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

TL;DR

This paper extends Noether's conservation laws in gauge theories, introduces an affine action for gravity, compares superpotentials, and applies these ideas to fluid vorticity, proposing a new forcing rule for helicity manipulation.

Contribution

It develops a gauge-invariant framework for conservation laws, introduces an affine action for gravity, and applies these concepts to fluid dynamics with a novel forcing rule.

Findings

Comparison of gravity superpotentials based on boundary conditions

Identification of subgroups responsible for bulk charges in fluids

Proposal of a new forcing rule for helicity control

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.