Integrability of Particle System around a Ring Source as the Newtonian Limit of a Black Ring

TL;DR

This paper demonstrates that the Newtonian limit of a black ring's geodesic system is integrable, revealing that chaos in the full relativistic system arises from relativistic effects rather than the underlying Newtonian potential.

Contribution

It shows that the Newtonian limit of a black ring admits separation of variables and a quadratic constant of motion, indicating integrability absent in the full relativistic case.

Findings

Newtonian limit admits separation of variables in spheroidal coordinates

A non-trivial quadratic constant of motion exists in the Newtonian limit

Geodesic chaos is attributed to relativistic effects

Abstract

The geodesic equation in the five-dimensional singly rotating black ring is non-integrable unlike the case of the Myers-Perry black hole. In the Newtonian limit of the black ring, its geodesic equation agrees with the equation of motion of a particle in the Newtonian potential due to a homogeneous ring gravitational source. In this paper, we show that the Newtonian equation of motion allows the separation of variables in the spheroidal coordinates, providing an non-trivial constant of motion quadratic in momenta. This shows that the Newtonian limit of a black ring recovers the symmetry of its geodesic system, and the geodesic chaos is caused by relativistic effects.