Pith. sign in

REVIEW 2 major objections 4 minor 5 cited by

This paper shows that the Gaussian curvature of the optical metric on a black hole's unstable photon orbit becomes multivalued during a first-order phase transition, mirroring the free-energy swallowtail and offering a purely geometric prob

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-05 05:36 UTC pith:V7NBB5KA

load-bearing objection The central identity is derivable in-paper and the math holds; the novelty is modest and the advertised 'heat-capacity-like divergence' is contradicted by the paper's own Eq. (46). the 2 major comments →

arxiv 2509.05103 v3 pith:V7NBB5KA submitted 2025-09-05 gr-qc hep-phhep-th

Gaussian curvature and Lyapunov exponent as probes of black hole phase transitions

Shi-Hao Zhang , Zi-Qiang Zhao , Zi-Yuan Li , Jing-Fei Zhang , Xin Zhang This is my paper
classification gr-qc hep-phhep-th PACS 04.70.-s
keywords Gaussian curvatureblack hole phase transitionsLyapunov exponentnull circular orbitsswallowtail structurespinodal regioncritical exponentAdS black holes
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that a purely geometric quantity—the Gaussian curvature K of the optical metric evaluated on a black hole's unstable photon orbit (the light ring)—carries direct information about black hole phase transitions. For a first-order transition, the paper claims, the curve K versus temperature T is multivalued over exactly the spinodal region where the Gibbs free energy shows its swallowtail structure, because three black hole branches coexist there. Numerical analysis of three spherically symmetric families (RN-AdS, Hayward-AdS, Hayward-Letelier-AdS) supports the claim, and the paper extends it by showing that the curvature jump ΔK scales as (T - Tc)^{1/2} near the critical point, matching the known Lyapunov-exponent and mean-field behavior. If correct, this means phase transitions can be diagnosed from intrinsic spacetime geometry alone, without computing thermodynamic potentials, and that the geometric probe and the chaotic-dynamics probe are two faces of the same underlying relation K = -λ².

Core claim

The central claim: intrinsic geometry encodes black hole phase transitions. For a spherically symmetric metric, the authors build the optical metric of null geodesics and evaluate its Gaussian curvature K at the unstable photon orbit (light ring), where K depends only on the metric functions and their derivatives. During a first-order phase transition, K as a function of temperature is multivalued inside the spinodal region (T1, T2), since the small, intermediate, and large black hole branches each carry their own curvature; this region coincides with the free-energy swallowtail. Via the identity K(r_LR) = -λ²(r_LR), the known multivaluedness and critical scaling of the Lyapunov exponent tra

What carries the argument

The central object is the Gaussian curvature K of the two-dimensional optical metric dt² = (1/f)(dr²/g + r² dφ²), evaluated at the light ring r_LR, the unstable null circular orbit defined by f'(r_LR) = 2f(r_LR)/r_LR. There K reduces to (g/2)(f'' - f'/r) at r_LR, a purely intrinsic quantity of the metric. The companion identity K(r_LR) = -λ²(r_LR) (Eq. 21), imported from Ref. [51], ties this curvature to the Lyapunov exponent of the orbit, so the geometric probe inherits the dynamical probe's multivaluedness at phase transitions and its critical exponent δ = 1/2. The light ring itself is located two independent ways—via the effective-potential condition (Appendix A) and via vanishing geodesi

Load-bearing premise

The load-bearing premise is the identity K(r_LR) = -λ²(r_LR), imported from Ref. [51] without re-derivation; if it fails for these metrics, the geometric probe loses its equivalence to the Lyapunov probe and its transferred critical exponent, though the numerical multivaluedness of K itself would stand.

