Energy localization invariance of tidal work in general relativity

2 1 2 EjkdIjk /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 ;EjkIjk , is the tidal work also ambiguous by this amount, and therefore is the formula dW/dt52 1 2 EjkdIjk /dt only valid unambiguously when integrated over time scales long compared to that for Ijk 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 2 1 2 EjkdIjk /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, Mo”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.