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 $(\alpha, \beta)$ 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 $\alpha=\beta=0$ the solution exhibits the vDVZ discontinuity but ordinary General Relativity is recovered deep inside the horizon due to the existence of electric charge. For $\alpha\neq0$ and $\beta=0$, 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 $\alpha$ and $\beta$ are both nonzero, there exist two branches of solutions depending on the positivity of $\beta$. When $\beta<0$, the strong coupling of the scalar graviton weakens gravity at distances smaller than the Vainshtein radius. However, when $\beta>0$ the metric factors exhibit only small corrections compared to the solutions obtained in General Relativity, and under a particular choice of $\beta=\alpha^2/6$ the standard Reissner-Nordstr\"om-de Sitter solution is recovered.