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. 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 be the tangent bundle over a manifold provided with bundle coordinates. A general linear connection on is represented by a connection tangent-valued form It is associated to a principal connection on the principal frame bundle of frames in the tangent spaces to whose structure group is a general linear group . Consequently, it can be treated as a gauge field. A pseudo-Riemannian metric on is defined as a global section of the quotient bundle, where 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, any linear connection on admits a splitting in the Christoffel symbols a nonmetricity tensor and a contorsion tensor where is the torsion tensor of. 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 and its torsion and non-metricity tensors. In particular, a metric-affine f gravity, whose Lagrangian is an arbitrary function of a scalar curvature of, is considered. A linear connection is called the metric connection for a pseudo-Riemannian metric if 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 Lorentzreduced subbundle of the frame bundle corresponding to a section of the quotient bundle. Restricted to metric connections, metric-affine gravitation theory comes to the above-mentioned Einstein – Cartan gravitation theory. At the same time, any linear connection defines a principal adapted connection on a Lorentz reduced subbundle by its restriction to a Lorentz subalgebra of a Lie algebra of a general linear group. For instance, the Dirac operator in metric-affine gravitation theory in the presence of a general linear connection is well defined, and it depends just of the adapted connection. 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.
