# 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

$\Gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\Gamma _{\lambda }{}^{\mu }{}_{\nu }{\dot {x}}^{\nu }{\dot {\partial }}_{\mu }).$ $\Gamma _{\mu \nu \alpha }=\{_{\mu \nu \alpha }\}+S_{\mu \nu \alpha }+{\frac {1}{2}}C_{\mu \nu \alpha }$ in the Christoffel symbols

$\{_{\mu \nu \alpha }\}=-{\frac {1}{2}}(\partial _{\mu }g_{\nu \alpha }+\partial _{\alpha }g_{\nu \mu }-\partial _{\nu }g_{\mu \alpha }),$ a non-metricity tensor

$C_{\mu \nu \alpha }=C_{\mu \alpha \nu }=\nabla _{\mu }^{\Gamma }g_{\nu \alpha }=\partial _{\mu }g_{\nu \alpha }+\Gamma _{\mu \nu \alpha }+\Gamma _{\mu \alpha \nu }$ and a contorsion tensor

$S_{\mu \nu \alpha }=-S_{\mu \alpha \nu }={\frac {1}{2}}(T_{\nu \mu \alpha }+T_{\nu \alpha \mu }+T_{\mu \nu \alpha }+C_{\alpha \nu \mu }-C_{\nu \alpha \mu }),$ where

$T_{\mu \nu \alpha }={\frac {1}{2}}(\Gamma _{\mu \nu \alpha }-\Gamma _{\alpha \nu \mu })$ 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

$\nabla _{\mu }^{\Gamma }g_{\nu \alpha }=0$ $\Gamma _{\mu \nu \alpha }=\{_{\mu \nu \alpha }\}+{\frac {1}{2}}(T_{\nu \mu \alpha }+T_{\nu \alpha \mu }+T_{\mu \nu \alpha }).$ 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.