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arxiv: 2607.01910 · v1 · pith:QKU2SIZYnew · submitted 2026-07-02 · 🌀 gr-qc · hep-th

Chaotic behaviors of particles around the black hole with an anisotropic matter immersed in a magnetic field

Khusan Alibekov , Hocheol Lee , Yovqochev Pahlavon , Bobomurat Ahmedov , Bum-Hoon Lee , Ahmadjon Abdujabbarov , Wonwoo Lee This is my paper

Pith reviewed 2026-07-03 08:37 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords black holesanisotropic mattermagnetic fieldschaotic dynamicsLyapunov exponentPoincaré sectionsEinstein-Maxwell equationsHarrison transformation
0
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The pith

Anisotropic matter suppresses local chaotic particle motion around black holes while external magnetic fields drive qualitative shifts in global chaos.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs an exact solution to the Einstein-Maxwell equations for a static black hole surrounded by anisotropic matter in an external magnetic field, obtained by applying the Harrison transformation to a seed metric. It establishes that raising the anisotropic matter parameter produces a systematic drop in the Lyapunov exponent, indicating reduced local chaos in particle trajectories. In contrast, changes to the magnetic field strength produce visible transitions between regular and chaotic orbits when examined through Poincaré sections. The two ingredients therefore exert distinct effects on the nonlinear gravitational-magnetic dynamics. This supplies a concrete framework for analyzing non-integrable motion in magnetized black-hole spacetimes.

Core claim

An exact solution describing a static black hole coexisting with anisotropic matter immersed in an external magnetic field is obtained via the Harrison transformation. An increase in the anisotropic matter parameter systematically suppresses the local chaotic behavior, as indicated by a reduction in the Lyapunov exponent. Conversely, variations in the external magnetic field lead to qualitative changes in global chaotic behavior, analyzed through Poincaré sections which demonstrate transitions between regular and chaotic trajectories resulting from the nonlinear gravitational-magnetic interactions.

What carries the argument

The exact Einstein-Maxwell spacetime generated by the Harrison transformation, which incorporates the anisotropic matter stress-energy tensor together with the external magnetic field.

If this is right

  • The anisotropic matter parameter and the magnetic field strength play distinct yet complementary roles in shaping chaotic particle dynamics.
  • The solution supplies a new theoretical framework for exploring non-integrable particle motion within magnetized black hole spacetimes.
  • The construction is relevant for probing black holes at galactic centers where magnetic fields may arise from plasma effects surrounding astrophysical black holes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Observed reductions in orbital chaos could serve as an indirect signature of anisotropic matter distributions near black holes.
  • The same solution technique could be applied to rotating seed metrics to examine whether the suppression of local chaos persists in Kerr-like geometries.
  • Varying the magnetic field might produce measurable changes in the fraction of captured versus scattered particles, affecting accretion flow patterns.

Load-bearing premise

The Harrison transformation applied to the chosen seed solution yields a physically valid exact solution of the Einstein-Maxwell equations that correctly incorporates the anisotropic matter stress-energy tensor.

What would settle it

Numerical integration of particle geodesics showing whether the Lyapunov exponent decreases monotonically with rising anisotropic parameter, or whether Poincaré sections exhibit the claimed transitions when the magnetic field strength is varied.

Figures

Figures reproduced from arXiv: 2607.01910 by Ahmadjon Abdujabbarov, Bobomurat Ahmedov, Bum-Hoon Lee, Hocheol Lee, Khusan Alibekov, Wonwoo Lee, Yovqochev Pahlavon.

