Axisymmetric generalization of zero-scalar-curvature solutions from the Schwarzschild metric via the Newman-Janis algorithm

We address a specific issue of the Newman-Janis algorithm: How to determine the general form of the complex transformation for the Schwarzschild metric and ensure that the resulting axisymmetric metric satisfies the zero-scalar-curvature condition, $R=0$. In this context, the zero-scalar-curvature condition acts as a constraint. Owing to this condition, we refer to the class of black holes as the ``Newman-Janis class of Schwarzschild black holes" in order to emphasize Newman-Janis algorithm's potential as a classification tool for axisymmetric black holes. The general complex transformation we derive not only generates the Kerr, Taub-NUT, and Kerr-Taub-NUT black holes under specific choices of parameters but also suggests the existence of additional axisymmetric black holes. Our findings open an alternative avenue using the Newman-Janis algorithm for the construction of new axisymmetric black holes.