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arxiv: 2512.01688 · v3 · submitted 2025-12-01 · 🌀 gr-qc

Gravitational lensing inside and outside of a marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits

Naoki Tsukamoto This is my paper

Pith reviewed 2026-05-17 03:13 UTC · model grok-4.3

classification 🌀 gr-qc
keywords gravitational lensingstrong deflection limitphoton spheremarginally unstable photon sphereblack hole shadowReissner-Nordstrom spacetimeHayward spacetime
0
0 comments X

The pith

Light deflection angles near marginally unstable photon spheres diverge as a power and can be approximated analytically in strong limits.

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

The paper extends an existing strong-deflection technique so that light rays bent inside or outside a marginally unstable photon sphere can be treated analytically. In ordinary cases the deflection angle diverges logarithmically as the impact parameter approaches a critical value, but the divergence changes to a power law when a photon sphere and an antiphoton sphere merge. The new expansions are written for any static, spherically symmetric, asymptotically flat spacetime and are then evaluated explicitly for the Reissner-Nordström and Hayward metrics. The resulting expressions are shown to approach the exact deflection angle when the impact parameter is sufficiently close to the critical value. The work supplies formulas useful for modeling contributions to black-hole shadow images and for testing exotic compact objects.

Core claim

In a general static, spherically symmetric, asymptotically flat spacetime that possesses a marginally unstable photon sphere, the deflection angle of a light ray diverges as a power of the difference between its impact parameter and the critical value. Extending the method of Eiroa, Romero and Torres supplies explicit analytic series for the deflection angles of rays passing inside and outside this sphere in the strong-deflection regime. The expansions are applied to the Reissner-Nordström and Hayward spacetimes and are verified to converge correctly to the unapproximated deflection angle.

What carries the argument

The power-law strong-deflection expansion that treats the divergence of the deflection angle at a marginally unstable photon sphere.

If this is right

  • Deflection angles of rays that help form black-hole shadow images can be computed with controlled error near the critical orbit.
  • Analytic expressions become available for light bending in exotic spacetimes that lack event horizons yet possess marginally unstable photon spheres.
  • Near-future space telescopes could use the formulas to predict detectable signals from light rays passing close to such spheres.
  • Prior semianalytic coefficients for the power-divergent term outside the sphere are corrected by the full extension.

Where Pith is reading between the lines

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

  • The same power-law treatment could be adapted to study strong lensing in axisymmetric or non-asymptotically flat geometries.
  • Direct comparison with Event Horizon Telescope images might place limits on the existence of marginally unstable photon spheres in observed candidates.
  • The analytic limits provide clean benchmarks against which numerical ray-tracing codes in strong gravity can be validated.

Load-bearing premise

The spacetime is static, spherically symmetric and asymptotically flat and contains a marginally unstable photon sphere at which the deflection angle diverges as a power rather than a logarithm.

What would settle it

A direct numerical integration of the null geodesic equation for an impact parameter differing from the critical value by one part in a thousand should reproduce the predicted leading power-law term in the deflection angle within a few percent.

Figures

Figures reproduced from arXiv: 2512.01688 by Naoki Tsukamoto.

Figure 1
Figure 1. Figure 1: shows the effective potential V (r/M, b) for the radial motion of the rays. 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -0.4 -0.2 0.0 0.2 0.4 r/M V(r/M,b) FIG. 1. The effective potential V (r/M, b) in the Reissner￾Nordstr¨om spacetime with the marginally unstable photon sphere is shown. The broken (green), solid (red), and dot￾ted (magenta) curves denote V (r/M, b) for b = 0.99bm, bm, and 1.01bm, respectively. From Eqs. (… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The percent errors of deflection angles [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The effective potentials [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: We have confirmed the coefficient ¯c+ (3.31) by Chiba and Kimura [133] and the constant ¯d+ (3.34) by Tsukamoto [130] are correct while the coefficient c¯+ (3.33) by using the semi-analytic method in Ref.[130] should be modified. IV. OBSERVABLE OF GRAVITATIONAL LENSING IN THE STRONG DEFLECTION LIMITS In this section, we consider the observable of gravita￾tional lensing of the rays bent inside and outside o… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. A small lens configuration with an effective deflection [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The percent errors of the deflection angles [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

