Bessel-Hagen currents for linearised Weyl-squared gravity
Abstract
We consider the Bessel-Hagen construction for linearised Weyl-squared gravity, whose action depends on the metric perturbation hμν through the linearised Weyl tensor C(1)μνρσ. This provides a curvature-based comparison with the Fierz-Pauli theory, in which the action contains first derivatives of hμν whereas the first local gauge-invariant field strength contains two, and the construction determines a gauge-invariant class of Noether currents without selecting a preferred gauge-invariant local energy-momentum tensor. Since the Weyl tensor is the irreducible spin-2 part of the curvature, it is of particular interest to ask whether an action built from it admits a local representative analogous to the Maxwell tensor. We show that the Bessel-Hagen construction extends to the conformal group, acting on the fields on a fixed Minkowski background, and yields a gauge-invariant conserved-current class for each generator. Nonetheless, no nonzero local, polynomial, symmetric rank-two tensor of mass dimension four, quadratic in hμν and with constant dimensionless coefficients and no explicit coordinate dependence, is both strictly invariant under the linearised diffeomorphism and Weyl gauge symmetries and conserved on the Bach shell. The gauge symmetries restrict such a tensor to depend on C(1) and its derivatives, while the dimension and field degree permit only terms quadratic in C(1) without further derivatives. Four-dimensional Weyl identities then reduce all candidates to pure traces, whose conservation requires their coefficients to vanish. Thus, building the action from a gauge-invariant curvature does not recover the electromagnetic result. A first-order gauge-covariant field strength may provide a route to such a representative, although its presence alone is not sufficient.
Versions
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BibTeX
@misc{hobson2026besselhagen,
title = {{Bessel-Hagen currents for linearised Weyl-squared gravity}},
author = {Michael Hobson and Will Barker and Anthony Lasenby},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.02912},
eprint = {2608.02912},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02912},
}
RIS
TY - UNPB
TI - Bessel-Hagen currents for linearised Weyl-squared gravity
AU - Michael Hobson
AU - Will Barker
AU - Anthony Lasenby
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2608.02912
UR - https://arxiv.org/abs/2608.02912
AB - We consider the Bessel-Hagen construction for linearised Weyl-squared gravity, whose action depends on the metric perturbation h_μν through the linearised Weyl tensor C^(1)_μνρσ. This provides a curvature-based comparison with the Fierz-Pauli theory, in which the action contains first derivatives of h_μν whereas the first local gauge-invariant field strength contains two, and the construction determines a gauge-invariant class of Noether currents without selecting a preferred gauge-invariant local energy-momentum tensor. Since the Weyl tensor is the irreducible spin-2 part of the curvature, it is of particular interest to ask whether an action built from it admits a local representative analogous to the Maxwell tensor. We show that the Bessel-Hagen construction extends to the conformal group, acting on the fields on a fixed Minkowski background, and yields a gauge-invariant conserved-current class for each generator. Nonetheless, no nonzero local, polynomial, symmetric rank-two tensor of mass dimension four, quadratic in h_μν and with constant dimensionless coefficients and no explicit coordinate dependence, is both strictly invariant under the linearised diffeomorphism and Weyl gauge symmetries and conserved on the Bach shell. The gauge symmetries restrict such a tensor to depend on C^(1) and its derivatives, while the dimension and field degree permit only terms quadratic in C^(1) without further derivatives. Four-dimensional Weyl identities then reduce all candidates to pure traces, whose conservation requires their coefficients to vanish. Thus, building the action from a gauge-invariant curvature does not recover the electromagnetic result. A first-order gauge-covariant field strength may provide a route to such a representative, although its presence alone is not sufficient.
ER -