Quasinormal modes of the electrically charged dilaton black hole

We sketch the results of calculations of the quasinormal frequencies of the electrically charged dilaton black hole. At the earlier phase of evaporation (Q is less than 0.7−0.8M), the dilaton black hole " rings " with the complex frequencies which differ negligibly from those of the Reissner-Nordström black hole. The spectrum of the frequencies weakly depends upon the dilaton coupling. When perturbing a black hole there appear damped oscillations with complex frequencies which are the eigenvalues of the wave equation satisfying the appropriate boundary conditions. Usually these are the requirements of purely outgoing waves near infinity and purely ingoing near the horizon. Both the complex part of the QN frequency (inversely proportional to the damping time) and the real one (representing the actual frequency of the oscillation) are independent of the initial perturbations and thereby characterize a black hole itself. The quasinormal spectrum of the neutron stars and black holes is intensively investigated now, since it is in the suggested range of the gravitational wave detectors (LIGO, VIRGO, GEO600, SPHERE) which are under construction. Frequencies of the quasinormal modes of the electrically charged BH were calculated in several papers long time ago (see [1] and references therein). Yet, on various ground, the main of which are suggestions of supergravity, one ascribes to a black hole a scalar (dilaton) field. The latter changes properties of a black hole, and it seems interesting to find out what will happen to the quasinormal spectrum when adding a dilaton charge to a black hole. Certainly, one should expect that for small charges of the electromagnetic and dilaton fields the spectrum will not differ seemingly from that of the R-N black hole, and, even though the black holes we see today, apparently, do not have large electric charge, the problem is of interest, since in charged environment electromagnetic waves will lead to gravitational ones thereby giving a simple model for studying of the conversing of gravitational energy into electromagnetic one and vice versa. We shall consider theories including coupling gravitational, electromagnetic and scalar fields with the action: