# Metric-affine gravitation theory

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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

$\Gamma_{\mu\nu\alpha}=\{_{\mu\nu\alpha}\} +S_{\mu\nu\alpha} +\frac12 C_{\mu\nu\alpha}$

in the Christoffel symbols

$\{_{\mu\nu\alpha}\}= -\frac12(\partial_\mu g_{\nu\alpha} + \partial_\alpha g_{\nu\mu}-\partial_\nu g_{\mu\alpha}),$

a non-metricity tensor

$C_{\mu\nu\alpha}=C_{\mu\alpha\nu}=\nabla^\Gamma_\mu g_{\nu\alpha}=\partial_\mu g_{\nu\alpha} +\Gamma_{\mu\nu\alpha} + \Gamma_{\mu\alpha\nu}$

and a contorsion tensor

$S_{\mu\nu\alpha}=-S_{\mu\alpha\nu}=\frac12(T_{\nu\mu\alpha} +T_{\nu\alpha\mu} + T_{\mu\nu\alpha}+ C_{\alpha\nu\mu} -C_{\nu\alpha\mu}),$

where

$T_{\mu\nu\alpha}=\frac12(\Gamma_{\mu\nu\alpha} - \Gamma_{\alpha\nu\mu})$

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 contains different terms expressed both in a curvature of a connection $\Gamma$ and its torsion and non-netricity 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

$\nabla^\Gamma_\mu g_{\nu\alpha}=0$

holds. A metric connection reads

$\Gamma_{\mu\nu\alpha}=\{_{\mu\nu\alpha}\} + \frac12(T_{\nu\mu\alpha} +T_{\nu\alpha\mu} + T_{\mu\nu\alpha}).$

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 $F^gX$ 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.

## 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