Generating rotating regular black hole solutions without complexification

Abstract: We drop the complexification procedure from the Newman-Janis algorithm and introduce more physical arguments and symmetry properties, and we show how one can generate regular and singular rotating black hole and non-black-hole solutions in Boyer-Lindquist coordinates. We focus on generic rotating regular black holes and show that they are regular on the Kerr-like ring but physical entities are undefined there. We show that rotating regular black holes have much smaller electric charges, and, with increasing charge, they turn into regular non-black-hole solutions well before their Kerr-Newman counterparts become naked singularities. No causality violations occur in the region inside a rotating regular black hole. The separability of the Hamilton-Jacobi equation for neutral particles is also carried out in the generic case, and the innermost boundaries of circular orbits for particles are briefly discussed. Other, but special, properties pertaining to the rotating regular counterpart of the Ay\'on-Beato--Garc\'{\i}a regular static black hole are also investigated.