Abstract
We employ the technique used in the classification of harmonic morphisms with one-dimensional fibers on four-dimensional Einstein manifolds (Pantilie in Commun Anal Geom 10:779–814, 2002) to give a simpler proof of the fact that the shear-free perfect fluids coupled to gravity are either irrotational or expansion free in the case when the equation of state is $$\rho =-3p$$ .
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