What would settle it

Compute K(r_LR) and λ(r_LR) independently for a black hole beyond the spherically symmetric class (for instance a rotating AdS solution) and test whether K = -λ² still holds; a counterexample severs the geometric-chaotic equivalence on which the critical-exponent argument depends. Closer to the paper's own setting: take a spherically symmetric black hole with a confirmed swallowtail and evaluate K(T) inside the spinodal region—if a single temperature yields fewer K branches than free-energy branches, the geometric signature is not universal.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • Gaussian curvature K(T) becomes a phase-transition diagnostic that needs no thermodynamic potential: multivalued K inside the spinodal region signals a first-order transition, and monotonic K rules one out.
  • The curvature jump between the large and small black hole branches, ΔK, behaves as an order parameter with the mean-field critical exponent 1/2, matching the Lyapunov-exponent order parameter in earlier studies.
  • Because K = -λ² on the light ring, any Lyapunov-exponent computation for null orbits in these spacetimes is also a Gaussian-curvature computation; the two probes are interchangeable.
  • At the second-order critical point, K exhibits heat-capacity-like divergent behavior, extending the geometric probe beyond first-order transitions (stated in the abstract).

Where Pith is reading between the lines

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

  • The construction is stated for spherical symmetry, but the optical-metric route to K should generalize to rotating black holes; if the multivaluedness persists on the equatorial light ring of Kerr-AdS, the geometric probe would cover a much wider class of AdS black holes.
  • If Eq. (21) survives testing beyond the metrics considered here, the entire Lyapunov-exponent literature for null orbits—order-parameter readings, MSS-bound checks—becomes convertible into geometric data wherever photon orbits are known.
  • A practical outgrowth could be a purely geometric criticality detector: a diverging ∂K/∂T at some T_c would flag a second-order transition in spacetimes whose thermodynamic description is incomplete or unknown.
  • The Hayward-Letelier example stacks a string cloud on nonlinear electrodynamics; testing more exotic matter distributions (quintessence, bumblebee fields) would show how well the spinodal-region coincidence survives.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proposes using the Gaussian curvature K of unstable null circular orbits (light rings), computed in the optical metric, as a geometric probe of black hole phase transitions. For RN-AdS, Hayward-AdS, and Hayward-Letelier-AdS black holes, the authors compute K as a function of temperature, find that K(T) is multivalued in the spinodal region where the free energy has a swallowtail, and show that this multivaluedness disappears when no phase transition occurs. Using the relation K = -lambda^2, they transfer known Lyapunov-exponent results to K and derive a critical exponent delta_K = 1/2 near the second-order critical point. The paper concludes that intrinsic spacetime geometry encodes black hole phase structure.

Significance. The direct numerical demonstration that the Gaussian curvature of light rings becomes multivalued exactly in the spinodal region is a genuinely useful and coordinate-invariant diagnostic: it does not require constructing thermodynamic potentials and works for three different spherically symmetric AdS black-hole families, including a Hayward-Letelier-AdS case not previously studied in this context. The plots and the explicit K(r_LR) formulas in Secs. II and III are clear and reproducible. However, the advertised 'heat-capacity-like divergence' of K is not supported by the paper's own scaling relation, and the 'order parameter-like' claim is not derived from the small/large branch discontinuity. These issues concern the framing and generality of the central claim rather than the multivaluedness result itself.

