nLab
first-order formulation of gravity
Contents
Idea
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 – essentially the Levi-Civita connection – 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 Palatini formalism for gravity.

The field strength of gravity – the Riemann tensor – is the curvature of Levi-Civita connection . Typically this is referred to a spin connection in this context.

Promoting this perspective form the Poincaré group to the super Poincaré group yields supergravity forulated 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 ).

geometric contextgauge group stabilizer subgroup local model space local geometry global geometry differential cohomology first order formulation of gravity differential geometry Lie group /algebraic group $G$ subgroup (monomorphism ) $H \hookrightarrow G$ quotient (“coset space ”) $G/H$ Klein geometry Cartan geometry Cartan connection
examples Euclidean group $Iso(d)$ rotation group $O(d)$ Cartesian space $\mathbb{R}^d$ Euclidean geometry Riemannian geometry affine connection Euclidean gravity
Poincaré group $Iso(d-1,1)$ Lorentz group $O(d-1,1)$ Minkowski spacetime $\mathbb{R}^{d-1,1}$ Lorentzian geometry pseudo-Riemannian geometry spin connection Einstein gravity
anti de Sitter group $O(d-1,2)$ $O(d-1,1)$ anti de Sitter spacetime $AdS^d$ AdS gravity
de Sitter group $O(d,1)$ $O(d-1,1)$ de Sitter spacetime $dS^d$ deSitter gravity
linear algebraic group parabolic subgroup /Borel subgroup flag variety parabolic geometry
conformal group $O(d,t+1)$ conformal parabolic subgroup Möbius space $S^{d,t}$ conformal geometry conformal connection conformal gravity
supergeometry super Lie group $G$ subgroup (monomorphism ) $H \hookrightarrow G$ quotient (“coset space ”) $G/H$ super Klein geometry super Cartan geometry Cartan superconnection
examples super Poincaré group spin group super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$ Lorentzian supergeometry supergeometry superconnection supergravity
super anti de Sitter group super anti de Sitter spacetime
higher differential geometry smooth 2-group $G$ 2-monomorphism $H \to G$ homotopy quotient $G//H$ Klein 2-geometry Cartan 2-geometry
cohesive ∞-group ∞-monomorphism (i.e. any homomorphism ) $H \to G$ homotopy quotient $G//H$ of ∞-action higher Klein geometry higher Cartan geometry higher Cartan connection
examples extended super Minkowski spacetime extended supergeometry higher supergravity : type II , heterotic , 11d

Properties
Local and global framing
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 .

References
A decent introduction is in sections 4 and 5 of

A detailed account is in section I.4.1 of

Revised on March 9, 2016 17:09:48
by

Urs Schreiber
(195.37.209.180)