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 $(x^{\mu },{\dot {x}}^{\mu })$. 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, 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 $\Gamma$ on $TX$ admits a splitting

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

in the Christoffel symbols

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

a nonmetricity tensor

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

and a contorsion tensor

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

where

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

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 contain different terms expressed both in a curvature of a connection $\Gamma$ 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 $R$ 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

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

holds. A metric connection reads

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

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 $F^{g}X$ 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 $F^{g}X$ 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.

See also

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

References

• Hehl, F.; McCrea, J.; Ne'eman, Y. (1995). "Metric-affine gauge theory of gravity: field equations". Physics Reports. 258 (1–2): 1–171. arXiv:gr-qc/9402012. doi:10.1016/0370-1573(94)00111-F. ISSN 0370-1573.
• Vitagliano, V.; Sotiriou, T.; Liberati, S. (2011). "The dynamics of metric-affine gravity". Annals of Physics. 326 (5): 1259–1273. arXiv:1008.0171. doi:10.1016/j.aop.2011.02.008.
• 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