nLab general relativity

Idea

The theory called the theory of general relativity is a classical field theory that in physics describes the field of gravity.

In general relativity, physical spacetime is modeled in terms of differential geometry as a Lorentzian manifold whose pseudo-Riemannian metric – or rather the Levi-Civita connection that corresponds to it – encodes the field of gravity.

The action functional describing the dynamics of this field is the Einstein-Hilbert action, in which the field of gravity enters in terms of the integral of the scalar curvature of the Levi-Civita connection over spacetime.

As is usual in classical field theory, the physically realized configurations – here: Levi-Civita connections – are those that extremize this functional. The Euler-Lagrange equations characterizing these extrema are the Einstein equations.

Historically the theory of general relativity was developed by Albert Einstein based on the theory known as special relativity. Given the conceptual simplicity of the Einstein-Hilbert action, there are several variations of his original version of the theory that are immediately obtained by adding certain terms to the action functional. One of these generalization is supergravity, which extends the theory from ordinary differential geometry to supergeometry.

Related concepts

• general covariance

• equivalence principle (physics)

• relativistic particle

• relativistic field theory

• spacetime

• black hole

• ADM mass

• Penrose-Hawking theorem

• cosmic censorship

References

General

Original articles:

• Albert Einstein, Marcel Grossmann: Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation, Teubner (1913) [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, ark:/13960/t13n7rw1f, spire:1410180]

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

• Theodore Frankel: Gravitational Curvature, Freeman, San Francisco (1979) [ark:13960/t58d7nn19]

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

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

• Thanu Padmanabhan, Gravitation – Foundations and Frontiers, Cambridge University Press (2012) [doi: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]

• 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]

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

• Kirill Krasnov, Formulations of General Relativity – Gravity, Spinors and Differential Forms, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2020) [doi:10.1017/9781108674652, taster:pdf]

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, pdf]

• Edward Witten, Light Rays, Singularities, and All That, Rev. Mod. Phys. 92 45004 (2020) [arXiv: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]

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]

• Shlomo Sternberg, Semi-Riemannian Geometry and General Relativity (2003) [pdf, ark:/13960/t5m927d2v]

• Shlomo Sternberg, Curvature in Mathematical Physics, Dover (2012) [ISBN:9780486478555]

Emphasis on mathematical physics

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

• José Natário, Mathematical relativity (arXiv:2003.02855)

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, oapen:20.500.12657/59217, pdf]

Introductory exposition:

• John Baez, The General Relativity Tutorial

See also

• Edward Witten, Light Rays, Singularities, and All That (arXiv:1901.03928)

See also the references at gravity.

In relation to thermodynamics

In relation to thermodynamics

• Patrick Iglesias-Zemmour, Jean-Marie Souriau Heat, cold and Geometry, in: M. Cahen et al (eds.) Differential geometry and mathematical physics, 37-68, D. Reidel 1983 (web, pdf)

  (on thermodynamics and general relativity)

History

• Peter Coles, Einstein, Eddington, and the 1919 Eclipse (arxIv:astro-ph/0102462)

  (on the experimental confirmation of general relativity)