Shear-free Perfect Fluids in General Relativity: Gravito-magnetic Spacetimes

We investigate shear-free, perfect fluid solutions of Einstein's field equations in which the perfect fluid satisfies a barotropic equation of state p = p(w) such that w + p ≠ 0. We find that if the electric part of the Weyl tensor (with respect to the fluid flow) vanishes and the spacetime is not conformally flat then the fluid volume expansion is zero but the vorticity is necessarily nonzero. In addition, we show that if p = −w/3 then necessarily either the fluid expansion is zero or the fluid vorticity is zero.