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

Curiel, Erik (2018) On Geometric Objects, the Non-Existence of a Gravitational Stress-Energy Tensor, and the Uniqueness of the Einstein Field Equation. [Preprint]

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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, 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. The main theorem proven underlying all the arguments
is considerably stronger than the standard result in the literature
used for the same purposes (Lovelock's theorem of 1972): it holds in
all dimensions (not only in four); it does not require an assumption
about the differential order of the desired concomitant of the
metric; and it has a more natural physical interpretation.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Curiel, Erikerik@strangebeautiful.com
Additional Information: Forthcoming in *Studies in History and Philosophy of Modern Physics*, 2018
Keywords: gravitational energy; stress-energy tensors; concomitants; jet bundles; principle of equivalence; geometric objects; Einstein field equation
Subjects: Specific Sciences > Physics > Classical Physics
Specific Sciences > Physics > Fields and Particles
Specific Sciences > Physics > Relativity Theory
Depositing User: Dr. Erik Curiel
Date Deposited: 29 Aug 2018 00:03
Last Modified: 29 Aug 2018 00:03
Item ID: 14980
Official URL: http://strangebeautiful.com/papers/curiel-nonexist...
Subjects: Specific Sciences > Physics > Classical Physics
Specific Sciences > Physics > Fields and Particles
Specific Sciences > Physics > Relativity Theory
Date: August 2018
URI: http://philsci-archive.pitt.edu/id/eprint/14980

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