Bessel-Hagen currents for the Fierz-Pauli action

TL;DR

This paper investigates the applicability of Bessel-Hagen's gauge-invariant energy-momentum tensor construction to the Fierz-Pauli spin-2 field, revealing fundamental limitations and proposing a gauge-invariant current class instead.

Contribution

It demonstrates that a direct local gauge-invariant energy-momentum tensor for the Fierz-Pauli field cannot exist, but a gauge-invariant class of Noether currents can be constructed.

Findings

No nonzero local quadratic tensor invariant under spin-2 gauge transformation.

Bessel-Hagen construction yields a gauge-invariant equivalence class of currents.

Changing gauge parameters affects currents only by trivial or field-equation terms.

Abstract

For electromagnetism in Minkowski spacetime, the Bessel-Hagen method gives a particularly direct Noetherian derivation of the standard gauge-invariant energy-momentum tensor. The key step is to supplement the form variation generated by an infinitesimal coordinate transformation with a compensating electromagnetic gauge transformation. In this paper we ask whether the same idea can be applied to the massless spin-2 field described by the Fierz-Pauli action. We first prove that no nonzero local tensor quadratic in first derivatives of the symmetric field $h_{\mu\nu}$ can be strictly invariant under the spin-2 gauge transformation $h_{\mu\nu}\mapsto h_{\mu\nu}+\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$; the direct electromagnetic analogue of the Bessel-Hagen construction therefore cannot exist. Once the inexact nature of the Fierz-Pauli gauge symmetry is treated correctly, however, the Bessel-Hagen construction does produce a gauge-invariant equivalence class of Noether currents. Changing the compensating spin-2 gauge parameter changes the current only by terms proportional to the Fierz-Pauli field equations; performing an independent spin-2 gauge transformation on $h_{\mu\nu}$ changes the current only by a trivial current given by the divergence of an antisymmetric superpotential plus field-equation terms. This provides the natural spin-2 analogue of Bessel-Hagen's electromagnetic construction, but only in the quotient space of conserved currents, and not as a preferred local gauge-invariant energy-momentum tensor.