Fractal basins and chaotic trajectories in multi-black-hole spacetimes

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

doi:10.1103/physrevd.50.r618

Cited by 93 publications

(118 citation statements)

References 23 publications

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“…Chandrasekhar [7] considered the Einstein-Maxwell two centre problem but was unable to integrate the equations of motion. Methods borrowed from dynamical systems theory were then used to prove that null and timelike geodesics of the Majumdar-Papapetrou spacetime were chaotic [8,9]. Here we extend these results to include null and timelike geodesics of the general Einstein-Maxwell-dilaton two centre spacetimes.…”

mentioning

confidence: 89%

“…The term "repellor" or "saddle" indicates that the orbits are unstable in some directions and stable in others. Uncovering these fractal structures in phase space provides a gauge invariant way of showing that the motion is chaotic [9,10] Of particular interest is the future invariant set. For unbound motion, the future invariant set correspond to those trajectories that approach the two centres from infinity with an impact parameter that allows them to take up everlasting periodic orbits.…”

Section: B Fractal Methodsmentioning

confidence: 99%

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A tale of two centres

Cornish

Gibbons

1997

Class. Quantum Grav.

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mentioning

confidence: 89%

Section: B Fractal Methodsmentioning

confidence: 99%

A tale of two centres

Cornish

Gibbons

1997

Class. Quantum Grav.

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“…Yet another kind of approach are the basin boundaries [25], which take advantage of the fractal geometry of a non-integrable system to detect the existence of chaos. The methods which use the curvature of a spacetime to search for chaos [15,26] are also Geometrical.…”

Section: Introductionmentioning

confidence: 99%

Adjusting chaotic indicators to curved spacetimes

Lukes-Gerakopoulos¹

2014

Phys. Rev. D

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In this work, chaotic indicators, which have been established in the framework of classical mechanics, are reformulated in the framework of general relativity in such a way that they are invariant under coordinate transformation. For achieving this, the prescription for reformulating mLCE given by [Y. Sota, S. Suzuki, and K.-I. Maeda, Classical Quantum Gravity 13, 1241 (1996)] is adopted. Thus, the geodesic deviation vector approach is applied, and the proper time is utilized as measure of time. Following the aforementioned prescription, the chaotic indicators FLI, MEGNO, GALI, and APLE are reformulated. In fact, FLI has been reformulated by adapting other prescriptions in the past, but not by adapting the Sota et al. one. By using one of these previous reformulations of FLI, an approximative expression giving MEGNO as function of FLI has been applied on non-integrable curved spacetimes in a recent work. In the present work the reformulation of MEGNO is provided by adjusting the definition of the indicator to the Sota et al. prescription. GALI, and APLE are reformulated in the framework of general relativity for the first time. All the reformulated indicators by the Sota et al. prescription are tested and compared for their efficiency to discern order from chaos.

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“…4 we see that the central basin of instability also has a fractal boundary. Fractal structures provide a gauge invariant signal of chaos in general relativity [4,12,13].…”

Section: R/mmentioning

confidence: 99%

“…It has previously been noted that a range of perturbations lead to chaotic dynamics. The perturbations considered include additional mass concentrations [1,2,3,4,5], magnetic fields [6], gravitational waves [7] and spin-orbit coupling [8]. Unlike the Kepler problem of Newtonian mechanics, essentially any perturbation of an isolated black hole spacetime will lead to chaotic orbits.…”

Section: Um-p-96/45mentioning

confidence: 99%

The black hole and the pea

Cornish

Frankel

1997

Phys. Rev. D

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Like a fairy-tale princess, trajectories around black holes can be sensitive to small disturbances. We describe how a small disturbance can lead to erratic orbits and an increased production of gravitational waves. 04.30.Db, 05.45.+b, 97.60.Lf UM-P-96/45The high degree of symmetry found in the spacetime of an isolated black hole leads to regular geodesic motion for orbiting bodies. Working on the assumption that small disturbances generally have small effects, it is often tacitly assumed that this idealised, textbook picture carries over to real astrophysical situations. Here we want to emphasise that the idealised picture is not stable against small perturbations, as the non-linearity of Einstein's equations tends to amplify small disturbances.The observation that small perturbations of an idealised black hole spacetime can lead to qualitative changes in the dynamics is not new. It has previously been noted that a range of perturbations lead to chaotic dynamics. The perturbations considered include additional mass concentrations [1,2,3,4,5], magnetic fields [6], gravitational waves [7] and spin-orbit coupling [8]. Unlike the Kepler problem of Newtonian mechanics, essentially any perturbation of an isolated black hole spacetime will lead to chaotic orbits. This is because even the most pristine black hole spacetime harbours the seeds of chaos in the form of isolated unstable orbits. A small perturbation causes these unstable orbits to break out and infest large regions of phase space. Note that experience with Newtonian systems is very misleading. For example, the Kepler problem has more integrals of motion than are needed for integrability. Keplerian systems are thus impervious to small perturbations. In contrast, black hole spacetimes are at the edge of chaos, just waiting for the proverbial butterfly to flap its wings.Once it is realised that typical black hole -satellite systems are chaotic in the strong field regime, we are forced to consider the consequences. One of the most immediate consequences is that there will be no such thing as the "last stable orbit" [9]. The boundary between stable and unstable orbits will be fractal, and there may be large fractal tendrils of unstable orbits invading what would have been stable territory in an ideal black hole spacetime. This feature will be important in determining when a binary system switches from inspiral to cataclysmic collapse [9]. Another important consequence will be the increased production of gravitational waves due to the erratic motion of chaotic orbits.In what follows we will illustrate both of these effects in a simple model system. While our model does not describe a real astrophysical situation, it does capture many of the salient features we expect to find in a relativistic binary system.For the purpose of illustration we will study the orbits of non-rotating satellites around an extreme RiessnerNordstrom black hole. Almost identical results hold for motion around a Schwarzschild black hole perturbed by an orbiting third body [3]. Extreme black ...

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Fractal basins and chaotic trajectories in multi-black-hole spacetimes

Cited by 93 publications

References 23 publications

A tale of two centres

A tale of two centres

Adjusting chaotic indicators to curved spacetimes

The black hole and the pea

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