On the construction of global models describing rotating bodies; Uniqueness of the exterior gravitational field

The problem of constructing global models describing isolated axially symmetric rotating bodies in equilibrium is analyzed. It is claimed that, whenever the global space-time is constructed by giving boundary data on the limiting surface of the body and integrating Einstein's equations on both inside and outside the body, the problem becomes overdetermined. Similarly, when the space-time describing the interior of the body is explicitly given, the problem of finding the exterior vacuum solution becomes overdetermined. We discuss in detail the procedure to be followed in order to construct the exterior vacuum field created by a given but arbitrary distribution of matter. Finally, the uniqueness of the exterior vacuum gravitational field is proven by exploiting the harmonic map formulation of the vacuum equations and the boundary conditions prescribed from the matching.