A Brief Remark on Energy Conditions and the Geroch-Jang Theorem

TL;DR

This paper examines the role of energy conditions in the Geroch-Jang theorem, showing through counterexamples that the theorem requires stronger energy conditions than previously thought.

Contribution

It provides explicit counterexamples demonstrating the necessity of stronger energy conditions for the Geroch-Jang theorem in General Relativity.

Findings

Weaker energy conditions are insufficient for the theorem.

Stronger energy conditions than previously assumed are necessary.

Counterexamples illustrate the precise requirements for the theorem.

Abstract

The status of the geodesic principle in General Relativity has been a topic of some interest in the recent literature on the foundations of spacetime theories. Part of this discussion has focused on the role that a certain energy condition plays in the proof of a theorem due to Bob Geroch and Pong-Soo Jang ["Motion of a Body in General Relativity." Journal of Mathematical Physics 16(1), (1975)] that can be taken to make precise the claim that the geodesic principle is a theorem, rather than a postulate, of General Relativity. In this brief note, I show, by explicit counterexample, that not only is a weaker energy condition than the one Geroch and Jang state insufficient to prove the theorem, but in fact a condition still stronger than the one that they assume is necessary.