On geometric objects, the non-existence of a gravitational stress-energy tensor, and the uniqueness of the Einstein field equation

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 66:90-102 (2009)
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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, which, en passant, provides a precise characterization of the idea of a "geometric object", 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. I conclude by showing that my results also imply that, under a few natural conditions, the Einstein field equation is the unique equation relating gravitational phenomena to spatiotemporal structure, and discuss how this relates to the non-localizability of gravitational stress-energy.



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Erik Curiel
Ludwig Maximilians Universität, München

Citations of this work

Two miracles of general relativity.James Read, Harvey R. Brown & Dennis Lehmkuhl - 2018 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 64:14-25.
The Mass of the Gravitational Field.Charles T. Sebens - 2022 - British Journal for the Philosophy of Science 73 (1):211-248.
A Primer on Energy Conditions.Erik Curiel - 2016 - In Dennis Lehmkuhl, Gregor Schiemann & Erhard Scholz (eds.), Towards a Theory of Spacetime Theories. New York, NY: Birkhauser. pp. 43-104.

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References found in this work

Space-Time-Matter.Hermann Weyl - 1922 - London,: E.P. Dutton and Company. Edited by Henry L. Brose.
Space-Time-Matter.Hermann Weyl & Henry L. Brose - 1953 - British Journal for the Philosophy of Science 3 (12):382-382.
Space-time structure.Erwin Schrödinger - 1950 - Cambridge [Eng.]: University Press.

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