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.


  • F.Hehl, J. McCrea, E. Mielke, Y. Ne'eman, Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilaton invariance, Physics Reports 258 (1995) 1-171; arXiv: gr-qc/9402012
  • V. Vitagliano, T. Sotiriou, S. Liberati, The dynamics of metric-affine gravity, Annals of Physics 326 (2011) 1259-1273; arXiv: 1008.0171
  • G. Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1869-1895; arXiv: 1110.1176
  • C. Karahan, A. Altas, D. Demir, Scalars, vectors and tensors from metric-affine gravity, General Relativity and Gravitation 45 (2013) 319-343; arXiv: 1110.5168

See also