On Tensorial Concomitants and the Non-Existence of a Gravitational Stress-Energy Tensor

TL;DR

This paper demonstrates that under natural conditions, a localized gravitational stress-energy tensor cannot exist in general relativity, implying gravitational energy is inherently non-local.

Contribution

It provides a novel tensorial analysis showing the non-existence of a gravitational stress-energy tensor in general relativity, challenging traditional folklore.

Findings

No tensor can represent gravitational stress-energy under certain conditions.

Gravitational energy in general relativity is necessarily non-local.

Analysis of jet bundles of Lorentz metrics offers new mathematical insights.

Abstract

The question of the existence of gravitational stress-energy in general relativity has exercised investigators in the field since the inception of the theory. Folklore has it that no adequate definition of a localized gravitational stress-energetic quantity can be given. Most arguments to that effect invoke one version or another of the Principle of Equivalence. I argue that not only are such arguments of necessity vague and hand-waving but, worse, are beside the point and do not address the heart of the issue. Based on a novel analysis of what it may mean for one tensor to depend in the proper way on another, I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of the bundle of Lorentz metrics over spacetime.