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 [math]\displaystyle{ X }[/math]. 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 [math]\displaystyle{ TX }[/math] be the tangent bundle over a manifold [math]\displaystyle{ X }[/math] provided with bundle coordinates [math]\displaystyle{ (x^\mu,\dot x^\mu) }[/math]. A general linear connection on [math]\displaystyle{ TX }[/math] is represented by a connection tangent-valued form
It is associated to a principal connection on the principal frame bundle [math]\displaystyle{ FX }[/math] of frames in the tangent spaces to [math]\displaystyle{ X }[/math] whose structure group is a general linear group [math]\displaystyle{ GL(4,\mathbb R) }[/math] . Consequently, it can be treated as a gauge field. A pseudo-Riemannian metric [math]\displaystyle{ g=g_{\mu\nu}dx^\mu\otimes dx^\nu }[/math] on [math]\displaystyle{ TX }[/math] is defined as a global section of the quotient bundle [math]\displaystyle{ FX/SO(1,3)\to X }[/math], where [math]\displaystyle{ SO(1,3) }[/math] 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 [math]\displaystyle{ g }[/math], any linear connection [math]\displaystyle{ \Gamma }[/math] on [math]\displaystyle{ TX }[/math] admits a splitting
in the Christoffel symbols
and a contorsion tensor
where
is the torsion tensor of [math]\displaystyle{ \Gamma }[/math].
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 [math]\displaystyle{ \Gamma }[/math] 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 [math]\displaystyle{ R }[/math] of [math]\displaystyle{ \Gamma }[/math], is considered.
A linear connection [math]\displaystyle{ \Gamma }[/math] is called the metric connection for a pseudo-Riemannian metric [math]\displaystyle{ g }[/math] if [math]\displaystyle{ g }[/math] is its integral section, i.e., the metricity condition
holds. A metric connection reads
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 [math]\displaystyle{ F^gX }[/math] of the frame bundle [math]\displaystyle{ FX }[/math] corresponding to a section [math]\displaystyle{ g }[/math] of the quotient bundle [math]\displaystyle{ FX/SO(1,3)\to X }[/math]. Restricted to metric connections, metric-affine gravitation theory comes to the above-mentioned Einstein – Cartan gravitation theory.
At the same time, any linear connection [math]\displaystyle{ \Gamma }[/math] defines a principal adapted connection [math]\displaystyle{ \Gamma^g }[/math] on a Lorentz reduced subbundle [math]\displaystyle{ F^gX }[/math] by its restriction to a Lorentz subalgebra of a Lie algebra of a general linear group [math]\displaystyle{ GL(4,\mathbb R) }[/math]. For instance, the Dirac operator in metric-affine gravitation theory in the presence of a general linear connection [math]\displaystyle{ \Gamma }[/math] is well defined, and it depends just of the adapted connection [math]\displaystyle{ \Gamma^g }[/math]. 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.