Metric-affine gravitation theory

In comparison with General Relativity, dynamic variables of metric-affine gravitation theory are both a pseudo-Riemannian metric and a general linear connection on a world manifold $X$ . Metric-affine gravitation theory has been suggested as a natural generalization of Einstein–Cartan theory of gravity with torsion where a linear connection obeys the condition that a covariant derivative of a metric equals zero.

Metric-affine gravitation theory straightforwardly comes from gauge gravitation theory where a general linear connection plays the role of a gauge field. Let $TX$ be the tangent bundle over a manifold $X$ provided with bundle coordinates $(x^{\mu },{\dot x}^{\mu })$ . A general linear connection on $TX$ is represented by a connection tangent-valued form

\Gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\Gamma _{\lambda }{}^{\mu }{}_{\nu }{\dot x}^{\nu }{\dot \partial }_{\mu }).

It is associated to a principal connection on the principal frame bundle $FX$ of frames in the tangent spaces to $X$ whose structure group is a general linear group $GL(4,{\mathbb R})$ . Consequently, it can be treated as a gauge field. A pseudo-Riemannian metric $g=g_{{\mu \nu }}dx^{\mu }\otimes dx^{\nu }$ on $TX$ is defined as a global section of the quotient bundle $FX/SO(1,3)\to X$ , where $SO(1,3)$ is the Lorentz group. Therefore, on can regard it as a classical Higgs field in gauge gravitation theory. Gauge symmetries of metric-affine gravitation theory are general covariant transformations.

It is essential that, given a pseudo-Riemannian metric $g$ , any linear connection $\Gamma$ on $TX$ admits a splitting

\Gamma _{{\mu \nu \alpha }}=\{_{{\mu \nu \alpha }}\}+S_{{\mu \nu \alpha }}+{\frac 12}C_{{\mu \nu \alpha }}

in the Christoffel symbols

\{_{{\mu \nu \alpha }}\}=-{\frac 12}(\partial _{\mu }g_{{\nu \alpha }}+\partial _{\alpha }g_{{\nu \mu }}-\partial _{\nu }g_{{\mu \alpha }}),

a nonmetricity tensor

C_{{\mu \nu \alpha }}=C_{{\mu \alpha \nu }}=\nabla _{\mu }^{\Gamma }g_{{\nu \alpha }}=\partial _{\mu }g_{{\nu \alpha }}+\Gamma _{{\mu \nu \alpha }}+\Gamma _{{\mu \alpha \nu }}

and a contorsion tensor

S_{{\mu \nu \alpha }}=-S_{{\mu \alpha \nu }}={\frac 12}(T_{{\nu \mu \alpha }}+T_{{\nu \alpha \mu }}+T_{{\mu \nu \alpha }}+C_{{\alpha \nu \mu }}-C_{{\nu \alpha \mu }}),

where

T_{{\mu \nu \alpha }}={\frac 12}(\Gamma _{{\mu \nu \alpha }}-\Gamma _{{\alpha \nu \mu }})

is the torsion tensor of $\Gamma$ .

Due to this splitting, metric-affine gravitation theory possesses a different collection of dynamic variables which are a pseudo-Riemannian metric, a non-metricity tensor and a torsion tensor. As a consequence, a Lagrangian of metric-affine gravitation theory can contain different terms expressed both in a curvature of a connection $\Gamma$ and its torsion and non-metricity tensors. In particular, a metric-affine f(R) gravity, whose Lagrangian is an arbitrary function of a scalar curvature $R$ of $\Gamma$ , is considered.

A linear connection $\Gamma$ is called the metric connection for a pseudo-Riemannian metric $g$ if $g$ is its integral section, i.e., the metricity condition

\nabla _{\mu }^{\Gamma }g_{{\nu \alpha }}=0

holds. A metric connection reads

\Gamma _{{\mu \nu \alpha }}=\{_{{\mu \nu \alpha }}\}+{\frac 12}(T_{{\nu \mu \alpha }}+T_{{\nu \alpha \mu }}+T_{{\mu \nu \alpha }}).

For instance, the Levi-Civita connection in General Relativity is a torsion-free metric connection.

A metric connection is associated to a principal connection on a Lorentz reduced subbundle $F^{g}X$ of the frame bundle $FX$ corresponding to a section $g$ of the quotient bundle $FX/SO(1,3)\to X$ . Restricted to metric connections, metric-affine gravitation theory comes to the above-mentioned Einstein – Cartan gravitation theory.

At the same time, any linear connection $\Gamma$ defines a principal adapted connection $\Gamma ^{g}$ on a Lorentz reduced subbundle $F^{g}X$ by its restriction to a Lorentz subalgebra of a Lie algebra of a general linear group $GL(4,{\mathbb R})$ . For instance, the Dirac operator in metric-affine gravitation theory in the presence of a general linear connection $\Gamma$ is well defined, and it depends just of the adapted connection $\Gamma ^{g}$ . Therefore, Einstein – Cartan gravitation theory can be formulated as the metric-affine one, without appealing to the metricity constraint.

In metric-affine gravitation theory, in comparison with the Einstein - Cartan one, a question on a matter source of a non-metricity tensor arises. It is so called hypermomentum, e.g., a Noether current of a scaling symmetry.

References

F.Hehl, J. McCrea, E. Mielke, Y. Ne'eman, Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilaton invariance, Physics Reports 258 (1995) 1-171; arXiv: gr-qc/9402012
V. Vitagliano, T. Sotiriou, S. Liberati, The dynamics of metric-affine gravity, Annals of Physics 326 (2011) 1259-1273; arXiv: 1008.0171
G. Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1869-1895; arXiv: 1110.1176
C. Karahan, A. Altas, D. Demir, Scalars, vectors and tensors from metric-affine gravity, General Relativity and Gravitation 45 (2013) 319-343; arXiv: 1110.5168

Metric-affine gravitation theory

References

See also