Rapid growth of superradiant instabilities for charged black holes in a cavity

Confined scalar fields, either by a mass term or by a mirror-like boundary condition, have unstable modes in the background of a Kerr black hole. Assuming a time dependence as $e^{-i\omega t}$, the growth time-scale of these unstable modes is set by the inverse of the (positive) imaginary part of the frequency, Im$(\omega)$, which reaches a maximum value of the order of Im$(\omega)M\sim 10^{-5}$, attained for a mirror-like boundary condition, where $M$ is the black hole mass. In this paper we study the minimally coupled Klein-Gordon equation for a charged scalar field in the background of a Reissner-Nordstr\"om black hole and show that the unstable modes, due to a mirror-like boundary condition, can grow several orders of magnitude faster than in the rotating case: we have obtained modes with up to Im$(\omega)M\sim 0.07$. We provide an understanding, based on an analytic approximation, to why the instability in the charged case has a smaller timescale than in the rotating case. This faster growth, together with the spherical symmetry, makes the charged case a promising model for studies of the fully non-linear development of superradiant instabilities.