We obtain the Misner-Sharp mass in 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 general 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. We make a concise discussion of the stability problem in frame of the massive gravity in consideration. 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.

• Received 30 March 2015

DOI:https://doi.org/10.1103/PhysRevD.92.024006