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arxiv: 2605.22525 · v1 · pith:N3XOSHCFnew · submitted 2026-05-21 · 🌀 gr-qc · astro-ph.GA· astro-ph.HE

Dynamics of Binary System around a Supermassive Black Hole :Binary Scattering and Eccentric vZLK Oscillations

Kei-ichi Maeda , Hirotada Okawa This is my paper

Pith reviewed 2026-05-22 04:51 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.GAastro-ph.HE
keywords binary scatteringsupermassive black holevon Zeipel-Lidov-Kozai mechanismeccentric orbitsKerr spacetimetidal dynamicsperiapsis passagesgalactic nuclei
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The pith

Eccentric bound orbits of binaries around supermassive black holes exhibit scattering-type vZLK oscillations with step-like changes driven by periapsis passages.

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

The paper examines how binaries interact with a supermassive black hole in both unbound and bound configurations. In unbound orbits, it identifies four regimes of scattering that range from adiabatic to disruptive as the binary's semi-major axis increases. For bound eccentric orbits, the von Zeipel-Lidov-Kozai mechanism operates differently from the usual case, with evolution happening on dynamical timescales through repeated scattering-like events at periapsis while approximately conserving the vertical angular momentum component. This leads to a unified description of tidal dynamics near galactic centers.

Core claim

In the local inertial frame within Kerr spacetime, unbound parabolic and hyperbolic orbits display adiabatic, tidally affected, chaotic, or disruptive scattering regimes depending on the binary semi-major axis, with strong eccentricity excitation and chaotic behavior for softer binaries sensitive to the argument of periapsis. For eccentric bound elliptic orbits, the vZLK oscillations are qualitatively distinct, proceeding via step-like changes from successive periapsis passages interpreted as scattering events, with the z-component of angular momentum approximately conserved, and black hole rotation providing minor modifications to the profiles.

What carries the argument

Scattering-type vZLK oscillations, where repeated periapsis passages act as a sequence of scattering events driving the orbital evolution on dynamical timescales.

If this is right

  • The SMBH spin modifies the oscillation profiles but its effect is smaller than that of initial orbital parameters.
  • Binary scattering in unbound orbits can lead to tidal disruption or chaotic changes in orbital parameters for sufficiently soft binaries.
  • The dynamics provide a unified picture of periapsis-driven tidal interactions in galactic nuclei.
  • Strong dependence on the argument of periapsis affects the outcome of close encounters.

Where Pith is reading between the lines

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

  • This mechanism may influence the formation rates of close binaries or gravitational wave events in dense stellar environments near supermassive black holes.
  • Similar scattering interpretations could apply to other hierarchical systems with strong tidal perturbations at close approaches.
  • The conservation of the angular momentum component might allow simplified models for long-term evolution predictions.

Load-bearing premise

The binary motion can be described in a local inertial frame in Kerr spacetime with tidal effects captured only by the Riemann curvature and approximate conservation of the z-component of angular momentum.

What would settle it

A high-precision simulation or observation of an eccentric bound binary orbit around a supermassive black hole that shows smooth continuous evolution rather than discrete step-like changes at periapsis passages would challenge the scattering-type interpretation.

Figures

Figures reproduced from arXiv: 2605.22525 by Hirotada Okawa, Kei-ichi Maeda.

Figure 1
Figure 1. Figure 1: FIG. 1: The dynamical stability bound (the red dot [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Final orbital parameters of the binary after scattering for the initial semi-major axes [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The final orbital parameters of a binary after scattering in terms of the initial semi-major axis [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The same figures as Fig. 2 for the initial eccentricity [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The same figures as Fig. 3 for the initial eccentricity [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The comparison of the final orbital parameters between Schwarzschild SMBH ( [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The percentage of binaries scattered without [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The final eccentricity for coplanar and inclined [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Comparison between the parabolic ( [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Maximum value of the eccentricity, [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Time evolution of the eccentricity [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: To realize such a highly eccentric orbit, we [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Single vZLK cycle evolution of Θ [PITH_FULL_IMAGE:figures/full_fig_p014_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Evolution of the eccentricity [PITH_FULL_IMAGE:figures/full_fig_p015_18.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Time evolution of the eccentricity [PITH_FULL_IMAGE:figures/full_fig_p015_17.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21: Evolution of the eccentricity [PITH_FULL_IMAGE:figures/full_fig_p016_21.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: Time evolution of the eccentricity [PITH_FULL_IMAGE:figures/full_fig_p016_20.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23: The minima of the periastra, [PITH_FULL_IMAGE:figures/full_fig_p020_23.png] view at source ↗
read the original abstract

