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 X. 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 TX be the tangent bundle over a manifold X provided with bundle coordinates  (x^\mu,\dot x^\mu). A general linear connection on TX is represented by a connection tangent-valued form

\Gamma=dx^\lambda\otimes(\partial_\lambda +\Gamma_\lambda{}^\mu{}_\nu\dot x^\nu\dot\partial_\mu).

It is associated to a principal connection on the principal frame bundle FX of frames in the tangent spaces to X whose structure group is a general linear group GL(4,\mathbb R) . Consequently, it can be treated as a gauge field. A pseudo-Riemannian metric g=g_{\mu\nu}dx^\mu\otimes dx^\nu on TX is defined as a global section of the quotient bundle FX/SO(1,3)\to X, where SO(1,3) 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 g, any linear connection \Gamma on TX admits a splitting

\Gamma_{\mu\nu\alpha}=\{_{\mu\nu\alpha}\} +S_{\mu\nu\alpha} +\frac12 C_{\mu\nu\alpha}

in the Christoffel symbols

\{_{\mu\nu\alpha}\}= -\frac12(\partial_\mu g_{\nu\alpha} + \partial_\alpha
g_{\nu\mu}-\partial_\nu g_{\mu\alpha}),

a non-metricity tensor

C_{\mu\nu\alpha}=C_{\mu\alpha\nu}=\nabla^\Gamma_\mu g_{\nu\alpha}=\partial_\mu g_{\nu\alpha} +\Gamma_{\mu\nu\alpha} + \Gamma_{\mu\alpha\nu}

and a contorsion tensor

S_{\mu\nu\alpha}=-S_{\mu\alpha\nu}=\frac12(T_{\nu\mu\alpha} +T_{\nu\alpha\mu} + T_{\mu\nu\alpha}+ C_{\alpha\nu\mu} -C_{\nu\alpha\mu}),


T_{\mu\nu\alpha}=\frac12(\Gamma_{\mu\nu\alpha} - \Gamma_{\alpha\nu\mu})

is the torsion tensor of \Gamma.

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 \Gamma 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 R of \Gamma, is considered.

A linear connection \Gamma is called the metric connection for a pseudo-Riemannian metric g if g is its integral section, i.e., the metricity condition

\nabla^\Gamma_\mu g_{\nu\alpha}=0

holds. A metric connection reads

\Gamma_{\mu\nu\alpha}=\{_{\mu\nu\alpha}\} + \frac12(T_{\nu\mu\alpha} +T_{\nu\alpha\mu} + T_{\mu\nu\alpha}).

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 F^gX of the frame bundle FX corresponding to a section g of the quotient bundle FX/SO(1,3)\to X. Restricted to metric connections, metric-affine gravitation theory comes to the above mentioned Einstein – Cartan gravitation theory.

At the same time, any linear connection \Gamma defines a principal adapted connection \Gamma^g on a Lorentz reduced subbundle F^gX by its restriction to a Lorentz subalgebra of a Lie algebra of a general linear group GL(4,\mathbb R). For instance, the Dirac operator in metric-affine gravitation theory in the presence of a general linear connection \Gamma is well defined, and it depends just of the adapted connection \Gamma^g. 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

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