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

TL;DR

This paper develops a method to transform metric perturbations from the Regge-Wheeler gauge to the harmonic gauge in Schwarzschild spacetime, addressing the gauge problem in gravitational self-force calculations.

Contribution

It formulates a Fourier-harmonic approach to compute gauge transformations, deriving decoupled radial equations and correcting previous typos in Zerilli's equations.

Findings

Derived simple second-order equations for odd parity gauge transformation

Established spin s=0 and 1 Teukolsky equations for even parity

Provided corrected equations for Zerilli's paper

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.