Einstein-Cartan formulation of Chern-Simons Lorentz-violating Gravity

TL;DR

This paper explores a modified Einstein-Cartan gravity theory incorporating a Lorentz-violating topological term, revealing conditions for solutions and their relation to Chern-Simons gravities, with implications for Lorentz symmetry breaking.

Contribution

It introduces a first-order Einstein-Cartan formulation with a Lorentz-violating term, connecting solutions to 2+1-dimensional Chern-Simons gravities and analyzing their relation to standard approaches.

Findings

Solutions of Einstein equations persist in the deformed theory.

Classical solutions correspond to 2+1-dimensional Chern-Simons gravities.

Standard and modified theories coincide at leading order.

Abstract

We consider a modification of the standard Einstein theory in four dimensions, alternative to R. Jackiw and S.-Y. Pi, Phys. Rev. D 68, 104012 (2003), since it is based on the first-order (Einstein-Cartan) approach to General Relativity, whose gauge structure is manifest. This is done by introducing an additional topological term in the action which becomes a Lorentz-violating term by virtue of the dependence of the coupling on the space-time point. We obtain a condition on the solutions of the Einstein equations, such that they persist in the deformed theory, and show that the solutions remarkably correspond to the classical solutions of a collection of independent 2+1-d (topological) Chern-Simons gravities. Finally, we study the relation with the standard second-order approach and argue that they both coincide to leading order in the modulus of the Lorentz-violating vector field.