Pseudotensor problem of gravitational energy-momentum and Noether's theorem revisited

TL;DR

This paper revisits Noether's theorem in the context of gravitational energy-momentum, clarifying misconceptions about pseudotensors and demonstrating that conserved quantities are scalars, with implications for gravitational energy localization.

Contribution

It clarifies the pseudotensor problem by linking it to misinterpretations of Noether's theorem and shows that gravitational energy-momentum conservation involves scalar quantities.

Findings

Pseudotensor problem results from misreading Noether's theorem.

All Noether's conserved quantities are scalars.

Non-localizability of gravitational energy-momentum is not due to the equivalence principle.

Abstract

Based on a general variational principle, Noether's theorem is revisited. It is shown that the so called pseudotensor problem of the gravitational energy-momentum is a result of mis-reading Noether's theorem, and in fact, all the Noether's conserved quantities are scalars. It is also shown, by using a counter-example, that the non-localizability of gravitational energy-momentum can not be attributed to the equivalence principle. As a direct consequence of variational principle, a generalized Hamilton-Jacobi equation for the Hamilton's principal functional is obtained.