Bessel-Hagen currents for the Fierz-Pauli action

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.

8 pages, no figures. Version accepted by PRD, containing minor edits

References in corpus (8)

• Localizing the Energy and Momentum of Linear Gravity
• Noether and Hilbert (metric) energy-momentum tensors are not, in general, equivalent
• GR as a classical spin-2 theory?
• Localized Energetics of Linear Gravity: Theoretical Development
• Localizing the Angular Momentum of Linear Gravity
• Canonical Noether and the energy-momentum non-uniqueness problem in linearized gravity
• Energy-momentum tensor and duality symmetry of linearized gravity in the Fierz formalism
• The general set of Noetherian energy-momentum tensors in linearized gravity: mathematical framework