Bessel-Hagen currents, reducibility charges, and local gravitational energy
Abstract
We consider whether the translation Noether current of a linearised metric spin-2 theory admits a strictly gauge-invariant local representative. For Fierz-Pauli and dynamically regular pure curvature-squared theories in four dimensions, we show that no such representative exists under the stated locality, polynomiality and covariance assumptions. A direct enumeration first excludes nonzero symmetric rank-two tensors at the dimension and derivative order of the quadratic action: in the curvature-squared case, the parity-even conservation kernel consists of the identically vanishing Lanczos--Lovelock combination, while both parity-odd coefficients vanish. Homogeneity of the actions in total derivative number then allows projection by field degree and derivative number, extending the exclusion to all finite-order local polynomial representatives with constant coefficients and no explicit coordinate dependence. The translation current nevertheless defines a nonzero conserved-current class. Independently, a constant translation is a Killing reducibility parameter of the linearised gauge symmetry and determines a nonzero linear surface-charge class, which is distinct from the quadratic energy current. A separate calculation shows why a direct compensator cancellation available in Maxwell theory fails for the metric spin-2 field, although this calculation is not needed for the current-class result. Maxwell theory, propagating free p-forms and Yang--Mills theory linearised about the trivial vacuum provide comparisons in which translations are not reducibility parameters and strictly gauge-invariant stress tensors exist. Gauss--Bonnet and pure flat-space R2 gravity lie outside the regular scope of the result, through variational triviality and dynamical degeneracy, respectively.
Related
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BibTeX
@misc{hobson2026besselhagen,
title = {{Bessel-Hagen currents, reducibility charges, and local gravitational energy}},
author = {Michael Hobson and Will Barker and Anthony Lasenby},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13700},
eprint = {2610.13700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13700},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bessel-Hagen currents, reducibility charges, and local gravitational energy
AU - Michael Hobson
AU - Will Barker
AU - Anthony Lasenby
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13700
UR - https://arxiv.org/abs/2610.13700
AB - We consider whether the translation Noether current of a linearised metric spin-2 theory admits a strictly gauge-invariant local representative. For Fierz-Pauli and dynamically regular pure curvature-squared theories in four dimensions, we show that no such representative exists under the stated locality, polynomiality and covariance assumptions. A direct enumeration first excludes nonzero symmetric rank-two tensors at the dimension and derivative order of the quadratic action: in the curvature-squared case, the parity-even conservation kernel consists of the identically vanishing Lanczos--Lovelock combination, while both parity-odd coefficients vanish. Homogeneity of the actions in total derivative number then allows projection by field degree and derivative number, extending the exclusion to all finite-order local polynomial representatives with constant coefficients and no explicit coordinate dependence. The translation current nevertheless defines a nonzero conserved-current class. Independently, a constant translation is a Killing reducibility parameter of the linearised gauge symmetry and determines a nonzero linear surface-charge class, which is distinct from the quadratic energy current. A separate calculation shows why a direct compensator cancellation available in Maxwell theory fails for the metric spin-2 field, although this calculation is not needed for the current-class result. Maxwell theory, propagating free p-forms and Yang--Mills theory linearised about the trivial vacuum provide comparisons in which translations are not reducibility parameters and strictly gauge-invariant stress tensors exist. Gauss--Bonnet and pure flat-space R^2 gravity lie outside the regular scope of the result, through variational triviality and dynamical degeneracy, respectively.
ER -