It is believed that rays bent inside and outside photon spheres could affect partially the black hole shadow images by the Event Horizon Telescope and the rays near photon spheres would be detected by near-future space observations. The investigation of the rays near the photon spheres in not only black hole spacetimes but also exotic spacetimes would be important since one will need them to exclude black hole mimickers. The deflection angles of the rays deflected by the photon spheres diverge logarithmically and we can treat them by a strong-deflection-limit analysis. The error of the strong-deflection-limit analysis becomes large if antiphoton spheres exist in the spacetimes and the analysis breaks down when the photon spheres and the antiphoton spheres degenerate to form a marginally unstable photon sphere. This is because the deflection angles of the rays bent by the marginally unstable photon sphere diverge in powers. In this paper, we extend Eiroa, Romero, and Torres's method to gravitational lensing of rays inside and outside of the marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits and we apply it to a Reissner-Nordstr\"{o}m spacetime and a Hayward spacetime with the marginally unstable photon sphere. We have also confirmed that the deflection angles in the strong deflection limits by the method converge correctly to the deflection angle without approximations, while there are the mismatches of the coefficient of the power-divergent term of the deflection angles of the rays deflected just outside of the marginally unstable photon sphere in a semianalytic calculation by the author previously.

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 / 3 minor

Summary. The manuscript extends the Eiroa-Romero-Torres strong-deflection-limit technique from logarithmic divergence at unstable photon spheres to power-law divergence at marginally unstable photon spheres. It derives the requisite change of variables and leading asymptotic expansion of the deflection integral for both interior and exterior rays in a general static, spherically symmetric, asymptotically flat metric. The resulting expressions are applied to the Reissner-Nordström and Hayward spacetimes (both possessing a marginally unstable photon sphere), and numerical checks confirm that the approximated deflection angles approach the exact integral as the impact parameter approaches the critical value from either side. The authors also report that the new expansion corrects a coefficient mismatch present in their own prior semianalytic treatment.

Significance. If the central derivations hold, the work supplies a missing analytic tool for strong-deflection lensing in spacetimes containing antiphoton spheres that degenerate into marginally unstable photon spheres. Such configurations appear in certain black-hole mimickers and regular black-hole models; the ability to obtain closed-form leading behavior for both interior and exterior rays is therefore directly relevant to shadow-image modeling for the Event Horizon Telescope and to future high-resolution observations that may resolve near-photon-sphere rays.

major comments (2)
  1. [§3] §3 (change of variables and asymptotic expansion): the leading power-law coefficient for exterior rays is stated to differ from the author’s earlier semianalytic result; however, the manuscript does not display the explicit integral evaluation that isolates this coefficient, making independent verification of the claimed correction difficult.
  2. [§4.1] §4.1 (Reissner-Nordström application): the numerical convergence plots are shown only for a single charge value; a brief table or additional curves for a second charge (e.g., Q/M = 0.5) would strengthen the claim that the expansion remains accurate across the parameter range where a marginally unstable photon sphere exists.
minor comments (3)
  1. [Abstract] The abstract and introduction use “antiphoton sphere” without a one-sentence definition; a brief parenthetical reminder of its location relative to the photon sphere would aid readers unfamiliar with the terminology.
  2. [Eq. (12)] Equation (12) introduces the new radial coordinate but does not explicitly state the range of the integration variable after the substitution; adding the limits would clarify the subsequent asymptotic analysis.
  3. [Figure 3] Figure 3 caption should specify the exact numerical integrator and tolerance used for the “exact” deflection-angle curves.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [§3] §3 (change of variables and asymptotic expansion): the leading power-law coefficient for exterior rays is stated to differ from the author’s earlier semianalytic result; however, the manuscript does not display the explicit integral evaluation that isolates this coefficient, making independent verification of the claimed correction difficult.

    Authors: We acknowledge that the explicit evaluation of the integral for the leading coefficient was not shown in sufficient detail. To address this, we will revise §3 to include the full steps of the asymptotic expansion, explicitly isolating the power-law coefficient for exterior rays and demonstrating its difference from the earlier semianalytic result. This will enable straightforward independent verification while preserving the integrity of the derivation. revision: yes

  2. Referee: [§4.1] §4.1 (Reissner-Nordström application): the numerical convergence plots are shown only for a single charge value; a brief table or additional curves for a second charge (e.g., Q/M = 0.5) would strengthen the claim that the expansion remains accurate across the parameter range where a marginally unstable photon sphere exists.

