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Contents

Idea

A field configuration of the physical theory of gravity on a spacetime $X$ is equivalently

• a vielbein field, hence a reduction of the structure group of the tangent bundle along $\mathbf{B} O(n-1,1) \to \mathbf{B}GL(n)$, defining a pseudo-Riemannian metric;

• a connection that is locally a Lie algebra-valued 1-form with values in the Poincare Lie algebra.

  $$(E, \Omega) : T X \to \mathfrak{iso}(d-1,1)$$

  such that this is a Cartan connection.

(This parameterization of the gravitational field is called the first-order formulation of gravity.) The component $E$ of the connection is the vielbein that encodes a pseudo-Riemannian metric $g = E \cdot E$ on $X$ and makes $X$ a pseudo-Riemannian manifold. Its quanta are the gravitons.

The “non-propagating field” $\Omega$ is the spin connection.

The action functional on the space of such connection which defines the classical field theory of gravity is the Einstein-Hilbert action.

More generally, supergravity is a gauge theory over a supermanifold $X$ for the super Poincare group. The field of supergravity is a Lie-algebra valued form with values in the super Poincare Lie algebra.

The additional fermionic field $\Psi$ is the gravitino field.

So the configuration space of gravity on some $X$ is essentially the moduli space of Riemannian metrics on $X$.

Details

for the moment see D'Auria-Fre formulation of supergravity for further details

Related concepts

• Einstein-Hilbert action, Einstein equation

• rubber-sheet analogy of gravity

• general covariance

• Penrose-Hawking theorem, cosmic censorship hypothesis

• weak gravity conjecture

• first order formulation of gravity

  • Plebanski formulation of gravity, gravity as a BF-theory, teleparallel gravity

  • Chern-Simons gravity

• Einstein-Maxwell theory, Einstein-Yang-Mills theory

• higher order curvature corrections

• supergravity

• D=2 gravity

  • Liouville theory

  • Jackiw-Teitelboim gravity

• D=5 gravity

• spacetime

  • black hole

  • gravitational wave

• gravitational entropy

  • Bekenstein-Hawking entropy

  • generalized second law of thermodynamics

• energy condition

• positive energy theorem

• asymptotic safety

• cosmology

  • standard model of cosmology

• MOND

• computer experiment: gevolution

References

General

Historical texts:

• Albert Einstein, Marcel Grossmann: Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation, Teubner (2013) [pdf]

• Leopold Infeld (ed.), Relativistic Theories of Gravitation, Proceedings of a conference held in Warsaw and Jablonna 1962, Pergamon Press (1964) [[pdf](https://cds.cern.ch/record/2282975/files/warsaw-1962.pdf)]

On the early history of the idea:

• John Earman, Clark Glymour: Lost in the tensors: Einstein’s struggles with covariance principles 1912–1916, Studies in History and Philosophy of Science Part A Volume 9, Issue 4, (1978) 251-278 [doi:10.1016/0039-3681(78)90008-0]

Monographs:

• Steven Weinberg: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley (1972, 2013) [[ISBN:978-0-471-92567-5](https://www.wiley.com/en-us/Gravitation+and+Cosmology%3A+Principles+and+Applications+of+the+General+Theory+of+Relativity-p-9780471925675), ark:/13960/t13n7rw1f, spire:1410180]

• Charles Misner, Kip Thorne, John Wheeler, Gravitation (1973)

• Theodore Frankel: Gravitational Curvature, Freeman, San Francisco (1979) [[ark:13960/t58d7nn19](https://archive.org/details/gravitationalcur0000fran/)]

• Robert Wald, General Relativity, University of Chicago Press (1984) [[doi:10.7208/chicago/9780226870373.001.0001](https://doi.org/10.7208/chicago/9780226870373.001.0001), pdf]

• Garth Warner: Mathematical Aspects of General Relativity, EPrint Collection, University Of Washington (2006) [[hdl:1773/2637](http://hdl.handle.net/1773/2637), pdf, pdf]

• Thanu Padmanabhan, Gravitation – Foundations and Frontiers, Cambridge University Press (2012) [[doi:10.1017/CBO9780511807787](https://doi.org/10.1017/CBO9780511807787), spire:852758, toc: pdf]

• Pietro Fré, Gravity, a Geometrical Course, Volume 1: Development of the Theory and Basic Physical Applications, Spinger (2013) [[doi:10.1007/978-94-007-5361-7](https://doi.org/10.1007/978-94-007-5361-7)]

• Pietro Fré, Gravity, a Geometrical Course, Volume 2: Black Holes, Cosmology and Introduction to Supergravity, Springer (2013) [[doi:10.1007/978-94-007-5443-0](https://doi.org/10.1007/978-94-007-5443-0)]

  on black holes, cosmology and the (D'Auria-Fré formulation of) supergravity

Background on pseudo-Riemannian geometry:

• Barrett O'Neill, Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics 103, Academic Press (1983) [[ISBN:9780125267403](https://shop.elsevier.com/books/semi-riemannian-geometry-with-applications-to-relativity/oneill/978-0-12-526740-3)]

• Shlomo Sternberg, Semi-Riemannian Geometry and General Relativity (2003) [[pdf](https://people.math.harvard.edu/~shlomo/docs/semi_riemannian_geometry.pdf), ark:/13960/t5m927d2v]

• Shlomo Sternberg, Curvature in Mathematical Physics, Dover (2012) [[ISBN:9780486478555](https://store.doverpublications.com/products/9780486478555)]

Lecture notes:

• Matthias Blau, Lecture notes on general relativity (web)

• Emil T. Akhmedov, Lectures on General Theory of Relativity (arXiv:1601.04996)

