Sourcing the Kerr Geometry

Abstract

The Kerr metric is a vacuum solution of the Einstein equations outside of a rotating black hole (BH), but what interior matter is actually rotating and sourcing the Kerr geometry? Here, a rotating exotic matter is described, which can source the Kerr geometry for the entire acceptable range of its spin parameter and be shown to saturate the radial null-energy condition at every point in the interior, while being free of any obvious pathologies. The rotating frozen star is introduced, whose compactness is controlled by a perturbative parameter and whose outer surface can be arbitrarily close to the horizon of a Kerr BH. The interior geometry modifies Kerr's such that there is neither an inner ergosphere nor an inner horizon. The geometry of each radial slice of the interior is a nearly null surface with the same geometry, but different radial size, as that of the would-be horizon on the outermost slice. The integral of the energy density leads to a rest mass that is equal to the irreducible mass of a Kerr BH, and the integral of the angular-momentum density confirms that the ratio of the angular momentum to the mass is equal to the Kerr spin parameter. Including the rotational energy in the standard way, the total gravitational mass and angular momentum of a Kerr BH with the same mass and spin parameters are obtained.