Spacetime Thermodynamics with Contorsion

TL;DR

This paper extends spacetime thermodynamics to Einstein-Cartan theory with torsion, deriving Einstein equations from thermodynamics and analyzing black hole laws in this context.

Contribution

It identifies a conserved energy-momentum tensor in Einstein-Cartan theory and generalizes Jacobson's thermodynamic derivation to include torsion.

Findings

Conserved energy-momentum tensor exists in Einstein-Cartan theory.

Jacobson's thermodynamic derivation applies with torsion.

Black hole laws are reviewed in the presence of torsion.

Abstract

We prove that a conserved effective energy-momentum tensor for Einstein-Cartan theory can be identified from the Noether identities of the matter Lagrangian, using the torsion field equations relating them. More precisely, a one-parameter family labelled by the Immirzi parameter. We use this result and the contorsion description to show that Jacobson's thermodynamical derivation of the Einstein equations follows as in the metric theory, namely from the equilibrium Clausius relation and the fact that a Killing horizon is metric-geodetic. Our derivation works for an arbitrary torsion field. In the course of our discussion we review the laws of black hole mechanics and their dependence on torsion.