  We obtain expressions for the shear and the vorticity tensors of
perfect-fluid spacetimes, in terms of the divergence of the Weyl tensor. For
such spacetimes, we prove that if the gradient of the energy density is
parallel to the velocity, then either the expansion rate is zero, or the
vorticity vanishes. This statement recalls the "shear-free conjecture" for a
perfect barotropic fluid: vanishing shear implies either vanishing expansion
rate or vanishing vorticity. Finally, we give a new condition for a perfect
fluid to be a Generalized Robinson-Walker spacetime.
