Algebraically special coplanar shear-free perfect fluids in general relativity

We investigate all algebraically special, not conformally flat, shear-free, isentropic (p(w), w + p ≠ 0), perfect fluid solutions of Einstein\u27s field equations. We show, using the GHP formalism, that if the repeated principle null direction of the Weyl tensor is coplanar with the fluid\u27s 4-velocity and vorticity vector (assumed nonzero), then the fluid\u27s expansion must vanish.