Energy localization invariance of tidal work in general relativity

Abstract

It is well known that when an external general relativistic (electric-type) tidal field Ejk(t) interacts with the evolving quadrupole moment Ijk(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=-1/2EjkdIjk/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 ∼EjkIjk, is the tidal work also ambiguous by this amount, and therefore is the formula dW/dt=-1/2EjkdIjk/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 -1/2EjkdIjk/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.