Kerr Black Hole in the Background of the Einstein Universe

The Vaidya-Einstein-Kerr (VEK) black hole which represents the spacetime of the Kerr black hole in a non-vacuum, asymptotically non-flat background is investigated. The energy-momentum tensor corresponding to this spacetime satisfies reasonable energy conditions. We study several properties of this black hole and compare and contrast them with those of the Kerr black hole. We investigate the effect of the background on the geometry of the event horizon by computing the equatorial and polar circumferences and determining the oblateness of the horizon. We find that the surface area of the VEK black hole gets nontrivially coupled to rotation in sharp contrast to the Kerr case. We show that the angular velocity of the VEK horizon goes up significantly as the background influence increases. By using the `equatorial tangential velocity' of the VEK horizon we classify the horizon and define the `limiting black hole' a generalization that contains the extreme Kerr black hole as a special case. Finally we investigate the Gaussian curvature and establish conditions for global embedding of the VEK black hole in Euclidean space.