### 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 [Formula Presented] 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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Journal | Physical Review D - Particles, Fields, Gravitation and Cosmology |

Volume | 67 |

Issue number | 10 |

DOIs | |

Publication status | Published - Jan 1 2003 |

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### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics
- Physics and Astronomy (miscellaneous)

### Cite this

**Gauge problem in the gravitational self-force : Harmonic gauge approach in the Schwarzschild background.** / Sago, Norichika; Nakano, Hiroyuki; Sasaki, Misao.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Gauge problem in the gravitational self-force

T2 - Harmonic gauge approach in the Schwarzschild background

AU - Sago, Norichika

AU - Nakano, Hiroyuki

AU - Sasaki, Misao

PY - 2003/1/1

Y1 - 2003/1/1

N2 - 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 [Formula Presented] 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.

AB - 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 [Formula Presented] 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.

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U2 - 10.1103/PhysRevD.67.104017

DO - 10.1103/PhysRevD.67.104017

M3 - Article

AN - SCOPUS:0037912176

VL - 67

JO - Physical review D: Particles and fields

JF - Physical review D: Particles and fields

SN - 1550-7998

IS - 10

ER -