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The standard argument for the uniqueness of the Einstein field equation isbased on Lovelock's Theorem, the relevant statement of which is restricted tofour dimensions. I prove a theorem similar to Lovelock's, with a physicallymodified assumption: that the geometric object representing curvature in theEinstein field equation ought to have the physical dimension of stress-energy.The theorem is stronger than Lovelock's in two ways: it holds in alldimensions, and so supports a generalized argument for uniqueness; it does notassume that the desired tensor depends on the metric only up second-orderpartial-derivatives, that condition being a consequence of the proof. This hasconsequences for understanding the nature of the cosmological constant andtheories of higher-dimensional gravity. Another consequence of the theorem isthat it makes precise the sense in which there can be no gravitationalstress-energy tensor in general relativity. Along the way, I prove a result ofsome interest about the second jet-bundle of the bundle of metrics over amanifold.

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