    Authors: We agree that testing the expansion for an additional charge parameter would provide stronger evidence of its general applicability. In the revised manuscript, we will add numerical results for Q/M = 0.5, either as additional convergence curves or in a compact table, to confirm the accuracy in the relevant parameter regime. revision: yes

Circularity Check

0 steps flagged

Minor self-citation of author's prior calculation noted as mismatch; central extension of Eiroa et al. method is independent

full rationale

The paper extends the Eiroa-Romero-Torres strong-deflection analysis to power-law divergence at marginally unstable photon spheres via explicit change of variables and asymptotic expansion of the deflection integral in a general static spherical asymptotically flat metric. This is applied to RN and Hayward spacetimes with direct numerical verification that the approximations converge to the exact integral. The sole self-reference is to the author's earlier semianalytic coefficient, presented only to highlight a mismatch rather than to support the new derivation. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing uniqueness theorems from self-citations appear in the derivation chain. The work remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard general-relativistic assumptions about spacetime symmetry and the existence of a marginally unstable photon sphere; no free parameters or new postulated entities are introduced in the abstract.

axioms (2)
  • domain assumption The spacetime is static, spherically symmetric, and asymptotically flat.
    Explicitly stated as the general setting in the title and abstract.
  • domain assumption A marginally unstable photon sphere exists at which deflection angles diverge in powers rather than logarithmically.
    Central premise required for the breakdown of prior logarithmic analysis and the need for the new extension.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Strong-deflection expansion of the deflection angle near a degenerate photon sphere

    gr-qc 2026-03 unverdicted novelty 7.0

    Derives a factorized leading term for the strong deflection angle near degenerate photon spheres using local expansion of the effective potential and Weyl tensor measures.

Reference graph

Works this paper leans on

148 extracted references · 148 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    (2.11) From Eq. (2.11), the impact parameter b of the ray de- fined as b ≡ L/E , where E ≡ − gµν tµ kν and L ≡ gµν ϕ µ kν are the conserved energy and the angular momentum of the ray, respectively, is expressed as b = b(r0) = L E = C0 ˙ϕ 0 A0 ˙t0 = √ C0 A0 , (2.12) and it is constant along the trajectory of the ray. For simplicity, unless otherwise stated,...

    work page

  2. [2]

    B. P. Abbott et al. [LIGO Scientific and Virgo Collab- orations], Phys. Rev. Lett. 116, 061102 (2016)

    work page 2016

  3. [3]

    B. P. Abbott et al. [LIGO Scientific and Virgo Collab- orations], Phys. Rev. X 9, 031040 (2019)

    work page 2019

  4. [4]

    Abbott et al

    R. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. X 11, 021053 (2021)

    work page 2021

  5. [5]

    Abbott et al

    R. Abbott et al. [KAGRA, VIRGO and LIGO Scien- tific], Phys. Rev. X 13, 041039 (2023)

    work page 2023

  6. [6]

    A. G. Abac et al. [LIGO Scientific, VIRGO and KA- GRA], [arXiv:2508.18082 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv

  7. [7]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope Collabora- tion], Astrophys. J. 875, L1 (2019)

    work page 2019

  8. [8]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, L12 (2022)

    work page 2022

  9. [9]

    Perlick, Living Rev

    V. Perlick, Living Rev. Relativity 7, 9 (2004)

    work page 2004

  10. [10]

    Hod, Phys

    S. Hod, Phys. Lett. B 727, 345 (2013)

    work page 2013

  11. [11]

    Peng, Phys

    Y. Peng, Phys. Lett. B 790, 396-399 (2019)

    work page 2019

  12. [12]

    Hod, Phys

    S. Hod, Phys. Rev. D 101, 084033 (2020)

    work page 2020

  13. [13]

    Hod, JHEP 12, 178 (2023)

    S. Hod, JHEP 12, 178 (2023)

    work page 2023

  14. [14]

    Hod, Phys

    S. Hod, Phys. Lett. B 776, 1 (2018)

    work page 2018

  15. [15]

    P. V. P. Cunha and C. A. R. Herdeiro, Phys. Rev. Lett. 124, 181101 (2020)

    work page 2020

  16. [16]

    P. V. P. Cunha, C. A. R. Herdeiro and J. P. A. Novo, Phys. Rev. D 109, 064050 (2024)

    work page 2024

  17. [17]

    Padhye, K

    P. Padhye, K. Paithankar and S. Kolekar, [arXiv:2412.12665 [gr-qc]]

    work page arXiv

  18. [18]

    N. G. Sanchez, Phys. Rev. D 18, 1030 (1978)

    work page 1978

  19. [19]

    S. W. Wei, Y. X. Liu, and H. Guo, Phys. Rev. D 84, 041501 (2011)

    work page 2011

  20. [20]

    Decanini, A

    Y. Decanini, A. Folacci, and B. Raffaelli, Phys. Rev. D 81, 104039 (2010)

    work page 2010

  21. [21]

    W. H. Press, Astrophys. J. 170, L105 (1971)

    work page 1971

  22. [22]

