Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Bounds on the minimum orbital periods of non-singular Hayward and Bardeen black holes

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Hayward and Bardeen regular black holes obey the conjectured universal bound $4\pi M \le T_{\min} \le 6\sqrt{3}\pi M$ on the fastest circular orbital period.

desk verdict A straightforward numerical check that Hayward and Bardeen black holes satisfy the conjectured orbital-period bounds; the result is likely right, but the evidence as shipped is incomplete. read the letter →

arxiv 2505.23000 v1 pith:PEIWPWPL submitted 2025-05-29 gr-qc hep-th

Guohua Liu , Yan Peng This is my paper
classification gr-qchep-th PACS 11.25.Tq04.70.Bw74.20.-z
keywords minimumorbitalperiodHaywardblackholeBardeenregularholesnullcircularorbitsuniversalboundshorizon
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tests a conjectured universal clock for black holes: for a black hole of mass $M$, the shortest possible period $T_{\min}$ of a light-speed circular orbit should obey $4\pi M \le T_{\min} \le 6\sqrt{3}\pi M$. The authors check this for two non-singular spacetimes, Hayward and Bardeen, whose central singularities have been smoothed by additional parameters $L$ and $g$. Using the null-orbit period formula and numerical minimization outside the outer horizon, they find that $T_{\min}$ falls inside the interval for both families, with the upper bound approached in the Schwarzschild limit $L=0$ or $g=0$. The result matters because it suggests the bounds are tied to the existence of a horizon, not to the singular interior, and so may be a generic feature of black hole spacetimes.

What carries the argument

The load-bearing object is the orbital-period function for circular null orbits in a static, spherically symmetric metric, $T(r)=2\pi r/\sqrt{f(r)}$, obtained by setting $ds^2=0$ for a full revolution $d\phi=2\pi$. The location of the minimum is controlled by $dT/dr=0$, and the allowed range of the regularity parameters $L$ and $g$ is cut off by the requirement that the outer horizon exist, i.e. that $f(r)=0$ still have a positive root. Numerical minimization over this horizon region is what converts the conjectured interval into a verified statement for these two spacetimes.

What would settle it

Take the horizon condition $f(r)=0$ for the Hayward metric, find the outer horizon $r_+$ for each allowed $L/M$, and minimize $T(r)=2\pi r/\sqrt{f(r)}$ over $r>r_+$ on a fine grid that includes values of $L$ just below the horizon-ending critical value; any $T_{\min}<4\pi M$ found this way would disprove the claimed bound, and the analogous scan with $g$ tests the Bardeen case.

Watch

Extended reading notes

Core claim

For the Hayward metric with $f(r)=1-2Mr^2/(r^3+2L^2M)$, the period of a full null circular orbit at radius $r$ is $T(r)=2\pi r/\sqrt{f(r)}$. The paper solves $dT/dr=0$ numerically and reports that $T_{\min}/M$ decreases as $L$ grows, with a numerical lower value about $30.9504$ and with the upper value approaching $6\sqrt{3}\pi\approx 32.6864$ as $L\to 0$. For the Bardeen metric, where $m(r)=M(r^2/(r^2+g^2))^{3/2}$, the same procedure gives $T_{\min}/M\approx 28.5784$ at the largest $g$ that still leaves a horizon, again above $4\pi$ and below the Schwarzschild upper limit. The paper therefore states that both regular black hole families obey $4\pi M \le T_{\min} \le 6\sqrt{3}\pi M$, and interprets the agreement as evidence that the bounds are properties of the horizon rather than the singularity.

Load-bearing premise

The claim depends on the numerical scans actually covering the whole black-hole region and correctly locating the critical $L$ and $g$ values at which the horizon disappears; if a narrow strip near those boundaries where $T_{\min}$ is smallest was missed, the lower bound $4\pi M$ could be violated.

