Conserved non-Noether charge in general relativity: Physical definition versus Noether’s second theorem

Abstract

In this paper, we make a close comparison of a covariant definition of an energy/ entropy in general relativity, recently proposed by a collaboration including the present authors, with existing definitions of energies such as the one from the pseudo-tensor and the quasi-local energy. We show that existing definitions of energies in general relativity are conserved charges from Noether’s second theorem for the general coordinate transformation, whose conservations are merely identities implied by the local symmetry and always hold without using equations of motion. Thus, none of the existing definitions in general relativity reflects the dynamical properties of the system, and the need for a physical definition of an energy. In contrast, our new definition of the energy/ entropy in general relativity is generically a conserved non-Noether charge and gives physically sensible results for various cases such as the black hole mass, the gravitational collapse and the expanding universe, while existing definitions sometimes lead to unphysical ones including zero and infinity. We conclude that our proposal is more physical than existing definitions of energies. Our proposal makes it possible to define almost uniquely the covariant and conserved energy/ entropy in general relativity, which brings some implications to future investigations.