Time evolution of superradiant instabilities for charged black holes in a cavity

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-Nordstr\"om background, with or without a mirror-like boundary condition. Initial data is taken to be either a Gaussian wave packet or a regularised (near the horizon) quasi-bound 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 (l=0) modes have even faster growth rates than the l=1 modes for `small' field charge. Thus, our study confirms that this setup is particularly promising for considering the non-linear 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.