Particles and black holes: time-domain integration of the equations of black-hole perturbation theory
Binary systems consisting of a solar mass compact object orbiting a supermassive black hole are a promising source of gravitational waves for space-based laser interferometers. Because of the small mass ratio involved, the system is amenable to a treatment using black hole perturbation theory, We present a covariant and gauge invariant formalism for the metric perturbations of a Schwarzschild black hole that accounts for the radiation emitted by a small orbiting object, The perturbations are simply described in terms of the Zerilli-Moncrief and Regge-Wheeler functions, and these obey simple inhomogeneous one-dimensional wave equations. The partial differential equations governing the evolution of these two functions are integrated numerically in the time domain using a corrected Lousto-Price algorithm. In this manner we obtain the gravitational waveforms associated with the motion of the small compact object, which is assumed to follow a geodesic of the Schwarzschild spacetime. We present a method for obtaining, from the gravitational waveforms, the fluxes of energy and angular momentum at infinity and through the event horizon. Astrophysical black holes, such as the ones residing at the centre of many galaxies, are likely to be rapidly rotating. To deal with these situations, we present a time-domain method of integration of the Teukolsky equation governing the evolution of the curvature perturbations of the Kerr black hole. We show that our method is both stable and quadratically convergent, and that it reproduces known predictions of the Teukolsky equation. We also comment on the difficulty of incorporating orbiting particles in the method.