On the Gauge Aspects of Gravity

TL;DR

This paper explores the gauge theoretical foundations of gravity, contrasting Einstein-Cartan and Einsteinian approaches, and discusses recent developments in metric-affine geometry and field equations within gauge gravity theories.

Contribution

It provides a comprehensive overview of gauge approaches to gravity, including the derivation of Einstein-Cartan theory and metric-affine frameworks, highlighting their equivalences and recent models.

Findings

Einstein-Cartan theory arises naturally from gauge theory of the Poincaré group.

Teleparallelism is equivalent to general relativity for spinless matter and electromagnetism.

Recent models extend metric-affine gauge frameworks for gravity.

Abstract

We give a short outline, in Sec.\ 2, of the historical development of the gauge idea as applied to internal ($U(1),\, SU(2),\dots$) and external ($R^4,\,SO(1,3),\dots$) symmetries and stress the fundamental importance of the corresponding conserved currents. In Sec.\ 3, experimental results with neutron interferometers in the gravitational field of the earth, as inter- preted by means of the equivalence principle, can be predicted by means of the Dirac equation in an accelerated and rotating reference frame. Using the Dirac equation in such a non-inertial frame, we describe how in a gauge- theoretical approach (see Table 1) the Einstein-Cartan theory, residing in a Riemann-Cartan spacetime encompassing torsion and curvature, arises as the simplest gravitational theory. This is set in contrast to the Einsteinian approach yielding general relativity in a Riemannian spacetime. In Secs.\ 4 and 5 we consider the conserved energy-momentum current of matter and gauge the associated translation subgroup. The Einsteinian teleparallelism theory which emerges is shown to be equivalent, for spinless matter and for electromagnetism, to general relativity. Having successfully gauged the translations, it is straightforward to gauge the four-dimensional affine group $R^4 \semidirect GL(4,R)$ or its Poincar\'e subgroup $R^4\semidirect SO(1,3)$. We briefly report on these results in Sec.\ 6 (metric-affine geometry) and in Sec.\ 7 (metric-affine field equations (\ref{zeroth}, \ref{first}, \ref{second})). Finally, in Sec.\ 8, we collect some models, currently under discussion, which bring life into the metric-affine gauge framework developed.