Relativistic helicity and link in Minkowski space-time

TL;DR

This paper formulates a relativistic helicity in Minkowski space-time that remains conserved under certain conditions, linking vortex filament topology with the conservation of a fundamental topological quantity in relativistic fluids.

Contribution

It introduces a new relativistic helicity in four-dimensional space-time that conserves in barotropic fluids, connecting vortex topology with relativistic fluid dynamics.

Findings

Relativistic helicity is conserved in barotropic fluids.

Vortex filaments link via the linking number in four-dimensional space.

Thermodynamic forces affect vortex filament states and their conservation.

Abstract

A relativistic helicity has been formulated in the four-dimensional Minkowski space-time. Whereas the relativistic distortion of space-time violates the conservation of the conventional helicity, the newly defined relativistic helicity conserves in a barotropic fluid or plasma, dictating a fundamental topological constraint. The relation between the helicity and the vortex-line topology has been delineated by analyzing the linking number of vortex filaments which are singular differential forms representing the pure states of Banach algebra. While the dimension of space-time is four, vortex filaments link, because vorticities are primarily 2-forms and the corresponding 2-chains link in four dimension; the relativistic helicity measures the linking number of vortex filaments that are proper-time cross-sections of the vorticity 2-chains. A thermodynamic force yields an additional term in the vorticity, by which the vortex filaments on a reference-time plane are no longer pure states. However, the vortex filaments on a proper-time plane remain to be pure states, if the thermodynamic force is exact (barotropic), thus, the linking number of vortex filaments conserves.