 + k 2 x 2 for x → ±∞, the metric (26) approaches asymptotically to the AdS
metric and, therefore, we reasonably would say that, for every b > 0, the metric (26)
describes a wormhole with a throat of size b connecting two separate asymptotically
AdS universes with negative scalar curvature RAdS = −12k 2 ; for these reasons, as
already mentioned in the Introduction section, we will refer to the metric (26) as
“AdS-AdS wormhole”, or more simply, “AdS wormhole”.
4 An Embedding for the AdS Wormhole
In this subsection, we show how to find a profile function F and an embedding

function z (introduced in Sect. 2) for the AdS wormhole. More precisely, we are
looking for two functions F and z satisfying the requirements (i)–(iv) under Eq. (15)
and such that the metric (7), (14), and (15) reduces to the AdS metric (26).
We start recalling that ρ(x) = F (z(x)) on the embedded hypersurface S = ι(t),
so that Eqs. (14), and (15) are equivalent to
r(x) = ρ(x) , (28)
ρ (x)2
q(x) = α(ρ(x)) z (x)2 + , (29)
α(ρ(x))4
F (z) := ρ(x(z)) , (30)
where x = x(z) is the inverse function of z = z(x) and α is as in (9).

If we know the metric coefficients q and r, Eqs. (28) and (29) become a system of
two differential equations in the unknown ρ = ρ(x), z = z(x), which is a trivially
solved setting
x 1 r (x̃)2
ρ(x) = r(x) , z(x) = q(x̃)2 − d x̃ (31)
0 α(r(x̃)) α(r(x̃))2
(note that in solving Eq. (29) we have required that z(0) = 0).
At this point, we think that it is necessary to spend some words about the use of
the parameter k that we have made until now. In Sect. 2, k ≡ kas = −RMk /12,
where RMk is the constant negative scalar curvature √ of the ambient space of the
embedding, while, in the present section, k ≡ kw = −RAdS /12, where RAdS is
the asymptotic constant negative scalar curvature of the AdS wormhole. Note that
the integral in Eq. (31) makes sense only if the functions q, r and α are such that
q(x)2 − r (x)2 /α(r(x)2 ≥ 0 for every x ∈ R; in the case of the AdS wormhole, i.e.
choosing the functions q and r as in Eq. (26) with k = kw and recalling the definition
of α in Eq. (9) with k = kas , this occurs whenever kas is chosen such that kas ≥ kw .
We would like to stress the fact that in any other case it is not possible to embed
the whole slice t of the AdS wormhole in an ambient space with constant curvature;
in particular, it is impossible to embed t in a four-dimensional flat space (defined
by the limit occurrence kas = 0) unless kw = 0 (which is the case of the EBMT
wormhole). From now on, we make the easiest choice possible k = kas = kw , so
that the second integral in Eq. (31) can be easily solved giving the following explicit
expression for the function z = z(x):
142 F. Cremona

b x
z(x) ≡ zAdS (x; k, b) = √ arcsinh ; (32)
1 + b2 k 2 b 1 + k 2 (x 2 + b2 )
this can be inverted so that Eq. (30) reads

√
1+b2 k 2 z
b cosh b
F (z) ≡ FAdS (z; k, b) = √ . (33)
1+b2 k 2 z
1 + b2 k 2 1 − cosh2 b
An elementary computation shows that, for all b > 0 and k > 0, the functions
z and F satisfy the conditions (i)–(iv) after Eq. (15). Note that Eqs. (32) and (33)
actually generalize Eq. (16) to the case k > 0 as zAdS (x; 0, b) = zEBMT (x; b) and
FAdS (x; 0, b) = FEBMT (x; b).
At this point, one might want to visualize the embedding of the AdS wormhole,
namely a three-dimensional picture of the embedded slice ι(t π2 ). Obviously, this
is not possible since the ambient space Mk is not flat, unless we settle for an
approximation. For example, we observe that in the limit case k → 0 or ρ → 0
the metric of the ambient space (9) tends to become flat. This means that the
bidimensional surface in R3 defined as
SAdS (k, b) := {(z, ρ, ϕ) : ρ = FAdS (z; k, b)} ⊂ R3
approaches to ι(t π2 ) ⊂ Mk in a region “suitably close to” the origin; this region
can be very large if k is very small. In Fig. 1, one can see the plots of the profile
function FAdS and the embedding diagram SAdS ⊂ R3 for a particular choice of the
parameters b and k.
Fig. 1 The profile function FAdS and the embedding diagram SAdS of the AdS wormhole with
b = 1 and k = 0.1
5 The Timelike and Null Geodesics Motion
5.1 General Spherically Symmetric Case
Let us firstly recall that the trajectory described by a free-falling particle or by a

light ray in a spacetime (M, g) is represented by a geodesic of M, i.e. a world line
τ → P(τ ) such that [13, 16]
∇ dP
= 0, (34)
dτ dτ
where ∇ is the covariant derivative defined by the Levi-Civita connection of the
metric g. Moreover, it is always possible to redefine the parameter τ in such a way
that, for all τ ,

dP dP 1 for a timelike geodesic (free-falling particle)
gP (τ ) , = −k , k=
dτ dτ 0 for a null geodesic (light ray).
(35)
In this section, we will study the geodesic motion in a four-dimensional spherically
symmetric spacetime defined in coordinate system (x μ ) := (t, x, θ, ϕ) ∈ R × R ×
(0, π ) × (0, 2π ) by the metric
1
g = −h(x)2 dt 2 + dx 2 + r(x)2 d2 ; (36)
h(x)2
each geodesic P in this spacetime is described locally by four functions of the

parameter τ
(x μ (P(τ ))) =: (x μ (τ )) =: (t (τ ), x(τ ), θ (τ ), ϕ(τ )) (37)
satisfying the geodesic equation (34) which locally reads
d 2xμ λ
μ dx dx
ν
+ λν = 0, (38)
dτ 2 dτ dτ
μ
where λν are the Christoffel symbols of the Levi-Civita connection of g.
Moreover, it can be proved [11] that the geodesic equations (38) are equivalent to
d ∂L ∂L
the Euler–Lagrange equations dτ ∂ ẋ μ − ∂x μ = 0, μ = 1, ..., 4, for the Lagrangian
1 h(x)2 2 1 r2 2
L(x μ , ẋ μ ) := gλν (x μ )ẋ λ ẋ ν = − t˙ + ẋ 2
+ θ̇ + sin2
θ ϕ̇ 2
;
2 2 2h(x)2 2
(39)
144 F. Cremona
these are satisfied if and only if the following system of four ordinary differential
equations hold:
d
h(x)2 t˙ = 0 , (40)
dτ

d 1
ẋ = h(x)r(x)r (x) θ̇ 2
+ sin2
θ ϕ̇ 2
− h(x)2 h (x)t˙2 , (41)
dτ h(x)2
d
r(x)2 θ̇ = r(x)2 sin θ cos θ ϕ̇ 2 , (42)
dτ
d d
r(x)2 ϕ̇ = r(x)2 ϕ̇ cos2 θ . (43)
dτ dτ
Before starting with the study of the system (40)–(43), let us summarize some
general and useful results about Lagrangian systems:
(i) In a time-independent n-dimensional Lagrangian system (L(q i , q̇ i ), q i =
q i (t), i = 1, ..., n), the total energy function defined by E := ∂∂q̇Li q̇ i − L is
conserved.
(ii) In the hypothesis of (i), the system of the n Euler–Lagrange equations for the
Lagrangian L is equivalent to the system made up of n − 1 Euler–Lagrange
equations and the conservation law E = const;
(iii) In the hypothesis of (i) and if the Lagrangian L is a quadratic function in the
generalized velocities q̇ i , it follows that E = L and the conservation law reads
L = const.
These results immediately apply to the Lagrangian (39); moreover, since for all τ

1 1 dP dP
L(x μ (τ ), ẋ μ (τ )) = gλν (x μ (τ ))ẋ λ (τ )ẋ ν (τ ) = gP(τ ) , ,
2 2 dτ dτ
recalling the position (35), we have that the conservation law becomes L = −k/2.
We are now ready to study the system (40)–(43), starting from the third equation
(42); obviously, this equation and the initial conditions
π
τ0 = 0 , θ (τ0 ) = , θ̇ (τ0 ) = 0 (44)
2
imply that θ (τ ) = π2 for every τ . Since it is always possible to redefine the

coordinates θ and ϕ and the parameter τ so that the previous conditions on θ are
true, from now on we assume (44); in this way, the four-dimensional system (40)–
(43) reduces to the three-dimensional system:
d
h(x)2 t˙ = 0 , (45)
dτ
d
r(x)2 ϕ̇ = 0 , (46)
dτ
h(x)2 2 1 r(x)2 2 k
− t˙ + 2
ẋ 2
+ ϕ̇ = − , (47)
2 2h(x) 2 2
where we have substituted the second equation (41) with the conservation law L =
−k/2, thanks to (i)–(iii) after Eq. (43) and the forthcoming remark.
Let us start with Eqs. (45) and (46); note that, hopefully performing the parameter
change τ → −τ , the vector dP/dτ can be regarded as future-oriented, so that one
can always suppose that t˙ > 0. Thanks to this remark, it is clear that Eqs. (45) and
(46) hold if and only if there exist two constants E ≥ −k/2 and L ∈ R [16] such
that
√
h(x)2 t˙ = k + 2E , r(x)2 ϕ̇ = L ; (48)
these two equations can be easily solved, leading to

√
τ k + 2E τ L
t (τ ) = d τ̃ + t (0) , ϕ(τ ) = d τ̃ + ϕ(0) . (49)
0 h(x(τ̃ ))2 0 r(x(τ̃ ))2
Note that it results from Eq. (48) that the two constants E and L are fully determined
by the initial data:
t˙(0)2 h(x(0))4 − k
E := , L = ϕ̇(0)r(x(0))2 .
2
It is easy to see that in the limit case of a particle moving slowly in a weak
gravitational potential (i.e. h(x) 1) it is possible to prove that E and L approach,
respectively, to the classical total energy and the angular momentum per unit rest
mass of the particle [16]; therefore, in the timelike case, one can interpret L and
E as a relativistic generalization of the total energy and the angular momentum
per unit rest mass of a free-falling particle and, in the null case, h̄L and h̄E as the
angular momentum and the total energy of a photon (recall however that in (5) we
have stipulated h̄ = 1).
Inserting Eq. (49) into the conservation law (47), we have that the reduced
Lagrangian system (45), (46), and (47) is equivalent to the dynamical system made
up of Eq. (49) and
1 2
ẋ + Veff (x) = E (50)
2
where we have defined the effective potential
146 F. Cremona
L2 h(x)2 k
Veff (x) := + h(x) 2
− 1 . (51)
2 r(x)2 2
Summing up, provided a suitable change of coordinates, the problem of finding the
qualitative behaviour of a timelike (k = 1) or a null (k = 0) geodesic in a spacetime
with a metric of the form (36) is reduced to studying its radial motion, which satisfies
Eqs. (50) and (51); since this radial motion is the same as the motion of a unit
mass particle of energy E in ordinary one-dimensional, nonrelativistic mechanics
moving in the effective potential Veff , in order to understand the qualitative features
of the geodesic motion one has to investigate the analytical properties of Veff in
dependence of the values of the parameters appearing in its definition (51).
5.2 The Case of the AdS Wormhole
Let us consider the AdS wormhole (which has obviously a metric of the form (36));
in this case, we have that the effective potential (51) reads

L2 1 k
Veff (x) ≡ Veff,AdS (x; b, k, k, L) := + k + k 2 (x 2 +b2 ) .
2
(52)
2 x +b
2 2 2
In the following, a complete analysis of the analytical properties of Veff will be

performed by varying the values of the parameters b, k, k, L in their respective
ranges; in this way, we can deduce some information about the geodesic motion
in the AdS wormhole.
We start from the limit case of the EBMT wormhole corresponding to the choice
k = 0: depending on the value of the angular momentum L, we encounter only
two qualitatively different situations: if L = 0, the potential Veff is identically null,
while, if L = 0, it possesses an asymptotically null “bell curve” shape with the
maximum L2 /(2b2 ) located in x = 0. This means that in the EBMT wormhole:
(i) There is no difference between timelike and null geodesics.
(ii) If E > L2 /(2b2 ), both particles and light rays heading towards the centre of the
wormhole will pass from one universe to the other and never come back (unless
they accelerate or are deviated).
We now focus on the AdS case, so suppose k > 0. Considering firstly the motion
of a light ray (k = 0), we have a situation similar to that of the EBMT wormhole:
if L = 0, the potential is again identically null, while if L = 0, the potential Veff
has once more a “bell curve” shape with the maximum of value L2 (k 2 + 1/b2 )/2
located in x = 0 and a non-null horizontal asymptote of value L2 k 2 /2. In Fig. 2, the
effective potential Veff and the total energy E have been plotted in three different
possible occurrences; in each case, the motion of a particular null geodesic P(τ ),
τ ∈ [0, τ1 ] has been computed numerically and plotted in the embedding diagram
of the AdS wormhole.
Fig. 2 Effective potential and embedding diagram of null geodesics in the AdS wormhole with
b = 1 and k = 0.1. (a) L = 3, E = 4.8. (b) L = 3, E = 3. (c) L = 3, E 4.545
Secondly, we consider the timelike geodesic motion (k = 1). If |L| ≤ b2 k, the

effective potential is a convex function with the minimum V0 := b2 k 2 /2 + L2 (k 2 +
1/b2 )/2 at x = 0; if |L| > b2 k, the effective potential has a “Mexican hat” shape,
with the local maximum of value V0 located in x = 0, the two local minima in
x = ± |L|/k − b2 both of value k|L|(1 + k|L|/2) and limits Veff (x) → +∞ for
x → ±∞. Therefore, in the AdS wormhole, the timelike geodesics:
(i) Orbit in a bounded region of the spacetime which depends on b, k, E, L and is
defined by

1
A : |x(A)| ≤ √ 2E − k 2 (L2 + 2b2 ) + 4E 2 +k 2 L2 (k 2 L2 − 4(E+1)) .
2k
(ii) If E > V0 , the particles pass from one universe to the other and keep oscillating
between them (unless they accelerate).
Figure 3 contains the plot of three possible effective potentials Veff and values of the
total energy E for the timelike geodesic motion; as in Fig. 2, for each possibility,
the motion of a particular null geodesic P(τ ), τ ∈ [0, τ1 ], has been computed
numerically and plotted in the embedding diagram of the AdS wormhole.
148 F. Cremona
Fig. 3 Effective potential and embedding diagram of timelike geodesics in the AdS wormhole
with b = 1 and k = 0.1. (a) L = 0.09, E = 0.1. (b) L = 0.3, E = 0.06. (c) L = 1, E = 0.15
6 Linear Instability of the EBMT Wormhole
6.1 Radial Perturbations of the EBMT Wormhole
In the first three subsections of this section, we focus on the EBMT wormhole
solution defined by Eqs. (21) and (22) and briefly review the deduction of the linear
instability of this configuration as it is performed in [5].
The starting point of this analysis is the introduction of a time-dependent
spherically symmetric perturbation of the EBMT wormhole solution; this means
that we consider a time-dependent spherically symmetric metric
g = −h(t, x)2 dt 2 + q(t, x)2 dx 2 + r(t, x)2 d2
defined by the positions
h(t, x) = 1 + δh(t, x) , q(t, x) = 1 + δq(t, u) , r(t, x) = x 2 + b2 + δr(t, x)

(53)
and a non-self-interactive phantom scalar field defined by

2 x
φ(t, x) = arctan + δφ(t, x) , V (φ) = 0 . (54)
κ b
With the purpose of simplifying the future computation, we stipulate that the
perturbation functions δh, δq, δr and δφ have the expressions1

t x
δh(t, x) := 0 , δq(s, u) := εQ , ,
b b

εb2 t x 2 t x
δr(t, x) := R , , δφ(t, x) := ε , , (55)
x 2 + b2 b b κ b b
where ε is a small real parameter and Q, R, : R × R → R are smooth,

dimensionless functions to be determined. Moreover, let us define the new variables
t x
s := , y := .
b b
on which the new perturbation functions Q, R and depend.
6.2 A Master Equation for R
The system obtained from the linearization of Einstein’s equations (18) and (19)
about the perturbed solution (53), (54), and (55) with respect to the parameter ε can
be not difficultly solved. Indeed, in [5], it is shown that this system implies that the
perturbation functions Q and have the following expression in dependence on R:
2R(s, y)
Q(s, y) = − 3
+ P0 (y) + s P1 (y) , (56)
(y 2 + 1) 2

Ry (s, y)
(s, y) = − y P0 (y) + s P1 (y) + C , (57)
1 + y2
where C ∈ R is an immaterial constant, since appears in the (linearized) field

equations only through its derivatives, and P0 , P1 : R → R are smooth functions
which depend on the set of initial data
Q0 (x) := Q(0, x) , R0 (x) := R(0, x) , Q1 (x) := Qs (0, x) , R1 (x) := Rs (0, x) , (58)
via the relation
1 Note that the choice δh(t, x):= 0 is not restrictive at all since it is always possible to find a gauge
in which this position holds, while the coefficient b2 /(x 2 + b2 ) in front of the function R has been
introduced in order to simplify the subsequent computation.
150 F. Cremona
2Ri (x)
Pi (x) = Qi (x) + 3
(i = 0, 1) . (59)
(1 + x 2 ) 2
Moreover, the perturbation function R has to satisfy the following equation:
(Rss + H R)(s, y) = J0 (y) + s J1 (y) , (60)
where
d2 3
H := − + V, V(y) := − (y ∈ R) , (61)
dy 2 (y 2 + 1)2

Ji (y) := − y 2 + 1 2 Pi (y) + y Pi,y (y) Pi as in Eq. (59) (i = 0, 1) . (62)
This wave-type equation for R is referred to as master equation; the source term J0 (y)+
s J1 (y) is fully determined by the functions P0 and P1 and therefore by the set of
initial data on Qi , Ri (i = 0, 1) (see Eq. (59)).
Taking advantage of some general considerations about Schrödinger-type opera-
tors in the framework of Hilbert spaces, one can prove2 that the operator H appearing
in the master equation is self-adjoint once restricted to the domain of functions
which are square-integrable together with their second derivatives; moreover, H
possesses a continuous spectrum which coincides with the interval [0, +∞) and a
discrete spectrum consisting of exactly one eigenvalue −E ; a numerical estimate for
E is given in [8]:
E 1.40 .
From now on, we call Y−E the normalized real eigenfunction for the eigenvalue −E .
6.3 Linear Instability of the EBMT Wormhole
The linear instability of the EBMT wormhole is proved finding at least one
solution of the linearized Einstein equations such that one or more corresponding
geometrical objects diverge in the large s limit. One can obtain the simplest solution
of this kind choosing the initial data
−2Y−E (y)
R0 (y) := Y−E (y) , R1 (y) := 0 , Q0 (y) = 3
, Q1 (y) := 0 .
(y 2 + 1) 2
2 For more details, see [5].

With these positions, from the general solution to the master equation (60) and from
Eqs. (56) and (57), we get
√ 2R(s, y) Ry (s, y)
R(s, y) = Y−E (y) cosh( Es) , Q(s, y) = − 3
, (s, y) = + C.
(y + 1)
2 2 y2 + 1
One can see that the linearized scalar curvature corresponding to this solution has
the expression

2b2 4ε x √ t
R=− + K cosh E + O(ε2 ) ,
(x 2 + b2 )2 b2 b b

1 E x
K(y) := − Y−E (y) + Y−E,y (y) . (63)
(y 2 + 1)7/2 (1 + y 2 )3/2 (1 + y 2 )5/2
The coefficient of ε of the scalar curvature R evaluated in x = 0 diverges

exponentially for t → ±∞; indeed, it can be proved that the above function K is
not identically zero and in particular
K(0) = (1 − E)Y−E (0) = 0 .
We would like to underline the fact that the above divergence does not depend on
the gauge chosen. Indeed, any smooth coordinate change (t, x) → (t˜, x̃), ε-close to
the identity transforms the linearized scalar curvature R = R(t, s) into a function
R̃ = R̃(t˜, x̃) whose coefficient of the first order in ε, at x̃ = 0, still diverges (again
exponentially) for t˜ → ±∞.
6.4 A Comparison with [2, 4, 8, 14]
Admittedly, the linear instability of the EBMT wormhole has been previously stated
in other works before [5], what is more in a gauge-invariant framework: in 2009,
González, Guzmán and Sarbach proved in [8] the linear instability of a general
wormhole solution of the Einstein-scalar field equations found by Bronnikov in
[1], which includes the EBMT wormhole as a special case; some years later,
in 2011, the linear instability result of Bronnikov’s solution was then extended
by Bronnikov himself, Fabris and Zhidenko in [4] to the whole class of static,
spherically symmetric non-self-interactive scalar field solutions with throats; let
us say that the same extension appears also in the recent survey [2]. However,
in contrast to the previously sketched approach of [5], the main idea of these
papers consists in fixing the radial coordinate and deriving a master equation for
the perturbation function of the scalar field: this is a wave-type equation whose
effective potential is singular at the throat (exactly where the derivative of the radial
coordinate becomes null). In order to deal with this singularity, the authors of [8]
have to introduce a suitable first order operator which, once applied to the function
152 F. Cremona
fulfilling the master equation, transforms the master equation into a regular one; this
method was then generalized and called “S-deformation method” in [4] (and also in
[2]). Let us add that the same approach is used also in [14] in order to prove the
linear instability of a multidimensional generalization of the EBMT wormhole to
spacetime dimension d + 1 (with d > 3).
Summing up, we want to stress the fact that the derivation of the linear
instability of the EBMT wormhole proposed by Cremona, Pirotta and Pizzocchero
introduces a novelty in the study of the stability of wormhole configurations: indeed,
the instability result is obtained without encountering any singularity and no S-
deformation of the master equation is necessary.
However, one could object that Cremona et al. do not make use of gauge-invariant
quantities from the very beginning of their derivation and have to prove a posteriori
the gauge invariance of the instability result. In order to respond to this criticism,
let us say that very recently, this problem has been overcome in [6], where a gauge-
invariant strategy is introduced of studying the linear stability of static, spherically
symmetric scalar field solutions with throats, obtaining a master equation which is
non-singular everywhere (provided that the scalar field does not have critical points).
Moreover, as this approach also includes the case of self-interactive scalar field, it
is used to prove for the first time the linear instability of the AdS wormhole.
Acknowledgments This work was supported by INdAM, Gruppo Nazionale per la Fisica
Matematica and Università degli Studi di Milano.
The author acknowledges prof. Livio Pizzocchero for his kind and professional support and
prof. Sergio Cacciatori for the opportunity of contributing to this publication.
References
1. K.A. Bronnikov, Scalar-tensor theory and scalar charge. Acta Phys. Polon. B4, 251–266 (1973)
2. K.A. Bronnikov, Scalar fields as sources for wormholes and regular black holes. Particles 1,
56–81 (2018). https://doi.org/10.3390/particles1010005
3. K.A. Bronnikov, J.C. Fabris, Regular phantom black holes. Phys. Rev. Lett. 96, 251101 (2006)
4. K.A. Bronnikov, J.C. Fabris, A. Zhidenko, On the stability of scalar-vacuum space-times. Eur.
Phys. J. C Particles Fields 71, 1791 (12 pp.) (2011)
5. F. Cremona, F. Pirotta, L. Pizzocchero, On the linear instability of the Ellis-Bronnikov-Morris-
Thorne wormhole. Gen. Relativ. Gravitat. 51, 19 (2019)
6. F. Cremona, L. Pizzocchero, O. Sarbach, Gauge-invariant spherical linear perturbations of
wormholes in Einstein gravity minimally coupled to a self-interacting phantom scalar field.
arXiv:1911.13103 [gr-qc] (2019)
7. H. Ellis, Ether flow through a drainhole: A particle model in general relativity. J. Math. Phys.
14, 104–118 (1973)
8. J.A. González, F.S. Guzmán, O. Sarbach, Instability of wormholes supported by a ghost scalar
field. I: Linear stability analysis. Classical and Quantum Gravity 26, 015010 (14 pp.) (2009)
9. J.B. Hartle, Gravity: An Introduction to Einstein’s General Relativity (Addison-Wesley, San
Francisco, 2003)
10. S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time (Cambridge University,
Cambridge, 1973)
11. C.W. Misner, K.S. Thorne, J.A. Wheeler Gravitation (W. H. Freeman, San Francisco, 1973)
12. M.S. Morris, K.S. Thorne, Wormholes in spacetime and their use for interstellar travel: a tool
for teaching general relativity. Am. J. Phys 56, 395–412 (1988)
13. M.M. Postnikov, The Variational Theory of Geodesics (Dover Publications, New York, 1983)
14. T. Torii, H. Shinkai Wormholes in higher dimensional space-time: Exact solutions and their
linear stability analysis. Phys. Rev. D88, 064027 (2013)
15. M. Visser, Lorentzian Wormholes: From Einstein to Hawking (Springer, New York, 1996)
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New Trends in the General Relativistic
Poynting–Robertson Effect Modeling
Vittorio De Falco
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
2 General Relativistic 3D PR Effect Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
2.1 Critical Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
3 Stability of the Critical Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4 Analytical form of the Rayleigh Dissipation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.1 Discussions of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
1 Introduction
The actual revolutionary discoveries occurred in the last years represented by the
detection of gravitational waves first from a binary black holes (BHs) [1] and then
from a neutron stars (NSs) [2] systems and the first imaging of the matter motion
around the supermassive BH in M87 Galaxy [3] constitute a strong motivation
to improve the actual theoretical models to validate Einstein theory or possible
extension of it when benchmarked with the observations. The motion of relatively
small-sized test particles, like dust grains or gas clouds, meteors, accretion disk
matter elements, around radiating sources located outside massive compact objects
is strongly affected by gravitational and radiation fields, and an important effect to
be taken into account is the general relativistic PR effect [4, 5].
This phenomenon occurs each time the radiation field invests the test particle,
raising up its temperature, which for the Stefan–Boltzmann law starts remitting
radiation. This process of absorption and remission of radiation generates a recoil
V. De Falco ()
Department of Physics, Scuola Superiore Meridionale, Largo San Marcellino, Naples, Italy
e-mail: vittorio.defalco-ssm@unina.it
156 V. De Falco
force opposite to the test body orbital motion. Such mechanism removes thus
very efficiently angular momentum and energy from the test particle, forcing it
to spiral inward or outward depending on the radiation field intensity. This effect
has been extensively studied in Newtonian gravity within Classical Mechanics
[4] and Special Relativity [5] and then applied in the Solar System [6]. Only in
2009–2011, this model has been proposed in General Relativity (GR) by Bini and
collaborators within the equatorial plane of the Ker spacetime [7, 8]. Recently, it
has been extended in the 3D space in Kerr metric [9–11]. One of the most evident
implications of such effect is the formation of stable structures, termed critical
hypersurfaces, around the compact object [12]. This phenomenon has been analyzed
also under a Lagrangian formulation [13–15]. The novel aspect of such approach
consists in the introduction of new techniques to deal with the nonlinearities in
gravity patterns based on two new fundamental aspects: (1) use of an integrating
factor to make closed differential forms [13] and (2) development of a new method
termed energy formalism, which permits to analytically determine the Rayleigh
potential associated with the radiation force [14, 15].
The chapter is structured as follows: in Sect. 2, the 3D model and its proprieties
are described; in Sect. 3, the stability of the critical hypersurfaces is discussed within
the Lyapunov theory; in Sect. 4, we analytically determine the Rayleigh dissipation
function by using the energy formalism; finally, in Sect. 5, the conclusions are
drawn.
2 General Relativistic 3D PR Effect Model
We consider a rotating compact object, whose geometry is described by the Kerr

metric. Using the signature (−, +, +, +) and geometrical units (c = G = 1),
the metric line element, ds 2 = gαβ dx α dx β , in Boyer–Lindquist coordinates,
parameterized by mass M and spin a, reads as [16]

2Mr 4Mra sin2 θ
ds = 2
− 1 dt 2 − dtdϕ + dr 2 +dθ 2 +ρ sin2 θ dϕ 2 , (1)

where ≡ r 2 +a 2 cos2 θ , ≡ r 2 −2Mr +a 2 , and ρ ≡ r 2 +a 2 +2Ma 2 r sin2 θ/.