Figure 1
Figure 1. Figure 1: Metric function f(r) versus r/M for eight values of ω (panel labels) and different values of K (colored curves). Horizons occur at f = 0 (dashed line). Parameters: M = 1. In [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Event horizon radius rH/M versus ω (left) and K (right) [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Parameter space (w, K) showing horizon structure regions. Parameters: M = 1, Bo = 0.1 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (color online). Magnetic field lines around a black hole immersed in an external magnetic field in the presence of an anisotropic matter field In Figs. 4, we present the magnetic field configuration (Eq. 13) for a black hole surrounded by an anisotropic matter field and immersed in an external magnetic field. At large radial distances the field lines remain nearly uniform, consistent with the asymptotic Me… view at source ↗
Figure 5
Figure 5. Figure 5: (color online). Veffp(r) − E2 for varying Bo ∈ [0, 0.25] at K = 0.01 for the left panel and for varying K ∈ [0, 0.025] at Bo = 0.1 for the right panel. Each row corresponds to w = 0.7, w = 1.5, and w = 2.0, respectively. In the left panel of the figure, the right side is clearly separated based on the Bo value. In [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Particle trajectories for w = 0.7 (top-left), w = 1.5 (top-right, middle-left), and w = 2.0 (middle-right). Color: time evolution [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (color online). λ vs. L for Bo ∈ [0, 0.25] (K = 0.01, q = 0.05) for left panels, while λ vs. L for K ∈ [0, 0.06] (Bo = 0.1, q = 0.05) for right panels. Dotted: Schwarzschild [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Chaotic-region map in the (Bo, K) plane for w ∈ {0.7, 1.0, 3/2, 2.0} (L = 5, q = 0.05). Colored: λ > 0. Grey hatched: no unstable orbit. Cyan dashed: Kcrit(w). We map the full (Bo, K) plane in [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Chaotic-region map in the (B, q) plane for K ∈ {0.0, 0.2, 0.4, 0.6}. Colored: λ > 0. We map the full (B, q) plane in [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Particle trajectories for w = 0.7 (top-left), w = 1.5 (top-right, middle-left), and w = 2.0 (middle-right). Color: time evolution [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Color map of λ(L, E), Bo = 0.1, q = 0.05; K ∈ [0, 1.4]. Iso-K curves overlaid (legend). Gold dash-dot: Emin(L). (a) w = 0.7, (b) w = 3/2, (c) w = 2.0. We investigate the Lyapunov exponent in the physical initial-condition space spanned by the angular momentum L and the orbital energy E. For each spacetime specified by fixed values of Bo, K, w, and q, the energy of the unstable circular orbit is obtained n… view at source ↗
Figure 12
Figure 12. Figure 12: Poincar´e section on the equatorial plane with the partic [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Minkowski–Bouligand (box-counting) dimension [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
read the original abstract

We present an exact solution to the Einstein-Maxwell equations that describes a static black hole coexisting with anisotropic matter immersed in an external magnetic field, obtained via the Harrison transformation. Our findings reveal that an increase in the anisotropic matter parameter systematically suppresses the local chaotic behavior, as indicated by a reduction in the Lyapunov exponent. Conversely, variations in the external magnetic field lead to qualitative changes in global chaotic behavior. This is analyzed through Poincar\'e sections, which demonstrate transitions between regular and chaotic trajectories resulting from the nonlinear gravitational-magnetic interactions. These factors play distinct yet complementary roles in shaping chaotic particle dynamics around black holes. This study would offer a new theoretical framework for exploring non-integrable particle motion within magnetized black hole spacetimes and for probing a black hole at the galactic center, where magnetic fields may arise from plasma effects surrounding astrophysical black holes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to construct an exact solution of the Einstein-Maxwell equations for a static black hole surrounded by anisotropic matter in an external magnetic field by applying the Harrison transformation to a suitable seed. It then studies timelike geodesic motion in this spacetime and reports that increasing the anisotropic-matter parameter systematically reduces the Lyapunov exponent (suppressing local chaos), while changes in the magnetic-field strength produce qualitative transitions between regular and chaotic orbits visible in Poincaré sections.

