Complete Integrability of Geodesic Motion in General Kerr-NUT-AdS Spacetimes

We explicitly exhibit n-1 constants of motion for geodesics in the general D-dimensional Kerr-NUT-AdS rotating black hole spacetime, arising from contractions of even powers of the 2-form obtained by contracting the geodesic velocity with the dual of the contraction of the velocity with the (D-2)-dimensional Killing-Yano tensor. These constants of motion are functionally independent of each other and of the D-n+1 constants of motion that arise from the metric and the D-n = [(D+1)/2] Killing vectors, making a total of D independent constants of motion in all dimensions D. The Poisson brackets of all pairs of these D constants are zero, so geodesic motion in these spacetimes is completely integrable.Comment: 4 pages. We have now found that the geodesic motion is not just integrable, but completely integrabl

Killing-Yano Tensors, Rank-2 Killing Tensors, and Conserved Quantities in Higher Dimensions

From the metric and one Killing-Yano tensor of rank D-2 in any D-dimensional spacetime with such a principal Killing-Yano tensor, we show how to generate k=[(D+1)/2] Killing-Yano tensors, of rank D-2j for all j=0,...,k-1, and k rank-2 Killing tensors, giving k constants of geodesic motion that are in involution. For the example of the Kerr-NUT-AdS spacetime (hep-th/0604125) with its principal Killing-Yano tensor (gr-qc/0610144), these constants and the constants from the k Killing vectors give D independent constants in involution, making the geodesic motion completely integrable (hep-th/0611083). The constants of motion are also related to the constants recently obtained in the separation of the Hamilton-Jacobi and Klein-Gordon equations (hep-th/0611245).Comment: 7 page

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

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.Comment: 33 pages, no figures, updated references and corrected typo