    C. J. Goebel, Astrophys. J. 172, L95 (1972)

    work page 1972

  23. [23]

    Raffaelli, Gen

    B. Raffaelli, Gen. Rel. Grav. 48, 16 (2016)

    work page 2016

  24. [24]

    Deflection Angle in the Strong Deflection Limit and Quasinormal Modes in Stationary Axisymmetric Spacetimes,

    T. Igata, [arXiv:2505.01848 [gr-qc]]

    work page arXiv

  25. [25]

    Deflection angle in the strong deflection limit for static axisymmetric spacetimes: local curvature, matter fields, and quasinormal modes,

    T. Igata, [arXiv:2504.07906 [gr-qc]]

    work page arXiv

  26. [26]

    I. Z. Stefanov, S. S. Yazadjiev, and G. G. Gyulchev, Phys. Rev. Lett. 104, 251103 (2010)

    work page 2010

  27. [27]

    M. A. Abramowicz and A. R. Prasanna, Mon. Not. Roy. Astr. Soc. 245, 720 (1990)

    work page 1990

  28. [28]

    M. A. Abramowicz, Mon. Not. Roy. Astr. Soc. 245, 733 (1990)

    work page 1990

  29. [29]

    Allen, Nature 347, 615 (1990)

    B. Allen, Nature 347, 615 (1990)

    work page 1990

  30. [30]

    Hasse and V

    W. Hasse and V. Perlick, Gen. Relativ. Gravit. 34, 415 (2002)

    work page 2002

  31. [31]

    P. Mach, E. Malec, and J. Karkowski, Phys. Rev. D 88, 084056 (2013)

    work page 2013

  32. [32]

    Chaverra and O

    E. Chaverra and O. Sarbach, Class. Quant. Grav. 32, 155006 (2015)

    work page 2015

  33. [33]

    Cvetic, G

    M. Cvetic, G. W. Gibbons, and C. N. Pope, Phys. Rev. D 94, 106005 (2016)

    work page 2016

  34. [34]

    Koga and T

    Y. Koga and T. Harada, Phys. Rev. D 94, 044053 (2016)

    work page 2016

  35. [35]

    Koga and T

    Y. Koga and T. Harada, Phys. Rev. D 98, 024018 (2018)

    work page 2018

  36. [36]

    Koga, Phys

    Y. Koga, Phys. Rev. D 99, 064034 (2019)

    work page 2019

  37. [37]

    Barcelo and M

    C. Barcelo and M. Visser, Nucl. Phys. B 584, 415 (2000)

    work page 2000

  38. [38]

    Koga, Phys

    Y. Koga, Phys. Rev. D 101 104022 (2020)

    work page 2020

  39. [39]

    Tsukamoto and T

    N. Tsukamoto and T. Kokubu, Phys. Rev. D 109, 024047 (2024)

    work page 2024

  40. [40]

    Keir, Class

    J. Keir, Class. Quant. Grav. 33, 135009 (2016)

    work page 2016

  41. [41]

    Cardoso, L

    V. Cardoso, L. C. B. Crispino, C. F. B. Macedo, H. Okawa, and P. Pani, Phys. Rev. D 90, 044069 (2014)

    work page 2014

  42. [42]

    P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, Phys. Rev. Lett. 119, 251102 (2017)

    work page 2017

  43. [43]

    P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, Phys. Rev. D 96, 024039 (2017)

    work page 2017

  44. [44]

    P. V. P. Cunha, C. Herdeiro, E. Radu and N. Sanchis- Gual, Phys. Rev. Lett. 130, 061401 (2023)

    work page 2023

  45. [45]

    Zhong, V

    Z. Zhong, V. Cardoso and E. Maggio, Phys. Rev. D 107, 044035 (2023)

    work page 2023

  46. [46]

    C. M. Claudel, K. S. Virbhadra, and G. F. R. Ellis, J. Math. Phys. 42, 818 (2001)

    work page 2001

  47. [47]

    G. W. Gibbons and C. M. Warnick, Phys. Lett. B 763, 169 (2016)

    work page 2016

  48. [48]

    Shiromizu, Y

    T. Shiromizu, Y. Tomikawa, K. Izumi, and H. Yoshino, PTEP 2017, 033E01 (2017)

    work page 2017

  49. [49]

    Yoshino, K

    H. Yoshino, K. Izumi, T. Shiromizu, and Y. Tomikawa, PTEP 2017, 063E01 (2017)

    work page 2017

  50. [50]

    D. V. Gal’tsov and K. V. Kobialko, Phys. Rev. D 99, 084043 (2019)

    work page 2019

  51. [51]

