Noether and Hilbert (metric) energy-momentum tensors are not, in general, equivalent

TL;DR

This paper demonstrates through a counterexample that the Noether and Hilbert energy-momentum tensors, often assumed equivalent, are generally not the same in complex theories like linearized Gauss-Bonnet gravity.

Contribution

It provides the first explicit counterexample showing the non-equivalence of Noether and Hilbert energy-momentum tensors in a gauge-invariant, symmetric, and conserved form.

Findings

Noether and Hilbert tensors differ in complex models

Linearized Gauss-Bonnet gravity has a unique, gauge-invariant energy-momentum tensor

Counterexample disproves the assumed general equivalence of the two methods

Abstract

Multiple methods for deriving the energy-momentum tensor for a physical theory exist in the literature. The most common methods are to use Noether's first theorem with the 4-parameter Poincar\'{e} translation, or to write the action in a curved spacetime and perform variation with respect to the metric tensor, then return to a Minkowski spacetime. These are referred to as the Noether and Hilbert (metric/ curved space/ variational) energy-momentum tensors, respectively. In electrodynamics and other simple models, the Noether and Hilbert methods yield the same result. Due to this fact, it is often asserted that these methods are generally equivalent for any theory considered, and that this gives physicists a freedom in using either method to derive an energy-momentum tensor depending on the problem at hand. $\dots$ For spin-2, the ideal candidate to check this equivalence for a more complicated model, there exist many energy-momentum tensors in the literature, none of which are gauge invariant, so it is not clear which expression one hopes to obtain from the Noether and Hilbert approaches unlike in the case of e.g. electrodynamics. It has been shown, however, that the linearized Gauss-Bonnet gravity model (second order derivatives, second rank tensor potential) has an energy-momentum tensor that is unique, gauge invariant, symmetric, conserved, and trace-free when derived from Noether's first theorem (all the same properties of the physical energy-momentum tensor of electrodynamics). This makes it the ideal candidate to check if the Noether and Hilbert methods coincide for a more complicated model. It is proven here using this model as a counterexample, by direct calculation, that the Noether and Hilbert energy-momentum tensors are not, in general, equivalent.