The determinant of the metric is g = −Σ 2 sin2 θ . The orthonormal frame adapted
to the zero angular momentum observers (ZAMOs) is [9, 10]
(∂t − N ϕ ∂ϕ ) ∂r ∂θ ∂ϕ
etˆ ≡ n = , er̂ = √ , eθ̂ = √ , eϕ̂ = √ , (2)
N grr gθθ gϕϕ
where N = (−g tt )−1/2 and N ϕ = gtϕ /gϕϕ . The nonzero ZAMO kinematical
quantities in the decomposition of the ZAMO congruence are acceleration a(n) =
New Trends in the General Relativistic Poynting–Robertson Effect Modeling 157
∇n n, expansion tensor along the ϕ̂-direction θϕ̂ (n), and the relative Lie curvature
vector k(Lie) (n) (see Table 1 in [9], for their explicit expressions).
The radiation field is modeled as a coherent flux of photons traveling along null
geodesics on the Kerr metric. The related stress–energy tensor is [9, 10]
T μν = I 2 k μ k ν , k μ kμ = 0, k μ ∇μ k ν = 0, (3)
where I is a parameter linked to the radiation field intensity and k is the photon
four-momentum field. Splitting k with respect to the ZAMO frame, we obtain [10]
k = E(n)[n + ν̂(k, n)], (4)

ν̂(k, n) = sin ζ sin β er̂ + cos ζ eθ̂ + sin ζ cos β eϕ̂ , (5)
where E(n) is the photon energy measured in the ZAMO frame, ν̂(k, n) is the
photon spatial unit relative velocity with respect to the ZAMOs, and β and ζ are
the two angles measured in the ZAMO frame in the azimuthal and polar directions,
respectively. The radiation field is governed by the two impact parameters (b, q),
associated, respectively, with the two emission angles (β, ζ ). The radiation field
photons are emitted from a spherical rigid surface having a radius R centered at the
origin of the Boyer–Lindquist coordinates and rotating with angular velocity .
The photon impact parameters are [10]

gtϕ + gϕϕ
b=− , q = b2 cot2 θ − a 2 cos2 θ . (6)
gtt + gtϕ r=R r=R
The related photon angles in the ZAMO frame are [10]
bN
cos β = √ , ζ = π/2. (7)
gϕϕ (1 + bN ϕ )
The parameter I has the following expression [10]:
I02
I2 = , (8)
2
r 2 + a 2 − ab − q + (b − a)2
where I0 is I evaluated at the emitting surface.

A test particle moves with a timelike four-velocity U and a spatial three-velocity
with respect to the ZAMO frames, ν(U, n), which both read as [10]
U = γ (U, n)[n + ν(U, n)], (9)

ν = ν(sin ψ sin αer̂ + cos ψeθ̂ + sin ψ cos αeϕ̂ ), (10)
158 V. De Falco
where γ (U, n) ≡ γ = 1/ 1 − ||ν(U, n)||2 is the Lorentz factor, ν = ||ν(U, n)||,

and γ (U, n) = γ . We have that ν represents the magnitude of the test particle spatial
velocity ν(U, n), α is the azimuthal angle of the vector ν(U, n) measured clockwise
from the positive ϕ̂ direction in the r̂ − ϕ̂ tangent plane in the ZAMO frame, and
ψ is the polar angle of the vector ν(U, n) measured from the axis orthogonal to the
r̂ − ϕ̂ tangent plane in the ZAMO frame.
We assume that the radiation test particle interaction occurs through the Thomson
scattering, characterized by a constant momentum transfer cross-section σ , inde-
pendent from direction and frequency of the radiation field. We can split the photon
four-momentum (4) in terms of the velocity U as [10]
k = E(U )[U + V̂(k, U )], (11)
where E(U ) is the photon energy measured by the test particle. The radiation force
can be written as [10]
2
F(rad) (U )α̂ ≡ −σ˜I (T α̂ β̂ U β̂ + U α̂ T μ̂ β̂ Uμ̂ U β̂ ) = σ̃ [IE(U )]2 V̂(k, U )α̂ , (12)
where m is the test particle mass and the term σ̃ [IE(U )]2 reads as [10]
A γ 2 (1 + bN ϕ )2 [1 − ν sin ψ cos(α − β)]2

σ̃ [IE(U )]2 = , (13)
2
N 2 r 2 + a 2 − ab − q + (b − a)2
with A = σ̃ [I0 Ep ]2 being the luminosity parameter, which can be equivalently

written as A/M = L/LEDD ∈ [0, 1] with L the emitted luminosity at infinity
and LEDD the Eddington luminosity, and Ep = −kt is the conserved photon
energy along the test particle trajectory. The terms V̂(k, U )α̂ are the radiation field
components, whose expressions are [10]
sin β
V̂ r̂ = − γ ν sin ψ sin α, V̂ θ̂ = −γ ν cos ψ, (14)
γ [1 − ν sin ψ cos(α − β)]

cos β sin ψ cos(α − β) − ν
V̂ ϕ̂ = − γ ν sin ψ cos α, V̂ tˆ = γ ν .
γ [1 − ν sin ψ cos(α − β)] 1 − ν sin ψ cos(α − β)
Collecting all the information together, it is possible to derive the resulting

equations of motion for a test particle moving in a 3D space, which are [10]
dν 1
=− sin α sin ψ a(n)r̂ + 2ν cos α sin ψ θ (n)r̂ ϕ̂ (15)
dτ γ
σ̃ [E(U )]2
+ cos ψ a(n)θ̂ + 2ν cos α sin ψ θ (n)θ̂ ϕ̂ + V̂ tˆ,
γ 3ν
dψ γ
= sin ψ a(n)θ̂ + k(Lie) (n)θ̂ ν 2 cos2 α + 2ν cos α sin2 ψ θ (n)θ̂ ϕ̂
dτ ν

− sin α cos ψ a(n)r̂ + k(Lie) (n)r̂ ν 2 + 2ν cos α sin ψ θ (n)r̂ ϕ̂ (16)
σ̃ [E(U )]2 tˆ θ̂

+ V̂ cos ψ − V̂ ν ,
γ ν 2 sin ψ
dα γ cos α
=− a(n)r̂ + 2θ (n)r̂ ϕ̂ ν cos α sin ψ + k(Lie) (n)r̂ ν 2 (17)
dτ ν sin ψ
σ̃ [E(U )]2 cos α
+k(Lie) (n)θ̂ ν 2 cos2 ψ sin α + V̂ r̂ − V̂ ϕ̂ tan α ,
γ ν sin ψ
dr γ ν sin α sin ψ
U r̂ ≡ = √ , (18)
dτ grr
dθ γ ν cos ψ
U θ̂ ≡ = √ , (19)
dτ gθθ
dϕ γ ν cos α sin ψ γ Nϕ
U ϕ̂ ≡ = √ − , (20)
dτ gϕϕ N
dt γ
U tˆ ≡ = , (21)
dτ N
where τ is the affine parameter along the test particle trajectory.
2.1 Critical Hypersurfaces
The dynamical system defined by Eqs. (15)–(20) exhibits a critical hypersurface

outside around the compact object, where there exists a balance among gravitational
and radiation forces, see Fig. 1. On such region, the test particle moves purely
circular with constant velocity (ν = const) with respect to the ZAMO frame
(α = 0, π ) and the polar axis orthogonal to the critical hypersurface (ψ = ±π/2).
These requirements entail dν/dτ = dα/dτ = 0, from which we have [10]
ν = cos β, (22)
r̂ r̂
a(n) + 2θ (n) ϕ̂ ν + k(Lie) (n) ν r̂ 2
(23)
A(1 + bN ϕ )2 sin3 β
= 2 ,
N 2γ r(crit) + a − ab − (crit) q + (b − a)
2 2 2
160 V. De Falco
Fig. 1 Left panel: critical hypersurfaces for = 0 and the luminosity parameters A =
0.5, 0.7, 0.8, 0.85, 0.87, 0.9 at a constant spin a = 0.9995. The respective critical radii in the
eq
equatorial plane are r(crit) ∼ 2.71M, 4.01M, 5.52M, 7.04M, 7.99M, 10.16M, while at poles they
pole
are r(crit) ∼ 2.97M, 4.65M, 6.56M, 8.38M, 9.48M, 11.9M. Right panel: critical hypersurfaces
for an NS (gray sphere) with = 0.031, R = 6M, and luminosity parameters A =
0.75, 0.78, 0.8, 0.85, 0.88 at a constant spin a = 0.41. The respective critical radii in the
eq
equatorial plane are r(crit) ∼ 8.88M, 10.61M, 12.05M, 17.26M, 22.43M , while at poles they
pole
are r(crit) ∼ 4.73M, 5.28M, 5.74M, 7.43M, 9.11M. The red arrow is the polar axis
where the first condition means that the test particle moves on the critical hyper-
surface with constant velocity equal to the azimuthal photon velocity, whereas the
second condition determines the critical radius r(crit) as a function of the polar angle
through an implicit equation, once A, a, R , and are assigned.
In general, we have dψ/dτ = 0 because the ψ angle changes during the test
particle motion on the critical hypersurface, having the so-called latitudinal drift.
This effect, occurring for the interplay of gravitational and radiation actions in the
polar direction, brings definitively the test particle on the equatorial plane [9, 10].
Only for ψ = θ = π/2, we have dψ/dτ = 0, corresponding to the equatorial ring.
However, we can have dψ/dτ = 0, also for a θ = θ̄ = π/2, having the so-called
suspended orbits. The condition for this last configuration for b = 0 reads as [10]
a(n)θ̂ + k(Lie) (n)θ̂ ν 2 + 2ν sin2 ψ θ (n)θ̂ ϕ̂

A(1 + bN ϕ )2 (1 − cos2 β sin ψ) cos β (24)
+ 2 = 0,
γ N2 r(crit) + a − ab − (crit) q + (b − a) tan ψ
2 2 2
which permits to be solved in terms of ψ. Instead, for b = 0, we obtain ψ = ±π/2

[9]. The critical points are either the suspended orbits or the equatorial ring, where
the test particle ends its motion. In Fig. 2, we display some selected test particle
trajectories to give an idea how the PR effect alters the matter motion surrounding a
radiation source around a compact object [10].
Fig. 2 Left panel: test particle trajectories around an NS of spin a = 0.41, radius R = 6M,
angular velocity = 0.031, and luminosity parameter A = 0.8, starting at the position
(r0 , θ0 ) = (15M, 10◦ ) with the initial velocity ν0 = 0.01 oriented in the azimuthal corotating
direction (orange) and oriented radially toward the emitting surface (red). Right panel: test particle
trajectories around an NS of spin a = 0.07, radius R = 6M, angular velocity = 0.005, and
luminosity parameter A = 0.85, starting at the position (r0 , θ0 ) = (15M, 10◦ ) with the initial
velocity ν0 = 0.01 oriented in the azimuthal corotating direction (orange) and oriented radially
toward the emitting surface (red). The black sphere corresponds to the emitting surface of the NS.
The blue–gray surface denotes the critical hypersurface
3 Stability of the Critical Hypersurfaces
To prove the stability of the critical hypersurfaces, we consider only those initial
configurations, where the test particle ends its motion on them without escaping at
infinity. Once the stability has been proven, it immediately follows that the critical
equatorial ring is a stable attractor (region where the test particle is attracted for
ending its motion), and the whole critical hypersurface is a basin of attraction [12].
Bini and collaborators have proved it only in the Schwarzschild case within the
linear stability theory (see Appendix in Ref. [8]). This method consists in linearizing
the dynamical system toward the critical points of the critical hypersurface and then
looking at its eigenvalues. Theoretically, such method is simple, but practically it
implies several calculations (especially in the Kerr case).
There is a simpler and more physical approach based on the Lyapunov theory.
The dynamical system (15)–(20), ẋ = f (x), is defined in the domain D, while
the critical hypersurface is defined by H. Let = Λ(x) be a real-valued function,
continuously differentiable in all points of D, and then Λ is a Lyapunov function for
ẋ = f (x) if it fulfills the following conditions:
(I) (x) > 0, ∀x ∈ D \ H; (25)

(II) (x0 ) = 0, ∀x0 ∈ H; (26)
(III) ˙
(x) ≡ ∇(x) · f (x) ≤ 0, ∀x ∈ D. (27)
162 V. De Falco
Once the Lyapunov function has been found for all points belonging to the critical
hypersurface H, a theorem due to Lyapunov assures that H is stable [12].
The advantage to use this approach relies on easily studying the behavior of a
dynamical system without knowing the analytical solution. The Lyapunov function
is not unique, and there is no fixed rules to determine it, indeed several times
one is guided by the physical intuitions. For the general relativistic PR effect,
three different Lyapunov functions have been determined. The proof that they are
Lyapunov function is based on expanding all the kinematic terms with respect to the
radius estimating thus their magnitude (see Ref. [12] for further details).
• The relative mechanical energy of the test particle with respect to the critical
hypersurface measured in the ZAMO frame is

m 2 2
1 1
K = ν − νcrit + (A − M) − , (28)
2 r rcrit
where νcrit (θ ) = [cos β]r=rcrit (θ) , which includes as a particular case the velocity
νeq = [cos β]r=rcrit (π/2) in the equatorial ring. Its derivative is
dν d(cos β)

A−M
K̇ = m sgn ν − cos β ν
2 2
− cos β − ṙ, (29)
dτ dτ r2
where sgn(x) is the signum function.

• The angular momentum of the test particle measured in the ZAMO frame is
L = m(rν sin ψ cos α − rcrit νcrit ). (30)
Its derivative is given by

d(νcrit ) dν
L̇ = m −ṙcrit νcrit − rcrit +r cos α sin ψ + ν(ṙ cos α sin ψ
dτ dτ (31)
−r sin α sin ψ α̇ + r sin α cos ψ ψ̇) .
• The Rayleigh dissipation function is (see Sect. 4 for its derivation and meaning)

Ecrit E
F = σ̃ I lg
2
− lg , (32)
Ep Ep
where Ep is the photon energy and E ≡ E(U ), defined as
Ep
E ≡ −kα U α = γ (1 + bN ϕ )[1 − ν sin ψ cos(α − β)]. (33)
N
Ecrit is the energy E evaluated on the critical hypersurface, given by
Ep |(sin β)crit | ϕ
Ecrit = [E]r=R ,α=0,π,ψ=±π/2,ν=νcrit = (1 + bNcrit ). (34)
Ncrit
Its derivative is

˙ Ecrit E 2 Ėcrit Ė
Ḟ = σ̃ (I ) lg
2 − lg + σ̃ I − . (35)
Ep Ep Ecrit E
In Fig. 3, we calculate a test particle orbit in the equatorial plane reaching the critical
hypersurface, and in the other panels, we show the three proposed functions (i.e.,
0.4
0.3
0.2 Ttouch
20
0.1
0
r sinφ (M)
0
0
−0.04 −0.02
−20
−20 0 20 1 10 100 1000 104 105

r cosφ (M) τ (M)
Ttouch Ttouch
6
1.5
4
1
2
0.5
0
0
0
0
−0.2 −0.15 −0.1 −0.05
−0.05
−0.1
1 10 100 1000 104 105 1 10 100 1000 104 105

τ (M) τ (M)
Fig. 3 We show a test particle orbit and the related three Lyapunov functions. Upper left panel:
test particle moving around a rotating compact object with mass M = 1, spin a = 0.3, luminosity
parameter A = 0.2, and photon impact parameter b = 0.√The test particle starts its motion at the
position (r0 , ϕ0 ) = (30M, 0) with velocity (ν0 , α0 ) = ( M/r0 , 0). The critical hypersurface is
a circle with radius r(crit) = 2.07M. The energy (see Eqs. (28) and (29), and upper right panel),
the angular momentum (see Eqs. (30) and (31), and lower left panel), and the Rayleigh potential
(see Eqs. (33) and (35), and lower right panel) together with their τ -derivatives are all expressed
in terms of the proper time τ . The dashed blue lines in all plots represent the proper time Ttouch at
which the test particle reaches the critical hypersurface, and it amounts to Ttouch = 2915M
164 V. De Falco
K, L, F) together with their derivatives (i.e., K̇, L̇, Ḟ), to graphically prove that
they verify the three proprieties to be Lyapunov functions.
It is important to note that the first two Lyapunov functions (energy and angular
momentum) are written using the classical definition, and not the general relativistic
expression, as instead it has been done with the third Lyapunov function. This is
not in contradiction with the definition of Lyapunov function, rather they are very
useful to carry out more easily the calculations. For example, even a mathematical
function with no direct physical meaning with the system under study but verifying
the conditions to be a Lyapunov function is a good candidate to prove the stability
of the critical hypersurfaces.
4 Analytical form of the Rayleigh Dissipation Function
We describe the energy formalism, which is the method used to derive the Rayleigh
potential [15]. The motion of the test particle occurs in M, a simply connected
domain (the region outside of the compact object including the event horizon).
We denote with T M the tangent bundle of M, whereas T ∗ M stands for the
cotangent bundle over M. Let ω : T M → T ∗ M be a smooth differential semi-
basic one-form. Defined X = (t, r, θ, ϕ) and U = (U t , U r , U θ , U ϕ ), the radiation
force components (12) are the components of the differential semi-basic one-form
ω(X, U ) = F(rad) (X, U )α dXα . We note that ω is closed under the vertical exterior
derivative dV if dV ω = 0. The local expression of this operator is
∂F
dV F = dXα . (36)
∂Uα
For the Poincaré lemma (generalized to the vertical differentiation), the closure
condition and the simply connected domain M guarantee that ω is exact. Therefore,
it exists a 0-form V (X, U ) ∈ C ∞ (T M, m), called primitive, such that −dV V = ω.
Due to the nonlinear dependence of the radiation force on the test particle
velocity field, the semi-basic one-form turns out to be not exact [13]. However,
the PR phenomenon exhibits the peculiar propriety according to which ω(X, U )
2
becomes exact through the introduction of the integrating factor μ = Ep /E [13].
β
Considering the energy E = −kβ U and substituting all the occurrences of E in
F(rad) (X, U )α , see Eq. (12), we obtain [14, 15]
F(rad) (X, U )α = −k α E(X, U ) + E(X, U )2 U α . (37)
Using the chain rule from the velocity to the energy derivative operator, we have
∂ (·) ∂ (·)
= −k α . (38)
∂Uα ∂E
Therefore, the V function satisfies the usual primitive condition [14, 15]
α ∂V
μF(rad) = kα . (39)
∂E
Such differential equation for V contains the k α factor, which represents an obstacle
for the integration process. To get rid of this term, we can consider the scalar product
of both members of Eq. (39) by Uα , which permits to obtain a more manageable
integral equation for V [14, 15], i.e.,

μF α
V =− dE + f (X, U ), (40)
E
where f is constant with respect to E, i.e., ∂f/∂E = 0 and V is still a function of

the local coordinates (X, U ). Integrating Eq. (40), the final result is [14, 15]

E 1 α
V = σ̃ I ln
2
+ Uα U + 1 . (41)
Ep 2
Equation (41) is consistent with the classical description [4, 5]. The PR effect
configures as the first dissipative nonlinear system in GR for which we know the
analytical form of the Rayleigh potential.
4.1 Discussions of the Results
The energy function E and the chain rule both represent the fundamental aspects
of the energy formalism, since they permit to simplify the demanding calculations
for the V primitive. We are able to substantially reduce the coordinates involved in
the calculations, passing from 4 initial parameters, represented by U , to one only,
i.e., the energy E. In particular, the f function occurring in Eq. (40) embodies our
ignorance about the analytic form of V as a function of the local coordinates (X, U ).
In some cases, as in our model, the f function can be determined by applying the
integration process for an exact differential one-form. Such method has also the
peculiar propriety, that it is independent from the considered metric, permitting to
be applied to generic metric theories of gravity, and for its generality and simplicity,
it can also be applied in different physical and mathematical fields.
The Rayleigh potential (40) is a valuable tool to investigate the proprieties of
the general relativistic PR effect and more in general the radiation processes in
high-energy astrophysics. This potential contains a logarithm of the energy, which
physically is interpreted as the absorbed energy from the test particle. Therefore, it
represents a new class of functions, never explored and discovered in the literature,
used to describe the radiation absorption processes. Another important implication
of the Rayleigh potential relies on the direct connection between theory and
166 V. De Falco
V
10
−2
a) b) c)
5
r sinϕ (M)
−4
0
−6
−5
−5 0 5 10 2 4 6 8 10 −1 −0.5 0 0.5 1
r cosϕ (M) r (M) cosϕ
−2
d) e) f)
−4
V
−6
0.1 1 10 1001000 104 −0.1 −0.05

. 0 0 0.1 0.2 0.3
t (M) r
.
rϕ
Fig. 4 Test particle trajectory with the Rayleigh potential V for mass M = 1 and spin a = 0.1,
luminosity parameter A = 0.1, and photon impact parameter b = 1. The test particle moves
in√the spatial equatorial plane with initial position (r0 , ϕ0 ) = (10M, 0) and velocity (ν0 , α0 ) =
( 1/10M, 0). (a) Test particle trajectory spiraling toward the BH and stopping on the critical
radius (red dashed line) r(crit) = 2.02M. The continuous green line is the event horizon radius
+
r(EH) = 1.99M. Rayleigh potential versus (b) radial coordinate, (c) azimuthal coordinate, (d) time
coordinate, (e) radial velocity, and (f) azimuthal velocity. The blue dashed line in panel (e) marks
the minimum value attained by the radial velocity, corresponding to ṙ = −0.13
observations. In Fig. 4, we show in panel (a) the test particle trajectory (what we can
observe) and in panels (b) − (f ) the Rayleigh potential in terms of the coordinates
r, ϕ, t, ṙ, ϕ̇, respectively (what comes from the theory). Therefore, observing the
test particle motion, it is possible to theoretically reconstruct the Rayleigh function;
vice versa new Rayleigh functions can be proposed to study then the dynamics and
see what we should observe (see Ref. [15] for details).
5 Conclusions
In this work, we have presented the fully general relativistic treatment of the 3D
PR effect in the Kerr geometry, which extends the previous works framed in the
2D equatorial plane of relativistic compact objects. The radiation field comes from
a rigidly rotating spherical source around the compact object. The emitted photons
are parametrized by two impact parameters (b, q), where b can be variable and
q depends on the value assumed by b and the polar angle θ , position occupied
by the test particle in the 3D space. The resulting equations of motion represent
a system of six coupled ordinary and highly nonlinear differential equations of first
order. The motion of test particles is strongly affected by PR effect together with
general relativistic effects. Such dynamical system admits the existence of critical
hypersurfaces, regions where the gravitational attraction is balanced by the radiation
forces.
We have presented the method to prove the stability of the critical hypersurfaces
by employing the Lyapunov functions. Such strategy permits to simplify the
calculations and to catch important physical aspects of the PR effect. Three different
Lyapunov functions have been proposed, all with a different and precise meaning.
The first two are deduced by the definition of the PR effect, which removes energy
and angular momentum from the test particle. The third example is less intuitive
because it is based on the Rayleigh dissipation function, determined by the use of
an integrating factor and the introduction of the energy formalism.
Such method revealed to be very useful for two reasons: (1) a substantial
reduction of the calculations from the 4 variables (i.e., the velocities U ) to only
one (i.e., the energy E) and (2) the obtained expression of the V potential as a
function of E suffices for the description of the dynamics, being very important
whenever the evaluation of f (X, U ) turns out to be too laborious. In this way, we
have obtained for the first time an analytical expression of the Rayleigh potential in
GR, and we have discovered a new class of functions, represented by the logarithms,
which physically describe the absorption processes in high-energy astrophysics.
As future projects, we plan to improve the actual theoretical assessments used to
treat the radiation field in some ingenue aspects, like the momentum transfer cross-
section will be not anymore constant, but it will depend on the angle and frequency
of the incoming radiation field, and the radiation field is not emitted anymore by
a point-like source, but from a finite extended source. We would like also to apply
this theoretical model to some astrophysical situations in accretion physics, like
accretion disk model, type-I X-ray burst, and photospheric radius expansion.
The new method to prove the stability of the critical hypersurfaces through
Lyapunov functions can be easily applied to any possible extensions of the general
relativistic PR effect model, naturally with the due modifications. Instead, the energy
formalism opens up new frontiers in the study of the dissipative systems in metric
theory of gravity and more broadly in other mathematical and physical research
fields thanks to its general structure and versatile applicability. It permits to acquire
more information on the mathematical structure and the physical meaning of the
problem under study, because as discussed in Fig. 4, it is incredibly evident the
profound connection between observations and theory.
Acknowledgments The author thanks the Silesian University in Opava and Gruppo Nazionale di
Fisica Matematica of Istituto Nazionale di Alta Matematica for support. The results contained in
the present paper have been partially presented at the summer school DOOMOSCHOOL 2019.
The author acknowledges the support of INFN sez. di Napoli, iniziative specifiche TEONGRAV.
168 V. De Falco
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Brief Overview of Numerical Relativity
Mario L. Gutierrez Abed
To my parents, my wife Laura, and my soon-to-be-born baby

boy Alessandro.
Contents
1 ADM Formalism of Numerical Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
1.1 ADM Variables and Adapted Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
1.2 ADM Evolution and Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
2 BSSN Formalism of Numerical Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
3 Further Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
3.1 Initial Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
3.2 Gauge Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
3.3 Potential Application to Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Mathematics Subject Classification (2000) Primary 83C05, 83F05; Secondary

