Gravitational Self-force in a Radiation Gauge

TL;DR

This work develops a practical framework to compute the gravitational self-force using a modified radiation gauge in Schwarzschild and Kerr spacetimes. By leveraging the NP Teukolsky equation and the Chrzanowski-Cohen-Kegeles Hertz-potential reconstruction, the authors express the metric perturbation outside the particle in terms of source-free Weyl scalars $\psi_0$ or $\psi_4$ and then restore the mass/angular-momentum changes to complete the perturbation. They formulate both analytic and numerical mode-sum renormalization strategies to extract a finite, physically meaningful self-force, and demonstrate parity properties and gauge relations that ensure the resulting motion matches the appropriate gauge (Lorenz-related) equations of motion. A detailed Schwarzschild circular-orbit case illustrates analytic and numerical consistency, validating the method and paving the way for Kerr applications. The approach promises efficient self-force calculations in a radiative gauge, with implications for accurate gravitational wave templates for EMRIs.

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 ($ψ_0$ or $ψ_4$) 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 $ψ_0$ 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).