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:

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

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 $GL(4,ℝ)$ . Consequently, it can be treated as a gauge field. A pseudo-Riemannian metric $g=g_{\mu \nu }d{x}^{\mu }\otimes d{x}^{\nu }$ 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 $\mathrm{\Gamma }$ on $TX$ admits a splitting

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

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

$C_{\mu \nu \alpha }=C_{\mu \alpha \nu }=\underset{\mu }{\overset{\mathrm{\Gamma }}{\mathrm{\nabla }}}{g}_{\nu \alpha }={\partial }_{\mu }{g}_{\nu \alpha }+{\mathrm{\Gamma }}_{\mu \nu \alpha }+{\mathrm{\Gamma }}_{\mu \alpha \nu }$

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

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

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

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 $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$, is considered.

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

$\underset{\mu }{\overset{\mathrm{\Gamma }}{\mathrm{\nabla }}}{g}_{\nu \alpha }=0$

holds. A metric connection reads

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

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

• Template:Cite journal
• Template:Cite journal
• G. Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1869–1895; Template:Arxiv
• C. Karahan, A. Altas, D. Demir, Scalars, vectors and tensors from metric-affine gravity, General Relativity and Gravitation 45 (2013) 319–343; Template:Arxiv