Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Massless scalar waves scattered by Frolov black holes are claimed to be governed by the unstable photon orbit, with the regular de Sitter-like core affecting absorption and scattering only through subleading corrections.

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 ↗

Frolov black hole scalar absorption and scattering are photon-sphere-dominated: rescaling with photon-sphere frequency and geometric cross section collapses the high-frequency spectra, and impact-parameter-matched Frolov, Reissner-Nordström, and Hayward black holes look nearly identical.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection Competent first computation of scalar absorption/scattering for Frolov black holes, but the photon-sphere-control claim is asserted on visual comparisons and needs quantitative residuals and convergence tests. the 4 major comments →

arxiv 2601.19364 v2 pith:BNLKRYDK submitted 2026-01-27 gr-qc

Absorption and scattering of massless scalar waves by Frolov black holes

Jining Tang , Yang Huang , Hongsheng Zhang This is my paper
classification gr-qc MSC 83C57 PACS 04.70.-s04.20.-q
keywords Frolov black holemassless scalar wavesabsorption cross sectionscattering cross sectionphoton sphereunstable null orbitsregular black holespartial-wave method
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.

The reading

This paper aims to establish that the high-frequency absorption and scattering of massless scalar waves by Frolov black holes—a regular deformation of Reissner–Nordström geometry—is set almost entirely by the unstable photon orbit rather than by the regular core. Using null-geodesic analysis and a partial-wave numerical solution of the scalar wave equation, it derives the geometric capture cross section and photon-sphere angular frequency, then shows that rescaling the absorption spectrum by these scales makes the high-frequency oscillations for different Frolov parameters collapse onto one curve. It also shows that Frolov, Reissner–Nordström, and Hayward black holes chosen to share the same critical or glory impact parameter produce nearly identical absorption and scattering spectra across a broad frequency range. If correct, massless-scalar wave observables are weak discriminators of the interior structure of regular black holes, and the photon sphere, not the core, supplies the dominant signature.

Core claim

The central claims are two. First, for the Frolov spacetime the geometric-optics absorption and its oscillatory fine structure are controlled by photon-sphere data: σ_geo = π b_c², Ω_c = 1/b_c, and the Lyapunov exponent Λ_c enter a sinc-type formula whose rescaled form σ_abs/σ_geo versus ω/Ω_c collapses the high-frequency oscillations across the Frolov parameter space. Second, along iso-impact-parameter lines in the horizon-admitting (ℓ², q²) plane, Reissner–Nordström, Hayward, and intermediate Frolov configurations with matched b_c (for absorption) or b_g (for glory scattering) yield cross sections that nearly coincide, including in the intermediate frequency/angle regime where overlap is n

What carries the argument

The unstable photon orbit of the Frolov spacetime and its derived quantities—the critical radius r_c, the critical impact parameter b_c = r_c/√f(r_c), the photon-sphere angular frequency Ω_c = 1/b_c, and the Lyapunov exponent Λ_c (the damping rate of the orbit)—form the machinery. These enter the geometric capture cross section σ_geo = π b_c² and the high-frequency absorption formula σ_hf ≈ σ_geo[1 − (8πΛ_c/Ω_c) e^{−πΛ_c/Ω_c} sinc(2π b_c ω)], which supplies the benchmark curve for interpreting the numerics. The paper also uses iso-impact-parameter comparisons in the (ℓ², q²) parameter space, where fixing r_h = 1 makes the dimensionless mass m(q,ℓ) vary, to test whether wave observables track

Load-bearing premise

The conclusions rest on treating the standard high-frequency sinc formula, built from photon-sphere data, as a valid benchmark for the Frolov geometry at all displayed frequencies; the paper does not re-derive this approximation for the Frolov radial equation, so a core-induced reflection not captured by the formula would weaken the claim.

