### 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 Schwarzschild background. 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 typographical errors in Zerilli's paper and present a set of corrected equations in Appendix A.

Original language | English |
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Article number | 104017 |

Journal | Physical Review D |

Volume | 68 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2003 |

### All Science Journal Classification (ASJC) codes

- Physics and Astronomy (miscellaneous)

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## Cite this

*Physical Review D*,

*68*(10), [104017]. https://doi.org/10.1103/PhysRevD.68.104017