Shear-free perfect fluids in general relativity II. Aligned, Petrov type III space-times

Petrov type III, shear-free, perfect fluid solutions of the Einstein field equations, with a barotropic equation of state p = p(w) satisfying w + p≢0, are investigated. It is shown that if the acceleration of the fluid is orthogonal to the two-spaces spanned by the repeated principal null direction of the Weyl tensor and the fluid four-velocity, or if the fluid four-velocity lies in the two-spaces spanned by the principal null directions of the Weyl tensor, then the fluid's volume expansion is zero.