83-01, 83-06
1 ADM Formalism of Numerical Relativity
We want to determine the dynamical evolution of a physical system governed by the

field equations of general relativity (GR); this will essentially boil down to formulate
the EFEs as a Cauchy problem with constraints. Analogous to classical dynamics,
where the evolution of a system is uniquely determined by the initial positions and
velocities of its constituents, we can evolve the gravitational field as determined by
specifying the metric quantities gab and ∂t gab at a given (initial) instant of time t.
This process can be simplified (for the sake of clarification) in two steps:
M. L. G. Abed ()
Newcastle University, Department of Mathematics, Statistics & Physics, Newcastle upon Tyne,
UK
e-mail: mariogutierrezabed@gmail.com
170 M. L. G. Abed
(i) Specify gab and ∂t gab (actually, it will be related quantities) everywhere on
some 3D spacelike hypersurface labelled by coordinate x 0 = t = constant.
(ii) Provided that we can obtain expressions for second-time derivatives of the 4-
metric gab at all points on the hypersurface from the EFEs, we then integrate
forward in time the metric quantities from step (i).
However, even though this seems like a straightforward proposal, one must first
define what is actually meant by “spatial hypersurface,” since in GR there is no
preferred timelike direction and, crucially, no global concept of time. This makes the
problem of solving the Einstein equations numerically substantially different from
other typical non-relativistic Cauchy problems. Moreover, it turns out that obtaining
the appropriate expressions for ∂t2 gab for such an integration is not so trivial. We
require a total of ten second derivatives which, at first glance at least, appear to be
achievable from the ten field equations (4) Gab = 8π T ab .1,2 However, note that by
the (contracted) Bianchi identities,3
0 = ∇b (4) Gab
= ∂0 (4) Ga0 + ∂i (4) Gai + (4) Gbc(4) bc
a
+ (4) Gab(4) bc
c
,
we get
∂t (4) Ga0 = −∂i (4) Gai − (4) Gbc(4) bc

a
− (4) Gab(4) bc
c
. (1)
Since no term on the RHS of (1) contains third-time derivatives (or higher), this
implies that there are no second-time derivatives contained in (4) Ga0 , and thus the
four equations
(4)
Ga0 = 8π T a0 (2)
do not yield any information whatsoever on how the fields evolve in time. Instead,
they function as four constraints on the initial data, i.e., four relations that must be
satisfied from the onset on the initial hypersurface at x 0 = t if we are to have
a physically meaningful system. Thus, we can see that the only true dynamical
(evolution) equations are encoded in the remaining six field equations
1 We adopt the usual convention in which 4D objects are distinguished from their 3D counterparts
by using a superscript (4). There are exceptions to this rule, where 3D objects are denoted by
different symbols from their 4D cousins (examples are Tab and gab whose 3D counterparts are
denoted by Sab and γab , respectively; these exceptions are merely a matter of convention, of course,
but they are widely used in the numerical relativity literature).
2 We will use geometric units (G = c = 1) throughout.
3 We also use standard index convention, whereby the letters a − h and o − z are used for 4D
spacetime indices that run from 0 to 3, whereas the letters i − n are reserved for 3D spatial
indices that run from 1 to 3. Lowercase Greek letters are reserved for components in a chosen
basis (see [27] for reference).
Brief Overview of Numerical Relativity 171
(4)
Gij = 8π T ij . (3)
We will see later on that certain projections of (2) and (3) onto the hypersurfaces
will indeed yield the desired constraint and evolution equations of the system.
Thus, in essence, we are after a method to formulate the evolution of a
gravitational field as a Cauchy problem; i.e., given adequate initial (and boundary)
conditions, the fundamental equations must predict the future (or past) evolution of
the system. However, an inconvenient hurdle that we immediately run into is that
in GR space and time are treated on equal footing, thus making the notion of “time
evolution” not as straightforward as our intuition dictates. Therefore, our first order
of business is to somehow split the roles of space and time in a clear manner (and by
this we do not mean “forget about GR and go back to Newtonian/Galilean gravity”).
It turns out that there is a special class of spacetimes, known as globally
hyperbolic spacetimes, that will allow us this sought-after time/space split. First,
recall that a Cauchy surface is a spacelike hypersurface embedded in an ambient
manifold M such that each causal curve without endpoint in M intersects once
and only once. Equivalently, a Cauchy surface for a spacetime M is an achronal
subspace ⊂ M (i.e., a subspace in which no two points are timelike-related)
which is met by every inextendible causal curve in M. Now, we properly define the
concept of global hyperbolicity:
Definition 1.1 A spacetime M is said to be globally hyperbolic if it admits a
Cauchy surface. Equivalently, M is globally hyperbolic if it satisfies the strongly
causal condition (i.e., if every p ∈ M has arbitrarily small neighbourhoods U in
which every causal curve with endpoints in U is entirely contained in U ) and if the
“causal diamonds” J + (p) ∩ J − (q) are compact for all p, q ∈ M.4
Global hyperbolicity is the standard condition in Lorentzian geometry that
ensures the existence of maximal causal geodesic segments. Physically, this con-
dition is closely connected to the issue of classical determinism and the strong
cosmic censorship conjecture. Ringström [24] Even though this is by no means a
condition satisfied a priori by all spacetimes, the 3 + 1 formalism assumes that all
physically reasonable spacetimes are of this type. This assumption is justified by
the desire to have “nice” chronological/causal features in our spacetime (i.e., no
grandfather paradox or any similar pathological behaviour). Moreover, the use of
global hyperbolicity allows us to foliate our full 4D spacetime M in such a way
that we can stack 3D spacelike Cauchy slices along a universal time axis, by virtue
of M having topology × R. This is certainly not the only way to foliate M, but
it is the most suitable one for the 3 + 1 formalism.
4 Here,we used the standard notation, where J + (p) = {q ∈ M | p ≤ q} and J − (p) = {q ∈ M |

q ≤ p} are the causal future and causal past, respectively, of p ∈ M.
172 M. L. G. Abed
1.1 ADM Variables and Adapted Coordinates
Given the foliation granted by the global hyperbolic condition described above, we
can now determine the geometry of the region of spacetime contained between two
adjacent hypersurfaces t and t+dt from just three basic ingredients:
(i) The 3D metric γij (metric induced on : γab ≡ ∗ gab , where : →
M is the embedding of into M) that measures proper distances within the
hypersurface itself:
dl 2 = γij dx i dx j .
The hypersurface is then said to be

– spacelike ⇐⇒ γab is positive definite; i.e., it has signature (+, +, +).

our case
– timelike ⇐⇒ γab is Lorentzian; i.e., it has signature (−, +, +).
– null ⇐⇒ γab is degenerate; i.e., it has signature (0, +, +).
(We will shortly justify why we express the spatial metric both as 3D object
(γij ) and as 4D object (γab ).)
(ii) The lapse of proper time between the hypersurfaces, as measured by observers
whose worldlines extend along the direction normal to the hypersurfaces (these
observers are usually referred to as Eulerian observers):
dτ = α(t, x i )dt,
where α is known as the lapse function (this is denoted as N by some

references, e.g., [15, 20]).
(iii) The relative velocity β i between the Eulerian observers:
i
xt+dt = xti − β i (t, x i )dt, for Eulerian observers.
This 3-vector β i measures how much the coordinates are shifted as we move
from one slice to the next, and it is therefore conventionally named as the shift
vector. (It is also denoted as N i in the literature.)
Note that, as alluded to earlier, the foliation of M is not unique, and neither is the
coordinates shift; α determines “how much slicing” is to be done, whilst β i dictates
how the spatial coordinates propagate from one hypersurface to the next. In fact, the
latitude to choose a lapse function and shift vector demonstrates the gauge freedom
that is inherent to the formulation of GR, a covariant theory.
From the universal time function t (given by the foliation), we can define the
future-pointing timelike unit normal na to the slice to be5
5 The minus sign is chosen to ensure that na is always future-pointing.

na ≡ −α∇ a t. (4)
We think of na as the 4-velocity of an Eulerian observer, i.e., an observer whose

worldline is always normal to the spatial slices. With this defined, we can see that
the three scalar quantities that yield the spatial components of the shift vector, β i ,
are given by

β i = −α n · ∇x i . (5)
These three scalar quantities can then be used to form a full 4-vector β a (orthogonal
to na , by construction) which, in the adapted 3 + 1 coordinates we are about to
introduce, will have components β μ = (0, β i ). Equipped with the unit normal and
the shift vector, we can also define a time vector t a given by
t a ≡ αna + β a , (6)
which is nothing but the vector tangent to the time lines, i.e., the congruence of lines
of constant spatial coordinates x i . The importance of this vector lies in the ability to
Lie drag the hypersurfaces along it: the spatial basis vectors e(i) a which are tangent
to a particular slice t (i.e., that satisfy ∇a t e(i) = 0) are Lie dragged along t a ,
a
Lt e(i)
a
= 0.
Remark 1.2 The normal evolution vector ma ≡ αna can also be used to Lie drag
the hypersurfaces (see, e.g., [15])
a whilst all remaining (spatial)
Now, since t a is aligned with the basis vector e(0)
coordinates remain constant along t a , we get the basis components
μ μ
t μ = e(0) = δ0 = (1, 0, 0, 0). (7)
This means that any Lie derivative along t a will reduce to a partial derivative with
respect to t: Lt = ∂t (we will use this later) Similarly, a straightforward derivation
shows that in these adapted coordinates we have
β μ = (0, β i ) (8a)
nμ = (α −1 , −α −1 β i ) (8b)
nμ = (−α, 0, 0, 0). (8c)
174 M. L. G. Abed
These quantities will appear in the 3 + 1 coordinates expression of the spatial metric
γab . To show this, we first need to introduce the spatial projection operator6
Pba ≡ δba + na nb , (9)
which we then apply (twice; once per index) to the spacetime metric gab to get

Pac Pbd gcd = δba + na nb δba + na nb gcd = gab + na nb .
Since the induced metric is merely a projection of the spacetime metric onto the
hypersurface, we have found an expression for our spatial metric:
γab = gab + na nb (10)
and similarly the inverse spatial metric
γ ab = g ac g bd γcd = g ab + na nb . (11)
Hence, γab is a projection tensor that discards components of 4D geometric objects

that lie along na ; it allows us to compute distances entirely within a slice .
Intuitively, γab first calculates the spacetime distance with gab and then kills off
the timelike contribution (normal to the spatial surface) with na nb .
Remark 1.3 From (10), we see that, if we raise only one index of the spatial metric
γab ,
γba = gba + na nb = δba + na nb ,
we find out that our projection operator is merely the spatial metric with one raised
index
Pba = γba .
Therefore, from now on, we will exclusively use γba to denote the spatial projection
operator.
Also, from (10) and from (8c), we can see that
6 This operator projects a 4D tensor onto a spatial slice. For instance, if we take an arbitrary 4-
vector v a and hit it with the projection operator,
Pba
va
−→ Pba v b ,

arbitrary, 4D purely spatial
we get a purely spatial object that lies entirely on a hypersurface.

γij = gij + ni nj = gij , (12)

=0
so that the spatial metric on is just the spatial part of the spacetime 4-metric
gab . Note also that, even though the covariant components do not necessarily vanish
(γ0μ = g0μ + n0 nμ = g0μ + n0 = g0μ + α 2 = 0, in general), any contribution
to the timelike direction can be safely ignored since na γab = 0. On the other hand,
timelike components of spatial contravariant tensors do vanish, so we must have
γ a0 = 0. Therefore, from (11), we get the components of the inverse spacetime
metric in these adapted coordinates:
g ab = γ ab − na nb
g 0a = −n0 na ⇒ g 00 = −α −2 & g 0i = α −2 β i
g ij = γ ij − ni nj = γ ij − (−α −1 β i )(−α −1 β j ) = γ ij − α −2 β i β j .
In matrix form,
−1/α 2 β i /α 2
g μν = . (13)
β /α γ − β i β j /α 2
j 2 ij
Now, by the condition g ab gbc = δca , we can invert (13) to write the spacetime metric
in 3 + 1 coordinates:
−α 2 + βk β k βi
gμν = . (14)
βj γij
The covariant components βi shown above come from lowering with the spatial
metric, i.e., βi = γik β k . We will always use the spatial metric to raise/lower indices
of spatial objects because γij and γ ij are inverses of each other in the adapted
coordinates:
γ ik γkj = (g ik + ni nk )(gkj + nk nj )
= g ik gkj + g ik nk nj + ni nk gkj + ni nk nk nj
= δji + ni nj +ni nj −ni nj = δji .

=0 =0 =0
Equation (14) shows that the line element of the full spacetime metric in 3 + 1
coordinates is given by

ds 2 = −α 2 + βi β i dt 2 + 2βi dtdx i + γij dx i dx j . (15)
176 M. L. G. Abed
1.2 ADM Evolution and Constraints
Using the projection operator (9) (which we now know is just γba ), we can define
the extrinsic curvature tensor to be7
Kab = −γac γbd ∇c nd . (16)
This quantity measures how much na varies as we move from point to point on
a particular slice , and in doing so it describes how is embedded in M.
Expanding (16), we get
Kab = −∇a nb − na nc ∇c nb , (17)
and, moreover, a straightforward calculation shows that
1
Kab = − Ln γab . (18)
2
From the latter and from the time vector (6), we get a natural time derivative of the
metric:
1 1
Kab = − Ln γab = − L t−β γab
2 2 α
1
=− Lt γab − Lβ γab
2α
1
=− ∂t γab − Lβ γab , (19)
2α
where on the last line we used the fact that, in the adapted coordinates, Lt reduces to
∂t . Thus, expanding the Lie derivative in (19) (and dropping timelike components),
we have an evolution equation of the spatial metric:
∂t γij = 2D(i βj ) − 2αKij . (20)
Here, D is the affine connection compatible with the 3D metric (i.e., Dc γab = 0),
which is furnished by projecting all indices present in a 4D covariant derivative ∇
onto ; that is, for a bc tensor T ,
Da Tji11...j
...ib
c
= γad γki11 . . . γkibb γj11 . . . γjcc ∇d Tk11...
...kb
c
. (21)
7 The minus sign is merely a convention in the NR community. In the cosmology community, the
sign is usually positive.

The evolution equation for the metric was not so difficult to derive; however, the
remaining evolution equation (of Kij ) and the constraint equations need a bit more
work. Given the brevity of this presentation, it would not be possible to show
derivations in great detail, but nevertheless it should (in theory at least) entice the
interested reader to derive all the results from scratch (this is the only true way to
learn anyhow).
We need to find a way to formulate the EFEs in 3 + 1 form, which is a task we
can accomplish by considering the following projections:
na nb ( ∗ [(4) ]Gab − 8π Tab ) = 0; (22a)

γca γdb ( ∗ [(4) ]Gab − 8π Tab ) = 0; (22b)
γcb na ( ∗ [(4) ]Gab − 8π Tab ) = 0. (22c)
(Note that all other projections vanish identically thanks to the symmetries of the
Riemann tensor.) These equations come about from projections of the 4D Riemann
tensor,
f g
γae γb γc γdh (4) Refgh = Rabcd + Kac Kbd − Kad Kcb ; (23a)
f g
γae γb γc nh(4) Refgh = Db Kac − Da Kbc ; (23b)
q 1
γa γbr nc nd (4) Rqcrd = Ln Kab + Da Db α + Kbc Kac . (23c)
α
These are the so-called Gauss–Codazzi, Codazzi–Mainardi, and Ricci equations,
respectively. Note how (23a) and (23b) depend exclusively on the spatial metric, the
extrinsic curvature, and their spatial derivatives; they will give rise to the constraint
equations (they can be thought of as integrability conditions allowing the embedding
of a 3D slice with data (γab , Kab ) inside the ambient spacetime manifold M). On
the other hand, the first term on the RHS of (23c) hints that we will get the evolution
equation for the extrinsic curvature from this equation.
In fact, expanding (23a) and doing some algebra, we end up with
R + K 2 − K ab Kab = 16πρ, (24)
where ρ is the total energy density as measured by a normal observer na ,
ρ ≡ na nb Tab . (25)
Dropping timelike components, we have the Hamiltonian constraint, which must be

satisfied on each slice of the foliation,
R + K 2 − K ij Kij = 16πρ. (26)

178 M. L. G. Abed
Similarly, from (23b), we get
Db Kab − Da K = 8π Sa , (27)
where we used the momentum density Sa as measured by a normal observer na ,
Sa ≡ −γab nc Tbc . (28)
Dropping timelike components and raising indices, we write (27) in its final form

Dj K ij − γ ij K = 8π S i , (29)
which is with the so-called momentum constraints that must also be satisfied on each
hypersurface. Lastly, from (23c), some very messy algebra yields
1
∂t Kab = α(Rab + KKab − 2Kac Kbc ) − 8π α Sab − γab (S − ρ)
2
− Da Db α + Lβ Kab , (30)
where Sab is the spatial stress, given from a projection of the 4D energy–momentum
tensor Tab ,
Sab ≡ γac γbd Tcd , (31)
and S is its trace,
S ≡ γ ab Sab = Saa . (32)
Then, since the entire content of spatial tensors is available from their spatial
components, we can write our results as
1
∂t Kij = α(Rij + KKij − 2Kik Kjk ) − 8π α Sij − γij (S − ρ)
2
− Di Dj α + β k Dk Kij + 2Kk(j Di) β k . (33)
And thus, we arrived at the evolution equation for the extrinsic curvature, our last
piece of the puzzle.
2 BSSN Formalism of Numerical Relativity
The 3 + 1 ADM (à la York) decomposition of the EFEs presented in Sect. 1 is

already, in theory at least, in a form suitable for evolution simulations on a computer.
Unfortunately, however, a lot more work needs to be done before we take a crack
at computing anything, since in practise one finds that this form of the 3 + 1
decomposition in fact results in large instabilities that develop during the simulation.
This issue is known to be mainly due to the fact that the equations in this form are
weakly hyperbolic rather than strongly hyperbolic, which means that they are not
well-posed.8 To get around this problem, many NR codes implement the so-called
BSSN (a.k.a. BSSNOK) formalism which, together with the “1 + log” slicing and the
“gamma-driver” gauge conditions (we will briefly discuss these in Sect. 3), does
admit a strongly hyperbolic formulation of the EFEs.
Whereas the standard ADM equations involve evolution equations for the raw
spatial metric and extrinsic curvature tensors, through the BSSN formalism we are
going to modify the equations by factoring out a conformal factor and introducing
three conformal connection coefficients ¯ i , reducing in this manner the evolution
equations to wave equations for the conformal metric components. In other words,
instead of the ADM data {γij , Kij }, the BSSN formalism splits γij into a conformal
factor χ and a conformally related metric γ̄ij , and it also splits Kij into its trace K
and a traceless part Aij . Moreover, three coefficients ¯ i of the conformal metric are
introduced as well. Then, it is these quantities that are evolved instead of the original
ADM ones . . . long story short, the dynamical variables for the BSSN system will
be given by
{χ , γ̄ij , Āij , K, ¯ i }. (34)
We will present each of these quantities and derive their evolution equations in this
section. Let us start by considering a conformal rescaling of the spatial metric of the
form
γij = χ −1 γ̄ij , (35)
where χ is some positive scaling factor called the conformal factor, and the
background auxiliary metric γ̄ij is known as the conformally related metric (or
simply conformal metric). It may seem unclear why we scaled the spatial metric
in this way, but let it suffice to say that this “trick” will actually yield a convenient
and tractable system for the EFEs. Besides the mathematical convenience that such
a conformal rescaling brings about, there is also the fact that equivalence classes
of conformally related manifolds share some geometric properties. For example,
(1) (2)
it can be shown that two strongly causal Lorentzian metrics gab and gab for some
8 For more on the concept of hyperbolicity, see the detailed analysis in [26].
180 M. L. G. Abed
manifold M determine the same future and past sets at all points (events) if and only
(1) (2)
if the two metrics are globally conformal, i.e., if gab = gab , for some smooth
∞ (1)
function ∈ C (M). In this case, both spacetimes (M, gab ) and (M, gab (2)
)
belong to the same conformal class and share the same causal structure.
A somewhat natural representative of a conformal equivalence class is a metric
γ̄ij whose determinant is equal to that of the flat metric δij in whatever coordinate
system we are using, i.e., γ̄ = δ. Thus, if we adopt a Cartesian coordinate
system, we can always enforce that our conformal representative must have unit
determinant, i.e., γ̄ = 1. Plugging this back into (35), we get
1 = det γ̄ij = det(χ γij ) = χ 3 detγij = χ 3 γ .

