A class of shear-free perfect fluids in general relativity. I

It has been conjectured that shear-free perfect fluids in general relativity, with an equation of state p=p(μ) and satisfying μ+p≠0, necessarily have either zero expansion or zero vorticity. We prove that this result holds in the restricted case when the fluid’s vorticity and acceleration are parallel. Specifically, we prove that if the vorticity is nonzero, the fluid’s volume expansion must vanish.