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

TL;DR

This paper proves a theorem establishing the uniqueness of the Einstein field equation across all dimensions, based on physically motivated assumptions, and explores implications for higher-dimensional gravity and gravitational stress-energy.

Contribution

It generalizes Lovelock's theorem to all dimensions without assuming second-order dependence on the metric, strengthening the case for Einstein's equation's uniqueness.

Findings

The theorem holds in all dimensions.

It clarifies the non-existence of gravitational stress-energy tensor.

Implications for the cosmological constant and higher-dimensional theories.

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.