≡γ
This would correspond to the choice χ = γ −1/3 , so that γij = γ 1/3 γ̄ij . Any
spatial metric γij in this conformal class yields the same value of γ̄ij . However,
note that, since the determinant γ is coordinate-dependent, the conformal factor
χ = γ −1/3 is not a scalar field. In fact, γ̄ij is not a tensor field, but rather a
tensor density of weight −2/3. To get around this issue of tensor densities, we could
introduce a background flat metric δij of Riemannian signature (+, +, +) and set
χ ≡ (γ /δ)−1/3 , so that χ becomes a scalar field in this manner, and we could
then use non-Cartesian coordinates. However, for our purposes of implementing
the standard BSSN formalism, it is convenient to stick to Cartesian coordinates;
to see the implementation of this extended BSSN formalism (where non-Cartesian
coordinates are used), the reader is referred to [15].
The conformal factor is one of the dynamical variables in the BSSN approach,
and as such we need an evolution equation for it. A straightforward calculation
shows that
2
∂t χ = χ (αK − ∂i β i ) + β i ∂i χ , (36)
3
where K is the trace of the extrinsic curvature,
K ≡ g ab Kab = γ ij Kij = Kii . (37)
(Note that the second equality holds because Kab is a purely spatial object.) Now,
before showing the evolution of the conformal metric (another BSSN dynamic
variable; cf. (34)), we need to briefly discuss the split of the extrinsic curvature
Kij into its trace K and its traceless part Aij ,
1
Kij = Aij + γij K. (38)
3
Just as we rescaled the spatial metric, in the BSSN formalism we shall also rescale
the traceless curvature Aij as9
Āij = χ Aij , (39)
which yields
1
Āij = χ Kij − γ̄ij K. (40)
3
In terms of these rescaled variables, it is straightforward to show that the evolution
of the conformal metric is given by
2
∂t γ̄ij = −2α Āij + β k ∂k γ̄ij + γ̄ik ∂j β k + γ̄kj ∂i β k − γ̄ij ∂k β k , (41)
3
where we had to use the fact that γ̄ij is a tensor density of weight −2/3 when
expanding the Lie derivative Lβ γ̄ij . (The Lie derivative of a tensor density of weight
ω is given by
Lx τ = [Lx τ ]ω=0 + ωτ ∂i x i ,
where the first term is the usual Lie derivative we would compute if τ had zero
weight (i.e., if τ was a tensor field rather than a tensor density).) Furthermore, a
somewhat involved calculation shows that the evolution of the trace of the extrinsic
curvature is given by
1
∂t K = α Āij Āij + K 2 + 4π α(ρ + S) − D 2 α + β i ∂i K, (42)
3
and another (yet even longer) computation yields the evolution of the conformal
traceless curvature,10
TF
∂t Āij = χ (αRij − 8π αSij − Di Dj α) − α(2Āik Ākj + Āij K) (43)
2
+ β k ∂k Āij + Āik ∂j β k + Ākj ∂i β k − Āij ∂k β k .
3
You may notice, however, that there is something off-putting on both (42) and (43);
we have covariant derivatives D of the lapse with respect to the physical metric
9 Note, however, that a different scaling for Aij is used when dealing with the initial data problem
(we briefly discuss initial data in Sect. 3).
10 Here, we use the notation [· · · ]TF to denote the trace-free part of whatever object lies inside the
brackets (e.g., [Kij ]TF = Aij ). In general, for a tensor T in a D-dimensional metric g, we have
[T ]TF = T − g/D Tr(T ).
182 M. L. G. Abed
γij as opposed to the BSSN conformal metric γ̄ij (moreover, (43) also has the 3D
Ricci tensor Rij of γij appearing in the expression). We correct these problems by
introducing the conformal connection D̄ of γ̄ij and writing the Christoffel symbols
of D in terms of those of D̄,
1
ji k = ¯ ji k − χ −1 δki ∂j χ + δji ∂k χ − γ̄j k γ̄ i ∂ χ . (44)
2
Using this relation, a quick calculation shows that
1
Di Dj α = D̄i D̄j α + 2D̄(i χ D̄j ) α − γ̄ij D̄k χ D̄ k α , (45)
2χ
which is to be inserted into both (42) and (43). Furthermore, we may show (after a
very long calculation) that the Ricci tensor can be split as
χ
Rij = R̄ij + Rij , (46)
where
1
R̄ij = − γ̄ k ∂k ∂ γ̄ij + γ̄k(i ∂j ) ¯ k + ¯ k ¯ (ij )k
2

m ¯
+ γ̄ k 2 ¯ k(i m ¯
j )m + ¯ ik j m (47a)
χ 1
Rij = D̄i D̄j (log χ ) + γ̄ij D̄k D̄ k (log χ )
2
1
+ D̄i (log χ )D̄j (log χ ) − γ̄ij D̄k (log χ )D̄ k (log χ ) . (47b)
4
Remark 2.1 In (47a), we used the conformal coefficients ¯ i ≡ γ̄ j k ¯ ji k . The reason

why we want to write R̄ij in this form is because, with the exception of the Laplacian
term γ̄ k ∂k ∂ γ̄ij , every other second derivative of the metric γ̄ij is being absorbed
into first derivatives of ¯ i . This in turn makes the BSSN equations more hyperbolic
(see, e.g., [26]).
Speaking of ¯ i , this is the only remaining BSSN dynamic variable for which we
need an evolution equation (cf. (34)). Here, we simply present it (this is another long
derivation; try it):
3 ij 2
∂t ¯ i = −2α Ā D̄j χ + D̄ i K + 8π S̄ i − ¯ ji k Āj k − 2Āij D̄j α (48)
2χ 3
2 1
+ β j ∂j ¯ i + γ̄ j k ∂j ∂k β i − ¯ j ∂j β i + ¯ i ∂j β j + γ̄ ij ∂j ∂k β k .
3 3
Lastly, we close out this section by writing the constraints in BSSN variables:
1 4 2 1
R̄ + 2 D̄ 2 (log χ ) − D̄k (log χ )D̄ k (log χ ) + K − Āij Āij = 16π ρ̄ (49)
2 3χ χ
3 ij 2
D̄j Āij − Ā D̄j χ − D̄ i K = 8π S̄ i , (50)
2χ 3
where we used the rescaling ρ̄ i ≡ χ −1 ρ and S̄ i ≡ χ −1 S i . Moreover, we used the

transformation law for the spatial Ricci scalar R,
1
R = χ R̄ + 2 χ D̄ 2 (log χ ) − χ D̄k (log χ )D̄ k (log χ ). (51)
2
Equations (49) and (50) are the Hamiltonian and momentum constraints, respec-
tively, in BSSN variables.
That was a very compact presentation of the BSSN formalism of numerical
general relativity. Admittedly, this formalism is not nearly as intuitive and straight-
forward as the ADM alternative that we presented in Sect. 1, but it is nevertheless a
much more robust formulation (numerically speaking). This is a running theme in
physics (and science in general): analytical and numerical implementations rarely
play fair ball with each other. The ADM formalism is important for historical (and
pedagogical) reasons, but it is nearly useless for practical purposes. We remark,
however, that BSSN is not by any means the only modern successful approach to
numerical relativistic studies; other flourishing alternatives such as the Generalised
Harmonic Coordinates with Constraint Damping (GHCD) [18, 22, 23] and Z4-like
[4, 6, 7, 9, 25] formalisms are just as good (and in some cases even superior) as
BSSN.
3 Further Considerations
Having presented two of the main formalisms of numerical relativity, we now turn
to a brief discussion of some further considerations that must be taken into account:
the initial data problem (Sect. 3.1) and gauge choice (Sect. 3.2). Yet another topic
that we have made no mention of (nor will we get into, since it is way beyond our
scope) is the actual numerical methods employed in the field of numerical relativity;
suffice it to say that the two most widely used methods used by the NR community
are the good old fashioned finite difference methods [21] and spectral methods
[16]. The reader is encouraged to study those references to get up to speed on the
numerical side of things. Last but not least, we close out this chapter by making
brief mention of some potential applications of NR outside of the usual realm of
black holes/neutron stars collisions (Sect. 3.3).
184 M. L. G. Abed
3.1 Initial Data
As we alluded to earlier, we are not free to impose whatever data we like on our
initial time slice; the initial data has to be chosen in such a way that the Hamiltonian
and momentum constraints are satisfied from the onset.11 That being said, the
constraints are just four equations that remove four degrees of freedom from the
total twelve degrees of freedom of the system {γij , Kij }. Moreover, there is no a
priori preference for which eight of the total data to use as free parameters and which
remaining four to use in solving the constraint equations. This initial data problem is
a difficult subject with a vast literature dedicated to it (see [13] or any of the standard
textbooks, e.g., Chapter 9 of [15]); the two most popular approaches to tackle this
problem are known as the conformal transverse-traceless (CTT) decomposition and
the conformal thin-sandwich (CTS) decomposition. Both of these methods provide
some guidance on how to choose which values will be free parameters and which
will be constrained data, although in the absence of significant symmetries many of
the choices are arbitrary.
3.2 Gauge Choice
Even though, in theory (i.e., analytically) all gauge choices should yield the same
physical result, as it is often the case numerical simulations do not always play nice.
Therefore, in order to achieve a long-term stable simulation, we need to specify the
right gauge (choice for α and β i ) and determine how these quantities will evolve in
coordinate time. Choosing static (i.e., time-independent) gauges is not a very good
idea, since we have no a priori knowledge of which functions will serve us better;
the best approach is to choose the lapse and shift dynamically as functions of the
evolving geometry.
One may naively think, for instance, that setting α = 1 would be an ideal
choice (certainly, our calculations would simplify quite a bit).12 Unfortunately,
however, this turns out to be a terrible pick: the acceleration of a normal observer
is given in terms of the lapse function as nb ∇b na = Da log α; thus, setting α = 1
yields a vanishing acceleration of normal observers (hence, the choice α = 1 is
usually referred to as geodesic slicing, since Eulerian observers are in free fall).
A detailed examination then shows that this almost always leads to a singularity;
thus, singularity-avoiding techniques such as maximal slicing (computationally
expensive) or 1 + log slicing (lower computational cost) must be employed. The
latter is the one that has been adopted by most modern NR codes; it is a generalised
11 In addition, for non-vacuum spacetimes, the matter distribution (ρ, S i ) may have constraints of
its own.
12 This corresponds to an evenly spaced slicing, so that coordinate time coincides with proper time
of Eulerian observers (recall that dτ = αdt).

hyperbolic slicing condition of Bona–Massó type [8] whose basic idea is to reduce
the lapse in regions where the curvature is particularly strong. In general, the so-
called alpha-driver condition is given by
∂t α = −ζ1 α ζ2 K + ζ3 β i ∂i α, (52)
with ζi being some positive scalar functions. From this equation, we get the 1 + log
slicing by fixing ζ1 = 2 and ζ2 = ζ3 = 1.
Similarly, we may also choose a vanishing shift vector (β i = 0), so that the
coordinates are not shifted as we move from slice to slice. This would also certainly
simplify matters, and it is in fact a common gauge choice that works well in certain
applications. However, in black hole spacetime simulations, if we use a vanishing
β i , the event horizon grows rapidly in coordinate space, due to the normal observers
falling in, which causes the computational domain to end up eventually trapped
inside the black hole (see [1]). Moreover, in order to counter the large field gradients
(or “slice stretching”)13 incurred in the presence of a black hole, a nonvanishing β i is
required [2]. To deal with this slice stretching issue, gauge conditions were designed
so that second-time derivatives of the shift are proportional to first-time derivatives
of the coefficients ¯ i (i.e., ∂t2 β i ∼ ∂t ¯ i ). In particular, a hyperbolic shift condition
∂t2 β i = η∂t ¯ i − ξ ∂t β i , (53)
where η and ξ are positive scalar fields, was introduced. This is the so-called
gamma-driver shift condition. We may then use an auxiliary vector field B i to
perform the usual trick of rewriting a second-order derivative in first order,
∂t B i = ϑ1 α ϑ2 ∂t ¯ i − 1 B i (54a)
∂t β i = 2 B i , (54b)
where we also rewrote the scalar fields η and ξ in terms of four damping parameters
ϑ1,2 and 1,2 that fine-tune the growth of the shift.
Once we have chosen our gauge, prescribed (and evolved, via the BSSN
formalism presented above) the initial data, one must turn to a last (and key) step:
interpret the obtained data in a gauge-independent way (e.g., one may wish to find
event horizons or extract gravitational wave signals).
13 3 + 1 simulations of black hole spacetimes without singularity excision and with singularity-
avoiding lapse and vanishing shift fail after an evolution time of around 30–40M due to the so-
called slice stretching (see [3]).
186 M. L. G. Abed
3.3 Potential Application to Cosmology
Thus far, we have only discussed purely geometric aspects of the 3 + 1 formulation
of GR, without discussing constraints or evolution of any potential matter field that
might be coupled to the EFEs; we now turn to this topic (for a thorough treatment,
the reader may consult, e.g., Chapter 6 of [15]). Let us focus our succinct discussion
in a scalar matter field φ minimally coupled to the EFEs. For such scalar field, the
Lagrangian is given by
1
LM = ∇a φ∇ a φ + V (φ), (55)
2
where V (φ) is the scalar potential, which may be decomposed as
1 2 2
V (φ) = m φ + Vint (φ), (56)
2
m being the mass of the field and Vint the interaction potential (since we are
interested in the minimally coupled case, the field is noninteracting; i.e., Vint = 0).
√
If we then add the associated scalar matter Lagrangian density LM = −gLM to
√
the gravitational Lagrangian density, namely LG = −gR, then minimisation of
the (modified) Einstein–Hilbert action

1
S= (LG + LM ) d4 x (57)
16π
leads to the EFEs with stress–energy tensor defined by
1
Tab = ∇a φ∇b φ − gab ∇c φ∇ c φ + 2V (φ) . (58)
2
The addition of this scalar field to our 3 + 1 formulation of GR opens the doors to
some interesting areas of study, for instance, inhomogeneous cosmological inflation
[11, 12, 19] and critical gravitational collapse [10, 11, 17].
Moreover, from (55), the Euler–Lagrange equations yield our equation of
motion, which coincides with the Klein–Gordon equation in curved spacetime:
dV (φ)
∇ 2φ = = m2 φ, (59)
dφ
where ∇ 2 = g ab ∇a ∇b . Since this equation of motion is of second order, it would

be useful to cast it into first order for integration purposes; we accomplish this with
the aid of new auxiliary variables and i given by14
14 Some references (e.g., [5]) define as the negative of ours; here, we follow the convention
in [11].
1
≡ ∂t φ − β i ∂ i φ (60)
α
i ≡ ∂i φ. (61)
Using these variables, (59) splits as
∂t φ = α + β i i (62a)
∂t i = β j ∂j i + j ∂i β j + α∂i + ∂i α (62b)
dV (φ)
∂t = β i ∂i + g ij α∂j i + j ∂i α − ijk k + α K + . (62c)
dφ
These equations must be solved in conjunction with the gravitational field’s 3 + 1

equations in order to determine the complete evolution of a spacetime containing
a scalar matter field. Note that (62a) is just the definition of (i.e., it is simply
a rewriting of (60)) and (62b) follows directly from (61) and (62a) by commuting
partial derivatives; the true equation of motion is in fact determined by (62c). The
constraint given by (61), namely,
Ri ≡ i − ∂i φ = 0, (63)
must be preserved by the system (62). Of course, if it is solved exactly, (62)

does guarantee the preservation of (63) throughout the evolution. The problem is
at the numerical level, where truncation errors can give the residual Ri nonzero
values; therefore, it is important to keep a close eye out on Ri (in addition to the
Hamiltonian and momentum constraints) during the evolution to make sure that we
are working with an accurate simulation.
Had we instead assumed homogeneity of the scalar field (i.e., ∇i φ = ∂i φ = 0),
then the equation of motion (59) for an FLRW metric would yield a relatively simple
second-order ODE for the evolution of the “inflaton” scalar field φ(t),
φ̈ + 3H φ̇ + m2 φ = 0, (64)
where, per usual notation, H (t) = ȧ/a represents Hubble’s constant, and a(t) is the
expansion parameter that appears in the FLRW metric. The much more complicated
problem of dealing with an inhomogeneous inflaton requires the full power of
numerical relativity, and it is currently a very active research area. For more on
this topic, the reader is referred to references such as [11, 12, 14, 19].
Acknowledgments This work was made possible by the Domoschool organisers. Special thanks
to them.
188 M. L. G. Abed
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to inhomogeneous initial conditions. JCAP 1709(09), 025 (2017)
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Length-Contraction in Curved Spacetime
Colin MacLaurin
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
2 Proper Frame Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
3 Length-Contraction and Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
3.1 Relative 3-Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
3.2 Length-Contraction Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
3.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
4 Volume Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.1 Metric Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.2 De-Contracted Volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
4.3 Contracted Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.1 Lorentz Boost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.2 Square Flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.3 Rotating Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.4 Schwarzschild Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7 DomoSchool Memories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
1 Introduction
Length-contraction is a staple of introductory special relativity, often encountered in

high school physics. However, it receives no mention in a typical general relativity
course nor in most curved spacetime literature. When first encountering the radial
length (1 − 2M/r)−1/2 dr in Schwarzschild spacetime, many students wonder what
C. MacLaurin ()
University of Queensland, Brisbane, QL, Australia
e-mail: colin.maclaurin@uqconnect.edu.au
192 C. MacLaurin
happens inside the horizon, where the coefficient becomes imaginary. In one case,
a lecturer explained that since there is no timelike Killing vector for r ≤ 2M there
is no preferred length, which is a solid answer though an incomplete one. How
does this relate to special relativity, where length depends on the relative motion
between observers? Certain aspects of this foundational topic of length deserve
careful reexamination.
Historically, length-contraction originated in Maxwell’s electromagnetic theory
and was developed by Heaviside, Fitzgerald, Lorentz, Larmor, Poincaré, Einstein,
and others. The Michelson-Morley experiment influenced many of these. Various
shape deformations were proposed, but the longitudinal contraction won out. This
contraction was relative, not fixed by the frame of the luminiferous ether (as
then assumed to exist). Yet its natural home, like the Lorentz transformation and
electromagnetism generally, was in the new ontology of Einstein’s relativity [1, §4].
In philosophy of relativity, the “ruler hypothesis” has received deserving atten-
tion. It states rulers achieve their intended purpose of measuring metric length. The
hypothesis is not presented as being entirely true, but rather a statement for critical
examination. Here, we will model a ruler by a spatial vector in the tangent space at
a single point in curved spacetime. This may be considered part of an orthonormal
frame. Equivalently, the ruler vector represents the orientation of a radar distance
device. But while the motivation and intuition is literal rods, technically we analyse
nothing more than the metric and several induced volume elements, including a
certain transformation between frames. I hope this is reassuring to the many critics
of the ruler concept in relativity; see also Sect. 6.
2 Proper Frame Measurement
This section relates proper distance within the local 3-dimensional space of a given
observer, to a coordinate . It summarises my earlier work, [6, §5] [7] with fresh
explanation, plus careful justification which many readers should skip.
At a given point in spacetime, suppose u is a 4-velocity (a unit, timelike, future-
pointing vector), ξ a unit vector orthogonal to u, and a scalar field differentiable
at that point. In particular:
u, ξ = 0, ξ , ξ = 1.
The angle brackets denote the metric scalar product, where in general:
a, b := gμν a μ bν = aμ bμ = g μν aμ bν ,
for any pair of rank-1 tensors. We interpret u as an observer, ξ as an idealised

“ruler”, and as a coordinate such as Schwarzschild r. We wish to compare the
ruler to the coordinate gradient d, to determine the interval spanned by the
ruler with proper length L = 1 say. To be more precise, we seek the ratio at a point.
Length-Contraction in Curved Spacetime 193
Fig. 1 A sphere with one tangent plane shown. For our later applications, think of this as a surface
within Minkowski or Schwarzschild spacetime. The solid curves are level sets θ = const. Within
the tangent plane, dθ is visualised as hyperplanes (dotted lines). Our “ruler” ξ is a vector within
the tangent space. Intuitively, it approximates a literal physical ruler’s extent along the manifold
itself (interpreting ξ as a line segment, then applying the exponential map), under minimal strain
from tidal forces, acceleration, etc. But when taken as a pointwise 1-dimensional volume element
on the manifold, it is precise
But the change in by proper length, in the ξ -direction, is d/dL = d, ξ .

Rearranging:1
1
dL = d. (1)
d, ξ
Formally, we treat both sides as covectors or 1-forms. Recall the gradient d,
equivalently written ∇, has components (d)μ = ∂/∂x μ in a coordinate basis.
There is an intuitive picture for the contraction d, ξ of a 1-form and vector
to return a scalar [12, §3.3] [8]. As in Fig. 1, d may be visualised as a set of
hyperplanes within a tangent space, or more roughly as hypersurfaces = const
within the manifold. Its action on a vector returns the number of hyperplanes
spanned by that vector. The tangent space and quantities therein are essentially a
linearisation or gradient at a point on the manifold, which is the motivation behind
ξ . Our approach is strictly local, which is more physically justified than claims of
1A negative coefficient is allowed. Also if d, ξ = 0, then is constant along the ξ direction,
so is of no use for demarcating this length. The analogy to Eq. (1) would be d = 0 but interpreted
as 1-dimensional only.
194 C. MacLaurin
“observer at infinity” measurement, as in Schwarzschild spacetime, for instance.

The idea is to piece together such local measurements into an overall quantity.
dL is equivalent to infinitesimal radar distance. In fact it is simply the metric
interval ds, restricted to the ruler direction! Just as the vector ξ represents a unit of
length, the 1-form ξ is a length element.2,3 So why bother, what is the use? The
equation relates a coordinate gradient to a physical, measurable quantity. This is
useful to find an object’s position and extent in terms of the mathematical description
of the theory—in particular, coordinates on spacetime.4 Similarly an orthonormal
tetrad, when interpreted as a rank-2 tensor, is nothing other than the metric in
disguise! [3, §J]. Such a tetrad, when expressed in coordinates, relates physical
quantities to coordinates. Our pair (u, ξ ), or even just ξ alone, forms a sort of proto-
tetrad. The orthonormal decomposition g = −u ⊗ u + ξ ⊗ ξ + · · · exhibits the
ruler’s part within the full metric tensor.
Traditional tensor analysis adopts the coordinate basis vectors as measurement
directions. Assuming ∂ is spacelike, ξ := (g )−1/2 ∂ is a unit vector, for which
√
Eq. (1) returns dL = g d. Similarly for the cobasis 1-form d, define ξ :=
(g )−1/2 (d) , then dL = (g )−1/2 d. The latter relies on being spacelike,
meaning d, d > 0. The problem is, neither ruler is orthogonal to u in general,
meaning these are not measurements in a given observer’s frame! To correct this,
use the spatially projected vector (d) + d, uu. This is orthogonal to u, hence
spatial or zero. If nonzero, normalise it and apply Eq. (1) to get:
1
dL,3−gradient = d. (2)
d, d + d, u2
If is taken as a coordinate, we may rewrite as components d, d = g

and d, u = u . This ruler direction is a natural definition of “the -direction”
relative to u. Recall the normal vector (d) is orthogonal to the level sets =
2 Technically, we define dL only on the 1-dimensional subspace of the cotangent space spanned
by ξ . This is because the rate d/dL is a derivative along the ruler direction specifically. With
this restriction understood, Eq. (1) is generally covariant. In fact dL is merely ξ , as seen from
contracting both with the single basis vector ξ . If using the notation ξ , it seems fine to treat it as
4-dimensional.
3 Compare H. Brown that “[rods and clocks] are not the analogue of thermometers of the spacetime
metric. They measure in the weaker sense that their behaviour correlates with aspects of spacetime
structure. . . ” [2]. Exceptions include Weyl’s theory and some recent alternate gravity theories [1].
4 Coordinates are useful for bookkeeping, to “chart” positions and build a global picture, an
extrinsic standard. In other cases they are an interim computational tool, such as for volume
integrals. It is often stated coordinates have no physical meaning, with Einstein quotes for support;
however, in the present context I prefer to nuance: no direct physical meaning, which is also
backed up by Einstein quotes such as “. . . immediate metrical significance”. Norton puts it well:
“In Einstein’s words the ‘[coordinate] system. . . has no physical reality.’ We might phrase this more
cautiously by saying that the coordinate system has no reality independent of the metric, for the
combination of coordinate system and metric certainly do represent aspects of physical reality”
[11, §5.5.3].
const. If is spacelike, (d) is the direction of greatest increase of , per length

of vector. Similarly the projected gradient gives the greatest increase of , amongst
any vectors orthogonal to u. It maximises d/dL and minimises dL/d, for any
vector in u’s 3-space. If d, u = 0, the measurement reduces to the traditional
quantity.
3 Length-Contraction and Measurement
This section considers another “observer” n, in whose frame the local 3-space of u is
length-contracted. We derive a 4-vector formalism for length-contraction, introduce
“de-contraction”, and apply these to length measurements. From now on, we mostly
term u “fluid” or “matter”, to distinguish from the frame doing the observing.
Shortly, n will be generalised to be timelike, null, or spacelike. It stands for “normal”
to a hypersurface on the manifold, or at least to a hyperplane in a tangent space.
3.1 Relative 3-Velocity
Recall the prototypical Lorentz boost t = γ (t − βx) and x = γ (x − βt)

between global inertial frames in Minkowski spacetime, where γ = (1 − β 2 )−1/2
is the Lorentz factor. There is an analogous transformation on 4-vectors rather than
coordinates, which easily accommodates arbitrary boost directions:
ˆ
n = γ (u + β n), −uˆ = γ (nˆ + βu).
In the following, n is timelike. In our notation, which is motivated by the usage

below, n ≡ β nˆ is the relative velocity of n according to u’s frame, and conversely
u ≡ β uˆ is the velocity of u in n’s frame, where β is the relative speed. The hats
signify unit vectors, so these are the boost directions. The boost sends u to n, and n
to −u [5, §4]. Note it is not an inverse boost; the plus signs above are because vectors
transform oppositely to coordinates. In curved spacetime, one applies a separate
local Lorentz boost at every point, which is a transformation within each tangent
space.
Supposing u and n are provided, we may recover:
γ = −u, n,
and hence:
n = γ −1 n − u.
196 C. MacLaurin
One can also show:
u = γ −1 u − n.
The relative velocity vectors are purely spatial, according to the observer doing the
measuring:
u, n = 0 = n, u.

