There Is No Gravitational Stress-Energy Tensor

          @misc{ pirsa_PIRSA:10100052,
            doi = {10.48660/10100052},
            url = {https://pirsa.org/10100052},
            author = {Curiel, Erik},
            keywords = {Quantum Foundations},
            language = {en},
            title = {There Is No Gravitational Stress-Energy Tensor},
            publisher = {Perimeter Institute},
            year = {2010},
            month = {oct},
            note = {PIRSA:10100052 see, \url{https://pirsa.org}}
          }
          

Abstract

The question of the existence of gravitational stress-energy in general relativity has exercised investigators in the field since the very 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 an 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.