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

Scalar fields confined either by a mass term or by a mirrorlike boundary condition have unstable modes in the background of a Kerr black hole. Assuming a time dependence as e^-i, the growth time scale of these unstable modes is set by the inverse of the (positive) imaginary part of the frequency, Im (), which reaches a maximum value of the order of Im () M10^-5, attained for a mirrorlike 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 mirrorlike boundary condition, can grow several orders of magnitude faster than in the rotating case: we have obtained modes with up to Im () M0. 07. We provide an understanding, based on an analytic approximation, of why the instability in the charged case has a shorter time scale 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 nonlinear development of superradiant instabilities.