Shear-free perfect fluids with a solenoidal electric curvature
  We prove that the vorticity or the expansion vanishes for any shear-free perfect fluid solution of the Einstein field equations where the pressure satisfies a barotropic equation of state and the spatial divergence of the electric part of the Weyl tensor is zero. 
Norbert Van den Bergh, John Carminati, Hamid Reza Karimian, Peter Huf
gr-qc