Higher-Dimensional Black Holes: Hidden Symmetries and Separation of Variables

TL;DR

The paper addresses hidden symmetries in rotating black holes by developing and leveraging a principal conformal Killing–Yano (CKY) tensor to generate a complete tower of Killing–Yano and Killing tensors. Starting from 4D flat space, it derives the Kerr–NUT–(A)dS metric and shows how the principal CKY tensor yields canonical coordinates and separability for the Hamilton–Jacobi, Klein–Gordon, and Dirac equations; this construction extends to higher dimensions, where the CKY tensor seeds a rich hierarchy of symmetry operators and ensures geodesic integrability. The main contributions are the formulation and proof of a general tower of hidden symmetries, the introduction of canonical coordinates tied to the principal CKY tensor, and the demonstration that these structures guarantee complete separability of key field equations in Kerr–NUT–(A)dS spacetimes across dimensions. This unifies the symmetry structure of higher-dimensional black holes with the well-known 4D Kerr case, enabling precise analysis of particle and field propagation and offering a framework to study stability and perturbations, though open questions remain for gravitational and higher-spin equations.

Abstract

In this paper, we discuss hidden symmetries in rotating black hole spacetimes. We start with an extended introduction which mainly summarizes results on hidden symmetries in four dimensions and introduces Killing and Killing-Yano tensors, objects responsible for hidden symmetries. We also demonstrate how starting with a principal CKY tensor (that is a closed non-degenerate conformal Killing-Yano 2-form) in 4D flat spacetime one can "generate" 4D Kerr-NUT-(A)dS solution and its hidden symmetries. After this we consider higher-dimensional Kerr-NUT-(A)dS metrics and demonstrate that they possess a principal CKY tensor which allows one to generate the whole tower of Killing-Yano and Killing tensors. These symmetries imply complete integrability of geodesic equations and complete separation of variables for the Hamilton-Jacobi, Klein-Gordon, and Dirac equations in the general Kerr-NUT-(A)dS metrics.