Decay of a charged scalar field around a black hole: Quasinormal modes of RN, RNAdS, and dilaton black holes

It is well known that the charged scalar perturbations of the Reissner-Nordstr\"om metric will decay slower at very late times than the neutral ones, thereby dominating in the late time signal. We show that, at the stage of quasinormal (QN) ringing, on the contrary, the neutral perturbations will decay slower for RN, RNAdS and dilaton black holes. The QN frequencies of the nearly extreme RN black hole have the same imaginary parts (damping times) for charged and neutral perturbations. An explanation of this fact is not clear but, possibly, is connected with the Choptuik scaling.