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

TL;DR

This paper tackles the gauge problem in the gravitational self-force by formulating a Regge–Wheeler gauge (RW) approach that regularizes the force through a mode-decomposed subtraction of the Detweiler–Whiting S part. By combining RWZ formalism for the full metric perturbation with a local Hadamard-style S-part expansion and a gauge transformation from the harmonic gauge, the authors compute the regularized self-force for a particle in a circular Schwarzschild orbit to first post-Newtonian (1PN) order. They find a finite radial self-force F^r_RW = 2μ^2/r_0^2 − 11μ^2M/r_0^3, recovering the Newtonian mass-correction term and confirming the absence of radiation-reaction effects at this order, while acknowledging a gauge ambiguity in the ℓ=1 even mode that requires further resolution. The work lays groundwork for general orbits and higher PN orders, and points toward extensions to Kerr via alternative formulations such as the radiative Green function or Weyl-scalar regularization.

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.