major comments (2)
  1. [Abstract; Sec. V; Eq. (46)] The abstract states that the Gaussian curvature exhibits a 'heat-capacity-like divergence' at the second-order phase transition point, and Sec. V says 'both K and lambda diverge with a critical exponent of 1/2'. These statements are contradicted by Eq. (46), which gives Delta K ~ (T-T_c)^{1/2}; this quantity vanishes as T -> T_c, and K(r_c) is finite. Only the derivative d(Delta K)/dT diverges. The divergence claim is therefore false as written and should be removed or explicitly reframed as a divergence of the derivative, analogous to heat capacity. The multivaluedness result is unaffected.
  2. [Sec. IV, Eqs. (38)-(46)] The critical-exponent derivation is built on lambda_+ - lambda_c, an expansion of one branch near the critical radius, but the text then identifies the order parameter as Delta lambda = lambda_l - lambda_s, the difference between coexisting large and small branches. These are not the same quantity, and Eq. (41) does not by itself establish the branch-difference scaling. The order-parameter claim in the abstract and Sec. V therefore needs an explicit argument that lambda_l - lambda_s (and likewise |K_l| - |K_s|) behaves as (T-T_c)^{1/2}, or the claim should be rephrased so that Delta K is not presented as a discontinuity order parameter.
minor comments (4)
  1. [Sec. II.C, Eq. (21)] The analytic argument that K inherits the multivaluedness of lambda relies on the known relation K = -lambda^2 from Ref. [51]. Since the paper aims to present K as an independent geometric probe, it would strengthen the presentation to show explicitly how Eqs. (10)-(12), (14), (17) and (19) combine to give Eq. (21). The numerical K plots already provide independent evidence, so this is a clarification rather than a blocking issue.
  2. [Sec. IV, Eq. (43)] The notation Delta K is defined as |K(r_+)|-|K(r_c)|, which is a one-branch quantity, while the text simultaneously invokes a small/large branch order parameter. Please define both quantities separately (e.g., Delta K_+ and Delta K_{l-s}) to avoid ambiguity.
  3. [Sec. III.A] The sentence 'we present only the behavior of the Gaussian curvature K_R for its light rings at T_p1' uses an unsubscripted T_p1; it should read \tilde{T}_{p1}. Similar typographical issues appear in Fig. 3 captions where T_p3 should be \tilde{T}_{p3}.
  4. [Abstract] The abstract says 'Numerical analysis of Hayward-Letelier-AdS black holes confirms the effectiveness', but the paper also analyzes RN-AdS and Hayward-AdS. Please list all three models, or phrase the sentence as covering the three families studied.

Circularity Check

0 steps flagged

No significant circularity; K = -λ² is derived in-paper and K multivaluedness is directly computed.

full rationale

The paper's central relation K(r_LR) = -λ²(r_LR) is not merely imported from Ref. [51]; it is derived in the text by combining Eqs. (14), (17), (19), and (12) (Sec. II C), and the algebra is straightforward. The multivaluedness of K in the spinodal region is established in two independent ways: directly from the optical-metric Gaussian curvature for each black hole (Figs. 1-3), and as a logical consequence of Eq. (21) plus previously known λ multivaluedness — an implication, not a circular definition. No parameter is fitted and then renamed a prediction. The critical-exponent result ΔK ~ (T-Tc)^{1/2} is a corollary of Eq. (21) and the λ expansion; it transfers the λ exponent to K but does not define K in terms of λ. The only substantive issue found is a correctness problem, not circularity: the abstract and Discussion claim a 'heat-capacity-like divergence' of K at the critical point, whereas Eq. (46) gives ΔK ~ (T-Tc)^{1/2} → 0; only d(ΔK)/dT diverges. This inconsistency should be corrected but does not make the derivation circular.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper introduces no new free parameters, since the specific charge and mass values in the figures are illustrative examples, not fitted. It relies on standard black hole thermodynamics, the optical metric construction, and the externally established relation K = -λ², which is load-bearing for the critical exponent result.

axioms (5)
  • domain assumption Black holes obey standard four laws of thermodynamics with F = M - TS and T = f'(r+)/4π.
    Invoked in Sec. II.A to define phase transitions and locate spinodal regions.
  • domain assumption The optical metric (ds²=0) restricted to the equatorial plane is a two-dimensional Riemannian manifold whose Gaussian curvature characterizes unstable null orbits.
    Used in Sec. II.B to define K and evaluate it at light rings.
  • domain assumption The relation K(r_LR) = -λ²(r_LR) (Eq. 21) holds for the spherically symmetric spacetimes studied, as established for null circular orbits in Ref. [51].
    Central to the critical-exponent derivation in Sec. IV and to the claimed equivalence of geometric and dynamical probes.
  • standard math The light ring condition f'(r_LR) = 2f(r_LR)/r_LR (Eq. 52) selects the unstable photon orbit.
    Derived in Appendix A from the effective potential; standard result.
  • standard math The critical point is determined by ∂T/∂r_+ = ∂²T/∂r_+² = 0 (Eq. 4).
    Standard condition for second-order critical point in black hole thermodynamics.

