Method to obtain shear-free two-fluid solutions of Einstein's equations

We use the Einstein equations, stated as an initial-value problem (3+1 formalism), to present a method for obtaining a class of solutions which may be interpreted as the gravitational field produced by a mixture of two perfect fluids. The four-velocity of one of the components is assumed to be a shear-free, irrotational, and geodesic vector field. The solutions are given up to a set of a hyperbolic quasilinear system