Gravitational radiation reaction for bound motion around a Schwarzschild black hole

Abstract

A particle of mass μ moves, in the absence of external forces, in the geometry of a nonrotating black hole of mass M. The system (black hole plus particle) emits gravitational waves, and the particle’s orbit evolves under radiation reaction. The aim of this paper is to calculate this evolution. Our calculations are carried out under the assumptions that μ/M≪1, that the orbit is bound, and that radiation reaction takes place over a time scale much longer than the orbital period. The bound orbits of the Schwarzschild spacetime can be fully characterized, apart from initial conditions, by two orbital parameters: the semi-latus rectum p, and the eccentricity e. These parameters are so defined that the turning points of the radial motion (the values of the Schwarzschild radial coordinate at which the radial component of the four-velocity vanishes) are given by ${\mathit{r}}_1$=pM/(1+e) and ${\mathit{r}}_2$=pM/(1-e). The units are such that G=c=1. We use the Teukolsky perturbation formalism to calculate the rates at which the gravitational waves generated by the orbiting particle remove energy and angular momentum from 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 reaction continually decreases p, in such a way that the particle eventually plunges inside the black hole. Plunging occurs when p becomes smaller than 6+2e. (Orbits for which pe do not have a turning point at r=${\mathit{r}}_1$). For weak-field, slow-motion orbits (which are characterized by large values of p), the radiation reaction decreases e also. However, for strong-field fast-motion orbits (small values of p), the radiation reaction increases the eccentricity if p is sufficiently close to its minimum value 6+2e. The change of sign of de/dt can be interpreted as a precursor effect to the eventual plunging of the orbit.