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-Nordstr\"om black holes in a cavity; we outline ways to explore this hypothesis.
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Dolan et al. (2015) studied this question.
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