J. Carminati et S. Cyganowski, SHEAR-FREE PERFECT FLUIDS IN GENERAL-RELATIVITY .4. PETROV TYPE-III SPACETIMES, Classical and quantum gravity, 14(5), 1997, pp. 1167-1181
It is shown that for any Petrov type III, shear-free, perfect-fluid so
lution of Einstein's field equations, in which the perfect fluid satis
fies a barotropic equation of state p = p(w) such that w + p not equal
0, the fluid volume expansion is zero. In addition, the fluid must ne
cessarily be rotating.