C. Cutler et al., GRAVITATIONAL-RADIATION REACTION FOR BOUND MOTION AROUND A SCHWARZSCHILD BLACK-HOLE, Physical review. D. Particles and fields, 50(6), 1994, pp. 3816-3835
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.