Energy localization invariance of tidal work in general relativity

It is well known that when an external general relativistic (electric-type) tidal field £,-(/) interacts with the evolving quadrupole moment Zy(0 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: d\Vldt = - £dljkldt. Thorne has posed the following question: In view of the fact that the gravitational interaction energy £int between the tidal field and the body is ambiguous by an amount ~Bj{L, is the tidal work also ambiguous by this amount, and therefore is the formula d\V/dt= -2£jkdZjk/dt only valid unambiguously when integrated over time scales long compared to that for Z, to change substantially? This paper completes a demonstration that the answer is no; dWIdt is not ambiguous in this way. More specifically, this paper shows that dWIdt is unambiguously given by -jtdlldt 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, M511er) 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.