The relative speed is recovered from β = 1 − γ −2 , or:
n, n = β 2 = u, u.
Our arrow notation imitates 3-vectors; however, we use 4-vector formalism.5 We

stress that n depends on both n and u, and likewise for u.
3.2 Length-Contraction Vector
We derive a 4-vector formalism for length-contraction. This algebraic approach is

useful for comparing measurements between frames.
In relativity, length-contraction results from relative motion between frames. In
Fig. 2, the observer n determines u’s 3-space to be contracted along the direction
of relative motion (the horizontal axis, which is aligned with u). This is due to the
relativity of simultaneity. This diagram is in principle standard, apart from the vector
Fig. 2 Spacetime diagram

showing length-contraction,
in the case of n timelike. The
observer u is holding a unit
ruler ξ , which sweeps out the
grey worldsheet over time.
The intersection with n’s
3-space is the vector ξ contr
5 3-vector treatments may equate spatial vectors with their boosted image, because the components
are the same in the boosted Minkowski coordinates.

notation. It is found in Schutz, for example, [12, §1.8] but appears absent from most
recent textbooks.
The figure applies to both inertial macroscopic objects in flat spacetime, or
locally in curved spacetime within a single tangent space. Either way, in Minkowski
coordinates aligned with the boost direction, the ruler is ξ μ = (βγ , γ , 0, 0), whereas
μ
the contracted ruler is ξcontr = (0, 1/γ , 0, 0). Note ξ contr is not the boost of ξ , nor is
it orthogonal projection onto n’s 3-space. These are all parallel—at least for purely
spatial vectors aligned with the boost direction—but have lengths 1/γ , 1, and γ ,
respectively, compare Jantzen, Carini, and Bini [5, Figure 1].
To obtain ξ contr as an abstract vector (not just components in an adapted
coordinate system), observe from the diagram we must add to ξ a multiple of u so
that the sum is orthogonal to n. Require ξ + λu, n = 0 say, with unique solution
λ = −ξ , n/u, n aside from some exceptional cases.6 Hence:
ξ , n
ξ contr = ξ − u. (3)
u, n
Fig. 3 Length-contraction, where the observer or normal n has arbitrary nature. The vertical
axis is u’s time, in contrast to Fig. 2, and n is a different vector in this case. For rulers aligned
with the boost or “tilt” direction, if the hyperplane normal n is the timelike vector shown, then
ξ contr is the spacelike vector, and vice versa. For an aligned ruler and n null, ξ contr = u + ξ ,
which is proportional to n. For unaligned rulers, the length-contracted ruler may remain spacelike,
irrespective of the sign of n, n
6 Ifn is orthogonal to both u and ξ , then the entire worldsheet lies in the hyperplane. If n is
orthogonal to u only, the intersection is the infinite line spanned by u.
198 C. MacLaurin
This holds for any ξ (orthogonal to u), not just those aligned with the boost
direction.7 It also holds for hypersurfaces of any character (hyperplanes, within a
tangent space): timelike, null, or spacelike. See Fig. 3. Equation (3) is unchanged
by rescaling n, so n need not be normalised, which is reassuring for the null case
especially. Hence n may be any nonzero vector. Note that while ξ is associated with
u, since u, ξ = 0, the contracted vector ξ contr is orthogonal to n, hence may be
spacelike, null, or even timelike. This generalises length-contraction to 4-vectors
and arbitrary “observer” hyperplanes.
Two sources deserve special commendation. Møller presents the only textbook
vector treatment of length-contraction I am aware of, though he uses 3-vectors [9,
§19]. Jantzen, Carini, and Bini define various projection and boost maps, including
a certain P (u, n)−1 which acts as: P (u, n)−1 ξ = Pu ξ + n, ξ u. This is equivalent
to Eq. (3). They also describe a scaled boost γ −1 B(n, u)ξ , which is equivalent only
for ξ aligned with the boost direction [5, §4]. Both sources assume n is timelike.
3.3 Measurement
The squared-norm ξ contr , ξ contr of the length-contracted vector may take any value
in (−∞, 1]. But physically we interpret it as the same ruler, just in a different frame.
It retains “unit” length in the various following senses. In the ruler’s proper frame
(that is, the frame u) it has unit length, since ξ , ξ = 1. If the ruler has tick marks or
labels like “1/10”, “2/10”, etc., then any frame reading them agrees the stated length
is 1, even though this does not correspond to metric distance in that frame. Further,
the mass or number of atoms is the same in any frame. Finally, any frame can correct
for the relative motion, to recover the proper length. Under this interpretation, the
ratio between tick mark (or “proper”) length and a coordinate gradient is:
1 1
dLticks = d = d. (4)
d, ξ contr d, ξ − u,,nn d, u
ξ
This is from substituting ξ contr from Eq. (3) directly into Eq. (1). That formula is
valid because the vector is “unit” in the sense considered here (tick mark length, not
metric length). Note the coefficient is not simply γ /d, ξ , in general.
One can also compare two rulers directly, not against a coordinate. Suppose ρ is
a ruler in some third frame. Then:
ξ contr , ρ contr
expresses their overlap in n’s frame. It accounts for both length-contraction, and
their co-alignment in n’s space.
7 We give an alternate derivation for the case of n timelike, for reassurance. Decompose ξ into
orthogonal components in the n and uˆ directions, plus a remainder. Remove the n-component,
divide the boost direction component by γ 2 , and preserve the remainder. Note that dividing the
uˆ component by γ 2 is equivalent to subtracting 1 − γ −2 = β 2 times u, ˆ from ξ . This leaves
ξ + ξ , nn − ξ , uu after absorbing the β’s, which simplifies to Eq. (3).
4 Volume Forms
There are at least three different volume elements an observer n might attribute to
a fluid u, or vice versa. These include the usual metric volume, a length-contracted
volume, and a “de-contracted” volume which corrects for length-contraction. As
previously we allow n to be timelike, null, or spacelike.
4.1 Metric Volume
Recall the volume form on some oriented region of a p-dimensional submanifold

is:

|g(p) | dx 1 ∧ · · · ∧ dx p , (5)
where the x i are coordinates on the submanifold, g(p) is the determinant of the
induced metric, and | · | its absolute value. Geometrically, at each point the
scalar |g(p) | is the volume of the p-dimensional parallelepiped spanned by the
coordinate basis vectors ∂i . To see this, first recall this parallelepiped has oriented
volume:
a ∧ b ∧ ··· ,
where we relabel the ∂i as a, b, etc. See the leftmost illustration in Fig. 4. This is
a p-vector, with “∧” the wedge product, so the expression is the sum of all signed
permutations of its elements. It seems convenient to omit the 1/p! normalisation
convention for our application. For a bivector, a ∧ b = a ⊗ b − b ⊗ a. Now the scalar
volume is formed from contracting the (dual of the) p-vector with itself, which is
given by the determinant of the metric products between the spanning sides:
Fig. 4 Parallelepipeds representing the metric volume, length de-contracted volume, and length-
contracted volume elements, respectively. All reside at a single point in curved spacetime. In the
diagrams p = 3, and the boost or tilt direction aligns roughly with “a”. In general, for each case the
wedge product of the p side vectors is a p-vector representing the oriented volume. Its magnitude is
the scalar volume. Pu is the spatial projector. This diagram is a more literal visualisation, in contrast
to the spacetime diagrams (Figs. 2 and 3) which are more abstract. Both pictures are insightful, and
complementary
200 C. MacLaurin

a, a a, b · · ·1/2

a ∧ b ∧ · · · , a ∧ b ∧ · · · 1/2
= b, a b, b · · · . (6)
. .. . .
.. . .
On the RHS, the inner | · | is called a Gram determinant, and the outer | · | is an
absolute value required by the Lorentzian spacetime signature. Since ∂ i , ∂j =
gij , the matrix entries are simply metric components, so this is indeed |g(p) | as
claimed. We interpret the parallelepiped as lying within the tangent space at a single
point in curved spacetime, which leads to an exact (as in precise) “infinitesimal”
volume.
4.2 De-Contracted Volume
To the observer (or hyperplane normal) n, the fluid u is length-contracted, so the

usual metric volume element on the hyperplane does not coincide with the proper
volume of fluid. A “de-contracted” volume element compensates for this.
Consider an arbitrary vector “a”, interpreted as a line segment. This vector
intersects part of the fluid’s worldline(s); however, in general the fluid will be length-
contracted along this direction through space and time. We wish to transform the
vector to one representing the proper length of fluid intersected. This is analogous
to recovering ξ from ξ contr , which is achieved by:
ξ = ξ contr + ξ contr , uu. (7)
To see this, evaluate the RHS, or inspect the previous length-contraction diagrams.
It is simply the usual spatial projection using P ab := g ab + ua ub . Call the map
ξ contr → ξ “de-contraction”. While length-contraction depends on both observers
n and u, the reverse process requires only u. (And whereas ξ contr is obtained by
a certain projection orthogonal to n, the recovery of ξ is an orthogonal projection
orthogonal to u.)
In the same way, the vector “a” is de-contracted as:
a → a + a, uu. (8)
We place no restriction on the resulting length, unlike for ξ . This vector recovers a
fluid frame quantity, locally.8 Its squared-length is:
8 Over a macroscopic region—a submanifold—each pointwise de-contraction is into u’s frame.

However, the overall result still depends on the choice of simultaneity, that is the “time” slice or
tilt of the submanifold. This matters if the fluid has intrinsic expansion or contraction over time.
The example in Sect. 5.4 considers this carefully.

a + a, uu, a + a, uu = a, a + a, u2 , (9)
which gives the proper length of material intersected by a. Note the vector “a” may
be spacelike, null, or timelike. But its de-contraction is orthogonal to the timelike
vector u, hence is spacelike or the zero vector.
Now consider a curve parametrised by some λ, which need not be √affine. Label
the tangent vector field a μ := dx μ /dλ. The usual metric length is |a, a| dλ.
However, at each point, “a” may have arbitrary spacetime direction, so the local
fluid element is length-contracted along this a-direction. (Here we need not specify
the “observer” n, as the curve itself sets the tilt. We know only that n, a = 0, in
principle.) Hence the proper length of fluid along the curve is:

L= a, a + a, u2 dλ.
This extends to higher dimensions. Consider a volume element a ∧ b ∧ · · ·

(typically, formed from coordinate basis vectors at a single point of a submanifold).
De-contract each of the sides, then recombine them using wedge products:
a + a, uu ∧ b + b, uu ∧ · · ·

= a ∧ b ∧ · · · + u ∧ u (a ∧ b ∧ · · · ) , (10)
after simplifying. The “” symbol means u is substituted “into” the leftmost
argument of: a∧b∧· · · ; for background see Vaz and da Rocha [13, §2.6]. The effect
is a spatial projection, returning a new parallelepiped within u’s space. Compare
Eq. (8), which is the 1-dimensional case. In particular, the expression vanishes upon
substituting u . The projected parallelepiped’s volume scalar represents the proper
volume of fluid contained in the original (coordinate basis) parallelepiped. It follows
by combining results similar to Eqs. (6) and (9):

a, a + a, u2 a, b + a, ub, u · · ·1/2

b, a + b, ua, u b, b + b, u2 · · ·
.
.. .. . .
. . .
No absolute value sign is required. This is the coefficient in the de-contracted

volume p-form: | · · · |1/2 dx 1 ∧ · · · ∧ dx p , in contrast to Eq. (5). If u lies within
the original volume element a ∧ b ∧ · · · , the de-contracted element vanishes.
Geometrically, this is because there is zero spatial length along the time direction,
and the p-volume of the remaining orthogonal (p − 1)-parallelepiped vanishes.
Another important special case is when either volume element includes the boost
direction. Then the fluid volume is simply the Lorentz factor times the metric
volume. But the power of our formalism is for volumes aligned with neither u nor
the boost (generally, “tilt”) direction.
202 C. MacLaurin
One could easily assume that since the spatial projection Pu a is onto u’s space,
it is a measurement by the observer u, in its own frame u. It is not, at least in the
following senses. It is acontr which represents the length determined in u’s frame (at
least for ‘a’ spacelike). Rather Pu a is used by n, it is the fluid length in n’s frame,
which corrects n’s volume element for length-contraction.
4.3 Contracted Volume
A third volume measure is a contraction of space, in contrast to the de-contraction

examined above. The fluid frame u might wonder, “To the observer n, my 3-space
is length-contracted. Which volume element would n attribute to my space, or
a subspace thereof?” In this interpretative choice, n does not adjust for length-
contraction. (The result is not just the metric volume in n’s space, at least not
directly, since the volume element does not lie orthogonal to n. It is only after length-
contraction that the volume element is orthogonal to n.) The square in Sect. 5.2 is
an intentionally simple example, to clarify the motivating idea.
Consider a parallelepiped a ∧ b ∧ · · · lying entirely in u’s space, so all sides are
orthogonal to u. To n, each side is length-contracted, for example:
a, n
acontr = a − u.
u, n
This is equivalent to Eq. (3), except we do not require “a” to have unit length. The
vector acontr may be timelike, null, or spacelike, since n may have any nature. The
p-vector obtained by wedging together the contracted sides represents the length-
contracted parallelepiped:
a, n b, n
a− u ∧ b− u ∧ ···
u, n u, n
1
= a ∧ b ∧ ··· − u ∧ n (a ∧ b ∧ · · · ) , (11)
u, n
after simplifying. See Fig. 4. This lies within the hyperplane orthogonal to n, as seen
by substituting n . Note it depends only on the combination a ∧ b ∧ · · · , and not the
individual sides; the same is true of the de-contracted volume in Eq. (10). The scalar
product of two length-contracted sides is, for example:
a, n b, n a, n b, n
a− u, b − u = a, b − , (12)
u, n u, n u, n2
using a, u = 0 = b, u. Hence the length-contracted volume scalar is, by analogy
with Eq. (6):
1/2

a, a − u
a,n2 a, b − a,n b,n · · ·
,n2 u,n2
b,n a,n
b, a − u,n2 b, b − b ,n2
,n2
· · · .
u
.. .. ..
. . .
Again, the double lines · mean absolute value of the determinant. The contracted
volume p-form is then · · · 1/2 dx 1 ∧· · ·∧dx p . A special case is when the boost or
tilt direction lies within the parallelepiped, then the volume scalar is just γ −1 times
the metric volume.
5 Examples
The Lorentz boost and square examples below are simple rectilinear motions used to
illustrate the concepts. The rotating disc seems paradoxical but is largely understood
in the literature. The Schwarzschild spacetime example is highly original.
5.1 Lorentz Boost
Consider again the prototypical Lorentz boost t = γ (t − βx) and x = γ (x − βt)

in Minkowski spacetime. Define two timelike observers, a “stationary” one with
u = ∂t and ruler ξ = ∂x , and a “moving” observer n = ∂t with ruler ρ = ∂x .
All components will be expressed in the unprimed coordinates (t, x, · · · ). Then
uμ = (1, 0, · · · ), ξ μ = (0, 1, · · · ), nμ = (γ , βγ , · · · ), ρ μ = (βγ , γ , · · · ),
(dx)μ = (0, 1, · · · ), and (dx )μ = (−βγ , γ , · · · ). From Eq. (1), the stationary
observer measures dL = dx, while the moving observer measures unit length
dL = dx , each using their own ruler. (Do not read anything into the repeated “dL”
notation, these quantities are distinct.) If the stationary observer compares their ruler
with the moving observer’s coordinate x , they find:
dL = γ −1 dx
in the proper frame, which is occasionally termed “length-expansion” because

the ruler takes up an interval x = γ , though of course we interpret this
philosophically as “really” the ruler having fixed unit length and the coordinate level
sets x = const being length-contracted. Likewise, the moving observer determines
dL = γ −1 dx. To the stationary frame, the moving ruler is ρcontr = (0, γ −1 , 0, 0),
μ
for which its tick mark readings claim:
dLticks = γ dx,
204 C. MacLaurin
so it occupies a shortened interval x = γ −1 . Yet also dLticks = dx since both

ruler and x -coordinate are length-contracted.
5.2 Square Flux
Continuing from the previous example, now consider a square spanned by the x and
z-axes, with oriented area the 2-vector:
∂x ∧ ∂z ,
which has proper area scalar 1. Consider a fluid with 4-velocity ũμ = (γ , βγ , 0, 0).
The square is like a window onto the fluid. In the “square frame”, the fluid is
length-contracted across this window, so it has intrinsic area greater than 1. The
boost direction is aligned with the square. Hence the fluid proper flux is γ , which
is straightforward once the scenario is understood. In terms of the formalism in
Sect. 4.2, the side ∂x has de-contraction vector:
(βγ 2 , γ 2 , 0, 0),
and ∂z needs no de-contraction. The above vector has length γ , and since the de-
contracted sides remain orthogonal, the area is simply their product γ . Note we did
not need to specify the “motion” of the square—that is, the 4-velocity or normal n
of an associated observer or frame—nor whether the square persists through time
(or space) at all! One only knows that in principle, such an n must be orthogonal,
hence be a linear combination of ∂t and ∂y .
But now assume the square is attached to the timelike observer ñ = ∂t . According
to the ũ frame, the square is length-contracted, so its area is reduced to γ −1 . (For this
contracted volume measure, we do not compensate for length-contraction, which
is a choice and definition.) Using the Sect. 4.3 formalism, the side ∂x effectively
becomes (β, 1, 0, 0), which has length γ −1 as expected. The contracted sides are
orthogonal, hence the contracted area scalar is simply their product γ −1 . This
square example helps to illustrate the conceptual motivation behind the three volume
elements.
5.3 Rotating Disc
Take Minkowski spacetime in polar coordinates (t, r, φ). In the coordinate or

“laboratory” frame n := ∂t , the disc rotates at dφ/dt = , a constant for all
particles on the disc. These particles u have:
uμ = γ (1, 0, ),
where γ = −n, u = (1 − 2 r 2 )−1/2 . The disc is bounded by r < −1 , to remain

subluminal. The natural ruler in the tangential direction, for a disc particle, is:
1
ξ μ = γ r, 0, ,
r
which is orthogonal to u and ∂r , and concurs with the maximal direction given in
Sect. 2. What is the disc circumference at given r? In one sense it is the Euclidean
value, according to the lab observer if they ignore length-contraction, in which case
the disk may as well be absent entirely.
A better answer is the disc’s proper circumference. In the lab frame, disc rulers
are length-contracted to:
μ 1
ξcontr = 1, 0, .
γr
Hence the unit rulers are shortened to γ −1 (recall this is a statement about vectors in
local tangent spaces; any ratios apply to “infinitesimal” measurements on the actual
disc). We have:
dLticks = γ r dφ,
meaning a greater length fits into a given φ coordinate interval than the Euclidean
value. Hence the proper circumference is L = 2π rγ ! This strange result is the
consensus view, which we thoroughly affirm.
Now consider the frame of a disc particle. In this frame, a disc ruler measures:
dL = γ −1 r dφ. (13)
Paradoxically, there is less ruler length in a given coordinate interval φ than the
Euclidean value, in this frame. Likely few sources state this result clearly. The result
holds locally but paradoxically does not lead to a total circumference 2π rγ −1 .
It may seem the problem is the vorticity of the disc particles, so no simultaneity
surface exists which is orthogonal to the worldlines, on any open neighbourhood of
the surface. See Fig. 5. (However, even for fluids without vorticity, the simultaneity
hypersurfaces do not align with the desired measurement direction, in general.)
For an integral curve of the field ξ at fixed r, it turns out a coordinate interval
φ = 2π γ 2 is needed to return to the starting particle on the disc, because the
particle has moved in the meantime. This recovers the L = 2π rγ global result.
Complications of this nature occur whenever the observer field n has vorticity.
While our formalism can handle this, a simpler and safer approach is to pick
a hypersurface or other submanifold, and “de-contract” it. That the fluid u has
vorticity is inconsequential, in this approach. For the circumference, an obvious
choice is the circle at fixed t and r. After de-contraction, the result is again
L = 2π rγ . Another safe approach is to define a “rotating” coordinate φ̃ := φ − t.
206 C. MacLaurin
Fig. 5 The failure of synchrony on the rotating disc, due to its rotation. The vertical axis is
Minkowski time t, and the loop is a circle at fixed r. Shown are the 4-velocity vectors of selected
disc particles, with each square representing the orthogonal subspace within each local tangent
space. A “circumference” curve tangent to all such planes does not close up but forms a helix in
spacetime. Hence the length measurement requires careful conceptual reasoning
It is comoving with the disc particle worldlines, since d φ̃, u = 0. Hence in any
observer frame n, the contracted ruler gives the same reading dLticks = γ r d φ̃, using
Eq. (4). Similarly one revolution is φ̃ = 2π, along any reasonable hypersurface.
Again, the proper circumference is 2π rγ .
5.4 Schwarzschild Spacetime
It is often claimed the properties of a hypersurface t = const of Schwarzschild

time are measurement by an “observer at infinity”. However, interpretations using
local observers are more physically justified. The normal vector field is (dt) ,
which is timelike for r > 2M. After normalising, these are the static observers
(1−2M/r)−1/2 ∂t , hence in the local observer field approach, t = const is physically
interpreted as the space of static observers.
But how do non-static observers measure Schwarzschild spacetime? For now,
assume the motion is restricted to the equatorial plane θ = π/2. Parametrise
observers u by their energy per mass e and angular momentum per mass , defined
from the usual Killing vector fields:
e := −u, ∂t , := u, ∂φ .
These are constant along geodesics, but either way:

2M −1 2M 2
uμ = e 1 − ,± e2 − 1− 1 + 2 , 0, 2 . (14)
r r r r
The “±” is an extra degree of freedom when outside the horizon but must be a
minus for r ≤ 2M. It is not obvious which choice of ruler orientation is best. The
coordinate gradient rulers yield measurements: [7]
1 r
dLr,3−gradient = dL(φ) =
dr, dφ,
e2 − (1 − 2M/r)2 /r 2 1 + 2 /r 2
(15)
from Eq. (2). The corresponding ruler vectors (omitted) are orthogonal for = 0,
but apparently not otherwise. The “θ” ruler is trivial, with measurement dL =
r dθ. For = 0 the motion is radial, and the above proper frame measurements
reduce to |e|−1 dr and r dφ. In fact these can be seen by inspection of the line
element expressed in generalised Gullstrand-Painlevé coordinates, as examined in
my
√ previous DomoSchool contribution [6]. For a static observer, additionally e =
1 − 2M/r, so the familiar radial length element (1 − 2M/r)−1/2 dr is recovered.
r is often called a “curvature coordinate”, which is accurate; however, a nuanced
description would mention spherically symmetric submanifolds, or observers with
zero angular momentum.
What is the 3-volume in Schwarzschild spacetime? If choosing an r = const
slice, the metric volume inside the horizon is infinite, unless the black hole formed
by collapse and later evaporates. One author obtains a radial distance ≈1077
lightyears for a solar mass black hole, despite a coordinate interval of just r =
2M ≈ 3 km. In our local observers approach, one must choose an observer field.
One reasonably natural choice is to fix e and as constants across all observers, after
checking the range of r is consistent. The presence of vorticity in the observer field
adds complication, as the rotating disc scenario showed. Hence we cannot naively
extrapolate the proper frame measurements in Eq. (15) to a total volume. Instead, it
is simpler to choose submanifolds and “de-contract” them.
Consider the circular curve parametrised by φ, at fixed t and r, where we allow
any r ∈ (0, ∞). The tangent vector is just ∂φ , which de-contracts (that is, spatially
projects orthogonal to the 4-velocity in Eq. (14)) to a vector of length 1 + 2 /r 2 r,
yielding a total circumference:
1 + 2 /r 2 2π r.
On the other hand, a great circle in the ∂θ direction has Euclidean circumference:
2π r,
208 C. MacLaurin
if misaligned with the fluid. (Imagine extending the field u by revolving the
equatorial plane about an axis passing through φ = 0, π. This also requires a certain
redefinition of for consistency. Choose the great circle at φ = π/2, 3π/2.)
Now consider a 2-sphere at fixed t and r, parametrised by θ and φ. The tangent
vector ∂θ is already de-contracted, has length r, and is orthogonal to the de-
contraction of ∂φ . Hence the 2-sphere area determined by u follows simply:
1 + 2 /r 2 4π r 2 ,
by invoking symmetry. Now the radial direction is less obvious. We seek the most
natural length corresponding to a given coordinate interval, say r ∈ (0, 2M].
However, the length depends on the curve chosen, even for our de-contracted length
element, intuitively because the tilt affects how many worldlines of u are intersected.
The natural choice is a radial curve orthogonal to u. We do not need to solve for it
explicitly, just its tangent:

e2 − (1 + 2 /r 2 )(1 − 2M/r)
a :=
μ
− , 1, 0, 0 ,
e(1 − 2M/r)
which is the unique vector orthogonal to u, dθ and dφ, up to scalar multiple. We

chose a r = 1, which
corresponds to a curve parametrised by r. This tangent vector
has norm |e|−1 1 + 2 /r 2 and is already de-contracted since it is orthogonal to u.
It follows the radial length element is:
dLradial = |e|−1 1 + 2 /r 2 dr. (16)
This is valid both locally in the proper frame u, and for our sought macroscopic
length (contrast Eq. (13)). For = 0 the coefficient diverges as r → 0+ , but this
is not the case for the “r-gradient” ruler (Eq. (15)), so not a physical disaster. An
antiderivative is:
1
Lradial = 1 + 2 /r 2 r − tanh−1 .
e 1 + 2 /r 2 r
For = 0 this also diverges as r → 0. But at least the 3-dimensional volume

remains finite, it turns out. The idea is to find a hypersurface whose tilt agrees
as closely with the fluid frames as possible. Because of the vorticity, it cannot be
orthogonal to the worldlines. We choose a spherically symmetric hypersurface, to
avoid the rotation, but one which aligns with the fluid’s space in the radial direction.
But this is just the one spanned by the vectors ∂θ , ∂φ , and “a” above, hence the
de-contracted 3-volume element is:
1 + 2 /r 2 2
r dr ∧ dθ ∧ dφ,
e
at the equator. Invoking symmetry, the integration is trivial, and the total scalar
volume between r → 0 and r = R ∈ (0, ∞) is:
1 4
π R 3 + 4π 2 R . (17)
e 3
The effect of angular momentum is to increase the proper volume, as for the rotating
disc. For = 0 and e = ±1, the 3-volume is Euclidean. This case was noticed
a century ago, from the line element in Gullstrand-Painlevé coordinates, but not
understood. The e = 0 case is distinct, with natural hypersurfaces at constant r <
2M, being topologically R × S 2 . Compare Fig. 5 in Ref. [6].
6 Discussion
In summary, we have introduced two new volume measures: a length-contracted

volume and a “de-contracted” volume, on submanifolds of spacetime. They are
derived from the usual metric volume, plus physical consideration of an (active)
transformation between two frames: a timelike “fluid” u, and an “observer” normal
n which may be timelike, null, or spacelike. The wedge product tool handles this
in a beautiful and elegant way. The results hold in arbitrary dimension, with the
expression “3-space” really meaning space in contrast to spacetime. The algebraic
formulation is simpler than reasoning from first principles alone. It applies to the
rotating disc, conveniently handling the vorticity which makes this scenario so
paradoxical. This is extended to Schwarzschild spacetime, returning the spatial
volume relative to a two-parameter family of fluid fields (Eq. (17)). In particular,
the radial distance is relative to the observer. Primarily our results are geometrical,
for which they are exact, and only secondarily a model of an extended physical
ruler.9
The very mention of length and volume measurement
brings to mind introductory
textbookmaterial,
such as the proper length ds of a curve, or generally the metric
volume |g(p) | d p x of a submanifold. Indeed the topic is foundational, but the
results herein result from years of careful thinking and refinement. We speculate
development was hindered by the overreach of various correct results, as follows.
9 The vectors like ξ and “a” are crude models, from a materials science or engineering perspective.
But a “constructive” approach incorporating all known physics would be extremely complicated.
It would include the stress-strain response of an extended material ruler to the undeformed
configuration (the metric length, as analysed in this work), using relativistic elasticity theory or
even our best quantum description of matter. The distortions from curvature, tidal forces, and
any external acceleration or forces on the ruler, may induce sound modes which depend on the
past history of the worldsheet. Its stress-energy has back-action on spacetime. Also a true ruler is
not 1-dimensional. Einstein was right, both to treat rods and clocks as fundamental, and to later
acknowledge the shortcomings [2]. Perhaps quantum fields would be effective rulers, using the
spreading of a wavepacket or the size of a hydrogen orbital.
210 C. MacLaurin
General covariance was a tremendous advance, but it does not imply coordinates
have no relation to physical meaning, at least when observers are specified.10,11
Born-rigid motion is very limited, and intrinsically rigid objects are unphysical, but
approximate rulers including radar devices work well in reasonable environments,
and length does not reduce to time measurement. The rotating disc is perplexing
but does not imply relativity can only accommodate pointlike—not extended—
objects. There are other contributing factors besides these “overreaches”. Alternate
voices including Gullstrand, Painlevé and Sagnac raised important questions, but
their (gravely mistaken) conclusion that relativity is self-contradictory surely did
not motivate others to sift out the gold. Yet perhaps the main reason is, in relativity
the topic of observers and physical measurement is under-promoted, despite some
excellent technical treatments [4, 5].
In future work we offer more diagrams and explanation, including separate
papers on black hole spacetimes, and a rotating disc in an axially symmetric
spacetime. The formalism applies to any abstract “fluid” where its proper flux is
significant (the spacelike hypersurface cases may be largely known already, but
presumably not lower-dimensional submanifolds with arbitrary tilt). Example flows
are the vector potential in Maxwell’s electromagnetism, or the Noether current in
various quantum theories. A signed volume scalar would accommodate multiple
crossings of the same fluid worldline over a chosen submanifold. There will be
connections with Stokes’ theorem. Another application is averaging or coarse-
graining of fluid quantities, within a different frame. A straightforward extension
would be length-contraction using simultaneity other than the usual Poincaré-
Einstein convention, or within theories without Lorentz invariance. We assumed
a field of observers, but many applications require extension of a single frame, such
as satellite radar distance in Solar System astrometry. Our results would form a
first-order approximation for an extended frame, but the error should be quantified.
7 DomoSchool Memories
Domodossola is stunning, including from Cima Lariè (2144m). :) Grazie to Sergio

