Chaotic particle motion around a homogeneous circular ring

TL;DR

This paper investigates the motion of test particles around a homogeneous circular ring in various dimensions, revealing the existence of stable orbits, integrability properties, and signs of chaos, with implications for black ring spacetimes.

Contribution

It analyzes particle dynamics in higher-dimensional rings, identifying stability, integrability, and chaos, and connects these findings to black ring spacetime geodesics.

Findings

No stable orbits in dimensions 6-10

Stable orbits exist in dimensions 3-5

Chaotic orbits appear in dimension 5

Abstract

We consider test particle motion in a gravitational field generated by a homogeneous circular ring placed in $n$-dimensional Euclidean space. We observe that there exist no stable stationary orbits in $n=6, 7, \ldots, 10$ but exist in $n=3, 4, 5$ and clarify the regions in which they appear. In $n=3$, we show that the separation of variables of the Hamilton-Jacobi equation does not occur though we find no signs of chaos for stable bound orbits. Since the system is integrable in $n=4$, no chaos appears. In $n=5$, we find some chaotic stable bound orbits. Therefore, this system is nonintegrable at least in $n=5$ and suggests that the timelike geodesic system in the corresponding black ring spacetimes is nonintegrable.