Significance. If the metric is verified to solve the Einstein-Maxwell system with the claimed stress-energy content, the work supplies a new exact magnetized black-hole spacetime that can be used to explore non-integrable particle dynamics. The separation of local (Lyapunov) and global (Poincaré) diagnostics is a useful distinction for assessing the complementary roles of anisotropic matter and magnetic fields, with potential relevance to galactic-center environments.

major comments (2)
  1. [Solution construction section] The section describing the solution construction: the Harrison transformation is applied to a seed already containing anisotropic matter; the manuscript must explicitly compute the resulting effective T_{\mu\nu} and demonstrate that it remains purely anisotropic plus electromagnetic, with no extraneous sources introduced by the transformation. This verification is load-bearing for every subsequent claim about particle motion.
  2. [Lyapunov exponent analysis] The section on Lyapunov-exponent computation: the exponent is stated to be obtained from the geodesic equations, yet no explicit form of the variational equations, integration method, or convergence/error analysis is supplied. Without these details the reported systematic suppression with the anisotropic parameter cannot be assessed.
minor comments (1)
  1. Notation for the anisotropic-matter parameter and magnetic-field strength should be introduced once with a clear table of symbols; repeated redefinition in later sections reduces readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [Solution construction section] The section describing the solution construction: the Harrison transformation is applied to a seed already containing anisotropic matter; the manuscript must explicitly compute the resulting effective T_{\mu\nu} and demonstrate that it remains purely anisotropic plus electromagnetic, with no extraneous sources introduced by the transformation. This verification is load-bearing for every subsequent claim about particle motion.

    Authors: We agree that an explicit post-transformation computation of the effective stress-energy tensor is necessary to confirm the solution's validity. In the revised manuscript we will add the full calculation of T_{\mu\nu}, demonstrating that the resulting tensor contains only the original anisotropic-matter contribution together with the electromagnetic field and introduces no extraneous sources. This verification will be placed immediately after the metric derivation so that all subsequent geodesic results rest on a rigorously verified spacetime. revision: yes

  2. Referee: [Lyapunov exponent analysis] The section on Lyapunov-exponent computation: the exponent is stated to be obtained from the geodesic equations, yet no explicit form of the variational equations, integration method, or convergence/error analysis is supplied. Without these details the reported systematic suppression with the anisotropic parameter cannot be assessed.

    Authors: We accept the referee's observation that the numerical procedure requires fuller documentation. The revised version will supply the explicit variational equations obtained by linearizing the geodesic equations, describe the integration algorithm (including the ODE solver and step-size control), and present a convergence study together with error estimates. These additions will allow independent assessment of the reported systematic decrease of the Lyapunov exponent with increasing anisotropic-matter parameter. revision: yes

Circularity Check

0 steps flagged

No circularity: metric derivation and chaos diagnostics are independent of target claims

full rationale

The paper constructs an exact solution via Harrison transformation applied to a seed metric containing anisotropic matter, then numerically extracts Lyapunov exponents and Poincaré sections from geodesic motion in the resulting spacetime. No step equates a fitted parameter to a 'prediction,' renames an input as output, or relies on a self-citation chain whose content is the target result. The suppression of chaos by the anisotropic parameter and magnetic-field effects on global behavior are computed outputs, not definitional inputs. The validity of the transformed stress-energy tensor is an external assumption (correctness risk), not a circular reduction within the derivation.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

Only the abstract is available. The solution rests on the validity of the Harrison transformation and the existence of a suitable seed metric; no free parameters are explicitly fitted in the abstract, but the anisotropic matter parameter and magnetic field strength function as tunable inputs whose effects are then measured.

free parameters (2)
  • anisotropic matter parameter
    Introduced to characterize the anisotropic stress-energy; its value is varied to observe suppression of the Lyapunov exponent.
  • external magnetic field strength
    Varied to produce qualitative changes in Poincaré sections.
axioms (1)
  • standard math Harrison transformation generates an exact solution to the Einstein-Maxwell system when applied to an appropriate seed metric
    Invoked in the first sentence of the abstract as the method yielding the new black-hole solution.

