nLab first-order formulation of gravity

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 – 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 context	gauge 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.

Related concepts

• Lorentzian geometry

• supergravity

• D'Auria-Fre formulation of supergravity

• gravity as a BF-theory.

• teleparallel gravity

References

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:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, section I.4.1 of Supergravity and Superstrings - A Geometric Perspective [doi:10.1142/0224, ch I.4: pdf]

For more in the case of supergravity:

• Bernard Julia, S. Silva, On first order formulations of supergravities, JHEP 0001 (2000) 026 [arXiv:hep-th/9911035, doi:10.1088/1126-6708/2000/01/026]

Formulation via BV-formalism and $L_\infty$-algebras:

• Marija Dimitrijević Ćirić, Grigorios Giotopoulos, Voja Radovanović, Richard J. Szabo, $L_\infty$-Algebras of Einstein-Cartan-Palatini Gravity, Journal of Mathematical Physics 61 (2020) 112502 [arXiv:2003.06173, doi:10.1063/5.0011344]

The equivalence of the phase space of Palatini-Cartan-Holst Lagrangian field theory with the Einstein-Hilbert version is established in

• Alberto Cattaneo, Michele Schiavina, The reduced phase space of Palatini-Cartan-Holst theory, Ann. Henri Poincaré (2019) 20: 445 (arXiv:1707.05351, doi:10.1007/s00023-018-0733-z)

Subtleties in extending to this to a local field theory (gluing pieces of spacetimes along their boundaries) are discussed in

• Alberto Cattaneo, Michele Schiavina, BV-BFV approach to General Relativity: Palatini-Cartan-Holst action (arXiv:1707.06328)

following

• Alberto Cattaneo, Michele Schiavina, BV-BFV approach to General Relativity, Einstein-Hilbert action, J. Math. Phys. 57, 023515 (2016) (arXiv:1509.05762)

see also:

• Giovanni Canepa, Alberto Cattaneo, Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism, Annales Henri Poincaré 25 (2024) 2585–2639 [arXiv:2202.08684, doi:10.1007/s00023-023-01360-8]

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:

• Alberto S. Cattaneo, Leon Menger, Michele Schiavina, Gravity with torsion as deformed BF theory [arXiv:2310.01877]

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]