What would settle it

Compute the absorption cross section for a near-extremal Frolov configuration where the core curvature scale is comparable to the photon-sphere radius, at frequencies around the effective-potential barrier peak, with partial waves pushed to numerical convergence; if σ_abs/σ_geo plotted against ωb_c does not fall onto the same sinc envelope as Reissner–Nordström and Hayward black holes with matched b_c, the photon-sphere-control claim is refuted. A direct detection of reflected waves from the regular core in the time-domain wave scattering would also break the one-way eikonal picture.

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

If this is right

  • If the photon-sphere control claim is right, massless-scalar absorption and scattering are weak probes of the regular core; distinguishing Frolov from Reissner–Nordström or Hayward geometries with these observables would require precision beyond the dominant oscillation pattern.
  • Any regular black hole with the same photon-sphere parameters would be expected to produce the same high-frequency absorption fine structure, making the rescaled spectrum a model-independent benchmark.
  • The r_h = 1 normalization means apparent trends of cross sections with charge and Hubble-length parameters partly reflect the variation of the normalized mass, so cross-normalization comparisons (r_h = 1 vs M = 1) must account for this to avoid misreading core effects.
  • The matched-impact-parameter degeneracy extends from absorption to glory scattering, so both observables are governed by the same null-orbit data in the intermediate-to-high frequency regime.

Where Pith is reading between the lines

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

  • The residual differences between the rescaled curves at intermediate frequencies, though subleading, may still encode the core length scale α; a systematic fit of σ_abs − σ_hf in ω could expose the de Sitter core's signature even when the dominant pattern does not.
  • If the same reasoning carries to higher-spin fields, their spin-curvature coupling near the photon sphere is likely to break the impact-parameter degeneracy, making spin a sharper probe of the regular interior than the scalar field.
  • The data collapse suggests a testable prediction for rotating regular black holes: the same photon-orbit rescaling applied separately to prograde and retrograde photon orbits should reproduce the collapse, with the core length scale entering only as subleading corrections.
  • Because no convergence test is shown for the l-truncations used (l=10 for absorption, l=50 for scattering), verifying that the reported collapse survives higher truncation at intermediate frequencies would strengthen the empirical case for photon-sphere control.
Share X Bluesky LinkedIn Reddit HN

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

4 major / 5 minor

Summary. The paper studies massless scalar absorption and scattering by static, spherically symmetric Frolov black holes. It derives null-geodesic photon-sphere data (critical impact parameter, orbital frequency, Lyapunov exponent), weak-deflection and glory approximations, the partial-wave radial equation and effective potential, and the standard low- and high-frequency limits. Numerical integration of the radial equation yields total and partial absorption cross sections and differential scattering cross sections over a broad frequency range for representative parameter pairs (q,l) in the rh=1 normalization. The paper's central claims are: (i) after rescaling by the geometric capture cross section and the photon-sphere angular frequency, the high-frequency absorption oscillations collapse across parameter space (Fig. 7); and (ii) Reissner-Nordström, Hayward, and Frolov black holes with matched critical or glory impact parameters produce nearly identical absorption and scattering spectra (Fig. 11), so that photon-sphere data control the dominant signatures while core effects are subleading.

Significance. If substantiated quantitatively, the paper would provide a useful demonstration of universality in wave scattering by regular black holes. The analysis is parameter-free in the sense that no amplitudes are fitted: photon-sphere data are derived from the metric, and the numerical spectra are compared with independent analytic limits (low-frequency area law, high-frequency sinc formula, glory approximation). The careful discussion of the rh=1 versus M=1 normalizations and the resulting reinterpretation of parameter trends is a genuine strength. However, the central claims currently rest on visual comparisons of rescaled and matched curves; the manuscript does not report convergence tests, error estimates, or quantitative residuals. The paper also does not provide code or data. These gaps are fixable and do not undermine the evident competence of the analytic and numerical work.

