TY - JOUR

T1 - Chaos in Schwarzschild spacetime

T2 - The motion of a spinning particle

AU - Suzuki, Shingo

AU - Maeda, Kei ichi

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1997

Y1 - 1997

N2 - We study the motion of a spinning test particle in Schwarzschild spacetime, analyzing the Poincaré map and the Lyapunov exponent. We find chaotic behavior for a particle with spin higher than some critical value (e.g., [Formula presented] for the total angular momentum [Formula presented], where [Formula presented] and [Formula presented] 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 five 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 [Formula presented] is large, some orbits are bounded and the “effective potentials” 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 [Formula presented] 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).

AB - We study the motion of a spinning test particle in Schwarzschild spacetime, analyzing the Poincaré map and the Lyapunov exponent. We find chaotic behavior for a particle with spin higher than some critical value (e.g., [Formula presented] for the total angular momentum [Formula presented], where [Formula presented] and [Formula presented] 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 five 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 [Formula presented] is large, some orbits are bounded and the “effective potentials” 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 [Formula presented] 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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U2 - 10.1103/PhysRevD.55.4848

DO - 10.1103/PhysRevD.55.4848

M3 - Article

AN - SCOPUS:0001693720

VL - 55

SP - 4848

EP - 4859

JO - Physical Review D - Particles, Fields, Gravitation and Cosmology

JF - Physical Review D - Particles, Fields, Gravitation and Cosmology

SN - 1550-7998

IS - 8

ER -