Currents and superpotentials in classical gauge-invariant theories: I. Local results with applications to perfect fluids and general relativity

Abstract

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 at a suitable singularity (in general, at infinity) of so-called superpotentials, namely local functions of the gauge fields (or more generally of the gauge forms). Some gauge-invariant bulk charges and current densities may subsist when distinguished one-dimensional subgroups are present. We shall study mostly local consequences of gauge invariance. Quite generally there exist local superpotentials analogous to those of Freud or Bergmann for general relativity. They are parametrized by infinitesimal gauge transformations, but are afflicted by topological ambiguities which one must handle on a case-by-case basis. The choice of variational principle: variables, surface terms and boundary conditions is crucial. As a first illustration we propose a new affine action that reduces to general relativity upon gauge fixing the dilatation (Weyl-1918-like) part of the connection and elimination of auxiliary fields. We can also reduce it by similar considerations either to the Palatini action or to the Cartan-Weyl moving frame action and compare the associated superpotentials. This illustrates the concept of Noether identities. We formulate a vanishing theorem for the superpotential and the current when there is a (Killing) global isometry or its generalization. We distinguish between asymptotic symmetries and symmetries defined in the bulk. A second and independent application is a geometrical reinterpretation of the convection of vorticity in barotropic non-viscous fluids first established by Helmholtz-Kelvin, Eckart and Ertel. In the homentropic case it can be seen to follow by a general theorem from the vanishing of the superpotential corresponding to the time-independent relabelling symmetry. The special diffeomorphism symmetry is, in the absence of dynamical gauge field and spin, associated with a vanishing internal transverse momentum flux density. We also consider the non-homentropic case. We identify the one-dimensional subgroups responsible for the bulk charges and thus propose an impulsive forcing for creating or destroying selectively helicity (respectively enstrophies) in odd (respectively even) dimensions. This is an example of a new and general forcing rule.