A connection between linearized Gauss-Bonnet gravity and classical electrodynamics II: Complete dual formulation

TL;DR

This paper explores the dual formulation of linearized Gauss-Bonnet gravity, revealing its analogy to Maxwell's theory and extending the concept of gauge invariance to higher spin gauge theories, thus deepening the connection between gravity and electrodynamics.

Contribution

It develops the dual formulation of linearized Gauss-Bonnet gravity and generalizes the gauge invariance concept to higher spin theories, establishing a unified framework.

Findings

Dual formulation of linearized Gauss-Bonnet gravity is analogous to Maxwell's homogeneous equations.

Complete gauge invariance is necessary for dual formulations, extending to higher spin theories.

Higher spin gauge theories can be described by Maxwell-like equations with gauge-invariant curvature tensors.

Abstract

In a recent publication a procedure was developed which can be used to derive completely gauge invariant models from general Lagrangian densities with $N$ order of derivatives and $M$ rank of tensor potential. This procedure was then used to show that unique models follow for each order, namely classical electrodynamics for $N = M = 1$ and linearized Gauss-Bonnet gravity for $N = M = 2$. In this article, the nature of the connection between these two well explored physical models is further investigated by means of an additional common property; a complete dual formulation. First we give a review of Gauss-Bonnet gravity and the dual formulation of classical electrodynamics. The dual formulation of linearized Gauss-Bonnet gravity is then developed. It is shown that the dual formulation of linearized Gauss-Bonnet gravity is analogous to the homogenous half of Maxwell's theory; both have equations of motion corresponding to the (second) Bianchi identity, built from the dual form of their respective field strength tensors. In order to have a dually symmetric counterpart analogous to the non-homogenous half of Maxwell's theory, the first invariant derived from the procedure in $N = M = 2$ can be introduced. The complete gauge invariance of a model with respect to Noether's first theorem, and not just the equation of motion, is a necessary condition for this dual formulation. We show that this result can be generalized to the higher spin gauge theories, where the spin-$n$ curvature tensors for all $N = M = n$ are the field strength tensors for each $n$. These completely gauge invariant models correspond to the Maxwell-like higher spin gauge theories whose equations of motion have been well explored in the literature.