A Simple Proof of the Uniqueness of the Einstein Field Equation in All Dimensions

Abstract

The standard argument for the uniqueness of the Einstein field equation is based on Lovelock's Theorem, the relevant statement of which is restricted to four dimensions. I prove a theorem similar to Lovelock's, with a physically modified assumption: that the geometric object representing curvature in the Einstein field equation ought to have the physical dimension of stress-energy. The theorem is stronger than Lovelock's in two ways: it holds in all dimensions, and so supports a generalized argument for uniqueness; it does not assume that the desired tensor depends on the metric only up second-order partial-derivatives, that condition being a consequence of the proof. This has consequences for understanding the nature of the cosmological constant and theories of higher-dimensional gravity. Another consequence of the theorem is that it makes precise the sense in which there can be no gravitational stress-energy tensor in general relativity. Along the way, I prove a result of some interest about the second jet-bundle of the bundle of metrics over a manifold.

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Erik Curiel
Ludwig Maximilians Universität, München

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On geometric objects, the non-existence of a gravitational stress-energy tensor, and the uniqueness of the Einstein field equation.Erik Curiel - 2009 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 66:90-102.

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