Free motion around black holes with discs or rings: between integrability and chaos - I

Semerák, O.; Suková, Petra

doi:10.1111/j.1365-2966.2009.16003.x

Cited by 66 publications

(66 citation statements)

References 109 publications

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“…The results of [11,12] point in the direction of the proof presented in this paper, whereas the results of [4,5,34,47] are mathematically inconsistent and the conclusions derived from them must be reformulated in light of the present approach. Since the stability criterion obtained and the adiabatic invariance analysis are both local, we can extend our formalism to black holes surrounded by razor-thin disks or rings [4,15,30,31,34], provided the metric outside the horizon has the form (5) and the orbit does not cross the horizon. In particular, the results obtained in [34,47] …”

Section: Discussionmentioning

confidence: 99%

“…is conserved for the Hamiltonian (30). The corresponding generating function for the vertical action-angle coordinates is given by…”

Section: A Motion Near a Stable Circular Orbitmentioning

confidence: 99%

“…Radial stability can also be analyzed by the relativistic generalization of Rayleigh's criterion [26][27][28]. Concerning razorthin disks, extensive numerical studies of the dynamics of timelike geodesic motion were performed in a wide class of spacetimes, including not only razor-thin disks, but also rings surrounding black holes [29][30][31][32][33]. Nevertheless, despite the many existing numerical results, the analytical description of motion around these structures is not yet fully developed.…”

Section: Introductionmentioning

confidence: 99%

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Vertical stability of circular orbits in relativistic razor-thin disks

2016

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During the last few decades, there has been a growing interest in exact solutions of Einstein equations describing razor-thin disks. Despite the progress in the area, the analytical study of geodesic motion crossing the disk plane in these systems is not yet so developed. In the present work, we propose a definite vertical stability criterion for circular equatorial timelike geodesics in static, axially symmetric thin disks, possibly surrounded by other structures preserving axial symmetry. It turns out that the strong energy condition for the disk stress-energy content is sufficient for vertical stability of these orbits. Moreover, adiabatic invariance of the vertical action variable gives us an approximate third integral of motion for oblique orbits which deviate slightly from the equatorial plane. Such new approximate third integral certainly points to a better understanding of the analytical properties of these orbits. The results presented here, derived for static spacetimes, may be a starting point to study the motion around rotating, stationary razor-thin disks. Our results also allow us to conjecture that the strong energy condition should be sufficient to assure transversal stability of periodic orbits for any singular timelike hypersurface, provided it is invariant under the geodesic flow.

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Section: Discussionmentioning

confidence: 99%

“…is conserved for the Hamiltonian (30). The corresponding generating function for the vertical action-angle coordinates is given by…”

Section: A Motion Near a Stable Circular Orbitmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Vertical stability of circular orbits in relativistic razor-thin disks

2016

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“…4.1. By the means of this approximative technique, FLI has already been applied in a few works (see, e.g., [21,32]), but FLI has not yet been tested by applying the geodesic deviation technique according to the author's knowledge.…”

Section: Flimentioning

confidence: 99%

“…For a regular orbit, MEGNO tends asymptotically to two, while if the orbit is chaotic it tends asymptotically to infinity. Recently, in the last article of the [21] series, the MEGNO was tested in curved spacetimes describing a Schwarzschild black hole surrounded by a thin disc or a ring. The authors of this article used the approximation given by Eq.…”

Section: Megnomentioning

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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Beyond Newtonian Dynamics of Planar CRTBP with Kerr—Like Primaries

Roychowdhury,

Banerjee

2023

Lecture Notes in Physics

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In this article, we shall discuss the dynamics of the classic planar circular restricted three-body problem (CRTBP) with spinning primaries in the context of a beyond-Newtonian approximation [27]. We shall begin by discussing the construction of the beyond-Newtonian potential in the so-called Fodor–Hoenselaers–Perjés procedure where we keep first order non-Newtonian contributions in both the mass and spin. Using this potential, we shall then discuss our model for a test particle of infinitesimal mass orbiting in the equatorial plane of the two primaries. The talk shall then discuss the dynamics as the system transitions from the Newtonian to the beyond-Newtonian regime. We shall then study the evolution and stability of the fixed points of the system as a function of the parameter $$\epsilon $$ ϵ with the dynamics of the particle analyzed using the Poincaré map of section and the Maximal Lyapunov Exponent as indicators of chaos. We shall establish that the intermediate values of $$\epsilon $$ ϵ seem to be the most chaotic for the two cases of primary mass ratios $$(=0.001, 0.5)$$ ( = 0.001 , 0.5 ) examined. We also conclude that the amount of chaos in the system remains higher than the Newtonian system as well as for the planar circular restricted three-body problem with Schwarzschild-like primaries for all non-zero values of $$\epsilon $$ ϵ .

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Free motion around black holes with discs or rings: between integrability and chaos - I

Cited by 66 publications

References 109 publications

Vertical stability of circular orbits in relativistic razor-thin disks

Vertical stability of circular orbits in relativistic razor-thin disks

Adjusting chaotic indicators to curved spacetimes

Beyond Newtonian Dynamics of Planar CRTBP with Kerr—Like Primaries

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