Energy localization invariance of tidal work in general relativity

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Abstract

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= - 1/2εjkjk/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=-1/2εjkjk/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 -1/2εjkjk/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 languageEnglish
Article number064013
Pages (from-to)640131-6401314
Number of pages5761184
JournalPhysical Review D - Particles, Fields, Gravitation and Cosmology
Volume63
Issue number6
StatePublished - 20 Aug 2001

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Ambiguous
General Relativity
relativity
invariance
Invariance
Energy
Conserved Quantity
Energy Transfer
Conservation Laws
Albert Einstein
energy
Time Scales
conservation laws
Valid
Moment
Generalise
Computing
quadrupoles
energy transfer
Arbitrary

Cite this

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title = "Energy localization invariance of tidal work in general relativity",
abstract = "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, M{\o}ller) 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.",
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Energy localization invariance of tidal work in general relativity. / Favata, Marc.

In: Physical Review D - Particles, Fields, Gravitation and Cosmology, Vol. 63, No. 6, 064013, 20.08.2001, p. 640131-6401314.

Research output: Contribution to journalArticleResearchpeer-review

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