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

TL;DR

This paper demonstrates how to generate multiple Killing-Yano tensors and Killing tensors from a principal tensor in higher-dimensional spacetimes, leading to complete integrability of geodesic motion and linking conserved quantities across different equations.

Contribution

It introduces a method to generate a hierarchy of Killing-Yano and Killing tensors from a principal tensor in higher dimensions, establishing their role in integrability.

Findings

Generated multiple Killing-Yano tensors from a principal tensor.

Established complete integrability of geodesic motion in Kerr-NUT-AdS spacetime.

Connected conserved quantities to separation constants in Hamilton-Jacobi and Klein-Gordon equations.

Abstract

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).