In theoretical physics and mathematical physics, a metric-affine gravitation theory is a mathematical formulation of the theory of gravity involving a general affine connection on the world manifold (that is, a general linear connection on the tangent bundle), in addition to a pseudo-Riemannian metric. This is in contrast to standard formulations of General Relativity, in which the affine connection is the Levi-Civita connection associated to the metric. Metric-affine gravitation theory has been suggested as a natural generalization of Einstein–Cartan theory, another alternative formulation of gravity in which the affine connection is allowed to have non-zero torsion but must still preserve the metric.
As a gauge theory
Metric-affine gravitation theory 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:
This connection 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.
Dynamical variables
Given a pseudo-Riemannian metric , any linear connection on admits a splitting
into 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 dynamical variables compared to General Relativity; namely, a pseudo-Riemannian metric, a non-metricity tensor and a torsion tensor. As a consequence, a Lagrangian of metric-affine gravitation theory can be expressed both in terms of the 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 of , is considered.
Metricity and comparison to Einstein–Cartan theory
An affine connection is called a metric connection for a pseudo-Riemannian metric if it preserves , i.e., the metricity condition
holds. A metric connection reads
For instance, the Levi-Civita connection in General Relativity is a metric connection; it is distinguished by being the unique such connection which is torsion-free.
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 the Lie algebra of the 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 only on the adapted connection . Therefore, Einstein–Cartan gravitation theory can be formulated as a metric-affine one, without appealing to the metricity constraint.
In metric-affine gravitation theory, in comparison with the Einstein–Cartan one, the role of the matter source of a non-metricity tensor is played by the so-called hypermomentum, e.g., a Noether current of a scaling symmetry.
See also Gauge gravitation theory Einstein–Cartan theory Affine gauge theory Classical unified field theories
References
G. Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1869–1895; C. Karahan, A. Altas, D. Demir, Scalars, vectors and tensors from metric-affine gravity, General Relativity and Gravitation 45 (2013) 319–343;