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Quantum theory
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The action functional of gravity was originally conceived as a functional on the space of pseudo-Riemannian metrics of a manifold $X$. Later on it was realized that it may alternatively naturally be thought of as a functional on the space of connections with values in the Poincaré Lie algebra – subject to the constraint that the component in the translation Lie algebra defines a vielbein field.
Mathematically this means that the field of gravity is modeled as a Cartan connection for the Lorentz group inside the Poincaré group. In physics this is known as the first order formalism or the Cartan moving frame method. The translation part of the Poincaré Lie algebra-connection is called the vielbein and the remainder the spin connection. The field strength of gravity – the Riemann tensor – is the curvature.
The reformulation of pseudo-Riemannian geometry in terms of Cartan geometry is suggestive of also re-rewriting the form of the Lagrangian density/action functional of the theory of gravity, even though this is logically an independent issue. In spacetime of dimension 3+1 one such alternative is known as the Palatini-Cartan-Holst action. That its phase space coincides with that induced by the Einstein-Hilbert action is due to Cattaneo-Schiavina 17a
Promoting the first-order formulation of gravity from the Poincaré group to the super Poincaré group yields supergravity formulated in super Cartan geometry. Promoting it further to the Lie n-algebra extensions of the super Poincaré group (from the brane scan/brane bouquet) yields type II supergravity, heterotic supergravity and 11-dimensional supergravity in higher Cartan geometry-formulation (D'Auria-Fré formulation of supergravity).
Discussion in the physics literature traditionally tends to ignore the global structure of spacetime manifolds and pretends that a vielbein field may be chosen globally, hence that spacetime admits a framing.
In general that is only valid locally, but it so happens that in the archetypical case of interest, namely for 4-dimensional globally hyperbolic spacetimes with orientable spatial slices, it is valid globally. See this remark at framed manifold for more.
See also at teleparallel gravity.
Introductions:
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, section I.4 of: Supergravity and Superstrings - A Geometric Perspective [doi:10.1142/0224, ch I.4: pdf]
Jorge Zanelli, sections 4 and 5 of: Lecture notes on Chern-Simons (super-)gravities. Second edition (February 2008), lectures at Geometric and topological methods for quantum field theory, Villa de Leyva (2001) [arXiv:hep-th/0502193, inspire:677203]
Pietro Fré, §5 in: Gravity, a Geometrical Course, Volume 1: Development of the Theory and Basic Physical Applications, Spinger (2013) [doi:10.1007/978-94-007-5361-7]
More explicitly understood via Cartan geometry:
Gabriel Catren, Geometrical Foundations of Cartan Gauge Gravity, Int. J. Geom. Methods in Modern Physics 12 04 (2015) 1530002 [arXiv:1407.7814, doi:10.1142/S0219887815300020]
Kirill Krasnov, §3 in: Formulations of General Relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2020) [doi:10.1017/9781108674652, taster:pdf]
and in view of supergravity:
For more in the case of supergravity:
Formulation via BV-formalism and $L_\infty$-algebras:
The equivalence of the phase space of Palatini-Cartan-Holst Lagrangian field theory with the Einstein-Hilbert version is established in
Subtleties in extending to this to a local field theory (gluing pieces of spacetimes along their boundaries) are discussed in
following
see also:
Reviewed in:
Alberto S. Cattaneo, Phase space for gravity with boundaries, Encyclopedia of Mathematical Physics (2023) [arXiv:2307.04666]
(following Kijowski & Tulczyjew (2005))
Relation to BF-theory:
See also:
See also:
Last revised on April 16, 2024 at 07:51:18. See the history of this page for a list of all contributions to it.