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A general class of solutions of Einstein's equation for a slowly rotatingfluid source, with supporting internal pressure, is matched using Lichnerowiczjunction conditions, to the Kerr metric up to and including first order termsin angular speed parameter. It is shown that the match applies to anypreviously known non-rotating fluid source made to rotate slowly for which azero pressure boundary surface exists. The method is applied to the dust sourceof Robertson-Walker and in outline to an interior solution due to McVittiedescribing gravitational collapse. The applicability of the method toadditional examples is transparent. The differential angular velocity of therotating systems is determined and the induced rotation of local inertial frameis exhibited

Slowly, Rotating Non-Stationary, Fluid Solutions of Einstein's Equations and Their Match to Kerr Empty Space-Time