Canonical Noether and the energy–momentum non-uniqueness problem in linearized gravity

Abstract

Recent research has highlighted the non-uniqueness problem of energy–momentum tensors in linearized gravity; many different tensors are published in the literature, yet for particular calculations a unique expression is required. It has been shown that (A) none of these spin-2 energy–momentum tensors are gauge invariant and (B) the Noether and Hilbert energy–momentum tensors are not, in general, equivalent; therefore uniqueness criteria is difficult to specify. Conventional wisdom states that the various published energy–momentum tensors for linearized gravity can be derived from the canonical Noether energy–momentum tensor of spin-2 Fierz-Pauli theory by adding ad-hoc ‘improvement’ terms (the divergence of a superpotential and terms proportional to the equations of motion), that these superpotentials are in some way unique or physically significant, and that this implies some meaningful connection to the Noether procedure. To explore this question of uniqueness, we consider the most general possible energy–momentum tensor for linearized gravity with free coefficients using the Fock method. We express this most general energy–momentum tensor as the canonical Noether tensor, supplemented by the divergence of a general superpotential plus all possible terms proportional to the equations of motion. We then derive systems of equations which we solve in order to prove several key results for spin-2 Fierz–Pauli theory, most notably that there are infinitely many conserved energy–momentum tensors derivable from the ‘improvement’ method, and there are infinitely many conserved symmetric energy–momentum tensors that follow from specifying the Belinfante superpotential alone. This disproves several recent claims that the Belinfante tensor is uniquely associated to the Hilbert tensor in spin-2 Fierz–Pauli theory. We give two new energy–momentum tensors of this form. Most importantly, since there are infinitely many energy–momentum tensors of this form, no meaningful or unique connection to Noether’s first theorem can be claimed by application of the canonical Noether ‘improvement’ method.