Regular Kerr black holes: junction conditions and the matter content across the ring

Abstract

Regular rotating black holes are usually described by a metric of the Kerr–Schild form with a particular mass function that is chosen to avoid the ring singularity of the Kerr metric and which approaches the Kerr metric at the asymptotic limit. However, as is well known, even for a class of well-behaved mass functions, the curvature scalars present a discontinuity in the equatorial plane at the ring. This discontinuity has been associated with the presence of a string of matter that joins the interior and exterior regions along the equatorial plane. By using the Darmois–Israel junction conditions, we analyze all four possible combinations of the normal vector orientations on each side of the ring, construct the complete stress-energy momentum tensor of the string, and interpret each resulting solution. We show that, out of the four possibilities, only one of the four models for the string solution at the ring yields the appropriate asymptotic geometry. In such a case, the string bears a fluid with nonzero pressure, but with a vanishing line energy density, and it does not rotate at all. Finally, taking an appropriate metric for the exterior region, we also discuss a different scenario in which the matter source at the ring is a rotating lightlike fluid.