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

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 ${\varphi }^a$, there is new invariant $I^{ab}=g^{\mu \nu }{\mathrm{\partial }}_{\mu }{\varphi }^a{\mathrm{\partial }}_{\nu }{\varphi }^b$ in the massive gravity, which adds to the ones usually encountered in general relativity (GR). In the unitary gauge ${\varphi }^a=x^{\mu }{\delta }_{\mu }^a$, 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.