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

${\displaystyle \Gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\Gamma _{\lambda }{}^{\mu }{}_{\nu }{\dot {x}}^{\nu }{\dot {\partial }}_{\mu }).}$

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

${\displaystyle \Gamma _{\mu \nu \alpha }=\{_{\mu \nu \alpha }\}+S_{\mu \nu \alpha }+{\frac {1}{2}}C_{\mu \nu \alpha }}$

in the Christoffel symbols

${\displaystyle \{_{\mu \nu \alpha }\}=-{\frac {1}{2}}(\partial _{\mu }g_{\nu \alpha }+\partial _{\alpha }g_{\nu \mu }-\partial _{\nu }g_{\mu \alpha }),}$

a nonmetricity tensor

${\displaystyle C_{\mu \nu \alpha }=C_{\mu \alpha \nu }=\nabla _{\mu }^{\Gamma }g_{\nu \alpha }=\partial _{\mu }g_{\nu \alpha }+\Gamma _{\mu \nu \alpha }+\Gamma _{\mu \alpha \nu }}$

and a contorsion tensor

${\displaystyle S_{\mu \nu \alpha }=-S_{\mu \alpha \nu }={\frac {1}{2}}(T_{\nu \mu \alpha }+T_{\nu \alpha \mu }+T_{\mu \nu \alpha }+C_{\alpha \nu \mu }-C_{\nu \alpha \mu }),}$

where

${\displaystyle T_{\mu \nu \alpha }={\frac {1}{2}}(\Gamma _{\mu \nu \alpha }-\Gamma _{\alpha \nu \mu })}$

is the torsion tensor of ${\displaystyle \Gamma }$.

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 contain different terms expressed both in a curvature of a connection ${\displaystyle \Gamma }$ 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 ${\displaystyle R}$ of ${\displaystyle \Gamma }$, is considered.

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

${\displaystyle \nabla _{\mu }^{\Gamma }g_{\nu \alpha }=0}$

holds. A metric connection reads

${\displaystyle \Gamma _{\mu \nu \alpha }=\{_{\mu \nu \alpha }\}+{\frac {1}{2}}(T_{\nu \mu \alpha }+T_{\nu \alpha \mu }+T_{\mu \nu \alpha }).}$

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

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

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