### Abstract

It is well known that when an external general relativistic (electric-type) tidal field ε_{jk}(t) interacts with the evolving quadrupole moment ι_{jk}(t) of an isolated body the tidal field does work on the body ("tidal work") - i.e., it transfers energy to the body - at a rate given by the same formula as in Newtonian theory: dW/dt= - 1/2ε_{jk}dι_{jk}/dt. Thorne has posed the following question: In view of the fact that the gravitational interaction energy E_{int} between the tidal field and the body is ambiguous by an amount ∼ει_{jk}, is the tidal work also ambiguous by this amount, and therefore is the formula dW/dt=-1/2ε_{jk}dι_{jk}/dt only valid unambiguously when integrated over time scales long compared to that for ι_{jk} to change substantially? This paper completes a demonstration that the answer is no; dW/dt is not ambiguous in this way. More specifically, this paper shows that dW/dt is unambiguously given by -1/2ε_{jk}dι_{jk}/dt independently of one's choice of how to localize gravitational energy in general relativity. This is proved by explicitly computing dW/dt using various gravitational stress-energy pseudotensors (Einstein, Landau-Lifshitz, Møller) as well as Bergmann's conserved quantities which generalize many of the pseudotensors to include an arbitrary function of position. A discussion is also given of the problem of formulating conservation laws in general relativity and the role played by the various pseudotensors.

Original language | English |
---|---|

Article number | 064013 |

Pages (from-to) | 640131-6401314 |

Number of pages | 5761184 |

Journal | Physical Review D - Particles, Fields, Gravitation and Cosmology |

Volume | 63 |

Issue number | 6 |

State | Published - 20 Aug 2001 |

### Fingerprint

### Cite this

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

*63*(6), 640131-6401314. [064013].

}

*Physical Review D - Particles, Fields, Gravitation and Cosmology*, vol. 63, no. 6, 064013, pp. 640131-6401314.

**Energy localization invariance of tidal work in general relativity.** / Favata, Marc.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Energy localization invariance of tidal work in general relativity

AU - Favata, Marc

PY - 2001/8/20

Y1 - 2001/8/20

N2 - It is well known that when an external general relativistic (electric-type) tidal field εjk(t) interacts with the evolving quadrupole moment ιjk(t) of an isolated body the tidal field does work on the body ("tidal work") - i.e., it transfers energy to the body - at a rate given by the same formula as in Newtonian theory: dW/dt= - 1/2εjkdιjk/dt. Thorne has posed the following question: In view of the fact that the gravitational interaction energy Eint between the tidal field and the body is ambiguous by an amount ∼ειjk, is the tidal work also ambiguous by this amount, and therefore is the formula dW/dt=-1/2εjkdιjk/dt only valid unambiguously when integrated over time scales long compared to that for ιjk to change substantially? This paper completes a demonstration that the answer is no; dW/dt is not ambiguous in this way. More specifically, this paper shows that dW/dt is unambiguously given by -1/2εjkdιjk/dt independently of one's choice of how to localize gravitational energy in general relativity. This is proved by explicitly computing dW/dt using various gravitational stress-energy pseudotensors (Einstein, Landau-Lifshitz, Møller) as well as Bergmann's conserved quantities which generalize many of the pseudotensors to include an arbitrary function of position. A discussion is also given of the problem of formulating conservation laws in general relativity and the role played by the various pseudotensors.

AB - It is well known that when an external general relativistic (electric-type) tidal field εjk(t) interacts with the evolving quadrupole moment ιjk(t) of an isolated body the tidal field does work on the body ("tidal work") - i.e., it transfers energy to the body - at a rate given by the same formula as in Newtonian theory: dW/dt= - 1/2εjkdιjk/dt. Thorne has posed the following question: In view of the fact that the gravitational interaction energy Eint between the tidal field and the body is ambiguous by an amount ∼ειjk, is the tidal work also ambiguous by this amount, and therefore is the formula dW/dt=-1/2εjkdιjk/dt only valid unambiguously when integrated over time scales long compared to that for ιjk to change substantially? This paper completes a demonstration that the answer is no; dW/dt is not ambiguous in this way. More specifically, this paper shows that dW/dt is unambiguously given by -1/2εjkdιjk/dt independently of one's choice of how to localize gravitational energy in general relativity. This is proved by explicitly computing dW/dt using various gravitational stress-energy pseudotensors (Einstein, Landau-Lifshitz, Møller) as well as Bergmann's conserved quantities which generalize many of the pseudotensors to include an arbitrary function of position. A discussion is also given of the problem of formulating conservation laws in general relativity and the role played by the various pseudotensors.

UR - http://www.scopus.com/inward/record.url?scp=0034904035&partnerID=8YFLogxK

M3 - Article

VL - 63

SP - 640131

EP - 6401314

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

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

SN - 1550-7998

IS - 6

M1 - 064013

ER -