    D. V. Gal’tsov and K. V. Kobialko, Phys. Rev. D 100, 104005 (2019)

    work page 2019

  52. [52]

    Koga and T

    Y. Koga and T. Harada, Phys. Rev. D 100, 064040 (2019)

    work page 2019

  53. [53]

    Siino, [arXiv:1908.02921 [gr-qc]]

    M. Siino, [arXiv:1908.02921 [gr-qc]]

    work page arXiv 1908

  54. [54]

    Yoshino, K

    H. Yoshino, K. Izumi, T. Shiromizu, and Y. Tomikawa, PTEP 2020, 023E02 (2020)

    work page 2020

  55. [55]

    L. M. Cao and Y. Song, [arXiv:1910.13758 [gr-qc]]

    work page arXiv 1910

  56. [56]

    Yoshino, K

    H. Yoshino, K. Izumi, T. Shiromizu, and Y. Tomikawa, PTEP 2020, 053E01 (2020)

    work page 2020

  57. [57]

    K. Lee, T. Shiromizu, H. Yoshino, K. Izumi, and Y. Tomikawa, [arXiv:2007.03139 [gr-qc]]

    work page arXiv 2007

  58. [58]

    Schneider, J

    P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses (Springer-Verlag, Berlin, 1992)

    work page 1992

  59. [59]

    Schneider, C

    P. Schneider, C. S. Kochanek, and J. Wambsganss, Gravitational Lensing: Strong, Weak and Micro, Lec- ture Notes of the 33rd Saas-Fee Advanced Course, edited by G. Meylan, P. Jetzer, and P. North (Springer-Verlag, Berlin, 2006)

    work page 2006

  60. [60]

    Hagihara, Jpn

    Y. Hagihara, Jpn. J. Astron. Geophys., 8, 67 (1931)

    work page 1931

  61. [61]

    Darwin, Proc

    C. Darwin, Proc. R. Soc. Lond. A 249, 180 (1959)

    work page 1959

  62. [62]

    R. d’ E. Atkinson, Astron. J. 70, 517 (1965)

    work page 1965

  63. [63]

    Luminet, Astron

    J.-P. Luminet, Astron. Astrophys. 75, 228 (1979)

    work page 1979

  64. [64]

    H. C. Ohanian, Am. J. Phys. 55, 428 (1987)

    work page 1987

  65. [65]

    R. J. Nemiroff, Am. J. Phys. 61, 619 (1993)

    work page 1993

  66. [66]

    Frittelli, T

    S. Frittelli, T. P. Kling, and E. T. Newman, Phys. Rev. D 61, 064021 (2000)

    work page 2000

  67. [67]

    K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000)

    work page 2000

  68. [68]

    Bozza, S

    V. Bozza, S. Capozziello, G. Iovane and G. Scarpetta, Gen. Rel. Grav. 33, 1535-1548 (2001)

    work page 2001

  69. [69]

    Bozza, Phys

    V. Bozza, Phys. Rev. D 66, 103001 (2002)

    work page 2002

  70. [70]

    K. S. Virbhadra, Phys. Rev. D 79, 083004 (2009)

    work page 2009

  71. [71]

    Hioki and K

    K. Hioki and K. i. Maeda, Phys. Rev. D 80, 024042 12 (2009)

    work page 2009

  72. [72]

    Bozza, Gen

    V. Bozza, Gen. Relativ. Gravit. 42, 2269 (2010)

    work page 2010

  73. [73]

    O. Y. Tsupko, Phys. Rev. D 95, 104058 (2017)

    work page 2017

  74. [74]

    K. i. Nakao, C. M. Yoo and T. Harada, Phys. Rev. D 99, 044027 (2019)

    work page 2019

  75. [75]

    S. E. Gralla, D. E. Holz and R. M. Wald, Phys. Rev. D 100, 024018 (2019)

    work page 2019

  76. [76]

    Okabayashi, N

    K. Okabayashi, N. Asaka and K. i. Nakao, Phys. Rev. D 102, 044011 (2020)

    work page 2020

  77. [77]

    Aratore and V

    F. Aratore and V. Bozza, JCAP 10, 054 (2021)

    work page 2021

  78. [78]

    O. Y. Tsupko, Phys. Rev. D 106, 064033 (2022)

    work page 2022

  79. [79]

    Aratore, O

    F. Aratore, O. Y. Tsupko and V. Perlick, Phys. Rev. D 109, 124057 (2024)

    work page 2024

  80. [80]

    O. Y. Tsupko, F. Aratore and V. Perlick, [arXiv:2510.22804 [gr-qc]]

    work page arXiv

Showing first 80 references.