pith-pipeline@v1.4.0-alltime-deepseek-medium · 10857 in / 18524 out tokens · 180858 ms · 2026-08-05T05:36:35.105717+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of Gaussian curvature and Lyapunov exponent as probes of black hole phase transitions." pith.science (2026). https://pith.science/paper/V7NBB5KA

@misc{pith2026250905103,
  author       = {Pith},
  title        = {Pith review of: Gaussian curvature and Lyapunov exponent as probes of black hole phase transitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V7NBB5KA}},
  note         = {Machine review of arXiv:2509.05103}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

First-order phase transitions of black holes have been extensively studied within thermodynamic frameworks, yet the corresponding evolution of spacetime geometric properties remains unclear. This paper establishes a purely differential geometric framework to probe such phase transitions by analyzing the curvature of unstable null orbits. Using the geodesic curvature of the null circular orbit in the optical metric to locate the light ring, we demonstrate that the corresponding Gaussian curvature $K$ serves as a direct geometric signature of the phase transition. During a first-order phase transition, the curve $K$ versus temperature $T$ exhibits a multivalued structure within the spinodal region, precisely mirroring the swallowtail behavior of the free energy. Numerical analysis of Hayward-Letelier-AdS black holes confirms the effectiveness of this geometric signature. Our work demonstrates that the intrinsic geometric quantities of spacetime encode the information of black hole phase transitions. These quantities serve as geometric probes of black hole phase transitions, while their discontinuity between the small and large black hole branches exhibits order parameter-like behavior. As an extension of this geometric probe, we also find that the Gaussian curvature exhibits a heat-capacity-like divergence at the second-order phase transition point. These results provide a purely geometric foundation for understanding the correspondence between thermodynamics and spacetime curvature in the null case.

Figures

Figures reproduced from arXiv: 2509.05103 by Jing-Fei Zhang, Shi-Hao Zhang, Xin Zhang, Zi-Qiang Zhao, Zi-Yuan Li.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Free energy [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Free energy [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Free energy [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 5 Pith papers

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

  1. Unifying topological, geometric, and complex classifications of black hole thermodynamics

    gr-qc 2026-04 unverdicted novelty 7.0

    Three classification schemes for black hole thermodynamics are equivalent, with the count of temperature extrema determining the class in each framework.

  2. Off-shell Hessian thermodynamic stability of higher-curvature black holes

    gr-qc 2026-06 unverdicted novelty 6.0

    An off-shell Hessian criterion H = S'_W(r_h) T'(r_h) governs thermodynamic stability of higher-curvature black holes, recovering the temperature-slope rule on physical branches and producing mean-field critical exponents.

  3. Unifying topological, geometric, and complex classifications of black hole thermodynamics

    gr-qc 2026-04 conditional novelty 6.0

    Black hole thermodynamic classifications based on temperature-curve extrema, topological winding numbers, and complex Riemann-surface foliations are equivalent; all reduce to counting the extrema of the temperature function.

  4. Phase Transitions with Lyapunov Exponents under Einstein and String Frames in Dilatonic Reissner--Nordstr\"om--AdS Black Holes

    gr-qc 2026-07 unverdicted novelty 5.0

    Lyapunov exponents of charged probes capture the same cusp and transition points in dilatonic RN-AdS phase structure across Einstein and string frames, even though the exponent values themselves depend on frame for ma...

  5. Synergy between CSST and third-generation gravitational-wave detectors: Inferring cosmological parameters using cross-correlation of dark sirens and galaxies

    astro-ph.CO 2025-10 unverdicted novelty 5.0

    Forecasts that cross-correlating 3G GW dark sirens with CSST photometric galaxies yields 1.04% precision on H0 and 2.04% on Omega_m while also constraining GW clustering bias.