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Reference graph

Works this paper leans on

92 extracted references · 63 canonical work pages · 43 internal anchors

  1. [1]

    A. M. Ghez, S. Salim, N. N. Weinberg, J. R. Lu, T. Do, J. K. Dunn, K . Matthews, M. Mor- ris, S. Yelda and E. E. Becklin, et al. Astrophys. J. 689, 1044-1062 (2008) [arXiv:0808.2870 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  2. [2]

    Monitoring stellar orbits around the Massive Black Hole in the Galactic Center

    S. Gillessen, F. Eisenhauer, S. Trippe, T. Alexander, R. Genzel, F . Martins and T. Ott, Astrophys. J. 692, 1075-1109 (2009) [arXiv:0810.4674 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  3. [3]

    R. D. Blandford and R. L. Znajek, Mon. Not. Roy. Astron. Soc. 179, 433-456 (1977)

    1977

  4. [4]

    M. D. Johnson, V. L. Fish, S. S. Doeleman, D. P. Marrone, R. L. P lambeck, J. F. C. Wardle, K. Akiyama, K. Asada, C. Beaudoin and L. Blackburn, et al. Science 350, no.6265, 1242- 1245 (2015) [arXiv:1512.01220 [astro-ph.HE]]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  5. [5]

    Blandford, D

    R. Blandford, D. Meier and A. Readhead, Ann. Rev. Astron. Ast rophys. 57, 467-509 (2019) [arXiv:1812.06025 [astro-ph.HE]]

    work page arXiv 2019

  6. [6]

    GW190521: A Binary Black Hole Merger with a Total Mass of 150M ⊙,

    R. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 125, no.10, 101102 (2020) [arXiv:2009.01075 [gr-qc]]

    work page arXiv 2020

  7. [7]

    First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L12 (2022) [arXiv:2311.08680 [astro-ph.HE]]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  8. [8]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astron. Astrophys. 704, A91 (2025) [arXiv:2509.24593 [astro-ph.HE]]

    work page arXiv 2025

  9. [9]

    P. G. S. Fernandes and V. Cardoso, Phys. Rev. Lett. 135, no.21, 211403 (2025) [arXiv:2507.04389 [gr-qc]]

    work page arXiv 2025

  10. [10]

    Schwarzschild, Sitzungsber

    K. Schwarzschild, Sitzungsber. Preuss. Akad. Wiss. Berlin (Ma th. Phys. ) 1916, 189-196 (1916)

    1916

  11. [11]

    R. P. Kerr, Phys. Rev. Lett. 11, 237-238 (1963)

    1963

  12. [12]

    R. P. Kerr, [arXiv:0706.1109 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv

  13. [13]

    Carter, Phys

    B. Carter, Phys. Rev. 174, 1559-1571 (1968)

    1968

  14. [14]

    Carter, Commun

    B. Carter, Commun. Math. Phys. 10, no.4, 280-310 (1968)

    1968

  15. [15]

    Walker and R

    M. Walker and R. Penrose, Commun. Math. Phys. 18, 265 (1970)

    1970

  16. [16]

    Benenti and M

    S. Benenti and M. Francaviglia, Gen. Rel. Grav. 10, no.1, 79-92 (1979)

    1979

  17. [17]

    Demianski and M

    M. Demianski and M. Francaviglia, Int. J. Theor. Phys. 19, no.9, 675-680 (1980). 27

    1980

  18. [18]

    J. M. Bardeen, 1973, in Black Holes, ed. C. DeWitt & B. S. DeWitt ( New York: Gordon & Breach), 215-240

    1973

  19. [19]

    I. G. Dymnikova, Sov. Phys. Usp. 29, 215 (1986)

    1986

  20. [20]

    Homoclinic Orbits around Spinning Black Holes I: Exact Solution for the Kerr Separatrix

    J. Levin and G. Perez-Giz, Phys. Rev. D 79, 124013 (2009) [arXiv:0811.3814 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  21. [21]