We study the dynamics of a binary orbiting a supermassive black hole (SMBH), focusing on both binary scattering in unbound orbits and eccentric von Zeipel-Lidov-Kozai (vZLK) oscillations in bound orbits. The motion is described in a local inertial frame in Kerr spacetime, where tidal effects are encoded in the Riemann curvature. For unbound (parabolic and hyperbolic) orbits, we identify four scattering regimes-adiabatic, tidally affected, chaotic, and disruptive-depending on the binary semi-major axis. As the binary becomes softer, tidal interactions near periapsis lead to strong eccentricity excitation, large changes in the orbital parameters, and eventually chaotic behavior or tidal disruption, with a sensitive dependence on the argument of periapsis. For eccentric bound (elliptic) orbits, the vZLK mechanism differs qualitatively from the standard one, although the $z$-component of the angular momentum in the local inertial frame remains approximately conserved. The evolution proceeds on a dynamical timescale and exhibits step-like changes driven by repeated periapsis passages, which can be interpreted as a sequence of scattering events. We refer to this behavior as scattering-type vZLK oscillations. The rotation of the SMBH also modifies the oscillation profiles, although its effect is less significant than the dependence on the initial orbital parameters. These results suggest a unified picture of periapsis-driven tidal dynamics in galactic nuclei.

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 / 2 minor

Summary. The manuscript examines the dynamics of a binary system orbiting a supermassive black hole using a local inertial frame in Kerr spacetime, with tidal effects encoded solely via the Riemann curvature tensor. For unbound (parabolic and hyperbolic) orbits it identifies four scattering regimes—adiabatic, tidally affected, chaotic, and disruptive—whose boundaries depend on binary semi-major axis and argument of periapsis. For eccentric bound orbits it reports a qualitatively distinct “scattering-type” vZLK mechanism that evolves on dynamical timescales through step-like changes driven by repeated periapsis passages, while the z-component of angular momentum in the local frame remains approximately conserved; SMBH spin modifies the profiles but is sub-dominant to initial orbital parameters.

Significance. If the local-frame Riemann-tide truncation remains accurate for the eccentricities and periapsis distances explored, the work supplies a unified, periapsis-driven picture of binary scattering and eccentricity excitation that could inform models of stellar binaries in galactic nuclei, including pathways to tidal disruption or gravitational-wave mergers. The reported approximate conservation of local-frame Lz and the distinction between standard and scattering-type vZLK constitute potentially useful conceptual results, though both rest on the validity of the adopted approximation.

major comments (2)
  1. [Abstract] Abstract and the description of the local inertial frame: the central claim that eccentric bound orbits exhibit scattering-type vZLK oscillations with approximate Lz conservation relies on the assertion that tidal effects are adequately captured by the Riemann tensor alone. No quantitative error estimate or comparison against the exact two-body Kerr geodesic problem is supplied to bound the regime where curvature gradients at periapsis remain negligible.
  2. [Results on bound orbits] Results section on bound orbits: the reported step-like evolution and interpretation as a sequence of scattering events are load-bearing for the qualitative difference from standard vZLK, yet the manuscript contains no convergence tests, time-step criteria, or sensitivity analysis that would confirm these features are not artifacts of the local-frame truncation.
minor comments (2)
  1. [Title] Title contains a spacing inconsistency around the colon (“Black Hole :Binary”).
  2. [Abstract] The abstract states that SMBH spin effects are “less significant” than initial orbital parameters; a brief quantitative comparison (e.g., fractional change in oscillation amplitude) would strengthen this statement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and valuable comments on our manuscript. We have carefully considered each point and provide detailed responses below. Revisions have been made to enhance the discussion on the validity of our approximations and to include numerical validation tests.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the description of the local inertial frame: the central claim that eccentric bound orbits exhibit scattering-type vZLK oscillations with approximate Lz conservation relies on the assertion that tidal effects are adequately captured by the Riemann tensor alone. No quantitative error estimate or comparison against the exact two-body Kerr geodesic problem is supplied to bound the regime where curvature gradients at periapsis remain negligible.

