Kerr-Schild transformation of the Benenti-Francaviglia metric

TL;DR

The paper addresses generating rotating, charged AdS-like black hole solutions within the Kerr–Schild framework by focusing on the degenerate Benenti–Francaviglia metrics that admit shear-free null geodesics. By enforcing Killing symmetry and circularity, it shows that Kerr–Schild deformations map back into the degenerate BF class through a simple replacement Q(q) → tilde{Q}(q), preserving geodesic separability and hidden symmetries. This approach yields a dyonic generalization of the Chong–Cvetič–Lu–Pope solution from an Einstein–scalar seed in four dimensions and extends naturally to five dimensions under suitable geodesic constraints. The work also connects conformal and nonconformal distortions within the same framework, highlighting a unified seed–deformation perspective for exact solutions and offering a systematic avenue to enlarge the landscape of higher-dimensional black hole metrics with preserved integrability properties.

Abstract

The Benenti-Francaviglia (BF) family of metrics provides the most general form of a spacetime metric that admits two mutually commuting Killing vectors and an irreducible Killing tensor. The geodesic equations for the BF family are thus completely integrable by separation of variables. Within this broad class, we explore the Kerr-Schild transformation of a degenerate subclass distinguished by the existence of a shear-free null geodesic congruence. By requiring the deformed metric to preserve the Killing symmetry and circularity, we demonstrate that the deformed metric again falls into the degenerate BF family, modulo the replacement of a single structure function. We apply the present algorithm to ${\cal N}=2$ gauged supergravity and obtain a dyonic generalization of the Chong-Cvetič-Lü-Pope rotating black hole solution, by taking the background metric to be a solution of the Einstein-scalar gravity. The present prescription extends to five dimensions, provided that the constant of geodesic motion associated with the extra Killing direction vanishes. The same reasoning applies to the case where the background degenerate BF metric is distorted in a (non)conformal manner. Our formalism offers a unified perspective on the relation between seed and deformed metrics in the Kerr-Schild construction.