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

Norichika Sago·Hiroyuki Nakano·Misao Sasaki
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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.

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Norichika Sago, Hiroyuki Nakano, Misao Sasaki. 2002-08-21. Gauge Problem in the Gravitational Self-Force I. Harmonic Gauge Approach in the Schwarzschild Background. https://doi.org/10.1103/physrevd.67.104017

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