Chaos in Geodesic Motion around a Black Ring

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Abstract

We study bound orbits of a free particle around a singly rotating black ring. We find there exists chaotic motion of a particle which is gravitationally bound to the black ring by using the Poincare map.

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A rotating black ring in five dimensions

Roberto Emparan, Harvey Reall (2002)

The vacuum Einstein equations in five dimensions are shown to admit a solution describing an asymptotically flat spacetime regular on and outside an event horizon of topology S^1 x S^2. It describes a rotating ``black ring''. This is the first example of an asymptotically flat vacuum solution with an event horizon of non-spherical topology. There is a range of values for the mass and angular momentum for which there exist two black ring solutions as well as a black hole solution. Therefore the uniqueness theorems valid in four dimensions do not have simple higher dimensional generalizations. It is suggested that increasing the spin of a five dimensional black hole beyond a critical value results in a transition to a black ring, which can have an arbitrarily large angular momentum for a given mass.

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Chaos in Schwarzschild Spacetime : The Motion of a Spinning Particle

Shingo Suzuki, Kei-ichi Maeda (1996)

We study the motion of a spinning test particle in Schwarzschild spacetime, analyzing the Poincar\'e map and the Lyapunov exponent. We find chaotic behavior for a particle with spin higher than some critical value (e.g. \(S_{cr} \sim 0.64 \mu M\) for the total angular momentum \(J=4 \mu M\)), where \(\mu\) and \(M\) are the masses of a particle and of a black hole, respectively. The inverse of the Lyapunov exponent in the most chaotic case is about three orbital periods, which suggests that chaos of a spinning particle may become important in some relativistic astrophysical phenomena. The ``effective potential" analysis enables us to classify the particle orbits into four types as follows. When the total angular momentum \(J\) is large, some orbits are bounded and the ``effective potential"s are classified into two types: (B1) one saddle point (unstable circular orbit) and one minimal point (stable circular orbit) on the equatorial plane exist for small spin; and (B2) two saddle points bifurcate from the equatorial plane and one minimal point remains on the equatorial plane for large spin. When \(J\) is small, no bound orbits exist and the potentials are classified into another two types: (U1) no extremal point is found for small spin; and (U2) one saddle point appears on the equatorial plane, which is unstable in the direction perpendicular to the equatorial plane, for large spin. The types (B1) and (U1) are the same as those for a spinless particle, but the potentials (B2) and (U2) are new types caused by spin-orbit coupling. The chaotic behavior is found only in the type (B2) potential. The ``heteroclinic orbit'', which could cause chaos, is also observed in type (B2).

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Particle Motion in the Rotating Black Ring Metric

James Hoskisson (2007)

In this paper, the equations of motion for geodesics in the neutral rotating Black Ring metric are derived and the separability of these equations is considered. The bulk of the paper is concerned with sets of solutions where the geodesic equations can be examined analytically - specifically geodesics confined to the axis of rotation, geodesics restricted to the equatorial plane, and geodesics that circle through the centre of the ring. The geodesics on the rotational axis behave like a particle in a potential well, while the geodesics confined to the equatorial plane mimic those of the Schwarzschild metric. It is shown that it is impossible to have circular orbits that pass through the ring, but some numerical results are presented which suggest that it is possible to have bound orbits that circle through the ring.