Editorial extensions

If this is right

  • If the bounds hold generally, the fastest full orbit around any Hayward or Bardeen black hole as measured from infinity lasts between $4\pi M$ and $6\sqrt{3}\pi M$.
  • The upper bound is reached only in the Schwarzschild limit, so adding the regularization parameters $L$ or $g$ shortens the minimum period but never lengthens it beyond the Schwarzschild value.
  • Because both spacetimes are non-singular, any future proof of these bounds would have to rely on horizon structure, not on the central singularity, narrowing the mechanism that could explain the universality.
  • The numerical lower values reported for Hayward and Bardeen sit strictly above $4\pi M$, so the lower bound is not saturated in these families.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension is to run the same $T(r)=2\pi r/\sqrt{f(r)}$ minimization on other regular black hole metrics with different smoothing mechanisms; if they also obey the interval, the horizon explanation would be harder to avoid.
  • Because the reported lower numerical minima are far above $4\pi M$, regularity may actually impose a stronger gap between the lower bound and its saturation; finding the minimal possible $T_{\min}$ among all horizon-having metrics would sharpen the conjecture.
  • The bound applies to light-speed test orbits, so an observational test would need a luminous process parked near the minimum radius, such as a photon ring or a high-frequency quasi-periodic oscillation; ordinary stellar orbits would not probe $T_{\min}$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. This paper tests a conjectured universal bound on the minimum orbital period of objects around black holes, 4πM ≤ Tmin ≤ 6√3πM, for two non-singular metrics: the Hayward black hole (parameter L) and the Bardeen black hole (parameter g). The authors derive the orbital period T(r) from the light-speed condition ds²=0, compute its derivative, and then use numerical minimization to find Tmin as a function of M and L or g. They report that Tmin/M decreases as L or g increases, that the lower bound 4π is never violated for parameter values admitting a horizon, and that the upper bound is attained in the Schwarzschild limit L=0 or g=0. The paper concludes that the conjectured bounds hold for these regular black holes and that the bounds may be tied to the existence of a horizon rather than to the central singularity.

Significance. If the numerical results are correct, the paper extends the universal-period conjecture to regular black holes, strengthening the interpretation that the bounds are horizon-related. The explicit formulas for T(r) in Eqs. (5) and (14) are simple and reduce correctly to the Schwarzschild case, and the analytical treatment of the L=0 and g=0 limits is sound. The main contribution is numerical verification, but the manuscript as shipped does not include code, data, convergence tests, or a detailed description of the numerical method. This limits reproducibility, and the internal inconsistency in the quoted numerical upper bound (32.6484 vs. 32.6864) needs correction. The central claim is plausible, but the paper's current form does not fully support it as a reproducible numerical result.

major comments (4)
  1. [Section II, after Eq. (8) and Figure 1] The numerical minimization method is not described: no grid resolution, sampling strategy, or convergence criteria are given. The reported lower bound Tmin/M = 30.9504 depends on a scan over the allowed black-hole parameter region, and the reader cannot verify that the scan reached the extremal horizon boundary L_c/M = 4/(3√3) ≈ 0.7698, where a double horizon exists. Please report the numerical algorithm and provide a convergence test or an independent check at the critical L_c to demonstrate that the global minimum was not missed.
  2. [Section II, p. 5, and Figure 1 caption] The text reports the numerical upper bound as 32.6484, while the figure caption and Section III give 32.6864 ≈ 6√3π. Since the exact Schwarzschild result is Tmin = 6√3πM ≈ 32.6864 M, these values are inconsistent. Please reconcile this discrepancy; the value 32.6484 appears to be a typo but must be corrected throughout the manuscript.
  3. [Sections II and III] The claim that Tmin/M decreases monotonically with L (or g) is supported only by 'extensive numerical analysis' with no data table or analytical argument. This monotonicity is used to assert that the upper bound is attained at L=0 or g=0. Please include a table of Tmin/M at representative parameter values (including near-critical values) or prove monotonicity analytically, so that the upper-bound statement is checkable.
  4. [General (numerical reproducibility)] The central result is a numerical verification, but the paper does not provide code, data files, or a specification of the root-finding/minimization algorithm. Without these, the quoted minima 30.9504 and 28.5784 cannot be independently reproduced. Please include the numerical code or, at a minimum, a detailed description of the algorithm, grid resolution, and convergence criteria as supplementary material.
minor comments (5)
  1. [Figure 2 caption] The caption says 'as a function of M and L' for the Bardeen black hole, but the relevant parameter is g. This should be corrected.
  2. [Equation (8)] The notation is inconsistent: the square root contains '2l²M' with a lowercase l, while the polynomial in the denominator uses '2L²M'. Please use one symbol throughout.
  3. [Section III, text near Eq. (13)] The phrase 'we transform the front equation into' should read 'the previous equation' or similar. Please proofread for typographical errors.
  4. [Sections II and III, interpretation of T(r)] The derivation uses ds²=0, so T(r) is the coordinate-time period of a circular null orbit. The paper should clarify this and state explicitly why the null-orbit period is the relevant quantity for the cited bounds on 'objects orbiting black holes'.
  5. [References [22,23]] These are arXiv identifiers; if the papers have been published in journals, please update the references to the published versions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the minimum-period computations are independent minimizations of explicit metric formulas against externally stated comparison thresholds.