major comments (4)
  1. [§IV.B, Fig. 7; §V] The 'data collapse' is only assessed visually. The variables σ_abs/σ_geo and x=ω/Ω_c are precisely the variables in which the imported sinc formula, Eq. (22), predicts a parameter-independent oscillation up to the slowly varying prefactor 8πΛ_c/Ω_c exp(-πΛ_c/Ω_c). Thus the collapse is, to a large extent, a restatement of Eq. (22) rather than an independent test of photon-sphere control. Please quantify: (i) the maximum/mean relative deviation among the rescaled curves over the displayed x range; (ii) the variation of the Eq. (22) prefactor across the parameter sets; (iii) the size of the residual relative to the numerical truncation error. Without such residuals, the claim that the high-frequency fine structure is 'largely governed by the unstable photon orbit' is not established at the quantitative level claimed.
  2. [§IV.A and §IV.D] The truncations l=10 for absorption and l=50 for scattering are asserted without convergence tests. This is load-bearing for Fig. 11: if the partial-wave sums are not fully converged at the displayed frequencies, the apparent overlap of matched spectra in the intermediate regime could be partly a truncation artifact. Please provide, for representative frequencies (e.g. ωrh=2, 4, 6), a convergence study in l_max for σ_abs and dσ/dΩ, including the YRW resummation at small angles, and state the estimated truncation error. This should be compared with the size of the claimed collapse residuals and matched-spectra differences.
  3. [§IV.D, Fig. 11] Matching b_c fixes σ_geo and the oscillation period, while matching b_g fixes the glory fringe spacing; hence the high-frequency agreement between the three geometries is substantially built into the matching procedure. The statement that the overlap in the intermediate regime is 'nontrivial' needs a quantitative definition. Please compute, for example, the L2 or maximum relative difference between the matched curves over ωrh∈[0,2] and over the displayed angular range, and compare these differences with those for unmatched parameter pairs (e.g. the curves in Figs. 6 and 10). Also show at least one additional matched pair at a different b_c or b_g value to demonstrate that the observed similarity is not a single tuned case.
  4. [Eqs. (22) and (39)] The high-frequency sinc formula is imported from Refs. [16,17] without derivation or a statement of the exact conditions under which it applies to the Frolov geometry, which has a de Sitter-like regular core. Since this formula is used as the benchmark for interpreting the numerical spectra and for motivating the collapse variables, the paper should either derive it in the Frolov setting via the complex-angular-momentum method, or explicitly cite the universality theorem and verify numerically that it reproduces the Frolov oscillations within residuals much smaller than the claimed subleading core effects. This is especially relevant at intermediate frequencies, where reflection from the regular core is not excluded a priori.
minor comments (5)
  1. [Fig. 10] The text and figure caption state ωrh=3, but the plot labels in the displayed figure appear to read 'ωrh = 1'. Please correct the inconsistency.
  2. [Fig. 4] The color-bar labels in the parameter-space plot are garbled (they appear as '5−1 0 1 62' etc.). The axis and color-bar labels should be cleaned up.
  3. [Eq. (44) and Fig. 5] The dependence of the dimensionless mass m(q,l) is central to the normalization discussion; a sentence explicitly stating that Fig. 5 uses Eq. (44) with the allowed domain from Eq. (43) would improve readability.
  4. [References] Ref. [44] is cited as an arXiv preprint; if the journal version has appeared, it should be updated. Also, some self-citations to very recent arXiv preprints (e.g. Refs. [49], [53]) may not yet be necessary for the core argument.
  5. [General notation] The notation x=ω/Ω_c is introduced in Fig. 7 and later used in the conclusion; please define it explicitly in the main text before first use, since the symbol x is otherwise used for the inverse-radius variable u=1/r in Sec. II.

