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 formula_1. 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 formula_2 be the tangent bundle over a manifold formula_1 provided  with bundle coordinates formula_4. A general linear connection on formula_2 is represented by a connection tangent-valued form It is associated to a principal connection on the principal frame bundle formula_7 of frames in the tangent spaces to formula_1 whose structure group is a general linear group formula_9 . Consequently, it can be treated as a gauge field. A pseudo-Riemannian metric formula_10 on formula_2 is defined as a global section of the quotient bundle formula_12, where formula_13 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 formula_14, any linear connection formula_15 on formula_2 admits a splitting in the Christoffel symbols a nonmetricity tensor and a contorsion tensor where is the torsion tensor of formula_15. 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 formula_15 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 formula_24 of formula_15, is considered. A linear connection formula_15 is called the "metric connection" for a pseudo-Riemannian metric formula_14 if formula_14 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 formula_31 of the frame bundle formula_7 corresponding to a section formula_14 of the quotient bundle formula_12. Restricted  to metric connections, metric-affine gravitation theory comes to the above-mentioned Einstein – Cartan gravitation theory. At the same time, any linear connection formula_15 defines a principal adapted connection formula_36 on a Lorentz reduced subbundle formula_31 by its restriction to a Lorentz subalgebra of a Lie algebra of a general linear group formula_9. For instance, the Dirac operator in metric-affine gravitation theory in the presence of a general linear connection formula_15 is well defined, and it depends just of the adapted connection formula_36. 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.