Gauge Problem in the Gravitational Self-Force II. First Post Newtonian Force under Regge-Wheeler Gauge

TL;DR

This paper develops a method to compute the gravitational self-force in the Regge-Wheeler gauge for particles orbiting black holes, and applies it to first post-Newtonian order, confirming mass correction results.

Contribution

It introduces a gauge transformation approach to derive the regularized self-force in the Regge-Wheeler gauge, extending previous harmonic gauge methods.

Findings

Successfully derived the self-force in Regge-Wheeler gauge.

Reproduced the correct mass correction at Newtonian order.

Provided a first post-Newtonian order calculation of the self-force.

Abstract

We discuss the gravitational self-force on a particle in a black hole space-time. For a point particle, the full (bare) self-force diverges. It is known that the metric perturbation induced by a particle can be divided into two parts, the direct part (or the S part) and the tail part (or the R part), in the harmonic gauge, and the regularized self-force is derived from the R part which is regular and satisfies the source-free perturbed Einstein equations. In this paper, we consider a gauge transformation from the harmonic gauge to the Regge-Wheeler gauge in which the full metric perturbation can be calculated, and present a method to derive the regularized self-force for a particle in circular orbit around a Schwarzschild black hole in the Regge-Wheeler gauge. As a first application of this method, we then calculate the self-force to first post-Newtonian order. We find the correction to the total mass of the system due to the presence of the particle is correctly reproduced in the force at the Newtonian order.