Circularity Check

0 steps flagged

No significant circularity: the derivation is self-contained, with photon-sphere data computed from the metric and the high-frequency benchmark imported from independent literature.

full rationale

The paper's central chain—null geodesics determine bc, bg, σgeo, Ωc, Λc; partial-wave numerics produce σabs and dσ/dΩ; rescaling by photon-sphere scales is then compared with the external sinc/glory benchmarks—does not reduce to its inputs by construction. No parameter is fitted to the cross sections; the photon-sphere quantities are derived from the Frolov metric, and Eq. (22) is taken from the independent Regge-pole literature (Ref. [16]) rather than from the authors' own prior work. The data-collapse in Fig. 7 is a visualization in variables whose leading eikonal dependence is indeed removed, but that is a test of whether the remaining oscillation period and amplitude match the universal formula, not a definitional identity. Similarly, the matched-bc/bg comparisons in Fig. 11 fix the leading geometric scale and glory fringe spacing by construction, and the paper itself acknowledges that asymptotic agreement is expected; the residual overlap in the intermediate regime is an empirical finding, not a circular derivation. The absence of quantitative residuals and convergence tests is an evidentiary limitation, not a circularity. Self-citations appear only as contextual references and are not load-bearing for the main conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No new particles, fields, forces, or geometric structures are introduced; the Frolov metric and photon-sphere quantities are pre-existing constructs. No free parameters are fitted to the computed observables: q and ℓ are input spacetime parameters, m is fixed by f(rh)=1, and matched impact parameters are imposed by matching conditions, not optimized. The assumptions are the standard wave-scattering framework plus the imported universal high-frequency formula.

axioms (5)
  • domain assumption Frolov metric (Eq. 2) is a valid static, spherically symmetric regular black hole with an event horizon; the parameter constraint Eq. (43) delimits horizon-bearing configurations.
    The paper takes Frolov's regular BH solution as its spacetime background and defines the parameter domain by horizon existence. This is model input, not proven in this paper.
  • standard math A minimally coupled massless scalar field obeys the Klein-Gordon equation, and its partial-wave decomposition with purely ingoing boundary conditions at the horizon yields the absorption and scattering cross sections.
    Standard separation of variables and S-matrix formalism used in Sec. III A, Eqs. (30)-(38).
  • domain assumption The high-frequency absorption cross section is given by the sinc/Regge-pole formula Eq. (22)/(39) with photon-sphere frequency Ωc and Lyapunov exponent Λc, taken from Ref. [16] and assumed applicable to Frolov.
    The data-collapse conclusion in Sec. IV B is read against this external universal formula; the paper does not re-derive it for regular cores.
  • standard math The classical cross sections σgeo=πbc², the weak-deflection expansion Eq. (25)/(28), and the glory approximation Eq. (29) apply to Frolov BHs.
    Standard geometrical optics and semiclassical scattering results used as benchmarks, Sec. II.
  • domain assumption The YRW resummation and l-cutoffs (l=10, 50) produce converged partial-wave series over the displayed frequency/angle ranges.
    No convergence proof or error estimate is given; relied on in all numerical spectra, Sec. IV A.

reviewed 2026-08-03 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Absorption and scattering of massless scalar waves by Frolov black holes." pith.science (2026). https://pith.science/paper/BNLKRYDK

@misc{pith2026260119364,
  author       = {Pith},
  title        = {Pith review of: Absorption and scattering of massless scalar waves by Frolov black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BNLKRYDK}},
  note         = {Machine review of arXiv:2601.19364}
}
Share X Bluesky LinkedIn Reddit HN
abstract

We study the absorption and scattering of massless scalar waves by Frolov black holes, a regular deformation of the Reissner--Nordstr\"om geometry. A null-geodesic analysis provides the photon-sphere radius and the critical impact parameters governing capture and glory scattering. Using a partial-wave approach, we compute total/partial absorption cross sections and differential scattering cross sections over a broad frequency range. When the absorption spectrum is rescaled by photon-sphere scales, the high-frequency oscillations exhibit a pronounced data collapse in the variables $\hat{\sigma}=\sigma_{\rm abs}/\sigma_{\rm geo}$ and $x=\omega/\Omega_c$, highlighting photon-sphere control of the absorption fine structure. We clarify the parameter dependence under the horizon-radius normalization and relate the apparent trends to the variation of the dimensionless mass across parameter space. Finally, comparing Frolov, Reissner--Nordstr\"om, and Hayward black holes with matched critical or glory impact parameters, we find that their absorption and scattering patterns can become remarkably close, indicating that in the intermediate-to-high frequency regime the dominant signatures are largely governed by the unstable photon orbit, while core effects enter as subleading corrections.

