M ay 2 00 2 On quasinormal modes of small Schwarzschild - Anti - de - Sitter black hole

We compute the quasinormal modes associated with decay of the massless scalar filed around a small Schwarzschild-Anti-de-Sitter black hole. The computations shows that when the horizon radius is much less than the anti-de-Sitter radius, the imaginary part of the frequency goes to zero as r d−2 + while the real part of ω decreases to its minimum and then goes to d − 1. Thus the quasinormal modes approach the usual AdS modes in the limit r + → 0. This agrees with suggestions of Horowitz et al (Phys.Rev. D62 024027 (2000)). Original interest to quasinormal modes of black holes arose since they are the characteristics of black holes which do not depend on initial perturbations and are functions of a black hole parameters only. At present, this interest is renewed since the QN frequencies are in the suggested region of the gravitational wave detectors which are under construction. In general QNM's are important in black holes dynamics and appear in such processes as collisions of two black holes, decay of different fields in a BH background. All this motivated the investigation of the QNM of black holes in asymptotically flat space-time (see [1] for a recent review). The quasinormal modes of asymptotically de-Sitter black holes were studied in [2],[3]. Recently an unexpected application of quasi-normal modes have appeared due to the AdS/CFT correspondence [4]: it proved out that a large black hole in AdS space corresponds to an approximately thermal state in the CFT, and, thereby, perturbation of the black hole corresponds to the perturbation of the above thermal state, while the decay of the perturbation can be associated with the return to thermal equilibrium. Thus the QN frequencies give us the thermalization timescale which is very hard to compute directly. Quasinormal modes for different types of perturbations of black holes in AdS space have been studied recently in a lot of papers [5]-[14]. Black holes are considered to be small (large), when its horizon radius is much smaller (larger) then the anti-de-Sitter radius. When computing the QN frequencies associated with the decay of massless scalar field in the background of small SAdS BH a striking conjecture with the black hole critical phenomena was found: ω Im is proportional to BH radius r + to high accuracy, and the slope of the line ω Im to the r + axis, 2.66, turned out to be very close to the …