A Note on General Relativity, Energy Conservation, and Noether’s Theorems

Abstract

The variational problem posed by Emmy Noether in her seminal 1918 paper leads to three theorems, two of which she presents in that paper and the third of which is due to F. Klein, also in 1918.1 The origins of these theorems lie in the discussions of Klein, Noether, D. Hilbert and A. Einstein over the status of energy conservation in generally covariant theories such as General Relativity. In this paper I will outline one thread of this discussion and show how the three theorems of Noether and Klein can be brought to bear. The particular thread of interest begins with Klein’s observation (in his response to Hilbert’s (1916) first note on the foundations of physics) that the energy conservation law associated with Hilbert’s energy vector is a mathematical identity, in constrast to the familiar energy conservation laws of mechanics which are not identities.2 These two aspects—the claim that energy conservation is an identity, and the claim that this marks a contrast with other theories—are picked up by Hilbert and by Einstein, and are the subject of this note.