Relabeling symmetry in relativistic fluids and plasmas

TL;DR

This paper derives a relativistic conservation law for canonical helicity from Noether's theorem using an action principle in Lagrangian coordinates, revealing why conventional helicity is not conserved in relativistic fluids and plasmas.

Contribution

It introduces a relativistic action principle for fluids and plasmas, deriving a new conservation law for relativistic canonical helicity and explaining the non-conservation of conventional helicity.

Findings

Derived relativistic canonical helicity conservation from Noether's theorem.

Explained the non-conservation of conventional helicity in relativistic settings.

Formulated a relativistic action principle for MHD and derived the relativistic cross helicity.

Abstract

The conservation of the recently formulated relativistic canonical helicity [Yoshida Z, Kawazura Y, and Yokoyama T 2014 J. Math. Phys. 55 043101] is derived from Noether's theorem by constructing an action principle on the relativistic Lagrangian coordinates (we obtain general cross helicities that include the helicity of the canonical vorticity). The conservation law is, then, explained by the relabeling symmetry pertinent to the Lagrangian label of fluid elements. Upon Eulerianizing the Noether current, the purely spatial volume integral on the Lagrangian coordinates is mapped to a space-time mixed three-dimensional integral on the four-dimensional Eulerian coordinates. The relativistic conservation law in the Eulerian coordinates is no longer represented by any divergence-free current; hence, it is not adequate to regard the relativistic helicity (represented by the Eulerian variables) as a Noether charge, and this stands the reason why the "conventional helicity" is no longer a constant of motion. We have also formulated a relativistic action principle of magnetohydrodynamics (MHD) on the Lagrangian coordinates, and have derived the relativistic MHD cross helicity.