On Black Holes in Massive Gravity

TL;DR

The paper shows that, in a ghost-free, resummed theory of massive gravity, horizon singularities that plague Schwarzschild-like BHs in unitary gauge can be avoided by allowing de Sitter asymptotics induced by a graviton mass $m$. It constructs exact Schwarzschild–de Sitter and Reissner–Nordström–de Sitter solutions in Gullstrand–Painlevé coordinates, accompanied by nontrivial Stückelberg backgrounds that keep the $I^{ab}$ invariant non-singular for a special parameter choice ($\beta=-\alpha^2/6$). The results demonstrate that BHs in this framework can be non-singular and self-accelerating, though the linear perturbations (vector and scalar modes) may be infinitely strongly coupled in the decoupling limit for these backgrounds, signaling potential stability challenges that require further study. Overall, the work provides concrete, analytic exemplars of non-singular BH configurations in massive gravity and highlights avenues to generalize beyond the specific parameter choices.

Abstract

In massive gravity the so-far-found black hole solutions 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 (dS) 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 GR, in the Schwarzschild coordinate system, a BH metric has to be accompanied by the Stückelberg fields with nontrivial backgrounds to prevent the horizons to convert into the singularities. We also find an analogous solution for a Reissner-Nordström BH on dS 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 loose their kinetic terms, however, we hope our solutions represent a broader class with better behaved perturbations.