Shear-free perfect fluids in general relativity .3. Petrov type III spacetimes

It is shown that for any Petrov type III, shear-free, perfect fluid solution of Einstein's field equations, in which the perfect fluid satisfies a barotropic equation of state p = p(w) such that w + p not equivalent to 0, the fluid volume expansion is zero, or the equation of state satisfies 9p double over dot(w + p) - 18p over dot(2) + 2 = 0. It follows that all such perfect fluids whose equation of state is the gamma law, must have zero fluid volume expansion.