Gravitational Radiation Reaction to a Particle Motion

TL;DR

The paper tackles gravitational radiation reaction for a small mass moving in a curved spacetime. It develops two complementary derivations: (i) a DeWitt–Brehme–style extension of electromagnetic self-force to gravity, and (ii) a rigorous matched asymptotic expansion with an internal Schwarzschild black hole and external perturbations, showing both yield the same equation of motion. The leading correction, of order $O(Gm)$, is determined entirely by the tail term $h_{(v)\mu\nu}$ of the metric perturbation, and the particle follows a geodesic of the regularized metric $\tilde{g}_{(v)\mu\nu}=g_{\mu\nu}+h_{(v)\mu\nu}$; this provides a principled interpretation of self-interaction in black hole perturbation theory and supports extending the framework to spinning bodies and Kerr backgrounds.

Abstract

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. In contrast to 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. 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.