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

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

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

It is essential that, given a pseudo-Riemannian metric [math]\displaystyle{ g }[/math], any linear connection [math]\displaystyle{ \Gamma }[/math] on [math]\displaystyle{ TX }[/math] admits a splitting

[math]\displaystyle{ \Gamma_{\mu\nu\alpha}=\\{_{\mu\nu\alpha}\\} +\frac12 C_{\mu\nu\alpha} + S_{\mu\nu\alpha} }[/math]

in the Christoffel symbols

[math]\displaystyle{ \\{_{\mu\nu\alpha}\\}= -\frac12(\partial_\mu g_{\nu\alpha} + \partial_\alpha g_{\nu\mu}-\partial_\nu g_{\mu\alpha}), }[/math]

a nonmetricity tensor

[math]\displaystyle{ 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} }[/math]

and a contorsion tensor

[math]\displaystyle{ 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}), }[/math]

where

[math]\displaystyle{ T_{\mu\nu\alpha}=\frac12(\Gamma_{\mu\nu\alpha} - \Gamma_{\alpha\nu\mu}) }[/math]

is the torsion tensor of [math]\displaystyle{ \Gamma }[/math].

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 [math]\displaystyle{ \Gamma }[/math] 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 [math]\displaystyle{ R }[/math] of [math]\displaystyle{ \Gamma }[/math], is considered.

A linear connection [math]\displaystyle{ \Gamma }[/math] is called the metric connection for a pseudo-Riemannian metric [math]\displaystyle{ g }[/math] if [math]\displaystyle{ g }[/math] is its integral section, i.e., the metricity condition

[math]\displaystyle{ \nabla^\Gamma_\mu g_{\nu\alpha}=0 }[/math]

holds. A metric connection reads

[math]\displaystyle{ \Gamma_{\mu\nu\alpha}=\\{_{\mu\nu\alpha}\\} + \frac12(T_{\nu\mu\alpha} +T_{\nu\alpha\mu} + T_{\mu\nu\alpha}). }[/math]

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

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

## See also

* Gauge gravitation theory
* Einstein–Cartan theory
* Affine gauge theory
* Classical unified field theories

## References

* 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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