black hole spacetimes | vanishing angular momentum | positive angular momentum |
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vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
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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 17
Promoting the first-ore 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.
A decent introduction is in
A detailed account is in
The equivalence of the phase space of Palatini-Cartan-Holst Lagrangian field theory with the Einstein-Hilbert version is established in