A study of the coupled gravitational and electromagnetic perturbations to the Reissner-Nordström black hole: the scattering matrix, energy conversion, and quasi-normal modes

The reflection and absorption, by the charged spherically symmetric Reissner-Nordström black hole, of an arbitrary superposition of gravitational and electromagnetic waves, with time dependence e iot and analyzed into spherical harmonics of various orders l , are expressed in terms of the complex reflection and transmission amplitudes (for incident waves) by two one-dimensional potential barriers. These amplitudes, expressed in terms of eight quantities (and composing the scattering matrix), are tabulated for various values of o,l (= 2, 3, and 6) and charge of the black hole. By virtue of the coupling of electromagnetic and gravitational perturbations by the charge of the black hole, the energy in an incident wave, which is purely gravitational, is, in part, reflected as electromagnetic waves; and conversely. This transformation of incident gravitational energy into electromagnetic energy (and vice versa) is expressed in terms of a conversion factor C and plotted in a series of graphs as a function of o for various values of l and the charge on the black hole Q * . Finally, the complex frequencies belonging to the quasi-normal modes (i.e., solutions of the underlying wave equations which correspond to purely outgoing waves at infinity and purely ingoing waves at the horizon) are tabulated. It is found that the imaginary part of these frequencies (which determine the damping of arbitrary initial perturbations of the black hole) is very nearly the same for all modes (with different l ’s) and Q * .