Cacciatori and Andrea Cottini for warm hospitality. It was a big privilege to meet
Donato Bini, world expert on observers and spacetime splitting: thank-you for
10 In Norton’s view, [10, §2] “That coordinate systems can be used to represent significant physical
content is not the modern view and it is tempting to think that no other view is possible. But that
narrowmindedness is quite incorrect”. Equation (1) is also relational, in that it depends on the ruler
orientation, which is in turn constrained by the observer 4-velocity u.
11 On the other hand, Schwarzschild t has been afforded too much meaning, as the (only) “time
at infinity”, the historical rejection of black hole collapse, and the repeated assertion by textbooks
that r is not metric distance but (1 − 2M/r)−1/2 dr is.
discussion and gelati. It was fun to use our coffee and meal tickets, exploring
various cafes and restaurants. Thanks to Orville Damaschke, Jiří Ryzner, and Bini
for tolerating my order of tea with pizza.
References
1. H. Brown, Physical Relativity: Space-Time Structure from a Dynamical Perspective (Oxford,

England, 2005)
2. H. Brown, The behaviour of rods and clocks in general relativity and the meaning of the metric
field, in Beyond Einstein, ed. by D. Rowe, T. Sauer, S. Walter (Springer, Berlin, 2018)
3. S. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Addison Wesley,
New York, 2004)
4. F. de Felice, D. Bini, Classical Measurements in Curved Space-Times (Cambridge Uninversity,
Cambridge, 2010)
5. R. Jantzen, P. Carini, D. Bini, The many faces of gravitoelectromagnetism. Ann. Phys. 215(1),
1–50 (1992)
6. C. MacLaurin, Schwarzschild Spacetime Under Generalised Gullstrand-Painlevé Slicing
(Birkhäuser, Cham, 2019)
7. C. MacLaurin, Clarifying Spatial Distance Measurement (World Scientific, Singapore, 2021).
Submitted, and book forthcoming
8. C. Misner, K. Thorne, J. Wheeler, Gravitation (W.H. Freeman and Co., New York, 1973)
9. C Møller, The Theory of Relativity (1952)
10. J. Norton, Einstein’s triumph over the spacetime coordinate system: A paper presented in honor
of roberto torretti, in Diálogos: Revista de filosofía de la Universidad de Puerto Rico (2002)
11. J. Norton, A Conjecture on Einstein, the Independent Reality of Spacetime Coordinate Systems
and the Disaster of 1913 (2005)
12. B. Schutz, A First Course in General Relativity (Cambridge University, Cambridge, 2009)
13. J. Vaz, R. da Rocha, An Introduction to Clifford Algebras and Spinors (Oxford, England, 2016)
Exact Solutions of
Einstein–Maxwell(-Dilaton) Equations
with Discrete Translational Symmetry
Jiří Ryzner and Martin Žofka
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
1.1 Majumdar–Papapetrou Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
2 Alternating Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
2.1 Constructing the Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
2.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
3 Uniform Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
3.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
4 Smooth Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
4.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
5 Uniform Reduced Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
5.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
6 Conclusions and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
Mathematics Subject Classification (2000) 83C15, 83C22, 83C57
1 Introduction
Einstein equations are non-linear set of differential equations, yet there exist exact
solutions, whose geometry and matter content lead to linearity. This is also the case
of the multiple extremal1 black hole solutions, which first appeared in the 1940s
due to the works of Majumdar [1] and Papapetrou [2], yet their interpretation had to
1 The term extremal refers to the fact that the black hole horizons are degenerate, which in this case
translates to their charges equal to their masses.
J. Ryzner () · M. Žofka

Institute of Theoretical Physics, Charles University, Prague, Czech Republic
e-mail: j8.ryzner@gmail.com; zofka@mbox.troja.mff.cuni.cz
214 J. Ryzner and M. Žofka
wait until the 1970s and the classical paper by Hartle and Hawking [3]. The family
of Majumdar–Papapetrou (MP) spacetimes is a manifestation of the curious fact
that the balance between electrostatic and gravitational forces in classical physics
is preserved even in general relativity. This also extends to the linearity of the
field equations and thus the principle of superposition holds in GR as well. In our
previous works on the subject, [4] and [5], we were interested in solutions exhibiting
axial and cylindrical symmetry, respectively. The cylindrically symmetric solution
was the field of an infinitely extended extremally charged string (ECS) located along
the axis of symmetry. One of the questions arising from the paper was whether it
is possible to produce the same field asymptotically far from the axis and due to
an infinite number of isolated, equidistantly distributed, identical point sources of
a mass equal to their charge. This was motivation for our next work [6], where
we investigated a few ways to construct such solutions. The interesting feature of
the resulting spacetimes is that it has a discrete translational symmetry everywhere
while asymptotically, it would be fully cylindrically symmetric. We now build on
this work, review some facts and show newly discovered features.
The chapter is organized as follows: in the first section, we review the general
MP solution in an arbitrary dimension with an arbitrary source. In Sect. 2, we look at
the first solution which is an infinite alternating “crystal” constructed of alternating
positive and negative charges. The solution is in the form of an infinite series of
functions: we review its convergence, asymptotics and derivatives to be able to infer
the symmetries of the spacetime and its interpretation. In Sect. 3, we study the case
of a uniform crystal composed of identical charges. In Sect. 4, we show the solution
of a smooth “crystal”, which is a solution of MP and charged dust. Section 5 presents
a way of constructing the 4D infinite crystal out of a closed-form 5D solution via
dimensional reduction. We conclude with some final remarks and open questions.
1.1 Majumdar–Papapetrou Solution
The metric D g of the Majumdar–Papapetrou solution in arbitrary dimension2 D =

n + 1, n ≥ 3, reads [7]
D
g = −U −2 dt 2 + n hij dx i dx j , (1)
where t is a time-like Killing coordinate, so that the metric is static with the function
U = U (x i ) only depending on Cartesian-like spatial coordinates3 x i . The spatial
metric n hij is conformally flat
2 The number of spacelike dimensions is denoted as n.

3 Latin indices range over 1, . . . , n and label only spatial components, and Greek indices are
0, . . . , n.
Exact Solutions of EM(-Dilaton) Eqs. with Discrete Translational Symmetry 215
2
n
h = U n−2 · n δij dx i dx j . (2)
These coordinates describe well the region above the horizons. The electromagnetic
potential A and the electromagnetic field tensor F read
dt U,i n
A = cn , F = dA = −cn dx i ∧ dt (3)
U U2
i=1

with cn = n−1
2(n−2) . The corresponding stress–energy tensor, T , is

1 μ F
T μν = F β F νβ − gμν , (4)
4π 4
where
⎡ −1
⎤2

n
⎢
∂ U (n−2)
⎥
F = Fμν F μν = cn2 (n − 2)2 ⎣ ⎦ (5)
∂x i
i=1
is the Maxwell scalar. The non-vanishing components of the stress–energy tensor

are
j 2−2n F j
16π T0 0 = F, 4π Ti = −cn2 U n−2 U,i U,j − δ . (6)
4 i
Einstein and Maxwell equations then have the form
R
Ricμν − gμν = 8π Tμν , ∇ν F μν = 4π J μ , ∇μ J μ = 0. (7)
2
The 4-current J μ due to the charge density (x j ) leads to a single Einstein–Maxwell
equation
(x j ) μ n
J μ = −cn √ δ0 ⇒ δ U = U,ii = −4π (x j ). (8)
−g
i=1
Here, g is the determinant of the metric gμν and δ denotes the flat-space
Laplacian.4 In case of Majumdar–Papapetrou, (x) is assumed to be a distribution
of point charges, which means that J μ = 0 away from the sources. One particular
solution, in which we are interested, is a multi-black hole spacetime of the form
√ ij
4 The Laplacian for the spatial metric h is defined as h f ≡ hij ∇i ∇j f = √1
h
hh fi ,j .

N
Mi
n
U (x) = 1 + n−2
, ri2 = (x a − xia )2 , (9)
i=1
ri a=1
with the corresponding charge current [7]
cn π 2 −1
n N
√ cn
−gJ = − δ U = n
0 Mi · n δ (x − xi ) . (10)
2 −1
4π
i=1
Here, Mi are constants, is the Gamma function and n δ is the n-dimensional Dirac
delta function. It can be shown that Mi is the mass and also charge of each black
hole, and for Mi > 0 the puncture located at ri = 0 looks like a point, but in fact
it represents the surface of a sphere Sn−1 of dimension n − 1 (and for Mi < 0
the surface ri = 0 corresponds to the location of a naked singularity). In D = 4,
there exists a coordinate transformation, which regularizes the metric at a (arbitrarily
chosen) horizon ri = 0 and the horizon is smooth [3]. However, in D > 4, this holds
only for a single black hole (N = 1). For N = 2, 3, it was shown that the horizon is
not smooth [8], while for a higher number of black holes the situation is still unclear.
2 Alternating Crystal
We aim to construct a solution, which would exhibit a discrete translational

symmetry along an axis due to which we shall call it a “crystal”. To achieve this,
we use linearity of the field equations in the MP spacetime. First, we construct
a spacetime with an anti-periodic potential, since it is easier to investigate the
convergence of the sums than in the case of a symmetric potential.
2.1 Constructing the Solution
We seek the function U in the form

Q
U = 1 + λχ , λ = , (11)
k
where k is the crystal lattice constant, Q is the charge (and mass) of each black hole
and χ corresponds to the classical potential. From (8), we know that χ has to satisfy
Laplace’s equation away from the sources
χ,ρ
χ,ρρ + + χ,zz = 0. (12)
ρ
Let us review the main results from [6]. We construct χ by the following ansatz:
∞

1 1 1
χ = χ0 + (−1)n χn , χ0 = , χn=0 = + , rn = ρ 2 + (z − n)2 . (13)
r rn r−n
n=1
The idea is to prove uniform convergence for 0 ≤ z ≤ 1/2 and then to extend the
definition of χ to other regions using symmetries of χ. One finds that

1 ∂χn ∂χn 1 ∂ 2 χn ∂ 2 χn 1
|χn | ∼ , ∼ ∼ 2, 2 ∼ 2 ∼ 3. (14)
n ∂ρ ∂z n ∂ρ ∂z n
This behaviour is independent of ρ which grants us uniform convergence of the

corresponding infinite sums, and we can also exchange derivatives with the infinite
summation. This ensures that χ constructed in this manner is a valid solution of
Laplace’s equation. We also get its symmetries:
χ (ρ, z) = χ (−ρ, z) = −χ (ρ, z + 1). (15)
The charge density can be expressed via the Dirac comb5 Ш distribution
1 Ш2 (z) − Ш2 (z − 1)
=− δ χ = δ(ρ). (16)
4π 2πρ
The function χ is periodic, and it is thus natural to ask for Fourier coefficients for a
fixed ρ. Since the potential diverges at r = 0 as 1/r, which is not integrable near the
origin, the coefficients will exist only for ρ > 0. Finding the coefficients directly is
hard, and we thus proceed in another way by assuming
∞

χ (ρ, z) = An (z)Bn (ρ), ρ > 0. (17)
n=1
We plug this into (12) and get

An (z) 1 Bn (ρ)
− αn2 = =− Bn (ρ) + . (18)
An (z) Bn ρ
5 Dirac comb is a periodic tempered distribution defined as
∞
1 2π int/T 1 2 2π nt
ШT (t) ≡ δ(t − nT ) = e = + cos .
T T T T
n∈Z n∈Z n=1
Here, αn is a separation constant. We see that An is solved by sines and cosines and
Bn is solved by Bessel functions. We use the symmetries of (15) and asymptotic
flatness in the ρ direction (e.g., from the Moore-Osgood theorem) and get
∞

χ= fl cos [αl z] K0 [αl ρ] , αl = π(2l − 1), ρ > 0. (19)
l=1
Here, K0 is the modified Bessel function,6 which diverges at ρ = 0 due to the non-
integrability of χ at the origin. We determine the unknown coefficients fl formally
from the charge enclosed in a cylinder of radius R and height 2h, which is aligned
with the z axis and centred at the origin:

R z=+h h
4π Q(R, h) = −2π χ,z z=−h ρ dρ − 2π (χ,ρ ρ)ρ=R dz. (20)
0 −h
We denote the first integral by q1 and the second one by q2 . The first term yields
∞

1
q1 = 2 fl sin (αl h) RK1 (αl R) − . (21)
αl
l=1
However, in the limit R → 0, this term vanishes. The second term gives
∞
h
q2 = fl cos [αl z] αl RK1 [αl R] dz. (22)
l=1 −h
We now write the charge using charge density (16) and determine the coefficients
h
lim 4π Q(R, h) = [Ш2 (z) − Ш2 (z − 1)] dz ⇒ fl = 4. (23)
R→0 −h
We see that the potential decays exponentially, and we determine the first leading
term by using the sum and integral inequality,7 which is independent of z, as χ,z →
6 Modified Bessel function of the second kind Kν is defined as

∞
Kν (x) ≡ exp (−xcosht) cosh (νt) dt, x > 0.
0
7 From the Newton integral test, we get the following inequality for integrable, non-increasing and
non-negative Cl :
∞ ∞ ∞
∞
∂Cl
Cl ≥ 0, ≤ 0 ∀l ≥ 1, Cl < ∞ ⇒ Cl dl ≤ Cl ≤ C1 + Cl dl.
∂l 1 1
l=1 l=1
0 for large ρ. We thus set z = 0 and get

∞
√ −αl ρ
1 πe
χ (ρ, z = 0) = Cl 1+O , Cl = 4 √ . (24)
ρ 2αl ρ
l=1
We use the integral inequality and get the following expression:

√ ∞ √
2e−πρ 1 1 2e−πρ 3 1
√ +O ≤ Cl ≤ √ 2+ +O .
ρ πρ ρ2 ρ 4πρ ρ2
l=1
(25)
This gives us strong bounds on the potential, although we do not get the leading
term precisely. From numerical computation, we see that the first leading term is

e−πρ e−πρ
χ (ρ, z = 0) = c √ + O , c ≈ 2.74. (26)
ρ ρ 3/2
We see that the potential decays faster than any multipole of an isolated system.
2.2 Geometry
In the cylindrical coordinates, the fields read
dt
g = −U −2 dt 2 + U 2 dρ 2 + ρ 2 dφ 2 + dz2 , A = , U = 1 + λχ . (27)
U
Note that χ is anti-periodic, but U is not. Near the origin, the potential behaves as a
single extremal Reissner–Nordström solution:
1 ζ̄ (3)
χ (r, θ ) = − 2 ln 2 − [1 − 3cos (2θ)] r 2 + O(r 4 ), r 1, (28)
r 8
where we use spherical coordinates r and θ . Therefore, it is possible to find a
suitable coordinate transformation, which regularizes the metric at r = 0. We apply
the transformation [3]
λ
v = t + W 2 (r), W (r) = 1 + − 2λ ln 2. (29)
r
The metric transforms to

dv 2 W2 W4
g=− + 2 dvdr + U 2
− dr 2 + U 2 r 2 d22 . (30)
U2 U2 U2
The coefficients near origin behave as
1 r2 r3
= + (λ ln 16 − 2) + O r 4 , (31)
U2 λ2 λ3
W2 3
2
= 1 + ζ (3) (3cos(2θ ) + 1) r 3 + O r 4 , (32)
U 4
W4 3
U2 − = − λ2 ζ (3) (3cos(2θ ) + 1) r + O r 2 . (33)
U2 2
Near the origin r ≈ 0, we have for the metric and its determinant
g = −2dvdr + λ2 d22 + O(r), g = −λ4 sin2 θ + O(r). (34)
We see that the metric is regular on the surface r = 0, which is a two-dimensional

sphere of radius λ, and it is possible to extend it to r < 0, where the metric takes
the form
ds 2 = −Ũ −2 dt 2 + Ũ 2 dr̃ 2 + r̃ 2 d2 , (35)
where r̃ ≥ 0 is the new radial coordinate and the functions read
2
Ũ = 1 + λχ̃, χ̃ (r̃, θ ) = χ (r̃, θ ) − . (36)
r̃
We can also see that λ → −λ corresponds to the r → r̃ region. The Maxwell

invariant expanded as a series for r 1 reads

2 8r(1 − λ ln 4) 20r 2 λ2 ln2 8 − 2λ ln 4 + 1
F = 2− + +O(r 3 ), r 1. (37)
λ λ3 λ4
For Kretschmann invariant, we have

8 64r(1 − λ ln 4) 336r 2 λ2 ln2 8 − 2λ ln 4 + 1
K= 4 − + + O(r 3 ), r 1.
λ λ5 λ6
(38)
We can thus see that the point r = 0 is in fact a non-singular surface of non-zero
area.
3 Uniform Crystal
In the previous section, we constructed a solution with alternating charges. The

alternating sign granted us uniform convergence and asymptotic flatness for large ρ;
however, the resulting potential χ is anti-periodic, and we always have singularities.
It is natural to ask about the existence of a superposition of individual charges with

potential 1/r for each of the charges distributed equidistantly along the z-axis.
We seek the function U in the form

Q
U = 1 + λϕ, λ = , (39)
k
where ϕ is a symmetric periodic potential. We will show that λ is the linear charge
(and mass) density to the first leading order of the weak-field limit and the spacetime
becomes the ECS spacetime for large ρ. We construct ϕ for 0 ≤ z ≤ 1/2 as
∞
ϕn 2
ϕ = ϕ0 + f (ρ) , ϕ0 = χ0 , ϕn=0 = χn − . (40)
f (ρ) n
n=1
Because ϕn goes as 1/n for large n, we need to regularize the terms to achieve
point-wise convergence. For finite ρ, the sum converges uniformly, but for large ρ
we need to introduce a suitable regulator function—we choose f = e2ρ . Then, the
individual terms and their derivatives read

−2ρ 1 4
e ϕn ≤ 3 + 2 , ∇ϕn = ∇χn . (41)
2n n
For the derivatives, we are allowed to take f out of the sum as we can use our
results from the alternating crystal (14), and we see that the potential ϕ and its first
and second derivatives converge uniformly. Then, we extend the potential for any z
via its symmetries
ϕ(ρ, z) = ϕ(−ρ, z) = ϕ(ρ, z + 1). (42)
The charge density can be expressed as
1 1
=− δ ϕ = Ш1 (z) δ(ρ). (43)
4π 2πρ
The function χ is periodic, and it is thus natural to ask for the Fourier coefficients
for a fixed ρ. Since the potential diverges at r = 0 as 1/r and also diverges for
large ρ (as can be seen, e.g., from the sum estimates for the first ρ derivative), the
coefficients will exist only for a positive finite ρ. We again search for the coefficients
via the ansatz
∞

ϕ(ρ, z) = f0 ln ρ + An (z)Bn (ρ), ρ > 0. (44)
n=1
The first term corresponds to the n = 0 case. We apply (42) and get
∞

ϕ = f0 ln ρ + fn cos [αn z] K0 [αn ρ] . (45)
n=1
We determine fl formally from the electric charge as in (20)

R z=+h h
4π Q(R, h) = −2π ϕ,z z=−h ρ dρ − 2π (ϕ,ρ ρ)ρ=R dz = 2π(q1 + q2 ).
0 −h
(46)
This time we get
∞

RK1 [αl R] 1
q1 = fl 2αl sin [αl h] − 2 . (47)
αl αl
l=1
We see that q1 vanishes in the limit R → 0. The second term gives

∞

h
q2 = −f0 + fl cos (αl z) αl RK1 (αl R) dz. (48)
−h l=1
We apply the limit R → 0, compare with charge density (43) and get
h
lim 4π Q(R, h) = Ш1 (z)dz ⇒ f0 = −2, fl = 4. (49)
R→0 −h
We explicitly identify the ECS contribution here, see [5].
3.2 Geometry
In cylindrical coordinates, the fields read
dt
g = −U −2 dt 2 + U 2 dρ 2 + ρ 2 dφ 2 + dz2 , A = , U = 1 + λϕ. (50)
U
Note that ϕ is periodic and so is U . Near the origin, the potential behaves as single
extremal Reissner–Nordström solution:
1 ζ̄ (3)
ϕ(r, θ ) = + [1 − 3cos (2θ)] r 2 + O(r 4 ), r 1. (51)
r 2
Therefore, we apply a similar transformation as in [3]
λ
v = t + W 2 (r), W (r) = 1 + . (52)
r
The metric in the new coordinates has the same form as in (30). The coefficients
near the origin behave as
1 r2 2r 3
2
= 2 + 3 + O r4 , (53)
U λ λ
W2
= 1 + ζ (3)(3cos(2θ ) + 1)r 3 + O r 4 , (54)
U2
W4
U2 − = −2λ2 ζ (3) (3cos(2θ ) + 1) r + O r 2 . (55)
U2
We see that the metric is regular at r = 0 and that it is possible to extend it to r < 0
as in (35), and we just formally replace (r, U ) → (r̃, Ũ ), where r̃ ≥ 0 is the new
radial coordinate and the new function Ũ reads
2
Ũ = 1 + λϕ̃, ϕ̃(r̃, θ ) = ϕ(r̃, θ ) − . (56)
r̃
4 Smooth Crystal
In the previous chapters, we constructed two unique solutions. The first one
consisted of alternating charges, which was useful for proving uniform convergence.
However, the range of the potential χ is R, which inevitably results in naked
singularities. At least we get an asymptotically flat spacetime at cylindrical infinity.
In the second solution, we summed only positive charges, but we had to regularize
the sum twice, and in the end, we also ended up with naked singularities. Therefore,
it is natural to ask—is there any solution, where we would get an extremal RN in the
vicinity of the sources which would not have any singularities? What is its behaviour
at infinity? Our idea is to construct a superposition of individual charges generating
the Yukawa potential, i.e., e−αr /r for each of the charges distributed equidistantly
along the z-axis. The constant α appearing in the potential determines its effective
range 1/α. Thanks to the exponential suppression, the uniform convergence will be
ensured. However, this potential is not a vacuum solution and involves charged dust.
We can easily modify the MP solution by adding dust. The stress–energy tensor is
then decomposed as
T μν = E μν + M μν , (57)
where E μν is the electromagnetic part

1 μ F
E μν = F β F νβ − gμν , (58)
4π 4
while M μν corresponds to the dust
(x j ) μ
Mμν = uμ uν , uμ = U δ0 . (59)
U3
Field equations for the metric, electromagnetic field and dust yield only a single
equation
δ U = −4π . (60)
We seek U in the form
U = 1 + λσ, (61)
where the potential σ satisfies (60) in cylindrical coordinates

σ,ρ
σ,ρρ + + σ,zz = −4π . (62)
ρ
We assume the potential σ to be of the following form:

∞
e−αrn e−αr−n
σ = σ0 + σn , σ0 = e−αr χ0 , σn=0 = + , α > 0. (63)
rn r−n
n=1
We use bounds for χn (14) and get

e−αn ∂σn ρ ∂σn α −αn
|σn | ∼ , ∼ ∼ e cosh(αz). (64)
n ∂z n ∂ρ n
For the second derivatives, we obtain

2 2
∂ σn
∼ α ∂σn , ∂ σn ∼ α ∂σn . (65)
∂z2 ∂z ∂ρ 2 ∂ρ
We see that we achieved uniform convergence of the potential up to its second

derivatives. The symmetries are the same as for the uniform crystal (42)
σ (ρ, z) = σ (−ρ, z) = σ (ρ, z + 1). (66)
The charge density can be expressed as
1 1 α2
=− δ σ = Ш1 (z) δ(ρ) − σ, (67)
4π 2πρ 4π
where we recognize the distributional and functional parts of the density. Let us now
seek the Fourier coefficients using the ansatz
∞

σ (ρ, z) = f0 K0 (αρ) + An (z)Bn (ρ), ρ > 0. (68)
n=1
The first term corresponds to n = 0. We get a separable set of equations for An and
Bn
An (z) ρBn (ρ) + Bn (ρ)

− βn2 = = α2 − . (69)
An (z) ρBn (ρ)
Here, βn is the separation constant. Using the symmetries (66), we have

∞

σ = f0 K0 (αρ) + fn cos [βn z] K0 [γn ρ] , γn = α 2 + βn2 . (70)
n=1
The coefficients are determined from the charge formally

R z=+h h
4π Q(R, h) = −2π σ,z z=−h ρ dρ − 2π (σ,ρ ρ)ρ=R dz = 2π(q1 + q2 ).
0 −h
(71)
The first term yields
∞

RK1 (γl R) 1
q1 = 2 fl βl sin [βl h] − 2 . (72)
γl γl
l=1
Again it vanishes for R → 0. The second term gives

