New Gauge Symmetry in Gravity and the Evanescent Role of Torsion

TL;DR

This paper introduces a new gauge symmetry in gravity that allows torsion to be reshuffled into curvature, offering multiple equivalent formulations of Einstein-Hilbert action in Riemann-Cartan spacetime.

Contribution

It reveals a novel gauge symmetry in gravity that enables arbitrary torsion gauge fixing and connects different formulations of Einstein-Hilbert action.

Findings

Torsion can be freely chosen via gauge fixing.

Multiple equivalent representations of gravitational action exist.

Long-distance gravitational effects are unaffected by matter spin configurations.

Abstract

If the Einstein-Hilbert action ${\cal L}_{\rm EH}\propto R$ is re-expressed in Riemann-Cartan spacetime using the gauge fields of translations, the vierbein field $h^\alpha{}_\mu$, and the gauge field of local Lorentz transformations, the spin connection $A_{\mu \alpha}{}^ \beta $, there exists a new gauge symmetry which permits reshuffling the torsion, partially or totally, into the Cartan curvature term of the Einstein tensor, and back, via a {\em new multivalued gauge transformation\/}. Torsion can be chosen at will by an arbitrary gauge fixing functional. There exist many equivalent ways of specifing the theory, for instance Einstein's traditional way where ${\cal L}_{\rm EH}$ is expressed completely in terms of the metric $g_{\mu \nu}=h^ \alpha {}_\mu h_ \alpha {}_ \nu $, and the torsion is zero, or Einstein's teleparallel formulation, where ${\cal L}_{\rm EH}$ is expressed in terms of the torsion tensor, or an infinity of intermediate ways. As far as the gravitational field in the far-zone of a celestial object is concerned, matter composed of spinning particles can be replaced by matter with only orbital angular momentum, without changing the long-distance forces, no matter which of the various new gauge representations is used.