Gravitational Collapse in Free Fall of a Slowly Rotating Star

The gravitational collapse of a slowly rotating star with small deviations from spherical symmetry is studied. The exterior metric is chosen to be the Kerr metric in synchronous coordinates discarding terms of order (a/r)². Interior geometry is constructed by adding an off-diagonal term in first order ofa to the exact solution of the non-rotating case. This term is determined in part by requiring the validity of the junction conditions at the star's surface and by demanding that the angular momentum of the source is equal toM A, in agreement with the value measured by a distant observer. The resulting stress-energy tensor describes a homogeneous, pressureless, ideal fluid which rotates nonuniformly relative to the synchronous frame which is no longer comoving the stellar matter.