Black hole solutions in de Rham-Gabadadze-Tolley massive gravity

TL;DR

This paper explores static spherically symmetric solutions in dRGT massive gravity, revealing seven solutions including black hole candidates, and discusses their horizon properties and singularity behavior in different gauges.

Contribution

The study derives a new class of solutions in dRGT gravity using a nonunitary gauge, expanding the known black hole solutions beyond the Schwarzschild and Reissner-Nordström types.

Findings

Seven static spherically symmetric solutions found, including Schwarzschild and Reissner-Nordström.

Some solutions possess event horizons with regular invariants, indicating potential black hole candidates.

Solutions exhibit different singularity behaviors depending on gauge choice and parameters.

Abstract

We present a detailed study of the static spherically symmetric solutions in de Rham-Gabadadze-Tolley (dRGT) theory. Since the diffeomorphism invariance can be restored by introducing the St\"{u}ckelberg fields $\phi^a$, there is new invariant $I^{ab}=g^{\mu\nu}\partial_{\mu}\phi^a\partial_\nu\phi^b$ in the massive gravity, which adds to the ones usually encountered in general relativity (GR). In the unitary gauge $\phi^a=x^\mu\delta_\mu^a$, any inverse metric $g^{\mu\nu}$ that has divergence including the coordinate singularity in GR would exhibit a singularity in the invariant $I^{ab}$. Therefore, there is no conventional Schwarzschild metric if we choose unitary gauge. In this paper, we obtain a self-consistent static spherically symmetric ansatz in the nonunitary gauge. Under this ansatz, we find that there are seven solutions including the Schwarzschild solution, Reissner-Nordstr\"{o}m solution and five other solutions. These solutions may possess an event horizon depending upon the physical parameters (Schwarzschild radius $r_s$, scalar charge $S$ and/or electric charge $Q$). If these solutions possess an event horizon, we show that the singularity of $I^{ab}$ is absent at the horizon. Therefore, these solutions may become candidates for black holes in dRGT.