Figures

Figures reproduced from arXiv: 2601.19364 by Hongsheng Zhang, Jining Tang, Yang Huang.

Figure 1
Figure 1. Figure 1: FIG. 1: Null geodesics for massless particles in the background of Frolov BHs with different [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The effective potential of a Frolov BH with the parameters [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Effective potential [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Parameter space of Frolov BHs in the [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: displays the behaviour of m as a function of q 2 and ℓ 2 within the region of parameter space that admits an event horizon, which we discussed in the last paragraph. One can see that m increases monotonically with both q and ℓ, so different choices of (q, ℓ) along the allowed domain correspond to Frolov BHs of different mass, even though the horizon radius is kept fixed. As a consequence, varying (q, ℓ) at… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Total absorption cross section [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Scaled total absorption cross section shown as [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Partial absorption cross sections [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Comparison between the numerical and the approximate analytical results for the DCS [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Differential scattering cross section for Frolov spacetime for different values of [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: ACSs and DCSs of Reissner–Nordström, Hayward and Frolov black holes with matched [PITH_FULL_IMAGE:figures/full_fig_p020_11.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 1 Pith paper

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

  1. Massive scalar field perturbations in noncommutative-geometry-inspired Schwarzschild black hole

    gr-qc 2026-04 conditional novelty 4.0

    Noncommutative parameter θ and scalar mass μ exert opposing influences on quasinormal modes and greybody factors in this modified black hole, with stability confirmed and extreme cases approaching classical Schwarzsch...

Reference graph

Works this paper leans on

53 extracted references · 2 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Gravitational waves: Volume 1: Theory and experiments , volume 1

    Michele Maggiore. Gravitational waves: Volume 1: Theory and experiments , volume 1. Oxford univer- sity press, 2008

    2008

  2. [2]

    Gravitational Waves: Astrophysics and Cosmology , volume 2

    Michele Maggiore. Gravitational Waves: Astrophysics and Cosmology , volume 2. Oxford University Press, 2008

    2008

  3. [3]

    Gravitational-wave physics and astronomy: An intro- duction to theory, experiment and data analysis

    Jolien DE Creighton and Warren G Anderson. Gravitational-wave physics and astronomy: An intro- duction to theory, experiment and data analysis . John Wiley & Sons, 2012

    2012

  4. [4]

    Physics, astrophysics and cosmology with gravitational waves

    Bangalore Suryanarayana Sathyaprakash and Bernard F Schutz. Physics, astrophysics and cosmology with gravitational waves. Living reviews in relativity , 12(1):2, 2009

    2009

  5. [5]

    Listening to the universe with gravitational-wave astronomy

    Scott A Hughes. Listening to the universe with gravitational-wave astronomy. Annals of Physics , 303(1):142–178, 2003. 22

    2003

  6. [6]

    Observation of gravitational waves from a binary black hole merger

    Benjamin P Abbott, Richard Abbott, Thomas D Abbott, Matthew R Abernathy, Fausto Acernese, Kendall Ackley, Carl Adams, Thomas Adams, Paolo Addesso, Rana X Adhikari, et al. Observation of gravitational waves from a binary black hole merger. Physical review letters , 116(6):061102, 2016

    2016

  7. [7]

    Scattering by black holes

    Nils Anderson and Bruce Jensen. Scattering by black holes. In Scattering, pages 1607–1626. Elsevier, 2002

    2002

  8. [8]

    Scattering theory of waves and particles

    Roger G Newton. Scattering theory of waves and particles . Springer Science & Business Media, 2013

    2013

  9. [9]

    Spherical gravitational waves and quasi-spherical waves scattered from black string in massive gravity

    Hongsheng Zhang and Yang Huang. Spherical gravitational waves and quasi-spherical waves scattered from black string in massive gravity. Journal of High Energy Physics , 2021(12):1–11, 2021

    2021

  10. [10]

