Axisymmetric generalization of zero-scalar-curvature solutions from the Schwarzschild metric via the Newman-Janis algorithm
Chen Lan, Zi-Xiao Liu, Yan-Gang Miao
TL;DR
The paper investigates generating axisymmetric black holes with vanishing scalar curvature through a generalized Newman-Janis algorithm (NJA) starting from Schwarzschild seeds. By formulating a non-complexified NJA within the Newman-Penrose framework and enforcing the constraint $R=0$, the authors derive general complex transformations parameterized by $\alpha(\theta)$ and $\beta(\theta)$ that yield axisymmetric spacetimes; Kerr, Taub-NUT, and Kerr-Taub-NUT appear as special cases, while new zero-scalar-curvature axisymmetric solutions are predicted. A concrete example with explicit metric components, horizons, and thermodynamic properties demonstrates a distinct axisymmetric black hole from Kerr. The approach positions NJA as both a classification tool and a constructive method for novel axisymmetric black holes, with potential extensions to gravity theories beyond general relativity such as Chern-Simons gravity and regular black hole models.
Abstract
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.