Charged black holes in nonlinear massive gravity

Abstract

We find static charged black hole solutions in nonlinear massive gravity. In the parameter space of two gravitational potential parameters $(\ensuremath{\alpha},\ensuremath{\beta})$ 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 $\ensuremath{\alpha}=\ensuremath{\beta}=0$ 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 $\ensuremath{\alpha}\ensuremath{\ne}0$ and $\ensuremath{\beta}=0$, 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 $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ are both nonzero, there exist two branches of solutions depending on the positivity of $\ensuremath{\beta}$. When $\ensuremath{\beta}<0$, the strong coupling of the scalar graviton weakens gravity at distances smaller than the Vainshtein radius. However, when $\ensuremath{\beta}>0$ the metric factors exhibit only small corrections compared to the solutions obtained in general relativity, and under a particular choice of $\ensuremath{\beta}={\ensuremath{\alpha}}^{2}/6$ the standard Reissner-Nordstr\"om-de Sitter solution is recovered.