    Universality of low energy absorption cross sections for black holes

    Sumit R Das, Gary Gibbons, and Samir D Mathur. Universality of low energy absorption cross sections for black holes. Physical Review Letters , 78(3):417, 1997

    1997

  11. [11]

    First m87 event horizon telescope results

    Event Horizon Telescope Collaboration. First m87 event horizon telescope results. i. the shadow of the supermassive black hole. Astrophysical Journal Letters , 875(1):L1, 2019

    2019

  12. [12]

    First sagittarius a* event horizon telescope results

    Event Horizon Telescope Collaboration. First sagittarius a* event horizon telescope results. i. the shadow of the supermassive black hole in the center of the milky way. Astrophysical Journal Letters , 930(2):L12, 2022

    2022

  13. [13]

    Poisson-arago spot for gravitational waves

    HongSheng Zhang and XiLong Fan. Poisson-arago spot for gravitational waves. Science China Physics, Mechanics & Astronomy , 64(12):120462, 2021

    2021

  14. [14]

    Low-frequency scalar absorption cross sections forstationary black holes

    Atsushi Higuchi. Low-frequency scalar absorption cross sections forstationary black holes. Classical and Quantum Gravity , 18(20):L139, 2001

    2001

  15. [15]

    Addendum to low-frequency scalar absorption cross sections for stationary black holes

    Atsushi Higuchi. Addendum to low-frequency scalar absorption cross sections for stationary black holes. Classical and Quantum Gravity , 19(3):599, 2002

    2002

  16. [16]

    Universality of high-energy absorption cross sections for black holes

    Yves Décanini, Gilles Esposito-Farese, and Antoine Folacci. Universality of high-energy absorption cross sections for black holes. Physical Review DParticles, Fields, Gravitation, and Cosmology , 83(4):044032, 2011

    2011

  17. [17]

    Absorption and emission spectra of a schwarzschild black hole

    Norma Sanchez. Absorption and emission spectra of a schwarzschild black hole. Physical Review D , 18(4):1030, 1978

    1978

  18. [18]

    Elastic scattering of waves by a black hole

    Norma Sánchez. Elastic scattering of waves by a black hole. Physical Review D , 18(6):1798, 1978

    1978

  19. [19]

    Futterman, F A Handler, and R A Matzner

    J A.H. Futterman, F A Handler, and R A Matzner. Scattering from black holes . Cambridge University Press,New York, NY, 12 1986

    1986

  20. [20]

    Scattering of massless scalar waves by reissner- nordström black holes

    Luís CB Crispino, Sam R Dolan, and Ednilton S Oliveira. Scattering of massless scalar waves by reissner- nordström black holes. Physical Review DParticles, Fields, Gravitation, and Cosmology , 79(6):064022, 2009

    2009

  21. [21]

    Scattering of massless scalar field by charged dilatonic black holes

    Yang Huang and Hongsheng Zhang. Scattering of massless scalar field by charged dilatonic black holes. The European Physical Journal C , 80(7):654, 2020

    2020

  22. [22]

    Spherical black holes with regular center: a review of existing models including a recent realization with gaussian sources

    Stefano Ansoldi. Spherical black holes with regular center: a review of existing models including a recent realization with gaussian sources. arXiv preprint arXiv:0802.0330 , 2008

    Pith/arXiv arXiv 2008

  23. [23]

    Regular black holes: a short topic review

    Chen Lan, Hao Yang, Yang Guo, and Yan-Gang Miao. Regular black holes: a short topic review. 23 International Journal of Theoretical Physics , 62(9):202, 2023

    2023

  24. [24]

    Regular black holes and black universes

    KA Bronnikov, H Dehnen, and VN Melnikov. Regular black holes and black universes. General Relativity and Gravitation , 39(7):973–987, 2007

    2007

  25. [25]

    Regular black holes

    Cosimo Bambi. Regular black holes . Springer, 2023

    2023

  26. [26]

