Generalized Vaidya Solutions and Misner-Sharp mass for $n$-dimensional massive gravity

TL;DR

This paper derives generalized Vaidya solutions in n-dimensional dRGT massive gravity, explores their thermodynamics, and introduces a generalized Misner-Sharp mass, showing the thermodynamic equilibrium nature of these solutions.

Contribution

It presents new generalized Vaidya and Vaidya-like solutions in n-dimensional massive gravity and develops a covariant form of the Misner-Sharp mass for these spacetimes.

Findings

Existence of generalized Vaidya solutions in massive gravity.

Validation of the Clausius relation on the apparent horizon.

Derivation of a covariant generalized Misner-Sharp mass.

Abstract

Dynamical solutions are always of interest to people in gravity theories. We derive a series of generalized Vaidya solutions in the $n$-dimensional de Rham-Gabadadze-Tolley (dRGT) massive gravity with a singular reference metric. Similar to the case of the Einstein gravity, the generalized Vaidya solution can describe shining/absorbing stars. Moreover, we also find a more general Vaidya-like solution by introducing a more generic matter field than the pure radiation in the original Vaidya spacetime. As a result, the above generalized Vaidya solution is naturally included in this Vaidya-like solution as a special case. We investigate the thermodynamics for this Vaidya-like spacetime by using the unified first law, and present the generalized Misner-Sharp mass. Our results show that the generalized Minser-Sharp mass does exist in this spacetime. In addition, the usual Clausius relation $\delta Q= TdS$ holds on the apparent horizon, which implicates that the massive gravity is in a thermodynamic equilibrium state. We find that the work density vanishes for the generalized Vaidya solution, while it appears in the more general Vaidya-like solution. Furthermore, the covariant generalized Minser-Sharp mass in the $n$-dimensional de Rham-Gabadadze-Tolley massive gravity is also derived by taking a general metric ansatz into account.