    Circular motion of neutral test particles in Reissner-Nordstr\"om spacetime

    D. Pugliese, H. Quevedo and R. Ruffini, Phys. Rev. D 83, 024021 (2011) [arXiv:1012.5411 [astro-ph.HE]]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  22. [22]

    Y. T. Li, C. Y. Wang, D. S. Lee and C. Y. Lin, Phys. Rev. D 108, no.4, 044010 (2023) [arXiv:2302.09471 [gr-qc]]

    work page arXiv 2023

  23. [23]

    Geodesic stability, Lyapunov exponents and quasinormal modes

    V. Cardoso, A. S. Miranda, E. Berti, H. Witek and V. T. Zanchin, Phys. Rev. D 79, no.6, 064016 (2009) [arXiv:0812.1806 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  24. [24]

    B. Gwak, N. Kan, B. H. Lee and H. Lee, JHEP 09, 026 (2022) [arXiv:2203.07298 [gr-qc]]

    work page arXiv 2022

  25. [25]

    Jeong, B

    S. Jeong, B. H. Lee, H. Lee and W. Lee, Phys. Rev. D 107, no.10, 104037 (2023) [arXiv:2301.12198 [gr-qc]]

    work page arXiv 2023

  26. [26]

    Y. Q. Lei and X. H. Ge, Phys. Rev. D 107, no.10, 10 (2023) [arXiv:2302.12812 [hep-th]]

    work page arXiv 2023

  27. [27]

    Lee and B

    H. Lee and B. Gwak, Phys. Rev. D 112, no.4, 046018 (2025) [arXiv:2506.00833 [gr-qc]]

    work page arXiv 2025

  28. [28]

    T. V. Targema, K. Bamba and U. Zafar, [arXiv:2605.26829 [hep-t h]]

    work page internal anchor Pith review Pith/arXiv arXiv

  29. [29]

    M. A. Melvin, Phys. Lett. 8, 65-70 (1964)

    1964

  30. [30]

    F. J. Ernst, J. Math. Phys. 17, no.1, 54-56 (1976)

    1976

  31. [31]

    F. J. Ernst and W. J. Wild, J. Math. Phys. 17, no.2, 182 (1976)

    1976

  32. [32]

    B. K. Harrison, J. Math. Phys. 9, no.11, 1744 (1968)

    1968

  33. [33]

    R. M. Wald, Phys. Rev. D 10, 1680-1685 (1974)

    1974

  34. [34]

    W. A. Hiscock, J. Math. Phys. 22, 1828 (1981)

    1981

  35. [35]

    K. S. Thorne, R. H. Price and D. A. Macdonald, BLACK HOLES: THE MEMBRANE PARADIGM, (Yale University Press, New Haven and London, 1986)

    1986

  36. [36]

    General Relativistic Electromagnetic Fields of a Slowly Rotating Magnetized Neutron Star. I. Formulation of the equations

    L. Rezzolla, B. J. Ahmedov and J. C. Miller, Mon. Not. Roy. Astro n. Soc. 322, 723 (2001) [arXiv:astro-ph/0011316 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  37. [37]

    Podolsky and H

    J. Podolsky and H. Ovcharenko, Phys. Rev. Lett. 135, no.18, 181401 (2025) [arXiv:2507.05199 [gr-qc]]. 28

    work page arXiv 2025

  38. [38]

    Black holes in the external Bertotti-Robinson-Bonnor-Melvin electromagnetic field,

    M. Astorino, Phys. Rev. D 112, no.10, 104077 (2025) [arXiv:2508.12908 [gr-qc]]

    work page arXiv 2025

  39. [39]

    V. V. Kiselev, Class. Quant. Grav. 20, 1187-1198 (2003) [arXiv:gr-qc/0210040 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  40. [40]

    Rotating black hole solutions with quintessential energy

    B. Toshmatov, Z. Stuchl ´ ık and B. Ahmedov, Eur. Phys. J. Plu s 132, no.2, 98 (2017) [arXiv:1512.01498 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  41. [41]

