Hidden Symmetries and Geodesics of Kerr spacetime in Kaluza-Klein Theory

TL;DR

This paper investigates the hidden symmetries and geodesic properties of Kerr black holes in Kaluza-Klein theory, revealing how extra dimensions influence orbital stability and the structure of spacetime.

Contribution

It demonstrates the separability of the Hamilton-Jacobi equation for massless geodesics and constructs explicit conformal Killing tensors in Kerr-Kaluza-Klein spacetime.

Findings

Complete separation occurs only for massless geodesics.

Extra dimension enlarges regions of stable and bounded orbits.

Analytical formulas for orbital frequencies and stability boundaries.

Abstract

The Kerr spacetime in Kaluza-Klein theory describes a rotating black hole in four dimensions from the Kaluza-Klein point of view and involves the signature of an extra dimension that shows up through the appearance of the electric and dilaton charges. In this paper, we study the separability properties of the Hamilton-Jacobi equation for geodesics and the associated hidden symmetries in the spacetime of the Kerr-Kaluza-Klein black hole. We show that the complete separation of variables occurs only for massless geodesics, implying the existence of hidden symmetries generated by a second rank conformal Killing tensor. Employing a simple procedure built up on an "effective" metric, which is conformally related to the original spacetime metric and admits a complete separability structure, we construct the explicit expression for the conformal Killing tensor. Next, we study the properties of the geodesic motion in the equatorial plane, focusing on the cases of static and rotating Kaluza-Klein black holes separately. In both cases, we obtain the defining equations for the boundaries of the regions of existence, boundedness and stability of the circular orbits as well as the analytical formulas for the orbital frequency, the radial and vertical epicyclic frequencies of the geodesic motion. Performing a detailed numerical analysis of these equations and frequencies, we show that the physical effect of the extra dimension amounts to the significant enlarging of the regions of existence, boundedness and stability towards the event horizon, regardless of the classes of orbits.