• Pietro Menotti, Lectures on gravitation (arXiv:1703.05155)

• Daniel Baumann, General Relativity (2021-24) [[pdf](http://cosmology.amsterdam/wp-content/uploads/2021/10/GR-Oct19.pdf), pdf]

• Edward Witten, Light Rays, Singularities, and All That, Rev. Mod. Phys. 92 45004 (2020) [[arXiv:1901.03928](https://arxiv.org/abs/1901.03928), doi:10.1103/RevModPhys.92.045004, lecture: video 1, 2, 3, 4, 5, 6, 7, 8, 9]

  (with focus on causal structure, the Penrose singularity theorem and related aspects)

• Kirill Krasnov, Formulations of General Relativity, PIRSA lecture series (Feb 2019) [part 1:doi:10.48660/19020079, 2:doi:10.48660/19020080, 3:doi:10.48660/19020081, 4:doi:10.48660/19020082]

• Peter Hayman: A Lean and Mean Introduction to Modern General Relativity [[arXiv:2412.08026](https://arxiv.org/abs/2412.08026)]

See also

• Alan Coley, Mathematical General Relativity (arXiv:1807.08628)

With focus on methods of conformal geometry (conformal boundaries, conformal compactification):

• Juan A. Valiente Kroon: Conformal Methods in General Relativity, Cambridge University Press (2017, 2022) [[doi:10.1017/9781009291309](https://doi.org/10.1017/9781009291309), oapen:20.500.12657/59217, pdf]

On gravity in relation to thermodynamics:

• Thanu Padmanabhan, Gravity and/is Thermodynamics, Current Science, 109 (2015) 2236-2242 [[doi:10.18520/v109/i12/2236-2242](https://doi.org/10.18520/v109/i12/2236-2242)]

• Thanu Padmanabhan, Exploring the Nature of Gravity, talk notes [[arXiv:1602.01474](https://arxiv.org/abs/1602.01474)]

Discussion of classical gravity via its perturbative quantum field theory:

• Günter Scharf, Quantum Gauge Theories -- A True Ghost Story, Wiley 2001

• Gustav Uhre Jakobsen, General Relativity from Quantum Field Theory (arXiv:2010.08839)

This way the theory of gravity based on the standard Einstein-Hilbert action may be regarded as just an effective quantum field theory, which makes some of its notorious problems be non-problems:

• John Donoghue, Introduction to the Effective Field Theory Description of Gravity (arXiv:gr-qc/9512024)

Relation of the first-order formulation of gravity to BF-theory:

• Alberto S. Cattaneo, Leon Menger, Michele Schiavina, Gravity with torsion as deformed BF theory [[arXiv:2310.01877](https://arxiv.org/abs/2310.01877)]

See also the references at general relativity.

Covariant phase space

The (reduced) covariant phase space of gravity (presented for instance by its BV-BRST complex, see there fore more details) is discussed for instance in

• Romeo Brunetti, Klaus Fredenhagen, Towards a Background Independent Formulation of Perturbative Quantum Gravity (arXiv:gr-qc/0603079)

which is surveyed in

• Katarzyna Rejzner, The BV formalism applied to classical gravity (pdf)

Careful discussion of observables in gravity is in

• Igor Khavkine, Local and gauge invariant observables in gravity, talk at Operator and Geometric Analysis on Quantum Theory, preprint arXiv:1503.03754

Further discussion of the phase space of gravity in first-order formulation via BV-BFV formalism:

• Michele Schiavina, BV-BFV approach to general relativity (2015) [[pdf](http://user.math.uzh.ch/cattaneo/schiavina.pdf), pdf]

• Alberto Cattaneo, Michele Schiavina, BV-BFV approach to General Relativity, Einstein-Hilbert action, J. Math. Phys. 57 023515 (2016) [[arXiv:1509.05762](https://arxiv.org/abs/1509.05762), doi:10.1063/1.4941410]

• Alberto Cattaneo, Michele Schiavina, The reduced phase space of Palatini-Cartan-Holst theory, Ann. Henri Poincaré 20 (2019) 445 [[arXiv:1707.05351](https://arxiv.org/abs/1707.05351), doi:10.1007/s00023-018-0733-z]

• Alberto Cattaneo, Michele Schiavina, BV-BFV approach to General Relativity: Palatini-Cartan-Holst action, Adv. Theor. Math. Phys. 23 (2019) 1801-1835 [[arXiv:1707.06328](https://arxiv.org/abs/1707.06328), doi:10.4310/ATMP.2019.v23.n8.a3]

• Alberto S. Cattaneo, Phase space for gravity with boundaries, in Encyclopedia of Mathematical Physics 2nd ed (2023) [[arXiv:2307.04666](https://arxiv.org/abs/2307.04666)]

  (following Kijowski & Tulczyjew (2005))

Discussion of flux-observables:

• Alberto S. Cattaneo, Alejandro Perez, A note on the Poisson bracket of 2d smeared fluxes in loop quantum gravity, Class. Quant. Grav. 34 (2017) 107001 [[arXiv:1611.08394](https://arxiv.org/abs/1611.08394), doi:10.1088/1361-6382/aa69b4]

Non-renormalizability

The result that gravity is not renormalizable is due to:

• Gerard 't Hooft, Martinus Veltman, One loop divergencies in the theory of gravitation, Ann. Inst. Poincar é 20 (1974) 69 (numdam:AIHPA_1974__20_1_69_0)

Review:

• Zvi Bern, Perturbative quantum gravity and its relation to gauge theory, Living reviews in relativity, www.livingreviews.org/Articles/Volume5/2002-5bern .
• Assaf Shomer, A pedagogical explanation for the non-renormalizability of gravity (arXiv:0709.3555)