Regular Kerr black holes: The fellowship of the string

In their interesting article entitled "Kerrr" black hole: The lord of the string [Phys. Lett. B bf 688, 82 (2010)], Smailagic and Spallucci report on a regular Kerr-like black hole spacetime with a string in the central region. The metric is of the Kerr-Schild form with a particular mass function, that is chosen in order to avoid the ring singularity of the Kerr metric. However, as it is known, even for a class of well-behaved mass functions, the curvature scalars present a discontinuity in the equatorial plane at the ring. The authors associate such a discontinuity with the presence of a string of matter and join the interior and exterior regions of the string by making a particular choice of the orientations of the normal vectors at each side of the ring. Differently from what was interpreted by the authors, we show here that the string is not rotating in the scenario of that work. We also show that the choice regarding the normal vector orientation leads to a geometry that is not compatible with the center of a rotating black hole in an asymptotically flat spacetime. We then consider all the four possible combinations of the normal vector orientation at each side of the ring and, by using the Darmois-Israel formalism, interpret each resulting solution. We show that, out of the four possibilities, only one of the four models for the thin shell solution at the ring yields the appropriate geometry. More importantly, this solution does not correspond to the one obtained by Smailagic and Spallucci. Finally, we also discuss a scenario where the string is indeed rotating.