On black holes in massive gravity

Abstract

In massive gravity, the black hole solutions found so far on Minkowski space happen to convert horizons into a certain type of singularities. Here, we explore whether these singularities can be avoided if space-time is not asymptotically Minkowskian. We find an exact analytic black hole (BH) solution, which evades the above problem by a transition at large scales to self-induced de Sitter space-time, with the curvature scale set by the graviton mass. This solution is similar to the ones discovered by Koyama, Niz, and Tasinato, and by Nieuwenhuizen, but differs in detail. The solution demonstrates that in massive general relativity, in the Schwarzschild coordinate system, a BH metric has to be accompanied by the St\"uckelberg fields with nontrivial backgrounds to prevent the horizons to convert into the singularities. We also find an analogous solution for a Reissner-Nordstr\"om BH on de Sitter space. A limitation of our approach is that we find the solutions only for specific values of the two free parameters of the theory, for which both the vector and scalar fluctuations lose their kinetic terms; however, we hope our solutions represent a broader class with better-behaved perturbations.