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