  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.
