Energy in Newtonian gravity

TL;DR

This paper explores the localization of energy in Newtonian gravity, proposing a compromise based on relativity and Noether's theorem, and analyzes a thought experiment illustrating gravitational energy transfer.

Contribution

It demonstrates that a positive definite gravitational energy contribution in vacuum is implied by both relativity and Noether's theorem with Dirichlet boundary conditions.

Findings

Energy in vacuum can be positive definite.

Noether's theorem supports the energy localization compromise.

Analysis of Bondi and McCrea's thought experiment illustrates gravitational energy transfer.

Abstract

In Newtonian gravity it is a moot question whether energy should be localized in the field or inside matter. An argument from relativity suggests a compromise in which the contribution from the field in vacuum is positive definite. We show that the same compromise is implied by Noether's theorem applied to a variational principle for perfect fluids, if we assume Dirichlet boundary conditions on the potential. We then analyse a thought experiment due to Bondi and McCrea that gives a clean example of inductive energy transfer by gravity. Some history of the problem is included