Reference graph

Works this paper leans on

58 extracted references · 14 canonical work pages · cited by 4 Pith papers · 3 internal anchors

  1. [1]

    S. W. Hawking, Commun. Math. Phys.43, 199 (1975), [Erratum: Commun.Math.Phys. 46, 206 (1976)]

    1975

  2. [2]

    J. D. Bekenstein, Phys. Rev. D7, 2333 (1973)

    1973

  3. [3]

    S. W. Hawking and D. N. Page, Commun. Math. Phys. 87, 577 (1983)

    1983

  4. [4]

    c” denotes the value of the Lya- punov exponent at the critical point (λ c =λ(˜rc)), and the subscript “+

    and a reentrant phase transition [5]. Furthermore, building on the holographic principle in black hole ther- modynamics, Maldacena established the AdS/CFT cor- respondence [6] in 1997. Further investigation within this duality showed that black holes as quantum thermody- namic systems exhibit chaos and that their Lyapunov exponents obey the MSS boundλ≤2πT...

    Pith/arXiv arXiv 1997

  5. [5]

    Kubiznak and R

    D. Kubiznak and R. B. Mann, JHEP07, 033 (2012), arXiv:1205.0559 [hep-th]

    Pith/arXiv arXiv 2012

  6. [6]

    Altamirano, D

    N. Altamirano, D. Kubiznak, and R. B. Mann, Phys. Rev. D88, 101502 (2013), arXiv:1306.5756 [hep-th]

    Pith/arXiv arXiv 2013

  7. [7]

    J. M. Maldacena, Adv. Theor. Math. Phys.2, 231 (1998), arXiv:hep-th/9711200

    Pith/arXiv arXiv 1998

  8. [8]

    S. H. Shenker and D. Stanford, JHEP3, 67 (2014), arXiv:1306.0622 [hep-th]

    Pith/arXiv arXiv 2014

  9. [9]

    S. H. Shenker and D. Stanford, JHEP3, 46 (2014), arXiv:1312.3296 [hep-th]

    Pith/arXiv arXiv 2014

  10. [10]

    Maldacena, S

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

    Pith/arXiv arXiv 2016

  11. [11]

    Hashimoto and N

    K. Hashimoto and N. Tanahashi, Phys. Rev. D95, 024007 (2017)

    work page 2017

  12. [12]

    Zhao, Y.-Z

    Q.-Q. Zhao, Y.-Z. Li, and H. L¨ u, Phys. Rev. D98, 124001 (2018)

    work page 2018

  13. [13]

    Dalui, B

    S. Dalui, B. R. Majhi, and P. Mishra, Phys. Lett. B788, 486 (2019), arXiv:1803.06527 [gr-qc]

    Pith/arXiv arXiv 2019

  14. [14]

    Lei, X.-H

    Y.-Q. Lei, X.-H. Ge, and C. Ran, Phys. Rev. D104, 046020 (2021)

    2021

  15. [15]

    Kan and B

    N. Kan and B. Gwak, Phys. Rev. D105, 026006 (2022)

    work page 2022

  16. [16]

    Cardoso, A

    V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Phys. Rev. D79, 064016 (2009)

    2009

  17. [17]

    G. Guo, P. Wang, H. Wu, and H. Yang, JHEP06, 060 (2022), arXiv:2112.14133 [gr-qc]

    Pith/arXiv arXiv 2022

  18. [18]

    Liu, D.-C

    Y. Liu, D.-C. Zou, and B. Wang, JHEP09, 179 (2014), arXiv:1405.2644 [hep-th]

    Pith/arXiv arXiv 2014

  19. [19]

    Mahapatra, JHEP04, 142 (2016), arXiv:1602.03007 [hep-th]

    S. Mahapatra, JHEP04, 142 (2016), arXiv:1602.03007 [hep-th]

    Pith/arXiv arXiv 2016

  20. [20]

    Chabab, H

    M. Chabab, H. El Moumni, S. Iraoui, and K. Masmar, Eur. Phys. J. C76, 676 (2016), arXiv:1606.08524 [hep- th]

    Pith/arXiv arXiv 2016

  21. [21]

    D.-C. Zou, Y. Liu, and R.-H. Yue, Eur. Phys. J. C77, 365 (2017), arXiv:1702.08118 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  22. [22]

