On the analytic extension of regular rotating black holes

We hereby focus on the analytic geodesic extension of several regular rotating black holes (RRBHs) obtained throughout the Newman-Janis algorithm starting from some popular spherically symmetric regular black holes. It turns out that if the metric is not an even function of Boyer-Lindquist radial coordinate r, similarly to the Kerr spacetime, the metric has to be extended to negative values of r to ensure the analyticity of the geodesic equations (and in turn of the geodesics). Therefore, some of the extended RRBHs considered in this paper, such as the rotating Hayward black hole, are geodetically incomplete because they are singular somewhere for r < 0, and non-analytic at the ring located in $(r = 0, {\theta = \pi/2})$. Conversely, other spacetimes, like nonlocal black holes, can be analytically extended to negative r. However, the real issue shows up at the ring, where, unfortunately, all the RRBHs studied in this paper fail to be regular. Indeed, at the ring, the Kretschmann invariant is finite but nonanalytic, while the higher derivative curvature invariants are divergent. In order to avoid such catastrophe, we propose a modification of the RRBHs in which the angular momentum is promoted to a function of the radial coordinate. According to our proposal, the angular momentum vanishes for $r\rightarrow 0$ and the ring shrinks to a point. Therefore, the regularity properties of the regular spherically symmetric black holes are recovered for $r\rightarrow 0$.