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

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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 ϕᵃ, there is new invariant Iab=gμν∂μϕᵃ∂νϕᵇ in the massive gravity, which adds to the ones usually encountered in general relativity (GR). In the unitary gauge ϕᵃ=xμδμᵃ, any inverse metric gμν that has divergence including the coordinate singularity in GR would exhibit a singularity in the invariant Iab. 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 Iab is absent at the horizon. Therefore, these solutions may become candidates for black holes in dRGT.