nLab
gravity

Idea

A field configuration of the physical theory of gravity on a spacetime $X$ is equivalently

• a vielbein field, hence a reduction of the structure group of the tangent bundle along $BO(n-1,1)\to B\mathrm{GL}(n)$, defining a pseudo-Riemannian metric;

• a connection that is locally a Lie algebra-valued 1-form with values in the Poincare Lie algebra.

  $$(E,\Omega ):TX\to \mathrm{𝔦𝔰𝔬}(d-1,1)$$(E, \Omega) : T X \to \mathfrak{iso}(d-1,1)

  such that this is a Cartan connection.

(This parameterization of the gravitational field is called the first-order formulation of gravity.) The component $E$ of the connection is the vielbein that encodes a pseudo-Riemannian metric $g=E\cdot E$ on $X$ and makes $X$ a pseudo-Riemannian manifold. Its quanta are the gravitons.

The non-propagating field? $\Omega$ is the spin connection.

The action functional on the space of such connection which defines the classical field theory of gravity is the Einstein-Hilbert action.

More generally, supergravity is a gauge theory over a supermanifold $X$ for the super Poincare group. The field of supergravity is a Lie-algebra valued form with values in the super Poincare Lie algebra.

$$(E,\Omega ,\Psi ):TX\to \mathrm{𝔰𝔦𝔰𝔬}(d-1,1)$$(E,\Omega, \Psi) : T X \to \mathfrak{siso}(d-1,1)

The additional fermionic field $\Psi$ is the gravitino field.

So the configuration space of gravity on some $X$ is essentially the moduli space of Riemannian metrics on $X$.

Details

for the moment see D'Auria-Fre formulation of supergravity for further details

Related concepts

• Einstein-Hilbert action, Einstein equation

• general covariance

• first order formulation of gravity

  • Plebanski formulation of gravity, gravity as a BF-theory, teleparallel gravity

  • Chern-Simons gravity

• Einstein-Maxwell theory, Einstein-Yang-Mills theory

• supergravity

• gravitational entropy

  • Bekenstein-Hawking entropy

  • generalized second law of thermodynamics

• cosmology

  • standard model of cosmology

• MOND

References

General

The theory of gravity based on the standard Einstein-Hilbert action may be regarded as just an effective quantum field theory, which makes some of its notorious problems be non-problems:

• John F. Donoghue, Introduction to the Effective Field Theory Description of Gravity (arXiv:gr-qc/9512024)

Covariant phase space

The (reduced) covariant phase space of gravity (presented for instance by its BV-BRST complex, see there fore more details) is discussed for instance in

• Romeo Brunetti, Klaus Fredenhagen, Towards a Background Independent Formulation of Perturbative Quantum Gravity (arXiv:gr-qc/0603079)

which is surveyed in

• Katarzyna Rejzner, The BV formalism applied to classical gravity (pdf)