    Phase transition and Quasinormal modes for Charged black holes in 4D Einstein-Gauss-Bonnet gravity

    M. Zhang, C.-M. Zhang, D.-C. Zou, and R.-H. Yue, Chin. Phys. C45, 045105 (2021), arXiv:2009.03096 [hep- th]. 9

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  23. [23]

    Zhao, X.-K

    Z.-Q. Zhao, X.-K. Zhang, and Z.-Y. Nie, JHEP02, 023 (2023), arXiv:2211.14762 [hep-th]

    Pith/arXiv arXiv 2023

  24. [24]

    Zhao, Z.-Y

    X. Zhao, Z.-Y. Nie, Z.-Q. Zhao, H.-B. Zeng, Y. Tian, and M. Baggioli, JHEP02, 184 (2024), arXiv:2311.08277 [hep-th]

    Pith/arXiv arXiv 2024

  25. [25]

    Zhao, Z.-Y

    Z.-Q. Zhao, Z.-Y. Nie, J.-F. Zhang, X. Zhang, and M. Baggioli, Eur. Phys. J. C85, 464 (2025), arXiv:2406.05345 [hep-th]

    Pith/arXiv arXiv 2025

  26. [26]

    Zhao, Z.-Y

    Z.-Q. Zhao, Z.-Y. Nie, J.-F. Zhang, and X. Zhang, (2025), arXiv:2504.04995 [gr-qc]

    arXiv 2025

  27. [27]

    Hou, Y.-Q

    Z.-Y. Hou, Y.-Q. Lei, and X.-H. Ge, (2025), arXiv:2508.00404 [hep-th]

    arXiv 2025

  28. [28]

    X. Guo, Y. Lu, B. Mu, and P. Wang, JHEP08, 153 (2022), arXiv:2205.02122 [gr-qc]

    Pith/arXiv arXiv 2022

  29. [29]

    S. Yang, J. Tao, B. Mu, and A. He, Journal of Cosmology and Astroparticle Physics2023, 045 (2023)

    work page 2023

  30. [30]

    X. Lyu, J. Tao, and P. Wang, Eur. Phys. J. C84, 974 (2024), arXiv:2312.11912 [gr-qc]

    Pith/arXiv arXiv 2024

  31. [31]

    Du, H.-F

    Y.-Z. Du, H.-F. Li, Y.-B. Ma, and Q. Gu, Eur. Phys. J. C85, 78 (2025), arXiv:2403.20083 [hep-th]

    Pith/arXiv arXiv 2025

  32. [32]

    Shukla, P

    B. Shukla, P. P. Das, D. Dudal, and S. Mahapatra, Phys. Rev. D110, 024068 (2024)

    work page 2024

  33. [33]

    N. J. Gogoi, S. Acharjee, and P. Phukon, Eur. Phys. J. C84, 1144 (2024), arXiv:2404.03947 [hep-th]

    Pith/arXiv arXiv 2024

  34. [34]

    D. Chen, C. Yang, and Y. Liu, Physics Letters B865, 139463 (2025)

    work page 2025

  35. [35]

    K. M. A. Karthik R., Dillirajan D. and K. Hegde, arXiv preprint arXiv:2504.12890 (2025)

    arXiv 2025

  36. [36]

    M. B. Awal and P. Phukon, arXiv preprint arXiv:2505.20800 (2025)

    Pith/arXiv arXiv 2025

  37. [37]

    C. Yang, C. Gao, D. Chen, and X. Zeng, (2025), arXiv:2506.21882 [hep-th]

    arXiv 2025

  38. [38]

    Wei and Y.-X

    S.-W. Wei and Y.-X. Liu, Phys. Rev. D97, 104027 (2018)

    2018

  39. [39]

    Wei, Y.-X

    S.-W. Wei, Y.-X. Liu, and Y.-Q. Wang, Phys. Rev. D 99, 044013 (2019), arXiv:1807.03455 [gr-qc]

    Pith/arXiv arXiv 2019

  40. [40]