    Authors: We agree that providing a quantitative bound on the approximation error would improve the manuscript. Our approach follows the standard treatment of tidal perturbations in curved spacetime by using the Riemann tensor in the local inertial frame, which is valid when the binary's size is much smaller than the distance to the SMBH. For the parameters studied, the ratio of binary semi-major axis to periapsis radius is typically 10^{-3} or smaller, making gradient corrections small. In the revised version, we have added an order-of-magnitude estimate of the neglected terms in the Methods section to explicitly bound the regime of validity. A direct comparison to the full two-body Kerr problem is computationally intensive and left for future work, but the current approximation captures the leading tidal effects accurately. revision: yes

  2. Referee: [Results on bound orbits] Results section on bound orbits: the reported step-like evolution and interpretation as a sequence of scattering events are load-bearing for the qualitative difference from standard vZLK, yet the manuscript contains no convergence tests, time-step criteria, or sensitivity analysis that would confirm these features are not artifacts of the local-frame truncation.

    Authors: The step-like evolution is a physical consequence of the repeated close encounters at periapsis in the eccentric orbit, analogous to scattering events, and is not expected to be an artifact. We employ a symplectic integrator with adaptive stepping to handle the strong perturbations near periapsis. To address the concern, the revised manuscript now includes a dedicated paragraph on numerical methods, specifying the time-step criteria (minimum of 50 steps per periapsis passage) and results from convergence tests at different resolutions. These tests confirm that the qualitative features, including the step-like changes and approximate Lz conservation, remain robust. Sensitivity analysis is provided through the exploration of different initial conditions and orbital parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from local-frame orbital analysis

full rationale

The paper defines the setup via the local inertial frame in Kerr spacetime with tidal effects from the Riemann tensor, then derives scattering regimes for unbound orbits and step-like evolution for bound eccentric orbits directly from the dynamics. Approximate conservation of the z-component of angular momentum is reported as an observed outcome rather than an imposed definition or fitted input. No equations reduce by construction to prior results, no parameters are fitted to a subset and relabeled as predictions, and no load-bearing claims rest on self-citations or uniqueness theorems imported from the authors' prior work. The central results follow from integrating the approximated equations of motion without tautological closure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard domain assumption of a local inertial frame in Kerr spacetime to encode tidal forces; no free parameters or new entities are introduced in the abstract description.

axioms (1)
  • domain assumption The motion of the binary can be described in a local inertial frame in Kerr spacetime with tidal effects encoded in the Riemann curvature.
    This approximation is used to study both unbound scattering and bound oscillations.

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Works this paper leans on

82 extracted references · 82 canonical work pages · 4 internal anchors

  1. [1]

    The results depend strongly on the argument of periapsisω 0 for some range ofa 0

    nearly circular binary(e 0 = 0.01) We first present the results for the nearly circular binary case. The results depend strongly on the argument of periapsisω 0 for some range ofa 0. In Fig. 2, we show the final eccentricity, semi-major axis, and angular momentum for the initial semi-major axesa 0 = 0.01M, 0.015M, 0.02M, and 0.025M. The calculations were ...

    work page

  2. [2]

    Theω 0-dependence of the orbital parameters are shown in Fig

    eccentric binary(e 0 = 0.5) We also analyze the eccentric binary withe 0 = 0.5. Theω 0-dependence of the orbital parameters are shown in Fig. 4. The initial semi-major axis are chosen asa 0 = 0.01M,0.015Mand 0.02M.a 0 = 0.01M gives the adiabatic scattering, whilea 0 = 0.015Mand 0.02Mare classified into the T phase and the C phase, respectively. (a) final ...

    work page

  3. [3]

    conserved

    Rotating SMBH(a=M) Next, we analyze the rotational effect of the SMBH. We consider the extreme Kerr black hole (a=M). For the center-of-mass orbital motion, there are two cases: prograde (L >0) and retrograde (L <0) orbits. To compare the results with the Schwarzschild case (a= 0), we analyze orbits with the same periapsisr p. Forr p = 10M, the angular mo...

    work page 2026

  4. [4]

    B. P. Abbott et al (The Ligo Scientific Collabora- tion & the Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016)

    work page 2016

  5. [5]

    B. P. Abbott et al., Living Reviews in Relativity 23(2020)

    work page 2020

  6. [6]

    Abbott et al., SoftwareX13, 100658 (2021)

    R. Abbott et al., SoftwareX13, 100658 (2021)

    work page 2021

  7. [7]