    Regular black hole in general relativity coupled to nonlinear electrodynamics

    Eloy Ayon-Beato and Alberto Garcia. Regular black hole in general relativity coupled to nonlinear electrodynamics. Physical review letters , 80(23):5056, 1998

    1998

  27. [27]

    The bardeen model as a nonlinear magnetic monopole

    Eloy Ayon-Beato and Alberto Garcıa. The bardeen model as a nonlinear magnetic monopole. Physics Letters B , 493(1-2):149–152, 2000

    2000

  28. [28]

    Regular black holes with a nonlinear electrodynamics source

    Leonardo Balart and Elias C Vagenas. Regular black holes with a nonlinear electrodynamics source. Physical Review D , 90(12):124045, 2014

    2014

  29. [29]

    Non-singular general relativistic gravitational collapse

    James Bardeen. Non-singular general relativistic gravitational collapse. In Proceedings of the 5th International Conference on Gravitation and the Theory of Relativity , page 87, 1968

    1968

  30. [30]

    Formation and evaporation of nonsingular black holes

    Sean A Hayward. Formation and evaporation of nonsingular black holes. Physical review letters , 96(3):031103, 2006

    2006

  31. [31]

    Absorption of planar massless scalar waves by bardeen regular black holes

    Caio FB Macedo and Luís CB Crispino. Absorption of planar massless scalar waves by bardeen regular black holes. Physical Review D , 90(6):064001, 2014

    2014

  32. [32]

    Scattering by regular black holes: planar massless scalar waves impinging upon a bardeen black hole

    Caio FB Macedo, Ednilton S de Oliveira, and Luís CB Crispino. Scattering by regular black holes: planar massless scalar waves impinging upon a bardeen black hole. Physical Review D , 92(2):024012, 2015

    2015

  33. [33]

    Notes on nonsingular models of black holes

    Valeri P Frolov. Notes on nonsingular models of black holes. Physical Review D , 94(10):104056, 2016

    2016

  34. [34]

    Regular black holes and the first law of black hole mechanics

    Sebastian Murk and Ioannis Soranidis. Regular black holes and the first law of black hole mechanics. Physical Review D , 108(4):044002, 2023

    2023

  35. [35]

    Quasinormal modes and ringdown waveforms of a frolov black hole

    Zhijun Song, Huajie Gong, Hai-Li Li, Guoyang Fu, Li-Gang Zhu, and Jian-Pin Wu. Quasinormal modes and ringdown waveforms of a frolov black hole. Communications in Theoretical Physics , 76(10):105401, 2024

    2024

  36. [36]

    Null geodesics, thermodynamics, weak gravitational lensing, and black hole shadow characteristics of a frolov regular black hole with constraints from eht observations

    Shubham Kala, Hemwati Nandan, Kush Maithani, Saswati Roy, and Amare Abebe. Null geodesics, thermodynamics, weak gravitational lensing, and black hole shadow characteristics of a frolov regular black hole with constraints from eht observations. The European Physical Journal Plus , 140:991, 2025

    2025

  37. [37]

    The mathematical theory of black holes , volume 69

    Subrahmanyan Chandrasekhar. The mathematical theory of black holes , volume 69. Oxford university press, 1998

    1998

  38. [38]

    Absorption cross section of regular black holes in scalar-tensor conformal gravity

    Yang Li and Yan-Gang Miao. Absorption cross section of regular black holes in scalar-tensor conformal gravity. Physical Review D , 105(4):044031, 2022

    2022

  39. [39]

    Dynamics of photons and shadows for black holes haired with parity-odd fields

    Yang Huang, Dao-Jun Liu, and Hongsheng Zhang. Dynamics of photons and shadows for black holes haired with parity-odd fields. Journal of High Energy Physics , 2025(11):1–18, 2025

    2025

  40. [40]

    Dynamics of null particles and shadow for general rotating black hole

    Kun Meng, Xi-Long Fan, Song Li, Wen-Biao Han, and Hongsheng Zhang. Dynamics of null particles and shadow for general rotating black hole. Journal of High Energy Physics , 2023(11):1–28, 2023

