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


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

in the Christoffel symbols


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

g_{\nu\mu}-\partial_\nu g_{\mu\alpha}), $$

a nonmetricity 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 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


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

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.