Gravitational radiation from a particle in circular orbit around a black hole. V. Black-hole absorption and tail corrections

Abstract

A particle of mass \ensuremath{\mu} moves on a circular orbit of a nonrotating black hole of mass M. Under the restrictions \ensuremath{\mu}/M\ensuremath{\ll}1 and v\ensuremath{\ll}1, where v is the orbital veloity (in units in which c=1), we consider the gravitational waves emitted by such a binary system. The framework is that of black-hole perturbation theory. We calculate E\ifmmode \dot{}\else \.{}\fi{}, the rate at which the gravitational waves remove energy from the system. The total energy loss is given by E\ifmmode \dot{}\else \.{}\fi{}=E\ifmmode \dot{}\else \.{}\fi{} $^{\mathrm{\ensuremath{\infty}}}$+E\ifmmode \dot{}\else \.{}\fi{} $^{\mathit{H}}$, where E\ifmmode \dot{}\else \.{}\fi{} $^{\mathrm{\ensuremath{\infty}}}$ denotes that part of the gravitational-wave energy which is carried off to infinity, while E\ifmmode \dot{}\else \.{}\fi{} $^{\mathit{H}}$ denotes the part which is absorbed by the black hole. We show that the black-hole absorption is a small effect: E\ifmmode \dot{}\else \.{}\fi{} $^{\mathit{H}}$/E\ifmmode \dot{}\else \.{}\fi{}\ensuremath{\simeq}${\mathit{v}}^{8}$. This is explained by the presence of a potential barrier in the vicinity of the black hole: Most of the waves propagating initially toward the black hole are reflected off the barrier; the black hole is therefore unable to absorb much. The black-hole absorption, and indeed any other effect resulting from imposing ingoing-wave boundary conditions at the event horizon, are sufficiently small to be irrelevant to the construction of matched filters for gravitational-wave measurements. To derive this result we extend the techniques previously developed by Poisson and Sasaki for integrating the Regge-Wheeler equation. The extension consists of an explicit consideration of the horizon boundary conditions, which were largely ignored in the previous work. Finally, we compare the wave generation formalism which derives from perturbation theory to the post-Newtonian formalism of Blanchet and Damour. Among other things we consider the corrections to the asymptotic gravitational-wave field which are due to wave-propagation (tail) effects. The results obtained using perturbation theory are identical to that of post-Newtonian theory.