    2023

  41. [41]

    Lensing and light rings of parity-odd rotating boson 24 stars

    Yang Huang, Dao-Jun Liu, and Hongsheng Zhang. Lensing and light rings of parity-odd rotating boson 24 stars. Science China Physics, Mechanics & Astronomy , 68(8):280411, 2025

    2025

  42. [42]

    Robert M. Wald. General Relativity. University of Chicago Press, Chicago, 1984

    1984

  43. [43]

    Geodesic stability, lyapunov exponents, and quasinormal modes

    Vitor Cardoso, Alex S Miranda, Emanuele Berti, Helvi Witek, and Vilson T Zanchin. Geodesic stability, lyapunov exponents, and quasinormal modes. Physical Review DParticles, Fields, Gravitation, and Cosmology, 79(6):064016, 2009

    2009

  44. [44]

    Geodesic analysis, absorption and scattering in the static hayward spacetime

    Marco AA de Paula, Luiz Leite, and Luís CB Crispino. Geodesic analysis, absorption and scattering in the static hayward spacetime. arXiv preprint arXiv:2311.15771 , 2023

    Pith/arXiv arXiv 2023

  45. [45]

    Glory scattering by black holes

    Richard A Matzner, Cécile DeWitte-Morette, Bruce Nelson, and Tian-Rong Zhang. Glory scattering by black holes. Physical Review D , 31(8):1869, 1985

    1985

  46. [46]

    Scattering properties of charged black holes in nonlinear and maxwells electrodynamics

    Marco AA de Paula, Luiz Leite, and Luís CB Crispino. Scattering properties of charged black holes in nonlinear and maxwells electrodynamics. The European Physical Journal Plus , 137(7):1–13, 2022

    2022

  47. [47]

    Phase-shift calculation of high-energy electron scat- tering

    DR Yennie, D_ G_ Ravenhall, and RN Wilson. Phase-shift calculation of high-energy electron scat- tering. Physical Review, 95(2):500, 1954

    1954

  48. [48]

    Fermion scattering by a schwarzschild black hole

    Sam Dolan, Chris Doran, and Anthony Lasenby. Fermion scattering by a schwarzschild black hole. Physical Review DParticles, Fields, Gravitation, and Cosmology , 74(6):064005, 2006

    2006

  49. [49]

    Rigorous calculation of scalar scattering in the schwarzschild background: The convergence of the partial-wave series and the poisson spot

    Zhao Li and Wen Zhao. Rigorous calculation of scalar scattering in the schwarzschild background: The convergence of the partial-wave series and the poisson spot. Physical Review D , 112(8):083030, 2025

    2025

  50. [50]

    Electrically charged black holes in linear and nonlinear electrodynamics: Geodesic analysis and scalar absorption

    Marco AA Paula, Luiz CS Leite, and Luis CB Crispino. Electrically charged black holes in linear and nonlinear electrodynamics: Geodesic analysis and scalar absorption. Physical Review D, 102(10):104033, 2020

    2020

  51. [51]

    Absorption and scattering of charged scalar waves by charged horndeski black hole

    Qian Li, Qianchuan Wang, and Junji Jia. Absorption and scattering of charged scalar waves by charged horndeski black hole. Physical Review D , 111(2):024059, 2025

    2025

  52. [52]

    Scattering of charged massive scalar waves by kerr-newman black holes

    Qian Li, Qianchuan Wang, and Junji Jia. Scattering of charged massive scalar waves by kerr-newman black holes. arXiv preprint arXiv:2511.21318 , 2025

    arXiv 2025

  53. [53]

    Gravitational lensing of gravitational waves: Spin-wave optics through black hole scattering

    Zhao Li, Shaoqi Hou, and Wen Zhao. Gravitational lensing of gravitational waves: Spin-wave optics through black hole scattering. arXiv preprint arXiv:2512.23933 , 2025. 25

    arXiv 2025

This paper was first reviewed by deepseek-v4-flash on August 3, 2026.