Formalism
Definition
Spacetime configurations
Properties
Spacetimes
black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
Quantum theory
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
A field configuration of the physical theory of gravity on a spacetime is equivalently
a vielbein field, hence a reduction of the structure group of the tangent bundle along , defining a pseudo-Riemannian metric;
a connection that is locally a Lie algebra-valued 1-form with values in the Poincare Lie algebra.
such that this is a Cartan connection.
(This parameterization of the gravitational field is called the first-order formulation of gravity.) The component of the connection is the vielbein that encodes a pseudo-Riemannian metric on and makes a pseudo-Riemannian manifold. Its quanta are the gravitons.
The “non-propagating field” 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 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 is the gravitino field.
So the configuration space of gravity on some is essentially the moduli space of Riemannian metrics on .
for the moment see D'Auria-Fre formulation of supergravity for further details
gravitational entropy
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]
On the early history of the idea:
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
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]
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)
See also
With focus on methods of conformal geometry (conformal boundaries, conformal compactification):
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]
Thanu Padmanabhan, Exploring the Nature of Gravity, talk notes [arXiv: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:
Relation of the first-order formulation of gravity to BF-theory:
See also the references at general relativity.
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
which is surveyed in
Careful discussion of observables in gravity is in
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, pdf]
Alberto Cattaneo, Michele Schiavina, BV-BFV approach to General Relativity, Einstein-Hilbert action, J. Math. Phys. 57 023515 (2016) [arXiv: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, 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, 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]
(following Kijowski & Tulczyjew (2005))
Discussion of flux-observables:
The result that gravity is not renormalizable is due to:
Review:
.
Last revised on September 25, 2024 at 09:03:13. See the history of this page for a list of all contributions to it.