Stability of black holes in Einstein-charged scalar field theory in a cavity

Can a black hole that suffers a superradiant instability evolve towards a “hairy” configuration which is stable? We address this question in the context of Einstein-charged scalar field theory. First, we describe a family of static black hole solutions which possess charged scalar-field hair confined within a mirror-like boundary. Next, we derive a set of equations which govern the linear, spherically symmetric perturbations of these hairy solutions. We present numerical evidence which suggests that, unlike the vacuum solutions, the (single-node) hairy solutions are stable under linear perturbations. Thus, it is plausible that stable hairy black holes represent the end point of the superradiant instability of electrically charged Reissner-Nordström black holes in a cavity; we outline ways to explore this hypothesis.