Jinmao Tower and Shanghai World Financial Center, Shanghai, China

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 Stückelberg fields φ a , there is new invariant I ab = g µν ∂µφ a ∂νφ b in the massive gravity, which adds to the ones usually encountered in general relativity (GR). In the unitary gauge φ a = x µ δ a µ , any inverse metric g µν 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-Nordström solution and five other solutions. These solutions may possess an event horizon depending upon the physical parameters (Schwarzschild radius rs, 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.

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