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

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.

12 pages, 2 figures

References in corpus (13)

• Bicycling Black Rings
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Cited by in corpus (4)

• Innermost stable circular orbits in Majumdar--Papapetrou dihole spacetime
• Particle dynamics in the Newtonian potential sourced by a homogeneous circular ring
• Effect of Multipole Moments in the Weak Field Limit of a Black Hole Plus Halo Potential
• Chaotic particle motion around a homogeneous circular ring