Gravitational Self-force in a Radiation Gauge

TL;DR

This paper introduces a method to compute the gravitational self-force in a radiation gauge for particles in Schwarzschild or Kerr spacetimes, extending previous results and addressing gauge regularity issues.

Contribution

It develops a new approach to determine the metric perturbation and self-force in a radiation gauge, overcoming regularity challenges and justifying the method through gauge comparisons.

Findings

The mode-sum method can renormalize the self-force in the radiation gauge.

The computed self-force yields correct equations of motion when compared to Lorenz gauge.

The approach addresses gauge regularity issues and validates the self-force calculation.

Abstract

In this, the first of two companion papers, we present a method for finding the gravitational self-force in a modified radiation gauge for a particle moving on a geodesic in a Schwarzschild or Kerr spacetime. An extension of an earlier result by Wald is used to show the spin-weight $\pm 2$ perturbed Weyl scalar ($\psi_0$ or $\psi_4$) determines the metric perturbation outside the particle up to a gauge transformation and an infinitesimal change in mass and angular momentum. A Hertz potential is used to construct the part of the retarded metric perturbation that involves no change in mass or angular momentum from $\psi_0$ in a radiation gauge. The metric perturbation is completed by adding changes in the mass and angular momentum of the background spacetime outside the radial coordinate $r_0$ of the particle in any convenient gauge. The resulting metric perturbation is singular on the trajectory of the particle and discontinuous across the sphere $r=r_0$. A mode-sum method can be used to renormalize the self-force, but the justification given in the published version of this paper \cite{sf2} referred to work by Sam Gralla \cite{gralla10} to justify the use of the renormalized self-force, and the radiation gauge we use does not satisfy the regularity conditions required by Gralla. Instead we show that the renormalized self-force, computed either from the retarded field for $r>r_0$ or for $r<r_0$ gives the correct equations of motion in a gauge smoothly related to a Lorenz gauge; and Pound et al. \cite{pmb13} argue that the average of the self-force obtained in the way described in our paper for $r>r_0$ and for $r<r_0$ gives the correct equation of motion for our gauge (what Pound et al. call the no-string gauge).