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

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.

[1] Metric-affine gravitation theory straightforwardly comes from gauge gravitation theory where a general linear connection plays the role of a gauge field.

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

[4] Consequently, it can be treated as a gauge field.

is defined as a global section of the quotient bundle

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.

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(R) gravity, whose Lagrangian is an arbitrary function of a scalar curvature

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.

Restricted to metric connections, metric-affine gravitation theory comes to the above-mentioned Einstein – Cartan gravitation theory.

defines a principal adapted connection

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.