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

TL;DR

This paper explores hidden symmetries in higher-dimensional rotating black hole spacetimes, showing how principal CKY tensors generate symmetries that enable complete integrability and separation of variables for key equations.

Contribution

It demonstrates the existence of principal CKY tensors in higher-dimensional Kerr-NUT-(A)dS metrics and their role in generating symmetries for integrability and variable separation.

Findings

Higher-dimensional Kerr-NUT-(A)dS metrics possess a principal CKY tensor.

Hidden symmetries lead to complete integrability of geodesic equations.

Separation of variables is possible for Hamilton-Jacobi, Klein-Gordon, and Dirac 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.