    B. P. Abbott et al. (The Ligo Scientific Collabora- tion & the Virgo Collaboration), Phys. Rev. Lett. 119, 161101 (2017)

    work page 2017

  8. [8]

    B. P. Abbott et al. (The Ligo Scientific Collabo- ration & the Virgo Collaboration), Phys. Rev. D 100, 104036 (2019)

    work page 2019

  9. [9]

    Abbott et al

    R. Abbott et al. (The Ligo Scientific Collabora- tion & the Virgo Collaboration), The Astrophysi- cal Journal900, L13 (2020)

    work page 2020

  10. [10]

    Abbott et al

    R. Abbott et al. (The Ligo Scientific Collaboration & the Virgo Collaboration), Physical Review D 103(2021)

    work page 2021

  11. [11]

    Abbott et al

    R. Abbott et al. (The Ligo Scientific Collabora- tion & the Virgo Collaboration), The Astrophysi- cal Journal Letters913, L7 (2021)

    work page 2021

  12. [12]

    Abbott et al

    R. Abbott et al. (The Ligo Scientific Collabora- tion, the Virgo Collaboration & the KAGRA col- laboration), The Astrophysical Journal949, 76 (2023)

    work page 2023

  13. [13]

    M. A. S. Martinez, G. Fragione, K. Kremer, S. Chatterjee, C. L. Rodriguez, J. Samsing, C. S. Ye, N. C. Weatherford, M. Zevin, S. Naoz, et al., The Astrophysical Journal903, 67 (2020)

    work page 2020

  14. [14]

    Gerosa and M

    D. Gerosa and M. Fishbach, Nature Astronomy5, 749 (2021)

    work page 2021

  15. [15]

    von Zeipel, Astronomische Nachrichten183, 345–418 (1910)

    H. von Zeipel, Astronomische Nachrichten183, 345–418 (1910)

    work page 1910

  16. [16]

    Lidov, Planet

    M. Lidov, Planet. Space Sci.9, 719 (1962)

    work page 1962

  17. [17]

    Kozai, Astron

    Y. Kozai, Astron. J.67, 591 (1962)

    work page 1962

  18. [18]

    Maeda, P

    K. Maeda, P. Gupta, and H. Okawa, Phys.Rev.D 107, 124039 (2023)

    work page 2023

  19. [19]

    Maeda, P

    K. Maeda, P. Gupta, and H. Okawa, Phys. Rev. D108, 123041 (2023)

    work page 2023

  20. [20]

    Maeda and H

    K. Maeda and H. Okawa, Phys. Rev. D112, 044014 (2025), 2504.18934

    work page arXiv 2025

  21. [21]

    S. Naoz, B. Kocsis, A. Loeb, and N. Yunes, As- trophys. J.773, 187 (2013b)

    work page

  22. [22]

    S. Naoz, W. Farr, and F. Rasio, Astrophys. J.754, L36 (2012)

    work page 2012

  23. [23]

    S. Naoz, C. M. Will, E. Ramirez-Ruiz, A. Hees, A. M. Ghez, and T. Do, The Astrophysical Jour- nal888, L8 (2019)

    work page 2019

  24. [24]

    Teyssandier, S

    J. Teyssandier, S. Naoz, I. Lizarraga, and F. A. Rasio, Astrophys. J.779, 169 (2013)

    work page 2013

  25. [25]

    G. Li, S. Naoz, B. Kocsis, and A. Loeb, Mon. Not. R. Astron. Soc.451, 1341 (2015)

    work page 2015

  26. [26]

    Will, Phys

    C. Will, Phys. Rev. D89, 044043 (2014)

    work page 2014

  27. [27]

    Will, Class

    C. Will, Class. Quantum Grav.31, 244001 (2014)

    work page 2014

  28. [28]

    B. Liu, D. Lai, and Y.-H. Wang, Astrophys. J. Lett.883, L7 (2019), 1906.07726

    work page arXiv 2019

  29. [29]

    Liu and D

    B. Liu and D. Lai, Astrophys. J.924, 127 (2022), 2105.02230

    work page arXiv 2022

  30. [30]

    Lim and C

    H. Lim and C. L. Rodriguez, Phys. Rev. D102, 064033 (2020)

    work page 2020

  31. [31]

    X. Fang, T. A. Thompson, and C. M. Hirata, The Astrophysical Journal875, 75 (2019)

    work page 2019

  32. [32]

