A Reissner-Nordström black hole (BH) is superradiantly unstable against spherical perturbations of a charged scalar field enclosed in a cavity, with a frequency lower than a critical value. We use numerical relativity techniques to follow the development of this unstable system—dubbed a charged BH bomb—into the nonlinear regime, solving the full Einstein-Maxwell-Klein-Gordon equations, in spherical symmetry. We show that (i) the process stops before all the charge is extracted from the BH, and (ii) the system settles down into a hairy BH: a charged horizon in equilibrium with a scalar field condensate, whose phase is oscillating at the (final) critical frequency. For a low scalar field charge $q$, the final state is approached smoothly and monotonically. For large $q$, however, the energy extraction overshoots, and an explosive phenomenon, akin to a bosenova, pushes some energy back into the BH. The charge extraction, by contrast, does not reverse.

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 $e^{-i\omega t}$, the growth time scale of these unstable modes is set by the inverse of the (positive) imaginary part of the frequency, $\mathrm{Im}(\omega )$, which reaches a maximum value of the order of $\mathrm{Im}(\omega )M\sim {10}^{-5}$, 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-Nordström 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 $\mathrm{Im}(\omega )M\sim 0.07$. 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.

We study the evolution of a massive scalar field surrounding a Schwarzschild black hole and find configurations that can survive for arbitrarily long times, provided the black hole or the scalar field mass is small enough. In particular, both ultralight scalar field dark matter around supermassive black holes and axionlike scalar fields around primordial black holes can survive for cosmological times. Moreover, these results are quite generic in the sense that fairly arbitrary initial data evolve, at late times, as a combination of those long-lived configurations.

Classical scalar fields have been proposed as possible candidates for the dark matter component of the universe. Given the fact that supermassive black holes seem to exist at the center of most galaxies, in order to be a viable candidate for the dark matter halo a scalar field configuration should be stable in the presence of a central black hole, or at least be able to survive for cosmological time scales. In the present work we consider a scalar field as a test field on a Schwarzschild background, and study under which conditions one can obtain long-lived configurations. We present a detailed study of the Klein-Gordon equation in the Schwarzschild space-time, both from an analytical and numerical point of view, and show that indeed there exist quasistationary solutions that can remain surrounding a black hole for large time scales.

Frequency domain studies have recently demonstrated that charged scalar fields exhibit fast growing superradiant instabilities when interacting with charged black holes in a cavity. Here, we present a time domain analysis of the long time evolution of test charged scalar field configurations on the Reissner-Nordström background, with or without a mirrorlike boundary condition. Initial data are taken to be either a Gaussian wave packet or a regularized (near the horizon) quasibound state. Then, Fourier transforming the data obtained in the evolution confirms the results obtained in the frequency domain analysis, in particular for the fast growing modes. We show that spherically symmetric ($\ell =0$) modes have even faster growth rates than the $\ell =1$ modes for a “small” field charge. Thus, our study confirms that this setup is particularly promising for considering the nonlinear development of the superradiant instability, since the fast growth makes the signal overcome the numerical error that dominates for small growth rates, and the analysis may be completely done in spherical symmetry.