Chaos and Fractals Around Black Holes

Dettmann, Carl P.; Frankel, N. E.; Cornish, NJ

doi:10.1142/s0218348x9500014x

Cited by 47 publications

(32 citation statements)

References 32 publications

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“…In this context, we have also experimented with norms reflecting the curvature of the spacetime, however, it appeared that for a given application the choice of the norm is not crucial (Kopáček et al 2010a). Although Lyapunov exponents within the relativistic framework are generally not invariant under coordinate transformations (Karas & Vokrouhlický 1992;Dettmann et al 1995), signs of the exponents are preserved (Motter 2003;Motter & Saa 2009). Therefore, the distinction between chaotic and regular orbits may be drawn invariantly.…”

Section: Maximal Lyapunov Characteristic Exponentmentioning

confidence: 99%

Inducing Chaos by Breaking Axial Symmetry in a Black Hole Magnetosphere

Kopáček

Karas

2014

ApJ

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While the motion of particles near a rotating, electrically neutral (Kerr), and charged (Kerr-Newman) black hole is always strictly regular, a perturbation in the gravitational or the electromagnetic field generally leads to chaos. The transition from regular to chaotic dynamics is relatively gradual if the system preserves axial symmetry, whereas non-axisymmetry induces chaos more efficiently. Here we study the development of chaos in an oblique (electrovacuum) magnetosphere of a magnetized black hole. Besides the strong gravity of the massive source represented by the Kerr metric, we consider the presence of a weak, ordered, large-scale magnetic field. An axially symmetric model consisting of a rotating black hole embedded in an aligned magnetic field is generalized by allowing an oblique direction of the field having a general inclination with respect to the rotation axis of the system. The inclination of the field acts as an additional perturbation to the motion of charged particles as it breaks the axial symmetry of the system and cancels the related integral of motion. The axial component of angular momentum is no longer conserved and the resulting system thus has three degrees of freedom. Our primary concern within this contribution is to find out how sensitive the system of bound particles is to the inclination of the field. We employ the method of the maximal Lyapunov exponent to distinguish between regular and chaotic orbits and to quantify their chaoticity. We find that even a small misalignment induces chaotic motion.

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Section: Maximal Lyapunov Characteristic Exponentmentioning

confidence: 99%

Inducing Chaos by Breaking Axial Symmetry in a Black Hole Magnetosphere

Kopáček

Karas

2014

ApJ

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“…Thus, the question arises whether the Lyapunov dimension is invariant under changes (see, e.g., [14], "Is the Dimension of Chaotic Attractors Invariant under Coordinate Changes? "; and [15][16][17]). Since the singular values and LEs are defined via the linearization of a system, the smooth changes of variables are of interest.…”

Section: Introductionmentioning

confidence: 96%

Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System

Леонов

Alexeeva

Kuznetsov

2015

Entropy

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In applied investigations, the invariance of the Lyapunov dimension under a diffeomorphism is often used. However, in the case of irregular linearization, this fact was not strictly considered in the classical works. In the present work, the invariance of the Lyapunov dimension under diffeomorphism is demonstrated in the general case. This fact is used to obtain the analytic exact upper bound of the Lyapunov dimension of an attractor of the Shimizu-Morioka system.

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“…1. Chaotic scattering in GR spacetimes has been observed and discussed in binary or multi-BH solutions (see, e.g., [45,46,[74][75][76][77][78][79][80][81][82][83][84]) and is well known in the context of manybody scattering in classical dynamics, for example, the scattering of charged particles off magnetic dipoles [85] and the three-body problem (see, e.g., [86]). KBHsSH or RBSs provide an example of chaos in geodesic motion on the background of a single compact object, which moreover solves a simple and well-defined matter model minimally coupled to GR.…”

Section: Introductionmentioning

confidence: 99%

Chaotic lensing around boson stars and Kerr black holes with scalar hair

et al. 2016

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In a recent paper [P. V. P. Cunha, C. A. R. Herdeiro, E. Radu, and H. F. Runarsson, Phys. Rev. Lett. 115, 211102 (2015).], it was shown that the lensing of light around rotating boson stars and Kerr black holes with scalar hair can exhibit chaotic patterns. Since no separation of variables is known (or expected) for geodesic motion on these backgrounds, we examine the 2D effective potentials for photon trajectories, to obtain a deeper understanding of this phenomenon. We find that the emergence of stable light rings on the background spacetimes allows the formation of "pockets" in one of the effective potentials, for open sets of impact parameters, leading to an effective trapping of some trajectories, dubbed "quasibound orbits." We conclude that pocket formation induces chaotic scattering, although not all chaotic orbits are associated to pockets. These and other features are illustrated in a gallery of examples, obtained with a new ray-tracing code, PYHOLE, which includes tools for a simple, simultaneous visualization of the effective potential, together with the spacetime trajectory, for any given point in a lensing image. An analysis of photon orbits allows us to further establish a positive correlation between photon orbits in chaotic regions and those with more than one turning point in the radial direction; we recall that the latter is not possible around Kerr black holes. Moreover, we observe that the existence of several light rings around a horizon (several fundamental orbits, including a stable one), is a central ingredient for the existence of multiple shadows of a single hairy black hole. We also exhibit the lensing and shadows by Kerr black holes with scalar hair, observed away from the equatorial plane, obtained with PYHOLE.

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Chaos and Fractals Around Black Holes

Cited by 47 publications

References 32 publications

Inducing Chaos by Breaking Axial Symmetry in a Black Hole Magnetosphere

Inducing Chaos by Breaking Axial Symmetry in a Black Hole Magnetosphere

Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System

Chaotic lensing around boson stars and Kerr black holes with scalar hair

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