    Simple Black Holes with Anisotropic Fluid

    I. Cho and H. C. Kim, Chin. Phys. C 43, no.2, 025101 (2019) [arXiv:1703.01103 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  42. [42]

    H. C. Kim, B. H. Lee, W. Lee and Y. Lee, Phys. Rev. D 101, no.6, 064067 (2020) [arXiv:1912.09709 [gr-qc]]

    work page arXiv 2020

  43. [43]

    H. C. Kim, B. H. Lee, W. Lee and Y. Lee, AIP Conf. Proc. 2874, no.1, 020008 (2024) [arXiv:2112.04131 [gr-qc]]

    work page arXiv 2024

  44. [44]

    H. C. Kim and W. Lee, Eur. Phys. J. C 85, no.11, 1245 (2025) [arXiv:2503.06961 [gr-qc]]

    work page arXiv 2025

  45. [45]

    Anisotropic matter and nonlinear electromagnetics black holes

    W. Lee and Y. S. Myung, Eur. Phys. J. C 86, no.5, 525 (2026) [arXiv:2603.01507 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  46. [46]

    M. Xu, J. Lu, Y. Liu and S. Wu, [arXiv:2606.29881 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv

  47. [47]

    G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752-2756 (1977)

    1977

  48. [48]

    S. W. Hawking and S. F. Ross, Phys. Rev. D 52, 5865-5876 (1995) [arXiv:hep-th/9504019 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  49. [49]

    Vigan` o, [arXiv:2211.00436 [gr-qc]]

    A. Vigan` o, [arXiv:2211.00436 [gr-qc]]

    work page arXiv

  50. [50]

    Lungu, M

    V. Lungu, M. A. Dariescu and C. Stelea, Phys. Rev. D 111 (2025) no.6, 064014 [arXiv:2405.14420 [gr-qc]]

    work page arXiv 2025

  51. [51]

    Carter, J

    B. Carter, J. Math. Phys. 10, 70-81 (1969)

    1969

  52. [52]

    V. P. Frolov, P. Krtous and D. Kubiznak, Living Rev. Rel. 20, no.1, 6 (2017) [arXiv:1705.05482 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  53. [53]

    B. H. Lee, W. Lee and Y. S. Myung, Phys. Rev. D 103, no.6, 064026 (2021) [arXiv:2101.04862 [gr-qc]]

    work page arXiv 2021

  54. [54]

    J. M. Bardeen, W. H. Press and S. A. Teukolsky, Astrophys. J . 178, 347 (1972)

    1972

  55. [55]

    Ferrari, L

    V. Ferrari, L. Gualtieri and P. Pani, General Relativity and its Applications , CRC Press, (2020)

    2020

  56. [56]

    Homoclinic Orbits around Spinning Black Holes II: The Phase Space Portrait

    G. Perez-Giz and J. Levin, Phys. Rev. D 79, 124014 (2009) [arXiv:0811.3815 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  57. [57]

    Computation of Lyapunov Exponents in General Relativity

    X. Wu and T. y. Huang, Phys. Lett. A 313, 77-81 (2003) [arXiv:gr-qc/0302118 [gr-qc]]. 29

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  58. [58]

    A bound on chaos

    J. Maldacena, S. H. Shenker and D. Stanford, JHEP 08, 106 (2016) [arXiv:1503.01409 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  59. [59]

    Universality in Chaos of Particle Motion near Black Hole Horizon

    K. Hashimoto and N. Tanahashi, Phys. Rev. D 95, no.2, 024007 (2017) [arXiv:1610.06070 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  60. [60]

    Falconer, Fractal Geometry: Mathematical Foundations and Applicati ons, John Wiley & Sons, Ltd, 2003

    K. Falconer, Fractal Geometry: Mathematical Foundations and Applicati ons, John Wiley & Sons, Ltd, 2003

    2003

  61. [61]