∞

h
q2 = f0 αRK1 (αR) + fl cos (βl z) γl RK1 (γl R) dz. (73)
−h l=1
The linear density here is more tricky, as we have two terms:

2π R
α2 R
λ(z) = lim
— (ρ, z)ρ dφ dρ = III1 (z) − limR→0 ρ(ρ, z)dρ.
R→0 0 0 2 0
(74)
λ(z) = III1 (z). We plug it in and determine

The second part vanishes, and we get —
the coefficients
h
lim 4π Q(R, h) = III1 (z)dz ⇒ f0 = 2, fl = 4. (75)
R→0 −h
4.2 Geometry
The metric and electromagnetic potential in cylindrical coordinates read
dt
g = −U −2 dt 2 + U 2 dρ 2 + ρ 2 dφ 2 + dz2 , A = , U = 1 + λσ. (76)
U
Both σ and U are periodic functions. It is no surprise that near the origin we retain
a single extremal Reissner–Nordström solution:
1 α2
σ (r, θ ) = − α − 2 ln 1 − e−α + r + O(r 2 ). (77)
r 2
We can regularize the metric at r = 0 by the transformation
λ
v = t + W 2 (r), W (r) = 1 − λα − 2λ ln 1 − e−α + . (78)
r

dv 2 W2 W4
g = − 2 + 2 2 dvdr + U − 2 dr 2 + U 2 r 2 d22 .
2
(79)
U U U
The coefficients near the origin behave as

1 r2 2r 3 λα + 2λ ln 1 − e−α − 1
= 2+ + O r4 , (80)
U2 λ λ3
W2
= 1 − α2r 2 + O r 3 , (81)
U2
W4
U2 − = 2λ2 α 2 + O (r) . (82)
U2
We see that the metric is regular here. Contrary to the uniform and alternating
crystals, we see that grr is not zero on the horizon. Near the origin r ≈ 0, the
metric and its determinant can be written as
g = −2dvdr + 2λ2 α 2 dr 2 + λ2 d22 + O(r), g = −λ4 sin2 θ + O(r). (83)
Under the horizon, the function U is replaced by Ũ , which takes the form
2
Ũ = 1 + λσ̃ , σ̃ (r̃, θ ) = σ (r̃, θ ) − , (84)
r̃
with r̃ ≥ 0 being the new radial coordinate.
5 Uniform Reduced Crystal
In the previous sections, we have constructed three solutions possessing a discrete

translational symmetry. However, the master function U was always expressed as
an infinite sum with no closed formula. In this section, we use a different approach
to get a solution with a closed formula.
We take a 5D crystal solution in the form

√
−2 3 dt
5
g = −U dt + U dρ + ρ dφ + ρ sin φ dξ + dz
2 2 2 2 2 2 2 2
, A=
5
. (85)
2 U
Here, U satisfies 4D Laplace’s equation and reads [9]
∞
1 π sinh(2πρ) M
U = 1 + μη, η = = ,μ = 2,
n=−∞
ρ 2 + (z − n)2 ρ cosh(2πρ) − cos(2π z) L
(86)
where M > 0 is mass of the individual black holes and L > 0 is spacing of the
grid. The metric is independent of the coordinate ξ , which we use for dimensional
reduction (for details, see [6]). We end up with 4D fields

ḡ = −U −2 dt 2 + U dρ 2 + ρ 2 dφ 2 + dz2 , = ρ U sin2 φ
4
(87)
√ √
3 dt 4 3 dt
4
A= , F = 2
∧ U,ρ dρ + U,z dz . (88)
2 U 2 U
Here, is an additional scalar field, which is produced as a result of the dimensional

reduction. From now on, we work only in 4D, so we drop the dimensional index 4.
The reduced quantities satisfy different equations than in the MP solutions:
U,ρ
U,ρρ + U,zz + 2 = 0, 3g = R. (89)
ρ
The Ricci scalar is no longer zero and equals the negative trace of the total stress–
energy tensor T
2 + U2
U,ρ ,z
F = R = −3 = −T . (90)
2U 3
The Fourier series of η reads
∞
π e−2π nρ
η= + cos(2π nz). (91)
ρ ρ
n=1
From the series expansion for large ρ, we see that the spacetime is cylinder-
asymptotically flat:
π
η= + O(ρ −2 ), ρ 1. (92)
ρ
5.2 Geometry
Near the origin, the potential behaves as
1 π2 π4
η(r, θ ) = + − [2cos(2θ ) − 1] r 2 + O(r 4 ), r 1. (93)
r2 3 45
The metric is apparently singular at r = 0, but this can be removed by the
following procedure. We follow up [10] and start with a transformation of the radial
coordinate:
√ dσ
r= σ , dr = , σ > 0. (94)
2σ 1/2
dt 2 U
ds 2 = − 2
+ dσ 2 + σ U dθ 2 + σ U sin2 θ dφ 2 . (95)
U 4σ
Now, we need to introduce another coordinate
dv± = dt ± [V (σ, θ )dσ + W (σ, θ )dθ] , (96)
where functions V and W read

σ
1 ∂V (σ , θ )
V (σ, θ ) = √ U 3/2 (σ, θ ), W (σ, θ ) = dσ . (97)
2 σ 0 ∂θ
This brings the metric to the form

dv 2
2dθdσ V W
ds 2 =− 2
− + σ U sin2 θdφ 2 ∓ (98)
σ >0 U U2

W dθ + V dσ W2
∓ 2dv + dθ 2
σ U − .
U2 U2
For σ 1, the metric coefficients behave as

2 √
μ3/2 π μ+3 μ 1
V = 2
+ + O σ 0 , W = − π 4 μ3/2 σ sin(2θ ) + O σ 2 .
2σ 4σ 15
(99)
Therefore, in the near-horizon limit, the metric becomes
√
1
ds 2 = ∓ √ dvdσ + μd2 + O σ . (100)
|σ |1,σ >0 μ
This is clearly non-singular at σ = 0. However, one needs to check whether the

horizon is smooth,8 which is the case here as proven in [10]. In the region σ < 0,
one repeats the process with functions
σ
1 3/2 ∂ Ṽ
Ũ = 1 + μη̃, Ṽ = √ Ũ , W̃ = dσ , (101)
2 |σ | 0 ∂θ
where the new potential reads

1 1
η̃(σ, θ ) = √ − . (102)
|σ | + m2 − 2m |σ |cosθ |σ |
m∈Z\{0}
Then, the metric reads

dv 2
2dθdσ Ṽ W̃
ds 2 =− − + σ Ũ sin2 θdφ 2 ∓ (103)
σ <0 Ũ 2 Ũ 2
8 For simple binary MP black holes in 5D, the horizon is not smooth. Thanks to the alignment of
all black holes in the crystal, the horizons of individual black holes are smooth.

W̃ dθ + Ṽ dσ W̃ 2
∓ 2dv + dθ 2
σ Ũ − .
Ũ 2 Ũ 2
One can check that the metric in region σ > 0 matches the one in region σ < 0:
√
1
ds 2 = ∓ √ dvdσ + μd2 + O −σ . (104)
|σ |1,σ <0 μ
6 Conclusions and Summary
In this proceedings contribution, we have studied the properties of spacetimes

exhibiting a discrete translational symmetry. The first solution consisted of alternat-
ing positive and negative charges located on a straight line. This had an impact on the
asymptotics—the potential decreased exponentially for large ρ. We thus obtained a
cylindrically symmetric asymptotic solution. However, the alternating signs of the
charges caused the potential to have both positive and negative values. This results
in naked singularities.
The second solution consisted only of positive charges. In the cylinder-
asymptotic region we obtained the ECS solution representing an extremally charged
string, as expected. However, due to the regularization, the range of the potential
contained both positive and negative values, and we ended up with singularities.
The third solution consisted of positive Yukawa-like charges forming thus a
“smooth crystal”, as the charge is distributed throughout the whole spacetime. Due
to that, we needed to include dust in the Einstein equations. Summing Yukawa
potentials resulted in an improved convergence of the sum, cylinder-asymptotic
flatness and no singularities, as the potential is always positive. Comparing the three
solutions, the smooth crystal is the best one of them regarding these criteria.
The fourth solution was constructed via dimensional reduction to obtain a closed
expression for the corresponding infinite sum. This resulted in an additional scalar
field, and the geometry was different from the corresponding 4D MP spacetime.
The resulting spacetime is cylinder-asymptotically flat, horizons are smooth and we
have no singularities.
We thus compared all four solutions and pointed out their similarities and
differences. Plots of potentials defining the solutions are shown in Figs. 1 and 2.
In our future work, we plan to try to include fluid with pressure. This modifies
Einstein equations as well as the geometry. The spatial 3D section is then conformal
to 3D spaces of constant curvature depending on the value of the pressure. It is
then natural to ask about the convergence of the sum appearing in the solution, its
asymptotics and the existence of naked singularities and horizons.
c d
Fig. 1 Conformal contour plot of (a) χ (alternating crystal), (b) ϕ (uniform crystal), (c) σ (smooth
crystal) (d) and η (uniform reduced crystal)
Fig. 2 Conformal plot of (a) χ (alternating crystal), (b) ϕ (uniform crystal), (c) σ (smooth crystal)
and (d) η (uniform reduced crystal)
Acknowledgments J.R. was supported by grant GAUK 80918. M.Ž. acknowledges support by
GACR 17-13525S.
Grants GAUK 80918, GACR 17-13525S.
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Exact Solutions of the Einstein Equations
for an Infinite Slab with Constant Energy
Density
Tereza Vardanyan and Alexander Yu. Kamenshchik
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
2 Einstein Equations for Spacetimes with Spatial Geometry Possessing Plane Symmetry . . . 237
3 Solution with Isotropic Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
4 Solution with Vanishing Tangential Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
1 Introduction
It is known that even in the absence of matter sources the Einstein equations of
General relativity can have very nontrivial solutions. Historically, the first such
solution was the external Schwarzschild solution for a static spherically symmetric
geometry [1]. It was extremely useful for the study of general relativistic corrections
to the Newtonian gravity and for the description of such effects as the precession
of the Mercury perihelion and the light deflection in the gravitational field. This
solution also opened a fruitful field of black hole physics. The Schwarzschild
T. Vardanyan ()
Dipartimento di Fisica e Astronomia, Università di Bologna and INFN, Bologna, Italy
e-mail: tereza.vardanyan@bo.infn.it
A. Yu. Kamenshchik
Dipartimento di Fisica e Astronomia, Università di Bologna and INFN, Bologna, Italy
L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow, Russia
e-mail: kamenshchik@bo.infn.it
236 T. Vardanyan and A.Yu. Kamenshchik
solution contains a genuine singularity in the centre of the spherical symmetry. To

avoid it and to describe real spherically symmetric objects like stars, Schwarzschild
also invented an internal solution [2] generated by a ball with constant energy
density and with isotropic pressure. At the boundary of the ball, the pressure
disappears and the external and internal solutions are matched. In this case, there
is no singularity in the centre of the ball. Later, more general spherically symmetric
geometries were studied in the papers by Tolman [3], Oppenheimer–Volkoff [4],
Buchdahl [5] and many others. Similar problems with cylindrical axial symmetry
were also studied (see e.g. [6] and the references therein). In paper [7], the question
of existence of solutions of the Einstein equations in the presence of concentrated
matter sources, described by the generalised functions (distributions), was studied.
It was shown that in contrast to the case of electrodynamics, where the charged ball
can be contracted to the point and the charge density becomes proportional to the
Dirac delta function while the Poisson equation is still valid, we cannot do it in the
General Relativity. The reason lies in the non-linearity of the Einstein equations. It
was shown in [7] that the solutions with distributional sources cannot exist for zero-
dimensional (point-like particles) and one-dimensional (strings) objects but can
exist for two-dimensional (shells) objects. This fact makes the study of geometries
possessing plane symmetries particularly interesting. Indeed, the plane-symmetric
solutions of the Einstein equations were also studied in the literature (see e.g. [8, 9]
and the references therein).
However, to our knowledge, exact static solutions of the Einstein equations, in
the spacetimes with plane symmetry in the presence of an infinite slab with a finite
thickness, were not studied. Thus, our objective in the present work was to find such
solutions with the matching between the geometry inside the slab and that outside
of it. Here, we would like to say that the first static solution in an empty spacetime
possessing plane symmetry is almost as old as the Schwarzschild solution. This is
the spatial Kasner solution [10] found in 1921 and its particular case—the Weyl–
Levi-Civita solution [11, 12], found even earlier. Hence, we wanted to find for the
case of plane symmetry some analog of matching between Schwarzschild external
and internal solutions. We have considered an infinite slab with a finite thickness and
a constant mass (energy) density and have found two particular solutions: one with
isotropic pressure and one for tangential pressure equal to zero. In both cases, we
require that all components of pressure vanish at the boundary of the slab, just like
in the case of the Schwarzschild internal solution. The structure of the chapter is the
following: in Sect. 2, we write down some general formulae for the spacetimes with
the spatial geometry possessing plane symmetry, in Sect. 3, we describe the solution
with isotropic pressure, while Sect. 4 is devoted to the solutions with vanishing
tangential pressure. Section 5 contains some concluding remarks.
The results presented here were published in [13].
Infinite Slab with Constant Energy Density 237
2 Einstein Equations for Spacetimes with Spatial Geometry

Possessing Plane Symmetry
Let us consider the metric with plane symmetry, where the metric coefficients
depend on one spatial coordinate x:
ds 2 = a 2 (x)dt 2 − dx 2 − b2 (x)dy 2 − c2 (x)dz2 . (1)
Before plunging into technical details connected with the search for the solutions
for the thick slab, let us recall briefly what is known about the empty spacetime
solutions and the solutions in the presence of thin shells.
For the metric (1) in the empty spacetime, we have two general solutions. One of
them is the Minkowski metric, where a = b = c = 1 and another one is the Kasner
solution [10] with
a(x) = a0 (x − x1 )p1 , b(x) = b0 (x − x1 )p2 , c(x) = c0 (x − x1 )p3 , (2)
where the Kasner indices p1 , p2 and p3 satisfy the equations
p1 + p2 + p3 = p12 + p22 + p32 = 1. (3)
The Kasner solution is more often used in a “cosmological form”:
ds 2 = dt 2 − a02 t 2p1 dx 2 − b02 t 2p2 dy 2 − c02 t 2p3 . (4)
This form of the Kasner metric was rediscovered in papers [14–16] and has played
an important role in cosmology. The study of Kasner dynamics in paper [16] has
led to the discovery of the oscillatory approach to the cosmological singularity [17],
known also as the Mixmaster universe [18]. The further development of this line
of research has brought the establishment of the connection between the chaotic
behaviour of the universe in superstring models and the infinite-dimensional Lie
algebras [19].
Coming back to the spatial form of the Kasner metric (2)–(3), one sees that the
requirement of symmetry in the plane between the y and z directions implies the
condition
p 2 = p3 . (5)
It is easy to see that there are two solutions of Eqs. (3) satisfying the condition (5).
One of them is the Rindler spacetime [20] with
p1 = 1, p2 = p3 = 0. (6)
It is well known that the Rindler spacetime represents a part of the Minkowski
spacetime rewritten in the coordinates connected with an accelerated observer.
There is a coordinate singularity (horizon) at x = x1 . Another solution is
1 2
p1 = − , p2 = p3 = . (7)
3 3
This particular solution was found by Weyl [11] and Levi-Civita [12] before
the work of Kasner.1 This solution describes a universe, where a real curvature
singularity is present at x = x1 .
The detailed account of the solutions in the presence of a thin plate with constant
energy density was given in paper [9]. These solutions have some distinguishing
features. First of all, the energy density of the plate and its tangential pressure
should both be proportional to the delta function, while the component of the
pressure perpendicular to the plate is equal to zero. Furthermore, the metric is
continuous everywhere, but its derivative has a finite jump at the location of the
plate. The spacetimes on the right and on the left from the plane are of the
type described above: Minkowski, Rindler or Weyl–Levi-Civita. The reflection
symmetry is present, i.e. the spacetimes on both sides are the same, if and only
if the energy density and the pressure are connected by the relation p = − 14 ρ or
ρ = 0. Otherwise, this symmetry is lost.
In the paper [9], the solutions in the presence of a finite-thickness slab were
also discussed. Some features of such solutions were analysed qualitatively or
numerically, but exact solutions were not found. One of these interesting features
is the absence of the reflection symmetry. Here, we obtain some exact solutions
manifesting this feature. Concerning the properties of the matter constituting the
slab, being inspired by the internal Schwarzschild solution [2], we assume that the
energy density is constant, while the pressure should disappear at the boundaries of
the slab.
Now, we write down some general formulae necessary for the metric with the
plane symmetry (1). The non-vanishing Christoffel symbols are
ttx = a a, yy
x
= −b b, zz
x
= −c c,
a y b c
t
tx = , yx = , zx
z
= . (8)
a b c
The components of the Ricci tensor are
1 In paper [16], a convenient parametrization of the Kasner indices was presented:
u 1+u u(1 + u)
p1 = − , p2 = , p3 = .
1 + u + u2 1 + u + u2 1 + u + u2
In terms of this parametrization, the Rindler solution corresponds to u = 0, while the Weyl–Levi-
Civita solution is given by u = 1.
a b a a c a
Rtt = a a + + ,
b c
a a b a c
Rtt = + + ,
a ab ac
a b c
Rxx = − − − ,
a b c
a b c
Rxx = + + ,
a b c
a b b b c b

Ryy = −b b − − ,
a c
y b a b b c
Ry = + + + ,
b ab bc
a c c b c c
Rzz = −c c − − ,
a b
c a c b c
Rzz = + + . (9)
c ac bc
The Ricci scalar is

a b c a b a c b c
R=2 + + + + + . (10)
a b c ab ac bc
The energy–momentum tensor for a fluid with isotropic pressure is
Tμν = (ρ + p(x))uμ uν − p(x)gμν , (11)
where we shall write
4k 2
ρ= = constant (12)
3
for convenience. Then,
ut = a, ux = uy = uz = 0. (13)
The equation
ν
Tμ;ν =0 (14)
for μ = x gives
a
p = − (ρ + p), (15)
a
where “prime” means the derivative with respect to x. The integration of Eq. (15)
gives
4k 2 p0
p=− + , (16)
3 a
where p0 is an arbitrary constant. The Einstein equations are
b c b c 4k 2
− − − = , (17)
b c bc 3
a b a c b c
+ + = p, (18)
ab ac bc
a c a c
+ + + = p, (19)
a c ac
a b a b
+ + + = p. (20)
a b ab
Introducing new functions
a b c
A= , B= , C= , (21)
a b c
we can rewrite the Einstein equations (17)–(20) as follows:
4k 2
− B − B 2 − C − C 2 − BC = , (22)
3
AB + AC + BC = p, (23)
A + A2 + C + C 2 + AC = p, (24)
A + A2 + B + B 2 + AB = p. (25)
3 Solution with Isotropic Pressure
In what follows, we shall consider only the solutions where the symmetry between
the directions along the coordinate axes y and z is present. Then,
B = C, (26)
and we obtain from Eq. (22)
4k 2
− 2B − 3B 2 = . (27)
3
Integrating this equation, we obtain
2
B = C = − k tan k(x + x0 ). (28)
3
Using the definitions (21), we obtain
2
b = b0 (cos k(x + x0 )) 3 , (29)
2
c = c0 (cos k(x + x0 )) 3 . (30)
Let us note that in order to not have singularities in the metric, we need to require
that
[−L + x0 , L + x0 ] ⊂ (−π/2, π/2), (31)
where x = −L and x = L are the locations of the boundary of the slab. Substituting
Eqs. (28) and (16) into Eq. (23), we obtain
a 4k 4k 2 4k 2 p0
− tan k(x + x0 ) + tan2 k(x + x0 ) = − + . (32)
a 3 9 3 a
This equation can be rewritten as
k 3p0
a − tan k(x + x0 )a − k cot k(x + x0 )a + cot k(x + x0 ) = 0. (33)
3 4k
The general solution of the corresponding homogeneous equation is
1
a(x) = a1 sin k(x + x0 )(cos k(x + x0 ))− 3 , (34)
where a1 is an integration constant. We shall look for the solution of the inhomoge-
neous equation (33) in the following form:
1
a(x) = ã(x) sin k(x + x0 )(cos k(x + x0 ))− 3 . (35)
Substituting the expression (35) into Eq. (33), we have

4
3p0 (cos k(x + x0 )) 3
ã = − . (36)
4k sin2 k(x + x0 )
Integrating by parts, we obtain
3p0 4
ã(x) = 2 cot k(x + x0 )(cos k(x + x0 )) 3
4k

p0 4
+ dx (cos k(x + x0 )) 3 + a2 , (37)
k
where a2 is an integration constant. Introducing a variable
u ≡ sin2 k(x + x0 ),
one can find that

p0 x 4 p0 1 7
dy(cos k(y + x0 )) = 2 B sin k(x + x0 ); ,
3 2
Sign[sin k(x + x0 )],
k −x0 2k 2 6
(38)
where the incomplete Euler function is defined as

x
B(x, r, s) ≡ duur−1 (1 − u)s−1 . (39)
0
Thus, the general solution of Eq. (33) is
3p0
a(x) = cos2 k(x + x0 )
4k 2

p0 1 1 7
+ 2 (cos k(x + x0 )) | sin k(x + x0 )|B sin k(x + x0 ); ,
3 2
2k 2 6
1
+a3 sin k(x + x0 )(cos k(x + x0 ))− 3 . (40)
Looking at the expression (16), we see that the disappearance of the pressure on the
boundary of the slab is equivalent to the requirement that
3p0
a(−L) = a(L) = . (41)
4k 2
On using Eq. (40), this condition can be rewritten as
3p0
− sin2 k(±L + x0 )
4k 2

p0 1 1 7
+ (cos k(±L + x 0 )) 3 | sin k(±L + x0 )|B sin2 k(±L + x0 ); ,
2k 2 2 6
1
+a3 k sin(±L + x0 )(cos k(±L + x0 ))− 3 = 0. (42)
Now, we have two free parameters x0 and a3 , which we can fix in such a way to
provide the disappearance of the pressure on the border of the slab. Let us first
choose
x0 = L. (43)
It guarantees that
3p0
a(−L) = (44)
4k 2
and, hence,
p(−L) = 0. (45)
With this choice of x0 , the requirement (31) becomes

π
2kL < . (46)
2
It is easy to see that if the inequality (46) is broken, the cosine is equal to zero at
some value of the coordinate x inside the slab and one encounters a singularity.
Now, substituting the value (43) into Eq. (42), we can choose the constant a3
requiring the disappearance of the pressure on the other border of the slab x = L.
This constant is
p0
a3 = 2 3 sin 2kL cos1/3 2kL − 2 cos2/3 2kL B(sin2 2kL; 1/2, 7/6) . (47)
4k
Finally, we can write
3p0
a(x) = cos2 k(x + L)
4k 2

p0 1 1 7
+ (cos k(x + L)) 3 sin k(x + L)B sin2 k(x + L); ,
2k 2 2 6
p0
+ 2 3 sin 2kL cos1/3 2kL − 2 cos2/3 2kL B(sin2 2kL; 1/2, 7/6)
4k
1
× sin k(x + L)(cos k(x + L))− 3 . (48)
Thus, we have obtained a complete solution of the Einstein equations in the slab,
where the energy density is constant and the pressure disappears on the boundary
between the slab and an empty space. Let us make some comments here. First,
the scale factors a, b and c and hence the metric coefficients are not even, and the
solution is not invariant with respect to the inversion
x → −x.
However, making the change x → −x , we obtain another solution of our equations.

It can be obtained also by choosing x0 = −L instead of x0 = L and by the
corresponding change of the expression for the coefficient a3 , which is reduced
to the change of the sign of the argument of the trigonometrical functions. There
is no qualitative difference between these two solutions. Thus, we shall study the
first one. Let us emphasise that the choice x0 = ±L is obligatory in order for the
pressure to vanish on both boundaries of the slab, and, hence, the asymmetry of
these two solutions is an essential feature of the problem. It arises in spite of the
initial symmetry of the Einstein equations and of the position of the slab. Thus, one
can speak about some kind of symmetry breaking phenomenon.
Let us consider the question of matching of the solutions in the slab with the
vacuum solutions outside the slab. Our solution inside the slab possesses symmetry
in the plane (y, z). Thus, we shall try to match it at x < −L and at x > L with one
of these three solutions: Minkowski, Rindler or Weyl–Levi-Civita (7).
Consider the plane x = −L. We shall require that
aext (−L) = a(−L), bext (−L) = b(−L), cext (−L) = c(−L),

aext (−L) = a (−L), bext

(−L) = b (−L), cext

(−L) = c (−L). (49)
Looking at the expressions (48), (29) and (30), we see that at x = −L the derivatives
of b and c disappear (provided x0 = L), while the derivative of a at this point is
different from zero. Thus, we should choose the Rindler geometry for x < −L
ds 2 = aR2 (x − xR )2 dt 2 − dx 2 − bR
2
(dy 2 + dz2 ). (50)
We can consider the analogous matching conditions at x = L. Here, the derivatives

of all three scale factors are non-vanishing. Thus, for x > L, we have a Weyl–Levi-
Civita solution
−2/3 2
ds 2 = aW
2
LC (x − xW LC ) dt − dx 2 − bW
2
LC (x − xW LC )
4/3
(dy 2 + dz2 ). (51)
Let us now discuss these matching conditions in more detail. On the plane x =
−L, we have
3p0
= aR (−L − xR ), (52)
4k 2
to match the scale factors (the subscript “R” means “Rindler”) and
a3 k = a R , (53)
where a3 is given by Eq. (47) to match their derivatives. It follows from Eqs. (52)
and (53) that
3p0
xR = −L − . (54)
4a3 k 3
Plotting (47) as a function of 2kL, we can see that for values smaller than 2kL ≈
1.05, a3 < 0 and thus
xR > −L. (55)
Therefore, there is no horizon for these values of kL.

