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 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 (xμ,x˙μ). A general linear connection on TX is represented by a connection tangent-valued form

Γ=dxλ⊗(∂λ+Γλμνx˙ν∂˙μ).

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,ℝ) . Consequently, it can be treated as a gauge field. A pseudo-Riemannian metric g=gμνdxμ⊗dxν on TX is defined as a global section of the quotient bundle FX/SO(1,3)->X, where SO(1,3) 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 g, any linear connection Γ on TX admits a splitting

Γμνα={μνα}+12Cμνα+Sμνα

in the Christoffel symbols

{μνα}=−12(∂μgνα+∂αgνμ−∂νgμα),

a nonmetricity tensor

Cμνα=Cμαν=∇μΓgνα=∂μgνα+Γμνα+Γμαν

and a contorsion tensor

Sμνα=−Sμαν=12(Tνμα+Tναμ+Tμνα+Cανμ−Cναμ),

where

Tμνα=12(Γμνα−Γανμ)

is the torsion tensor of Γ.

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 Γ 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 R of Γ, is considered.

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

∇μΓgνα=0

holds. A metric connection reads

Γμνα={μνα}+12(Tνμα+Tναμ+Tμνα).

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 FgX of the frame bundle FX corresponding to a section g of the quotient bundle FX/SO(1,3)->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 Γ defines a principal adapted connection Γg on a Lorentz reduced subbundle FgX by its restriction to a Lorentz subalgebra of a Lie algebra of a general linear group GL(4,ℝ). For instance, the Dirac operator in metric-affine gravitation theory in the presence of a general linear connection Γ is well defined, and it depends just of the adapted connection Γg. 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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