Misner-Sharp Mass and the Unified First Law in Massive Gravity

TL;DR

This paper derives the Misner-Sharp mass in four-dimensional massive gravity using the inverse unified first law method, confirming its conservation and thermodynamic consistency in various spacetime settings, including FRW universes.

Contribution

It introduces a method to obtain the Misner-Sharp mass in massive gravity and demonstrates its conservation and thermodynamic properties without additional constraints.

Findings

Misner-Sharp mass is conserved in massive gravity with a common reference metric.

Massive gravity can achieve thermodynamic equilibrium in FRW spacetime.

No extra constraints are imposed by the existence of the Misner-Sharp mass in this context.

Abstract

We obtain the Misner-Sharp mass in the massive gravity for a four dimensional spacetime with a two dimensional maximally symmetric subspace via the inverse unified first law method. Significantly, the stress energy is conserved in this case with a widely used reference metric. Based on this property we confirm the derived Misner-Sharp mass by the conserved charge method. We find that the existence of the Misner-sharp mass in this case does not lead to extra constraint for the massive gravity, which is notable in modified gravities. In addition, as a special case, we also investigate the Misner-Sharp mass in the static spacetime. Especially, we take the FRW universe into account for investigating the thermodynamics of the massive gravity. The result shows that the massive gravity can be in thermodynamic equilibrium, which fills in the gap in the previous studies of thermodynamics in the massive gravity.