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 $g=g_dx^\mu\otimes dx^\nu$ 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, one 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

: $\Gamma_=\ +\frac12 C_ + S_$

in the Christoffel symbols

: $\= -\frac12(\partial_\mu g_ + \partial_\alpha g_-\partial_\nu g_),$

a nonmetricity tensor

: $C_=C_=\nabla^\Gamma_\mu g_=\partial_\mu g_ +\Gamma_ + \Gamma_$

and a contorsion tensor

: $S_=-S_=\frac12(T_ +T_ + T_+ C_ -C_),$

where

: $T_=\frac12(\Gamma_ - \Gamma_)$

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

: $\nabla^\Gamma_\mu g_=0$

holds. A metric connection reads

: $\Gamma_=\ + \frac12(T_ +T_ + T_).$

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 $F^gX$ 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 $F^gX$ 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.

See also

*Gauge gravitation theory
*Einstein–Cartan theory
* Affine gauge theory
*Classical unified field theories

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;
* V. Vitagliano, T. Sotiriou, S. Liberati, The dynamics of metric-affine gravity, ''Annals of Physics'' 326 (2011) 1259-1273;
* G. Sardanashvily, Classical gauge gravitation theory, ''Int. J. Geom. Methods Mod. Phys.'' 8 (2011) 1869-1895;
* 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

Theories of gravity