nLab first-order formulation of gravity





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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 XX. 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).

geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d


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.



More explicitly understood via Cartan geometry:

and in view of supergravity:

For more in the case of supergravity:

Formulation via BV-formalism and L 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


see also:

Reviewed in:

Relation to BF-theory:

See also:

  • J. Fernando Barbero G., Juan Margalef-Bentabol, Valle Varo, Eduardo J.S. Villaseñor, Covariant phase space for gravity with boundaries: metric vs tetrad formulations (arXiv:2103.06362)

See also:

  • Fernando T. Brandt, J. Frenkel, S. Martins-Filho, D. G. C. McKeon, Quantization of Einstein-Cartan theory in the first order form [arXiv:2401.16343]

Last revised on April 16, 2024 at 07:51:18. See the history of this page for a list of all contributions to it.