    Zhang, S.-Z

    M. Zhang, S.-Z. Han, J. Jiang, and W.-B. Liu, Phys. Rev. D99, 065016 (2019), arXiv:1903.08293 [hep-th]

    Pith/arXiv arXiv 2019

  41. [41]

    Xu, H.-M

    Y.-M. Xu, H.-M. Wang, Y.-X. Liu, and S.-W. Wei, Phys. Rev. D100, 104044 (2019), arXiv:1906.03334 [gr-qc]

    Pith/arXiv arXiv 2019

  42. [42]

    H. Li, Y. Chen, and S.-J. Zhang, Nucl. Phys. B954, 114975 (2020), arXiv:1908.09570 [hep-th]

    Pith/arXiv arXiv 2020

  43. [43]

    Naveena Kumara, C

    A. Naveena Kumara, C. L. Ahmed Rizwan, S. Punacha, K. M. Ajith, and M. S. Ali, Phys. Rev. D102, 084059 (2020), arXiv:1912.11909 [gr-qc]

    Pith/arXiv arXiv 2020

  44. [44]

    Du, H.-F

    Y.-Z. Du, H.-F. Li, F. Liu, and L.-C. Zhang, JHEP01, 137 (2023), arXiv:2204.01007 [hep-th]

    Pith/arXiv arXiv 2023

  45. [45]

    Kumar, S

    A. Kumar, S. G. Ghosh, and A. Wang, Phys. Dark Univ. 46, 101608 (2024)

    2024

  46. [46]

    Yang, S.-P

    S.-J. Yang, S.-P. Wu, S.-W. Wei, and Y.-X. Liu, arXiv preprint arXiv:2505.20860 (2025), arXiv:2505.20860 [gr- qc]

    arXiv 2025

  47. [47]

    Qiao and M

    C.-K. Qiao and M. Li, Phys. Rev. D106, L021501 (2022), arXiv:2204.07297 [gr-qc]

    Pith/arXiv arXiv 2022

  48. [48]

    Chern,Differential Geometry(Peking University Press, 2006)

    W.-H. Chern,Differential Geometry(Peking University Press, 2006)

    work page 2006

  49. [49]

    Qiao, Eur

    C.-K. Qiao, Eur. Phys. J. C85, 191 (2025), arXiv:2407.14035 [gr-qc]

    Pith/arXiv arXiv 2025

  50. [50]

    Z. Li, G. Zhang, and A. ¨Ovg¨ un, Phys. Rev. D101, 124058 (2020), arXiv:2006.13047 [gr-qc]

    Pith/arXiv arXiv 2020

  51. [51]

    Qiao and M

    C.-K. Qiao and M. Zhou, JCAP12, 005 (2023), arXiv:2212.13311 [gr-qc]

    Pith/arXiv arXiv 2023

  52. [52]

    Bounds for Lyapunov exponent of circular light orbits in black holes

    E. Gallo and T. M¨ adler, Eur. Phys. J. C85, 299 (2025), arXiv:2412.10328 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  53. [53]

    Bhattacharyya, W

    A. Bhattacharyya, W. Chemissany, S. S. Haque, J. Mu- rugan, and B. Yan, SciPost Phys. Core4, 002 (2021), arXiv:2007.01232 [hep-th]

    Pith/arXiv arXiv 2021

  54. [54]

    J., inConference Proceedings of GR5(1968)

    B. J., inConference Proceedings of GR5(1968)

    work page 1968

  55. [55]

    S. A. Hayward, Phys. Rev. Lett.96, 031103 (2006)

    2006

  56. [56]

    Kumar, A

    A. Kumar, A. Sood, S. G. Ghosh, and A. Beesham, Particles7, 1017 (2024)

    work page 2024

  57. [57]

    Banerjee and D

    R. Banerjee and D. Roychowdhury, Phys. Rev. D85, 104043 (2012)

    work page 2012

  58. [58]

    Banerjee and D

    R. Banerjee and D. Roychowdhury, Phys. Rev. D85, 044040 (2012)

    work page 2012