The Weyl tensor and shear-free perfect fluids

It is proved that a necessary and sufficient condition for a shear-free perfect fluid to be irrotational is that the Weyl tensor be pure electric type. For shear-free isentropic flow with unit tangent uα, we find the conservation law ∇α(n1/3iω uα) =0, where i is the relativistic specific enthalpy, n is the conserved particle number density, and ω is the vorticity scalar.