Energy Localization Invariance of Tidal Work In General Relativity

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It is well known that when an external general relativistic (electric-type) tidal field (Formula presented) interacts with the evolving quadrupole moment (Formula presented) 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: (Formula presented) Thorne has posed the following question: In view of the fact that the gravitational interaction energy (Formula presented) between the tidal field and the body is ambiguous by an amount (Formula presented) is the tidal work also ambiguous by this amount, and therefore is the formula (Formula presented) only valid unambiguously when integrated over time scales long compared to that for (Formula presented) to change substantially? This paper completes a demonstration that the answer is no; (Formula presented) is not ambiguous in this way. More specifically, this paper shows that (Formula presented) is unambiguously given by (Formula presented) independently of one’s choice of how to localize gravitational energy in general relativity. This is proved by explicitly computing (Formula presented) 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.



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