ar X iv : g r - qc / 0 21 01 05 v 2 3 1 O ct 2 00 2 Massive charged scalar field in a Reissner - Nordstrom black hole background : quasinormal ringing

We compute characteristic (quasinormal) frequencies corresponding to decay of a massive charged scalar field in a Reissner-Nordstrom black hole background. It proves that, contrary to the behavior at very late times, at the stage of quasinormal ringing the neutral perturbations will damp slower than the charged ones. In the limit of the extremal black hole the damping rate of charged and neutral perturbations coincides. Possible connection of this with the critical collapse in a massive scalar electrodynamics is discussed. Black hole (BH) quasinormal modes (QNM's) govern the decay of perturbations at intermediate times and are important when studying dynamics of black holes and external fields around them. Now, the interest in quasinormal modes is stipulated by three their features. The first is connected with possibility of observing of QNM's, and of obtaining the " footprint " of a black hole, with the help of a new generation of gravitational antennae which are under construction (see [1] for a review). The second concerns the Anti-de-Sitter/Conformal Field Theory (AdS/CFT) correspondence [2], from which it follows that a large black hole in AdS space corresponds to an approximately thermal states in the CFT. Thus the decay of the BH perturbations can be associated with the return to thermal equilibrium of the perturbed state in CFT [3]. Note that in AdS space, quasinormal modes of black holes govern the decay of a field at late times as well, since there is no power-low tails and the decay is always exponential (see [3] and references therein). The third feature is related to possible connection of BH QNM's in some space-time geometries with the Choptuik scaling [3], [4], [5], [6]. When Horowitz and Hubeny calculated massless scalar quasinormal modes for a four-dimensional SAdS black hole they found that for intermediate black holes, i.e. when the horizon radius is of order the Anti-de Sitter radius (see also [7]), the striking approximate relation takes place ω Im ∼ 1 γ r + , (1) where r + is the horizon radius, γ is the Choptuik scaling parameter for a black hole being formed from a collapse of a scalar field in free space. Then, exact connection between the Choptuik scaling parameter and the imaginary part of the QNF's was obtained for the