On the theories of the interacting perturbations of the Reissner-Nordström black hole

Abstract

A complete account of the Hamiltonian approach to the coupled perturbations of the Reissner-Nordström black hole, initiated by Moncrief, is given. All Hamiltonian equations are expressed explicitly in suitable forms; the metric and electromagnetic field perturbations are found in terms of Moncrief's gauge invariant canonical variables in the Regge-Wheeler gauge. The basic (both tetrad and coordinate) gauge invariant scalars occurring in the perturbation studies based on the Newman-Penrose formalism are then related to Moncrief's variables. The strikingly simple relations obtained enable us to show that the fundamental pair of decoupled equations, derived recently within the Newman-Penrose formalism by Chandrasekhar, can be cast into gauge invariant form, and that it can be obtained from Moncrief's formalism.

It is demonstrated how the fundamental equations, supplemented by another combination of the Newman — Penrose equations, generalize the Bardeen-Press equations for uncoupled electromagnetic and gravitational perturbations of the Schwarzschild black hole.

The odd and the even parityl=1 perturbations are also considered in detail. In the Appendix the relations to Zerilli's work on coupled perturbations of the Reissner-Nordström black hole are given.