Gauge Problem in the Gravitational Self-Force I. Harmonic Gauge Approach in the Schwarzschild Background

TL;DR

This work tackles the gauge problem in the gravitational self-force by linking the Regge-Wheeler gauge perturbations to the harmonic gauge, where the self-force resides in the tail part of the metric perturbation. The authors formulate a direct approach that expands the metric perturbation in Fourier-harmonics and reduces the problem to solving gauge transformation equations for the radial functions that connect RW to harmonic gauge. They derive decoupled radial equations: a simple second-order equation for the odd-parity sector, a spin-0 Teukolsky equation for the even-scalar part, and a spin-1 Teukolsky system for the even-vector part, which can be solved via Green functions and the Mano-Suzuki-Takasugi method. This framework provides a principled route to obtain harmonic-gauge metric perturbations and hence the self-force; it also outlines connections to Detweiler-Whiting S/R decomposition and discusses challenges for extending to Kerr, including practical difficulties from double Green-function integrals and gauge choices. The appendix corrects typos in Zerilli’s RWZ formalism, illustrating the care needed in implementing the RWZ equations within this gauge-problem context.

Abstract

The metric perturbation induced by a particle in the Schwarzschild background is usually calculated in the Regge-Wheeler (RW) gauge, whereas the gravitational self-force is known to be given by the tail part of the metric perturbation in the harmonic gauge. Thus, to identify the gravitational self-force correctly in a specified gauge, it is necessary to find out a gauge transformation that connects these two gauges. This is called the gauge problem. As a direct approach to solve the gauge problem, we formulate a method to calculate the metric perturbation in the harmonic gauge on the Schwarzshild backgound. We apply the Fourier-harmonic expansion to the metric perturbation and reduce the problem to the gauge transformation of the Fourier-harmonic coefficients (radial functions) from the RW gauge to the harmonic gauge. We derive a set of decoupled radial equations for the gauge transformation. These equations are found to have a simple second-order form for the odd parity part and the forms of spin $s=0$ and 1 Teukolsky equations for the even parity part. As a by-product, we correct typos in Zerilli's paper and present a set of corrected equations in Appendix.