Conservation of energy-momentum of matter as the basis for the gauge theory of gravitation

TL;DR

This paper explores the gauge theory of gravity based on energy-momentum conservation, extending from translation invariance to the full Poincaré group, and reviews the Einstein-Cartan theory and its potential developments.

Contribution

It extends the gauge approach to gravity by incorporating the full Poincaré group, including angular momentum, building on the Einstein-Cartan framework.

Findings

The theory is equivalent to general relativity with torsion.

Gauging the full Poincaré group leads to a richer gravitational theory.

Potential for further theoretical developments in gravitational gauge theories.

Abstract

According to Yang \& Mills (1954), a {\it conserved} current and a related rigid (`global') symmetry lie at the foundations of gauge theory. When the rigid symmetry is extended to a {\it local} one, a so-called gauge symmetry, a new interaction emerges as gauge potential $A$; its field strength is $F\sim {\rm curl} A$. In gravity, the conservation of the energy-momentum current of matter and the rigid translation symmetry in the Minkowski space of special relativity lie at the foundations of a gravitational gauge theory. If the translation invariance is made local, a gravitational potential $\vartheta$ arises together with its field strength $T\sim {\rm curl}\,\vartheta$. Thereby the Minkowski space deforms into a Weitzenb\"ock space with nonvanishing torsion $T$ but vanishing curvature. The corresponding theory is reviewed and its equivalence to general relativity pointed out. Since translations form a subgroup of the Poincar\'e group, the group of motion of special relativity, one ought to straightforwardly extend the gauging of the translations to the gauging of full Poincar\'e group thereby also including the conservation law of the {\it angular momentum} current. The emerging Poincar\'e gauge (theory of) gravity, starting from the viable Einstein-Cartan theory of 1961, will be shortly reviewed and its prospects for further developments assessed.