Noether's Theorems and Energy in General Relativity

Abstract: This paper has three main aims: first, to give a pedagogical introduction to Noether's two theorems and their implications for energy conservation in general relativity, which was a central point of discussion between Hilbert, Klein, Noether and Einstein. Second, it introduces and compares two proposals for gravitational energy and momentum, one of which is very influential in physics: and, so far as I know, neither of the two has been discussed in the philosophical literature. Third, it assesses these proposals in connection with recent philosophical discussions of energy and momentum in general relativity. After briefly reviewing the debates about energy conservation between Hilbert, Klein, Noether and Einstein, I give Noether's two theorems. I show that Einstein's gravitational energy-momentum pseudo-tensor, including its superpotential, is fixed, through Noether's theorem, by the boundary terms in the action. That is, the freedom to add an arbitrary superpotential to the gravitational pseudo-tensor corresponds to the freedom to add boundary terms to the action without changing the equations of motion. This freedom is fixed in the same way for both problems. I also review two proposals for energy and momentum in GR, of which one is a quasi-local alternative to the local expressions, and the other builds on Einstein's local pseudo-tensor approach. I discuss the recent philosophical literature on the conservation of energy and momentum in general relativity, and I assess and compare the two proposals in the light of this literature: especially, in light of questions about diffeomorphism invariance and background-independence.