Metric-affine gravitation theory
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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, on 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 non-metricity tensor
and a contorsion tensor
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 contains different terms expressed both in a curvature of a connection and its torsion and non-netricity tensors. In particular, a metric-affine f(R) 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 Lorentz reduced 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.
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