Hidden Symmetries of Higher-Dimensional Rotating Black Holes

TL;DR

This thesis explores the hidden symmetries of higher-dimensional rotating black holes, revealing their integrability, separability of key equations, and the unique algebraic properties of their metrics, advancing understanding in string theory and higher-dimensional gravity.

Contribution

It demonstrates that the most general higher-dimensional Kerr-NUT-(A)dS spacetime admits a principal conformal Killing-Yano tensor, leading to complete integrability and separability of fundamental equations.

Findings

Existence of a principal conformal Killing-Yano tensor in these spacetimes

Complete integrability of geodesic equations in Kerr-NUT-(A)dS spacetime

Separable Hamilton-Jacobi, Klein-Gordon, and string equations

Abstract

In this thesis we study higher-dimensional rotating black holes. Such black holes are widely discussed in string theory and brane-world models at present. We demonstrate that even the most general known Kerr-NUT-(A)dS spacetime, describing the general rotating higher-dimensional asymptotically (anti) de Sitter black hole with NUT parameters, is in many aspects similar to its four-dimensional counterpart. Namely, we show that it admits a fundamental hidden symmetry associated with the principal conformal Killing-Yano tensor. Such a tensor generates towers of hidden and explicit symmetries. The tower of Killing tensors is responsible for the existence of irreducible, quadratic in momenta, conserved integrals of geodesic motion. These integrals, together with the integrals corresponding to the tower of explicit symmetries, make geodesic equations in the Kerr-NUT-(A)dS spacetime completely integrable. We further demonstrate that in this spacetime the Hamilton-Jacobi, Klein-Gordon, and stationary string equations allow complete separation of variables and the problem of finding parallel-propagated frames reduces to the set of the first order ordinary differential equations. Moreover, we show that the Kerr-NUT-(A)dS spacetime is the most general Einstein space which possesses all these properties. We also explicitly derive the most general (off-shell) canonical metric admitting the principal conformal Killing-Yano tensor and demonstrate that such a metric is necessarily of the special algebraic type D of the higher-dimensional algebraic classification. The results presented in this thesis describe the new and complete picture of the relationship of hidden symmetries and rotating black holes in higher dimensions.