Integrability of Some Charged Rotating Supergravity Black Hole Solutions in Four and Five Dimensions

TL;DR

This paper demonstrates the integrability of geodesic flow and scalar field equations in certain charged rotating supergravity black hole spacetimes in four and five dimensions, using Killing tensors and separation of variables.

Contribution

It constructs explicit Killing tensors for these solutions, proving integrability and separability of equations of motion and scalar fields in these complex backgrounds.

Findings

Killing tensors enable separation of Hamilton-Jacobi equation

Scalar Klein-Gordon equation is separable in these spacetimes

Results apply to a broad class of supergravity black hole solutions

Abstract

We study the integrability of geodesic flow in the background of some recently discovered charged rotating solutions of supergravity in four and five dimensions. Specifically, we work with the gauged multicharge Taub-NUT-Kerr-(Anti) de Sitter metric in four dimensions, and the $U(1)^3$ gauged charged-Kerr-(Anti) de Sitter black hole solution of N = 2 supergravity in five dimensions. We explicitly construct the Killing tensors that permit separation of the Hamilton-Jacobi equation in these spacetimes. These results prove integrability for a large class of previously known supergravity solutions, including several BPS solitonic states. We also derive first-order equations of motion for particles in these backgrounds and examine some of their properties. Finally, we also examine the Klein-Gordon equation for a scalar field in these spacetimes and demonstrate separability.