A connection between linearized Gauss-Bonnet gravity and classical electrodynamics is found by developing a procedure which can be used to derive completely gauge invariant models. The procedure involves building the most general Lagrangian for a particular order of derivatives (N ) and rank of tensor potential (M ), then solving such that the model is completely gauge invariant (the Lagrangian density, equation of motion and energy-momentum tensor are all gauge invariant). In the case of N = 1 order of derivatives and M = 1 rank of tensor potential, electrodynamics is uniquely derived from the procedure. In the case of N = 2 order of derivatives and M = 2 rank of symmetric tensor potential, linearized Gauss-Bonnet gravity is uniquely derived from the procedure. The natural outcome of the models for classical electrodynamics and linearized Gauss-Bonnet gravity from a common set of rules provides an interesting connection between two well explored physical models.

Multiple methods for deriving the energy-momentum tensor for a physical theory exist in the literature. The most common methods are to use Noether's first theorem with the 4-parameter Poincaré translation, or to write the action in a curved spacetime and perform variation with respect to the metric tensor,

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.

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Dedicated to the late Bessel-Hagen, who when alive had his habilitation thesis thrown into the sea, and even now must feel as if his work was lying somewhere on the seabed.