### 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 |
---|---|

Article number | 104017 |

Journal | Physical Review D |

Volume | 68 |

Issue number | 10 |

DOIs | |

Publication status | Published - Dec 1 2003 |

Externally published | Yes |

### Fingerprint

### All Science Journal Classification (ASJC) codes

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

### Cite this

*Physical Review D*,

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

**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

*Physical Review D*, vol. 68, no. 10, 104017. https://doi.org/10.1103/PhysRevD.68.104017

}

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 - 2003/12/1

Y1 - 2003/12/1

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 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.

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 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.

UR - http://www.scopus.com/inward/record.url?scp=33750388091&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750388091&partnerID=8YFLogxK

U2 - 10.1103/PhysRevD.68.104017

DO - 10.1103/PhysRevD.68.104017

M3 - Article

AN - SCOPUS:33750388091

VL - 68

JO - Physical Review D - Particles, Fields, Gravitation and Cosmology

JF - Physical Review D - Particles, Fields, Gravitation and Cosmology

SN - 1550-7998

IS - 10

M1 - 104017

ER -