Currents and Superpotentials in classical gauge invariant theories I. Local results with applications to Perfect Fluids and General Relativity

TL;DR

This work develops a local Noether framework for gauge-invariant theories, showing that bulk charges are replaced by fluxes of local superpotentials and that physical charges arise from chosen one-dimensional gauge subgroups and boundary conditions. It introduces an affine first-order gravity action, clarifies how GR emerges under appropriate gauge fixing, and compares the resulting gravitation superpotentials with standard constructions like Komar, Freud, and Katz. The paper also extends the formalism to fluids, revealing how relabeling symmetries yield conserved currents and proposing a forcing rule to manipulate helicity and enstrophy. Together, these results illuminate how variational principles, boundary terms, and gauge choices shape physically meaningful charges in gravity and fluid dynamics, with broad implications for quasi-local charges and asymptotic fluxes.

Abstract

E. Noether's general analysis of conservation laws has to be completed in a Lagrangian theory with local gauge invariance. Bulk charges are replaced by fluxes of superpotentials. Gauge invariant bulk charges may subsist when distinguished one-dimensional subgroups are present. As a first illustration we propose a new {\it Affine action} that reduces to General Relativity upon gauge fixing the dilatation (Weyl 1918 like) part of the connection and elimination of auxiliary fields. It allows a comparison of most gravity superpotentials and we discuss their selection by the choice of boundary conditions. A second and independent application is a geometrical reinterpretation of the convection of vorticity in barotropic nonviscous fluids. We identify the one-dimensional subgroups responsible for the bulk charges and thus propose an impulsive forcing for creating or destroying selectively helicity. This is an example of a new and general Forcing Rule.