    Kormendy and D

    J. Kormendy and D. Richstone, Ann. Rev. Astron. Astrophys . 33, 581 (1995)

    1995

  62. [62]

    V. C. Rubin and W. K. Ford, Jr., Astrophys. J. 159, 379-403 (1970)

    1970

  63. [63]

    S. J. Sin, Phys. Rev. D 50, 3650-3654 (1994) [arXiv:hep-ph/9205208 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  64. [64]

    J. w. Lee and I. g. Koh, Phys. Rev. D 53, 2236-2239 (1996) [arXiv:hep-ph/9507385 [hep- ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  65. [65]

    W. Hu, R. Barkana and A. Gruzinov, Phys. Rev. Lett. 85, 1158-1161 (2000) [arXiv:astro- ph/0003365 [astro-ph]]

    work page arXiv 2000

  66. [66]

    J. W. Lee and S. Lim, JCAP 01, 007 (2010) [arXiv:0812.1342 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  67. [67]

    L. Hui, J. P. Ostriker, S. Tremaine and E. Witten, Phys. Rev. D 95, no.4, 043541 (2017) [arXiv:1610.08297 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  68. [68]

    S. Kang, A. Kar and S. Scopel, JCAP 11, 077 (2023) [arXiv:2308.13203 [hep-ph]]

    work page arXiv 2023

  69. [69]

    D. P. Marrone, J. M. Moran, J. H. Zhao and R. Rao, Astrophys . J. Lett. 654, L57-L60 (2006) [arXiv:astro-ph/0611791 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  70. [70]

    R. P. Eatough, H. Falcke, R. Karuppusamy, K. J. Lee, D. J. Ch ampion, E. F. Keane, G. Desvignes, D. H. F. M. Schnitzeler, L. G. Spitler and M. Kramer, et al. Nature 501, 391-394 (2013) [arXiv:1308.3147 [astro-ph.GA]]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  71. [71]

    Bicak and V

    J. Bicak and V. Janis, Mon. Not. R. astr. Soc. 212, 899 (1985)

    1985

  72. [72]

    A. N. Aliev and D. V. Galtsov, Sov. Phys. Usp. 32, 75 (1989)

    1989

  73. [73]

    Karas and D

    V. Karas and D. Vokroulflicky, Gen. Relativ. Gravit. 24, 729-743 (1992)

    1992

  74. [74]

    M. Wang, S. Chen and J. Jing, Eur. Phys. J. C 77, no.4, 208 (2017) [arXiv:1605.09506 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  75. [75]

    Chaotic motion of neutral and charged particles in the magnetized Ernst-Schwarzschild spacetime

    D. Li and X. Wu, Eur. Phys. J. Plus 134, no.3, 96 (2019) [arXiv:1803.02119 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  76. [76]

    Chaos in pp-wave spacetimes

    J. Podolsky and K. Vesely, Phys. Rev. D 58, 081501 (1998) [arXiv:gr-qc/9805078 [gr-qc]]. 30

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  77. [77]

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

    O. Semerak and P. Sukova, Mon. Not. Roy. Astron. Soc. 404, 545-574 (2010) [arXiv:1211.4106 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  78. [78]

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

    O. Semerak and P. Sukova, Mon. Not. Roy. Astron. Soc. 425, 2455-2476 (2012) [arXiv:1211.4107 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  79. [79]

    Acceleration of charged particles due to chaotic scattering in the combined black hole gravitational field and asymptotically uniform magnetic field

    Z. Stuchl ´ ık and M. Koloˇ s, Eur. Phys. J. C76, no.1, 32 (2016) [arXiv:1511.02936 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  80. [80]

    Radiation reaction of charged particles orbiting magnetized Schwarzschild black hole

    A. Tursunov, M. Koloˇ s, Z. Stuchl ´ ık and D. V. Gal’tsov, Astrophys. J. 861, no.1, 2 (2018) [arXiv:1803.09682 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

Showing first 80 references.