In a recent Letter [Sanchis-Gual et al., Phys. Rev. Lett. 116, 141101 (2016)], we presented numerical relativity simulations, solving the full Einstein–Maxwell–Klein-Gordon equations, of superradiantly unstable Reissner-Nordström black holes (BHs), enclosed in a cavity. Low frequency, spherical perturbations of a charged scalar field trigger this instability. The system’s evolution was followed into the nonlinear regime, until it relaxed into an equilibrium configuration, found to be a hairy BH: a charged horizon in equilibrium with a scalar field condensate, whose phase is oscillating at the (final) critical frequency. Here, we investigate the impact of adding self-interactions to the scalar field. In particular, we find sufficiently large self-interactions suppress the exponential growth phase, known from linear theory, and promote a nonmonotonic behavior of the scalar field energy. Furthermore, we discuss in detail the influence of the various parameters in this model: the initial BH charge, the initial scalar perturbation, the scalar field charge, the mass, and the position of the cavity’s boundary (mirror). We also investigate the “explosive” nonlinear regime previously reported to be akin to a bosenova. A mode analysis shows that the “explosions” can be interpreted as the decay into the BH of modes that exit the superradiant regime.

• Received 26 July 2016

DOI:https://doi.org/10.1103/PhysRevD.94.044061