At the boundary x = L, it is more convenient to write down the conditions of
matching of the tangential scale factors b:
b0 (cos 2kL)2/3 = bW LC (L − xW LC )2/3 , (56)
2 2
− b0 k(cos 2kL)−1/3 sin 2kL = bW LC (L − xW LC )−1/3 . (57)
3 3
From these two equations, we easily find that
1
xW LC = L + cot 2kL. (58)
k
Provided the condition (46), we see that xW LC is necessarily bigger than L, and
we cannot avoid having a singularity in the space on the right side of the slab,
at least not if the energy density ρ of the slab is positive. To obtain the solution
for the case ρ < 0, we can replace k by ik in the solution that we already have.
Then, trigonometric functions turn into hyperbolic ones, and the expression (58) is
replaced by
1
xW LC = L − coth 2kL. (59)
k
The above expression is smaller than L; therefore, there is no singularity. In the case
of an infinitely thin slab, the conclusion that the singularity is unavoidable for ρ > 0
was obtained in [9].
The expression for xW LC given by Eq. (58) guarantees the satisfaction of the
matching conditions also for the scale factor a and its derivative. It follows from the
fact that for both the Weyl–Levi-Civita solution and our internal solution,
a 1 b
(L) = − (L), (60)
a 2b
which in turn follows from Eq. (23) and from the disappearance of the pressure on
the border of the slab.
As we mentioned earlier, the solution (48) is not invariant with respect to the
inversion of the coordinate x . However, for a particular value of kL, one can have an
even solution, invariant with respect to this inversion. Indeed, we can transform the
general solution for the scale factor a (40) into an even function of x by putting a3 =
0 and x0 = 0. Then, also b(x) and c(x) become even. One can check numerically that
at kL ≈ 1.05 the expression for a at the boundaries x = ±L is such that the pressure
disappears. The argument of the trigonometric functions runs between −1.05 >
− π2 and 1.05 < π2 , the cosine is always different from zero and the singularity
does not arise. Besides, at both boundaries, the derivatives of the scale factors are
different from zero. Hence, in both half-spaces outside the slab, this solution should
be matched with the Weyl–Levi-Civita solutions. Let us stress once again that this
symmetric solution is a very particular one, arising at some special value of kL,
while generally we have a pair of solutions, each of which is not symmetric with
respect to the reflection x → −x , instead this reflection transforms one solution into
another and vice versa. One can trace here an analogy with a well-known case of
two-well potential, which is often considered at the introducing of the spontaneous
symmetry breaking phenomenon in quantum field theory (see e.g. [21])
V (φ) = (φ 2 − φ02 )2 ,
which is symmetric with respect to φ → −φ , while its minimum values φ = ±φ0

are not symmetric.
4 Solution with Vanishing Tangential Pressure
In the preceding section, we have considered a situation where the tangential pres-
sure coincides with the transversal pressure, just like in the internal Schwarzschild
solution [2]. In the case of the Schwarzschild spherically symmetric geometry,
such a choice is obligatory because otherwise the pressure becomes infinite in the
centre of the ball and a non-singular internal solution does not exist (unless it is
assumed that radial pressure is identically zero and tangential pressure does not
vanish at the boundary; see [22]). However, it is not obvious that in the case of the
plane symmetry the situation is the same. Let us consider a more general energy–
momentum tensor
y
Ttt = ρ, Txx = −px , Ty = −py , Tzz = −pz . (61)
Then, the energy–momentum tensor conservation condition (14) takes the following
form:
px + A(ρ + px ) + B(px − py ) + C(px − pz ) = 0. (62)
In our case, B = C and, hence, py = pz . We shall consider the case, where the
tangential pressure py = pz = 0. Now, Eq. (62) becomes
p + A(ρ + p) + 2Bp = 0, (63)
where p ≡ px . We have two unknown functions: p and A. However, it is not

convenient to try to find the relation between these functions using Eq. (63). It is
better to take Eq. (25) with the vanishing right-hand side:
A + A2 + B + B 2 + AB = 0. (64)
The function B still satisfies (27) and (28); using (28), we can rewrite (64) in terms
of the function a :

2 4 2 2 2 k2
a − tan k(x + x0 )a + k tan k(x + x0 ) − a = 0. (65)
3 3 3 cos2 k(x + x0 )
Looking for the solution of these second-order linear differential equation in the
form
a(x)(cos k(x + x0 ))α (sin k(x + x0 ))β ekγ (x+x0 ) , (66)
we find two sets of the parameters giving the solution of Eq. (65):
1 1
α = − , β = 0, γ = √ ,
3 3
1 1
α = − , β = 0, γ = − √ . (67)
3 3
Thus, the general solution of Eq. (65) is
√1 k(x+x0 ) − √1 k(x+x0 )
a(x) = (cos k(x + x0 ))−1/3 (a4 e 3 + a5 e 3 ). (68)
Now, we find
√1 k(x+x ) − √1 k(x+x )
a k k a4 e 3 0
− a5 e 3 0
A= = tan k(x + x0 ) + √ 1 1
. (69)
a 3 3 a e √3 k(x+x0 ) + a e− √3 k(x+x0 )
4 5
Substituting this expression into Eq. (23), we find the transversal pressure
√1 k(x+x ) − √1 k(x+x )
4k 2 a4 e 3 0
− a5 e 3 0
p = − √ tan k(x + x0 ) 1 1
. (70)
3 3 √ k(x+x0 ) − √ k(x+x0 )
a e 3 4 +a e 3 5
In order to have the pressure vanishing at x = −L, we can again choose x0 = L.

Then, fixing
4kL
√
a5 = a4 e 3 , (71)
we have the pressure vanishing also at x = L. Finally, we have
4k 2 k
p = √ tan k(x + L) tanh √ (L − x), (72)
3 3 3
and
1
a(x) = a6 (cos k(x + L))−1/3 cosh √ k(x − L). (73)
3
For x > L, this solution should be matched with the Weyl–Levi-Civita solution with
the same value of the parameter xW LC as in the previous section. For x < −L, the
obtained solution is matched with the Rindler solution with
√
3 coth 2kL
√
3
xR = −L + . (74)
k
It is easy to see that as long as 2kL < π/2 the internal metric is regular and the
pressure (72) is finite everywhere in the slab. Thus, in contrast to the case of the
Schwarzschild geometry, we have here a non-singular internal solution with an
anisotropic pressure, namely with the pressure whose tangential components are
identically equal to zero.
5 Concluding Remarks
We have found two static solutions for an infinite slab of finite thickness immersed in
the spacetime with plane symmetry. How are these solutions related to the solutions
of a matter source localised on an infinitely thin plane? First of all, let us note that
our solutions are non-singular inside the slab if the condition (46) is satisfied. If we
introduce the notion of the energy of the unit square of the slab M :
8k 2 L
M = 2ρL = , (75)
3
then the condition (46) becomes
π2
L< . (76)
12M
Thus, if we fix the value of M and begin squeezing the slab, diminishing L, we
do not encounter anything similar to the Buchdahl limit for spherically symmetric
configurations [5]. In other words, if the relation (76) is satisfied at some value of
L0 , it remains satisfied at all finite values of L < L0 . On the other hand, if we
π2
start increasing the thickness of the slab, then at the value L = 12M a singularity
arises inside the slab. Moreover, in the case considered in Sect. 4, the pressure also
becomes infinite.
What happens when L → 0? Obviously, the energy density will tend to the delta
function
ρL→0 → Mδ(x). (77)
As was discussed in paper [9], the tangential pressure should also tend to infinity
to maintain the validity of the energy–momentum conservation equation (14). In
our solution presented in Sect. 4, the tangential pressure is identically zero. One
can show, using Eqs. (16) and (42), that in the solution with an isotropic pressure
presented in Sect. 3, the pressure in the slab is limited by the value p ≈ M 2 when
L → 0. Thus, while both of these solutions are well defined at any arbitrary small,
but finite value of the thickness parameter L, there is not a smooth transition to an
infinitely thin plane configuration for these two solutions. However, these solutions
represent some particular configurations acceptable from a physical point of view.
Let us emphasise once again that we did not fix some particular equation of state
for the matter filling our slab. We simply required that the energy density on the
slab is constant and that the pressure disappears at the boundaries of the slab. These
conditions are the same used in the Schwarzschild internal solution [2]. Then, we
considered two particular additional conditions: one of them requires the isotropy
of the pressure, just like in the Schwarzschild solution [2], and another requires the
disappearance of the tangential pressure in all the slab. For both these requirements,
we have found exact solutions. In principle, one can imagine the existence of a
solution where the transversal and tangential pressures are different functions of the
coordinate x , vanishing on the borders of the slab. Then, one cannot exclude that for
some solutions of this kind a smooth transition to the localised matter configurations
is possible.
There is also another problem here: it would be interesting to find matter
distributions, which imply the existence of solutions of the Einstein equations which
are matched in the empty regions of the space with the general spatial Kasner
solutions (2) and (3) with p2 = p3 . We hope to attack these problems in a future
work [23].
Acknowledgments We are grateful to R. Casadio, J. Ovalle and G. Venturi for useful discussions.
References
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Emergence of Classicality from an
Inhomogeneous Universe
Adamantia Zampeli
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
3 Quantum Dynamics and Classical Emergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
1 Introduction
More than a century now, we know that the world is fundamentally quantum
mechanical; yet our everyday experience fools us with a classical world. How do
these two pictures reconcile? Indeed, if one accepts the thesis that the world is
fundamentally quantum mechanical, the natural question to ask is how classical
world we experience in our everyday life emerges from the quantum structures.
There must be a mechanism for this to happen; but more importantly, what are
the requirements for a system to be considered classical? Here, we will adopt
the position that this transition from quantum to classical happens through the
mechanism of decoherence, which destroys the interference terms between different
quantum states. Usually, this happens through the interaction of the system with an
environment. In quantum cosmology, on which we focus our considerations here,
the universe is a closed system, and the role of the environment is played by inho-
Prepared for the proceedings of the 2nd Domoschool, Domodossola.
A. Zampeli ()
Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, Prague,
Czech Republic
e-mail: azampeli@phys.uoa.gr
252 A. Zampeli
mogeneous degrees of freedom acting as perturbations in an overall homogeneous

background (e.g. [1, 2]). We are interested to explore a different path, by starting
with a genuinely inhomogeneous spacetime and investigating the emergence of
classicality due to the presence of symmetries.
Our starting point is the Szekeres spacetime, which is a dust-type D exact solution
of general relativity. This classical model has attracted the interest since it is an
alternative not only to the Friedman–Lemaitre–Robertson–Walker (FLRW) model
but also of the Bianchi I (Kasner). The first currently serves as the “standard”
cosmological model, while the latter as a model for the dynamics close to the
singularity [3]. It is therefore clear that the class of models described by the Szekeres
spacetimes can provide useful information for many properties of physically
interesting models and possible new effects which might appear due to quantum
gravity.
There are two criteria we consider to examine the emergence of classicality: (i)
Hartle’s criterion that states that predictions in quantum cosmology are possible
when there is a peak on the configuration space, which accordingly indicates
a correlation between conjugate momenta on the phase space [4, 5] and (ii)
decoherence, which is quantified in the condition that the sum of the non-diagonal
terms of the reduced density matrix is much smaller than the sum of its diagonal
terms [1, 6]. To this end, we define the reduced density matrix by tracing out the
constant of motion related to the classical symmetry in question.
In the following sections, we first summarise the previous results regarding the
classical symmetries of a reduced Lagrangian for the Szekeres spacetime and the
solution of the quantum equations. Then, we proceed to check whether the first but
mainly the second criterion holds for a reduced density matrix defined as described
above. In the last section, we discuss the results and the connection with the current
cosmological observations about the homogeneity of the universe.
2 Preliminary Results
The general spacetime element for the Szekeres solution is [7]
ds 2 = −dt 2 + e2A(t,x,y,z) dx 2 + e2B(t,z,y,z) (dy 2 + dz2 ), (1)
where the functions A(t, x, y, z) and B(t, x, y, z) can be specified by the solution
of the Einstein equations with energy–momentum tensor of the dust,
Gμν = Tμν
(D)
. (2)
Instead of the metric variables, we choose the physical variables, since we can take
advantage of the fact that in these solutions the two components of the electric part
of the Weyl tensor and the two components of the shear for the observer denoted
by a time-like 4−vector uμ are equal, respectively. In these variables, the evolution
Emergence of Classicality from an Inhomogeneous Universe 253
equations take the form [8]
ρ̇ + θρ = 0, (3a)
θ2 1
θ̇ + + 6σ 2 + ρ = 0, (3b)
3 2
2
σ̇ − σ 2 + θ σ + E = 0, (3c)
3
1
Ė + 3Eσ + θ E + ρσ = 0, (3d)
2
where˙ = uμ ∇μ and the energy density is defined as ρ = T μν uμ uν . The constraint
equation is
θ2 (3) R
− 3σ 2 + = ρ. (4)
3 2
We can find a second-order differential system by solving equation (3a) with
respect to θ and Eq. (3d) with respect to σ and replacing the results to the other
two. The new system, which we omit to write, can be further simplified through the
transformation
6 v
ρ= , E= , (5)
u3 (1 − uv ) u4 uv −1
thus taking the simplified form
2v
v̈ − = 0, (6)
u3
1
ü + 2 = 0. (7)
u
These correspondingly can be seen as the Euler–Lagrange equations of the follow-
ing Lagrangian function:
v
L = u̇v̇ − (8)
u2
It is interesting to note the initial spacetime, despite the degeneracy between the
different components of the Weyl tensor and the shear has no Killing vector
field and it can only be said that these solutions are locally axisymmetric [9].
However, this reduced Lagrangian possess generalised symmetries which facilitates
the quantisation of this system [10]. The symmetries of this Lagrangian are
v
h = u̇v̇ − , (9a)
u2
254 A. Zampeli
2
I0 = u̇2 − , (9b)
u
and in Appendix, it is shown that their presence is due to the existence of two trivial
Killing tensor fields on the configuration space of the (u, v) variables. It is clear that
the first equation is the Hamiltonian function, and thus it plays the role of “energy”
of the reduced system. The stability analysis of this system [8] showed that there are
two exact solutions when h = 0 and I0 = 0 of the form u(t) = u0 z−1 , v(t) = v0 t 2/3
and u(t) = u0 t 2/3 , v(t) = v0 t 2/3 . The first solution corresponds to an unstable
critical point for the dynamical system (6) while the latter to a stable one [8].
3 Quantum Dynamics and Classical Emergence
We now turn to the quantum dynamics which arises by turning to quantum operators
the fundamental variables on the phase space and the classical observables to self-
adjoint operators. Then, we find the following eigen equations [10]:
v
−∂uv + 2 = h, (10)
u

2
∂vv + = −I0 , (11)
u
which are the quantum analogues of (9). We note in passing that, contrary to what
happens for gravitational systems, which are constrained due to the presence of the
arbitrary functions (the lapse function and the shift vector), here the dynamics of
the reduced system is not constrained. We now limit ourselves to the case h = 0, in
which the wave function takes the form
√
u 2 + I0 u 2 + I0 u
(I0 , u, v) = √ 1 cos v + 2 sin v .
2 + I0 u u u
(12)
If we select the constants such that 2 = i1 = C, the wave function is written in
polar form as
√
C u 2 + I0 u
(I0 , u, v) = √ exp i v , (13)
2 + I0 u u
where C is a constant. In [10], we performed the Bohmian analysis for this wave
function as well as for general values of the constant h. It was shown that the
quantum potential, which is given by
(q i )
Q(q i ) = − , (14)
2(q i )
where (q i ) is the amplitude of the wave function (13), q i are the variables of the
configuration space, in this case (u, v) and the Laplacian for this space, becomes
zero. Since the quantum potential appears in the Hamilton–Jacobi equation as an
additional term arising due to quantum effects, its value becoming zero means
that the classical dynamics emerges from these quantum solutions. This can also
be attributed to the fact that the variables qi = u, v and their conjugate momenta
which are defined as pi = ∇i S are highly correlated. Indeed, following the analysis
of [11], where it was shown that for the case of WKB-type wave functions strong
correlations between the variables and their conjugate momenta on the phase space
lead to strong peaks of the wave function and to classicality. We can conclude that
the first criterion for the emergence of classicality as introduced by Hartle is satisfied
[4, 5].
We are now interested to check whether the second criterion holds, which is
decoherence; this is the destruction of the interference terms between different
systems due to correlations [12] and happens due to the interaction between
subsystems. One plays the role of environment, which has infinite degrees of
freedom and is of no interest in the analysis. Therefore, it is traced out, keeping
only the degrees of freedom of the system under physical interest. In cosmology,
the environment is usually inhomogeneous degrees of freedom of some scalar field.
In our case, though, we are interested to examine decoherence in relation to the
existence of a symmetry. The induction of decoherence due to symmetries has been
discussed before elsewhere, e.g. [13, 14]. In order to quantify this effect, we will
define the reduced density matrix as
α
ρ red (ui , vj , uk , vl ) = (ui , vj ) (uk , vl )| = DI0 ∗ (ui , vj , I0 )(uk , vl , I0 ),
0
(15)
i.e. by tracing out the symmetry constant I0 . If we insert the polar form of the wave
function, it becomes
α
kl =
ρijred dI0 ∗ (ui , I0 )(uk , I0 )e−i(S(ui ,vj ,I0 )−S(uk ,vl ,I0 )) . (16)
0
The condition for decoherence is that the sum of the real part of the non-diagonal
elements of this matrix should be much smaller than the sum of the diagonal
elements [6]

| Re ρ red (ui , uj )| < ρ red (ui , uj ). (17)
i=j i=j
256 A. Zampeli
The diagonal elements are the ones with

α α
ij =
ρijred dI0 ∗ (ui , vj )(ui , vj ) = dI0 |(ui , I0 )|2 , (18)
0 0
which, after substituting the explicit form of the solution, become
α |C|2 u αu
red
ρdiag = dI0 = |C|2 ln(1 + ) (19)
0 2 + I0 u 2
and depend only on u. For the non-diagonal elements, we are interested in the
behaviour of their real part. These are given by i = k and/or j = l . Their real
part is given by
α
Re ρ red (ui , vj , uk , vl , I0 ) = dI0 ∗ (ui , I0 )(uk , I0 ) cos(Sij − Skl ) . (20)
0
This expression is always bounded, i.e., it always satisfies the relationship

α
| Re ρ red (ui , vj , uk , vl , I0 )| ≤ dI0 ∗ (ui , I0 )(uk , I0 ) (21)
0
with the equality holding for the diagonal elements when Sij = Skl . The right-hand
side can be calculated, and it is equal to
⎛ ⎞
ui αuj + 1 + ui uj (αui + 2) αuj + 2 + uj
|C|2 ln ⎝ √ ⎠ (22)
2 ui uj + ui + uj
from which we recover the relation (18) for i = j . The relation we wish to show
that holds for all the range of values of u is the sum of the corresponding term, i.e.
|2 Re ρ red (u1 , u2 )| < (ρ red (u1 , u1 ) + ρ red (u2 , u2 )) (23)
since Re ρ red (u1 , u2 ) = Reρ red (u2 , u1 /), which is written in our case as
⎛ ⎞
ui uj (αui + 2) kuj + 2 + ui αuj + 1 + uj αu
2 ln ⎝ √ ⎠ < ln αui + 1 + ln j
+1 .
2 ui uj + ui + uj 2 2
(24)
This relation is true for every α > 0 and positive values of the configuration
variables, which is of our interest, and therefore we do have decoherence for this
case induced by the presence of symmetry.
4 Discussion
We studied the quantum solution of an inhomogeneous gravitational model which

is an exact classical solution. We showed that the presence of symmetry can satisfy
two criteria for the emergence of classicality for the particular case of h = 0. Instead
of separating our system to environment and subsystem, we traced out over the
classical constant of motion I0 . We examined the possibility that the interference
terms are destroyed due to the existence of symmetries, and we found that this can
indeed be the case. It is a known fact that symmetries can lead to decoherence and
this can also be manifest formally through the existence of superselection rules.
These are rules prohibiting the existence of pure states which are superpositions of
states that belong to different coherent subspaces of the Hilbert space [14].
These considerations strengthen the results in [10] and give further motivation to
study possible quantum effects at the low-energy limit coming from inhomogeneous
spacetimes.
Appendix: The Killing Tensors of the Lagrangian
The metric on the configuration space is

01
Gμν = . (A.1)
10
The Killing fields are
ξ1 = ∂u , ξ2 = ∂v , ξ3 = u∂u − v∂v . (A.2)
The (trivial) Killing tensors constructed by these Killing vector fields are found by
the relation
1
K= ξi ⊗ ξj + ξj ⊗ ξi (A.3)
2
and have the form

10
K1 = (A.4)
00
and

01
K2 = . (A.5)
10
258 A. Zampeli
The conserved quantities are given by Ki = Kiμν pμ pν , and one can see that K1
corresponds to Eq. (7), while K2 to the energy, and thus can be associated with the
constants of motion considered in the main text as K1 → I0 and K2 → h.
Acknowledgments I would like to thank the organisers of the 2nd Domoschool for the high
level of lectures and their kind hospitality. During the school, I was benefited from discussions
with Profs. Sergio Cacciatori, Vittorio Gorini and Alexander Kamenshchik. I also thank Drs.
Andronikos Paliathanasis, Georgios Pavlou and Otakar Svitek for suggestions and corrections of
the manuscript. Finally, I acknowledge the financial support from the Albert Einstein Center.
References
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org/10.1103/PhysRevD.39.2912
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cosmology at the onset of inflation. Nucl. Phys. B551, 374–396 (1999). https://doi.org/10.
1016/S0550-3213(99)00208-4, gr-qc/9812043
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in the relativistic cosmology. Adv. Phys. 19, 525–573 (1970). https://doi.org/10.1080/
00018737000101171
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03263-3_6

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Tutorials, Schools, and Workshops in the

Sergio Luigi Cacciatori

Einstein Equations: Local

ISSN 2522-0969 ISSN 2522-0977 (electronic)

Adamantia Zampeli (Charles University in Prague): Emergence of classicality

Como, Italy Sergio Luigi Cacciatori

Part I Main Lectures

Exact Solutions of Einstein–Maxwell(-Dilaton) Equations with

2010 Mathematics Subject Classification Primary 83C99; Secondary 53A05

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 3

2 Isometric Embedding of Spheres into R3,1

Let be a closed surface that is topologically a sphere. Let σ denote a smooth

H3−κ 2 = {(x, t) | t 2 − |x|2 = κ −2 , t > 0, x ∈ R3 } ⊂ R3,1 .

In [37], Wang–Yau provided an embedding theorem that allows one to prescribe

3 Surface Hamiltonian and Its Properties

3.1 Surface Hamiltonian

Let be a closed spacelike 2-surface in a spacetime S. Suppose = ∂ for some

where X is tangential to and N is a scalar function on . The surface Hamiltonian

X = X, ν ν + T . (3)

Here, T is the projection of X (hence of T ) to the tangent space of . Then,

(K − (trK)g)(X, ν ) = K(T , ν ) − (tr K)X, ν , (4)

where tr K is the trace of K restricted to . The mean curvature vector H of in

H = (tr K)n − H ν . (5)

J = −(tr K)ν + H n , (6)

J , T = −NH − X, ν tr K. (7)

In terms of J , H(T , n ) takes the form of

αν (·) = ∇ (·) n , ν , (9)

where ∇ denotes the spacetime connection.

ν = cosh φ ν̃ + sinh φ ñ and n = sinh φ ν̃ + cosh φ ñ

for some function φ : → R. Then, ∀ v ∈ T , and it is readily checked

∇ v ν, n = ∇ v ν̃, ñ − v, ∇φ. (12)

In terms of αν (·) = ∇ (·) n, ν, one has

αν (·) = αν̃ (·) + dφ. (13)

As a result, by (8) and (13),

Here, ∇ denotes the gradient on .

3.2 Graphical Surfaces in R3,1

{ν0 , n0 } ⊂ (T )⊥ and ν0 ⊥ n0 . (17)

˜ q̂) = Ĥ (q̂) and

where {eα }α=1,2 is an orthonormal frame in T at q. Plug in T0 = Nn0 + T , and

Ĥ (q̂) = −H0 , ν0 − N −1 ∇ (T −T ) ν0 , n0 , (21)

Next, suppose τ : → R is the time function on with respect to T0 and ζ .

and is the graph of t = τ over . ˆ By abuse of nation, τ can be viewed as a

It follows from (15) and (22) that

(See equation (3.4) in [37].)

3.3 Some Related Inequalities

Suppose is a graph of t = τ over a convex surface ˆ ⊂ R3 = {t = 0}. Let ||

Let T0 = ∂t . By (23) and (8),

Suppose ⊂ H3−1 ⊂ R3,1 . Let be the domain bounded by in H3−1 . By (8),

if has positive mean curvature and is star-shaped in H3−1 , then

Note that, by (14) and (23),

Here, τ = t| , the time function on , ˆ is the projection of in R3 , and ψ

4 Construction of the Wang–Yau Energy

In a Hamiltonian approach toward defining energy for ⊂ S, one considers the

4.1 The Choice of {T , n}

Suppose ⊂ S, and let σ denote the metric on . Given an isometric embedding

a relation satisfied by T0 = Nn0 + T0 in R3,1 . Here, T is the pullback of T0 to

Viewing T and T0 as observers of in S and R3,1 , respectively, one may further

Without losing generality, taking T0 = ∂t , and writing ι = (x1 , x2 , x3 , τ ), then

H0 = (x1 , x2 , x3 , τ ) and H0 , T0 = −τ. (33)

In what follows, one assumes the mean curvature vector H of in S is spacelike.

H0 = (x1 , x2 , x3 , τ ) and H0 , T0 = −τ. (33)

+ (φ − φ0 )τ + τ div[αν (·) − ᾱν (·)] dσ.

Identifying with ˆ via x → (x, f (x)), one can view f as a function on .

Here, : → ˜ is the map with (x) = (x, f (x)).