Energy Localization Invariance of Tidal Work in General Relativity

TL;DR

This paper demonstrates that the tidal work rate in general relativity is unambiguously defined and independent of the gravitational energy localization method, resolving a longstanding ambiguity in gravitational energy transfer calculations.

Contribution

It proves that the tidal work rate is invariant under different gravitational energy pseudotensors, clarifying the energy transfer in general relativistic tidal interactions.

Findings

Tidal work rate dW/dt is unambiguous and equals -1/2 E dI/dt.

The result holds regardless of the gravitational energy localization method used.

The paper discusses the implications for conservation laws in general relativity.

Abstract

It is well known that, when an external general relativistic (electric-type) tidal field E(t) interacts with the evolving quadrupole moment I(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 = -1/2 E dI/dt. Thorne has posed the following question: In view of the fact that the gravitational interaction energy between the tidal field and the body is ambiguous by an amount of order E(t)I(t), is the tidal work also ambiguous by this amount, and therefore is the formula dW/dt = -1/2 E dI/dt only valid unambiguously when integrated over timescales long compared to that for I(t) 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 -1/2 E dI/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, Moller) 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.