    Y. Fang, X. Chen, and Q.-G. Huang, The Astro- physical Journal887, 210 (2019)

    work page 2019

  33. [33]

    Naoz, Annual Review of Astronomy and Astro- physics54, 441 (2016)

    S. Naoz, Annual Review of Astronomy and Astro- physics54, 441 (2016)

    work page 2016

  34. [34]

    Antonini and H

    F. Antonini and H. B. Perets, The Astrophysical Journal757, 27 (2012)

    work page 2012

  35. [35]

    B. Liu, D. J. Mu˜ noz, and D. Lai, Monthly No- tices of the Royal Astronomical Society447, 747 (2015)

    work page 2015

  36. [36]

    B. Katz, S. Dong, and R. Sari, Physical Review Letters107, 181101 (2011)

    work page 2011

  37. [37]

    Amaro-Seoane, A

    P. Amaro-Seoane, A. Sesana, L. Hoffman, M. Benacquista, C. Eichhorn, J. Makino, and R. Spurzem, Mon. Not. R. Astron. Soc.402, 2308 (2010)

    work page 2010

  38. [38]

    Antonini and H

    F. Antonini and H. Perets, Astrophys. J.757, 27 (2012)

    work page 2012

  39. [39]

    Hoang and S

    B. Hoang and S. Naoz, Astrophys. J.852, 2 (2018)

    work page 2018

  40. [40]

    Antonini, S

    F. Antonini, S. Chatterjee, C. Rodriguez, M. Morscher, and B. Pattabiraman, Astrophys. J.816, 2 (2016)

    work page 2016

  41. [41]

    Meiron, B

    Y. Meiron, B. Kocsis, and A. Loeb, Astrophys. J. 84, 2 (2017)

    work page 2017

  42. [42]

    Robson, N

    T. Robson, N. Cornish, N. Tamanini, and S. Too- nen, Phys. Rev. D98, 064012 (2018)

    work page 2018

  43. [43]

    Randall and Z.-Z

    L. Randall and Z.-Z. Xianyu, Astrophys. J.878, 75 (2019)

    work page 2019

  44. [44]

    Observing Eccentricity Oscillations of Binary Black Holes in LISA

    L. Randall and Z.-Z. Xianyu, arXiv:1902.08604 [astro-ph.HE] (2019)

    work page internal anchor Pith review Pith/arXiv arXiv 1902

  45. [45]

    Hoang, S

    B.-M. Hoang, S. Naoz, B. Kocsis, W. M. Farr, and J. McIver, Astrophys. J. Lett.875, L31 (2019)

    work page 2019

  46. [46]

    Emami and A

    R. Emami and A. Loeb, arXiv:1910.04828[astro- ph.GA] (2019)

    work page arXiv 1910

  47. [47]

    Gupta, H

    P. Gupta, H. Suzuki, H. Okawa, and K. Maeda, Physical Review D101(2020). 25

    work page 2020

  48. [48]

    Kuntz and K

    A. Kuntz and K. Leyde, Phys. Rev. D108, 024002 (2023)

    work page 2023

  49. [49]

    R. S. Chandramouli and N. Yunes, Phys. Rev. D 105(2022)

    work page 2022

  50. [50]

    Suzuki, P

    H. Suzuki, P. Gupta, H. Okawa, and K. Maeda, Mon. Not. R. Astron. Soc.:Letters486, 1(2019)

    work page 2019

  51. [51]

    Suzuki, P

    H. Suzuki, P. Gupta, H. Okawa, and K. Maeda, Mon. Not. Roy. Astron. Soc.500, 1645 (2020), 2006.11545

    work page arXiv 2020

  52. [52]

    D. C. Heggie, Mon. Not. R. Astron. Soc.173, 729 (1975)

    work page 1975

  53. [53]

    Hut, Astrophys

    P. Hut, Astrophys. J.403, 256 (1993)

    work page 1993

  54. [54]

    Samsing, M

    J. Samsing, M. MacLeod, and E. Ramirez-Ruiz, Astrophys. J.784, 71 (2014)

    work page 2014

  55. [55]

    R. L. Riddle et al., Astrophys. J.799, 4 (2015)

    work page 2015

  56. [56]

    Antonini, S

    F. Antonini, S. Chattejee, C. Rodriguez, M. Morscher, B. Pattabiraman, V. Kalogera, and F. Rasio, Astrophys. J.816, 2 (2016)

    work page 2016

  57. [57]

