Charged black holes in nonlinear massive gravity

We find static charged black hole solutions in nonlinear massive gravity. In the parameter space of two gravitational potential parameters (α,β) we show that below the Compton wavelength the black hole solutions reduce to that of Reissner-Nordstr\"om via the Vainshtein mechanism in the weak-field limit. In the simplest case with α=β=0 the solution exhibits the van Dam-Veltman-Zakharov discontinuity but ordinary general relativity is recovered deep inside the horizon due to the existence of electric charge, though this case is observationally excluded. For α≠0 and β=0, the post-Newtonian parameter of the charged black hole evolves to that of general relativity via the Vainshtein mechanism within a macroscopic distance; however, a logarithmic correction to the metric factor of the time coordinate is obtained. When α and β are both nonzero, there exist two branches of solutions depending on the positivity of β. When β<0, the strong coupling of the scalar graviton weakens gravity at distances smaller than the Vainshtein radius. However, when β>0 the metric factors exhibit only small corrections compared to the solutions obtained in general relativity, and under a particular choice of β=α²/6 the standard Reissner-Nordstr\"om-de Sitter solution is recovered.