GRAVITATIONAL-RADIATION REACTION FOR BOUND MOTION AROUND A SCHWARZSCHILD BLACK-HOLE

A particle of mass mu moves, in the absence of external forces, in the geometry of a nonrotating black hole of mass M. The system (black hol e plus particle) emits gravitational waves, and the particle's orbit e volves under radiation reaction. The aim of this paper is to calculate this evolution. Our calculations are carried out under the assumption s that mu/M << 1, that the orbit is bound, and that radiation reaction takes place over a time scale much longer than the orbital period. Th e bound orbits of the Schwarzschild spacetime can be fully characteriz ed, apart from initial conditions, by two orbital parameters: the semi -latus rectum p, and the eccentricity e. These parameters are so defin ed that the turning points of the radial motion (the values of the Sch warzschild radial coordinate at which the radial component of the four -velocity vanishes) are given by r(1) = pM/(1 + e) and r(2) = pM/(1- e ). The units are such that G = c = 1. We use the Teukolsky perturbatio n formalism to calculate the rates at which the gravitational waves ge nerated by the orbiting particle remove energy and angular momentum fr om the system. These are then related to the rates of change of p and e, which determine the orbital evolution. We find that the radiation r eaction continually decreases p, in such a way that the particle event ually plunges inside the black hole. Plunging occurs when p becomes sm aller than 6 + 2e. (Orbits for which p < 6 + 2e do not have a turning point at r = r(1).) For weak-field, slow-motion orbits (which are char acterized by large values of p), the radiation reaction decreases e al so. However, for strong-field, fast-motion orbits (small values of p), the radiation reaction increases the eccentricity if p is sufficientl y close to its minimum value 6 + 2e. The change of sign of de/dt can b e interpreted as a precursor effect to the eventual plunging of the or bit.