Gravitational radiation reaction to a particle motion

A small mass particle traveling in a curved spacetime is known to trace a background geodesic in the lowest order approximation with respect to the particle mass. In this paper, we discuss the leading order correction to the equation of motion of the particle, which presumably describes the effect of gravitational radiation reaction. We derive the equation of motion in two different ways. The first one is an extension of the well-known formalism by DeWitt and Brehme developed for deriving the equation of motion of an electrically charged particle. Constructing the conserved rank-two symmetric tensor, and integrating it over the interior of the world tube surrounding the orbit, we derive the equation of motion. Although the calculation in this approach is straightforward, it contains less rigorous points. In contrast with the electromagnetic case, in which there are two different charges, i.e., the electric charge and the mass, the gravitational counterpart has only one charge. This fact prevents us from using the same renormalization scheme that was used in the electromagnetic case. In order to overcome this difficulty, we put an ansatz in evaluating the integral of the conserved tensor on a three spatial volume which defines the momentum of the small particle. To make clear the subtlety in the first approach, we then consider the asymptotic matching of two different schemes: i.e., the internal scheme in which the small particle is represented by a spherically symmetric black hole with tidal perturbations and the external scheme in which the metric is given by small perturbations on the given background geometry. The equation of motion is obtained from the consistency condition of the matching. We find that in both ways the same equation of motion is obtained. The resulting equation of motion is analogous to that derived in the electromagnetic case. We discuss implications of this equation of motion.