On Geometric Objects, the Non-Existence of a Gravitational Stress-Energy Tensor, and the Uniqueness of the EinsteinudField Equation

摘要

The question of the existence of gravitational stress-energy inud general relativity has exercised investigators in the field sinceud the inception of the theory. Folklore has it that no adequateud definition of a localized gravitational stress-energetic quantityud can be given. Most arguments to that effect invoke one version orud another of the Principle of Equivalence. I argue that not only areud such arguments of necessity vague and hand-waving but, worse, areud beside the point and do not address the heart of the issue. Basedud on a novel analysis of what it may mean for one tensor to depend inud the proper way on another, which, en passant, provides aud precise characterization of the idea of a "geometric object", Iud prove that, under certain natural conditions, there can be no tensorud whose interpretation could be that it represents gravitationalud stress-energy in general relativity. It follows that gravitationalud energy, such as it is in general relativity, is necessarilyud non-local. Along the way, I prove a result of some interest in ownud right about the structure of the associated jet bundles of theud bundle of Lorentz metrics over spacetime. I conclude by showingud that my results also imply that, under a few natural conditions, theud Einstein field equation is the unique equation relatingud gravitational phenomena to spatiotemporal structure, and discuss howud this relates to the non-localizability of gravitationalud stress-energy.