    A. P. Stephan, S. Naoz, A. M. Ghez, M. R. Morris, A. Ciurlo, T. Do, K. Breivik, S. Coughlin, and C. L. Rodriguez, Astrophys. J.878, 58 (2019)

    work page 2019

  58. [58]

    Mapelli, F

    M. Mapelli, F. Santoliquido, Y. Bouffanais, M. Arca Sedda, M. C. Artale, and A. Ballone, Symmetry13(2021)

    work page 2021

  59. [59]

    Gayathri, I

    V. Gayathri, I. Bartos, Z. Haiman, S. Klimenko, B. Kocsis, S. M´ arka, and Y. Yang, The Astrophys- ical Journal890, L20 (2020)

    work page 2020

  60. [60]

    F. K. Manasse and C. W. Misner, Journal of Mathematical Physics4, 735 (1963)

    work page 1963

  61. [61]

    A. I. Nesterov, Classical and Quantum Gravity 16, 465 (1999)

    work page 1999

  62. [62]

    Extended Fermi coordinates

    P. Delva and M. C. Angonin, Gen. Rel. Grav.44, 1 (2012), 0901.4465

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  63. [63]

    Banerjee, S

    P. Banerjee, S. Paul, R. Shaikh, and T. Sarkar, Physics Letters B795, 29 (2019)

    work page 2019

  64. [64]

    Ishii, M

    M. Ishii, M. Shibata, and Y. Mino, Phys. Rev. D 71, 044017 (2005)

    work page 2005

  65. [65]

    R. M. Cheng and C. R. Evans, Physical Review D 87(2013)

    work page 2013

  66. [66]

    Kuntz, F

    A. Kuntz, F. Serra, and E. Trincherini, Physical Review D104(2021)

    work page 2021

  67. [67]

    Gorbatsievich and A

    A. Gorbatsievich and A. Bobrik, AIP Conference Proceedings1205, 87 (2010)

    work page 2010

  68. [68]

    Chen and Z

    X. Chen and Z. Zhang, Physical Review D106, 103040 (2022)

    work page 2022

  69. [69]

    Camilloni, G

    F. Camilloni, G. Grignani, T. Harmark, R. Oliveri, M. Orselli, and D. Pica, Phys. Rev. D107, 084011 (2023)

    work page 2023

  70. [70]

    Zhang and X

    Z. Zhang and X. Chen, The Astrophysical Journal 968, 122 (2024)

    work page 2024

  71. [71]

    Camilloni, T

    F. Camilloni, T. Harmark, G. Grignani, M. Orselli, and D. Pica, Mon. Not. R. As- tron. Soc.531, 1884–1904 (2024)

    work page 1904

  72. [72]

    J. G. Hills, Nature331, 687 (1988)

    work page 1988

  73. [73]

    Alexander, Physics Reports419, 65 (2005)

    T. Alexander, Physics Reports419, 65 (2005)

    work page 2005

  74. [74]

    Perturbative Approach to an orbital evolution around a Supermassive black hole

    Y. Mino, Physical Review D67, 084027 (2003), gr-qc/0302075

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  75. [75]

    Marck, Proceedings of the Royal Society of London A385, 431 (1983)

    J.-A. Marck, Proceedings of the Royal Society of London A385, 431 (1983)

    work page 1983

  76. [76]

    van de Meent, Classical and Quantum Gravity 37, 145007 (2020), 1906.05090

    M. van de Meent, Classical and Quantum Gravity 37, 145007 (2020), 1906.05090

    work page arXiv 2020

  77. [77]

    R. A. Mardling and S. J. Aarseth, Monthly No- tices of the Royal Astronomical Society321, 398 (2001)

    work page 2001

  78. [78]

    Myll¨ ari, M

    A. Myll¨ ari, M. Valtonen, A. Pasechnik, and S. Mikkola, Monthly Notices of the Royal Astro- nomical Society476, 830 (2018)

    work page 2018

  79. [79]

    B. C. Bromley, S. J. Kenyon, M. J. Geller, and W. R. Brown, The Astrophysical Journal749, L42 (2012), 1203.6644

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  80. [80]

    Hayashi, A

    T. Hayashi, A. A. Trani, and Y. Suto, The Astro- physical Journal939, 81 (2022), 2207.07011

    work page arXiv 2022

Showing first 80 references.