full rationale

The paper derives T(r) directly from the Hayward metric (Eq. 5) and Bardeen metric (Eq. 14) using the null circular-orbit condition ds^2=0, then minimizes it over r outside the horizon. The conjectured bounds 4πM ≤ Tmin ≤ 6√3πM (Eqs. 9 and 15) enter only as comparison thresholds; they are not substituted into the equations whose roots are found. The upper bound is independently recovered in the Schwarzschild limit L=g=0 via Eq. (7), Tmin=6√3πM. The lower-bound check is purely numerical, and although the paper does not report grid resolution or the horizon-boundary algorithm (a reproducibility weakness), this is not circular: the quoted minima 30.9504 and 28.5784 for M=1 are far above 4π≈12.57 and would not be affected by assuming the bound. Refs. [22] and [23] are self-citations by coauthor Yan Peng, but the verification here does not presuppose their results; it is a self-contained calculation against fixed numerical thresholds. Hence no load-bearing step reduces by construction to its input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; L and g are the model parameters being scanned, and M is the mass scale. The main hidden inputs are standard GR assumptions about null geodesics and the numerical completeness of the scans.

assumptions (4)
  • domain assumption The Hayward and Bardeen metrics (Eqs. 1 and 10) accurately describe regular black hole spacetimes.
    Adopted from prior literature; the paper does not derive these solutions from an action or matter model.
  • standard math For circular null orbits in static spherically symmetric metrics, the orbital period is obtained by setting ds²=0, and the orbit condition is equivalent to minimizing T(r).
    Standard GR result used implicitly in Secs. II and III; not explicitly derived from the geodesic equation.
  • ad hoc to paper The numerical scans over M,L and M,g are complete enough to find the global minimum and the horizon-existence boundary.
    The paper states results from 'extensive numerical analysis' without specifying grid resolution, root-finding method, or convergence tests.
  • standard math M denotes the asymptotic ADM mass in both models, so the bound's mass parameter matches the metric's mass parameter.
    Required for comparing Tmin/M to 4π and 6√3π; standard for these metrics.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bounds on the minimum orbital periods of non-singular Hayward and Bardeen black holes." pith.science (2026). https://pith.science/paper/PEIWPWPL

@misc{pith2026250523000,
  author       = {Pith},
  title        = {Pith review of: Bounds on the minimum orbital periods of non-singular Hayward and Bardeen black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PEIWPWPL}},
  note         = {Machine review of arXiv:2505.23000}
}
abstract

Based on previous studies, universal bounds $4\pi M \leqslant T_{min} \leqslant 6\sqrt{3}\pi M$ were conjectured to be characteristic properties of black hole spacetimes, where $M$ represents the mass of black holes and $T_{min}$ is the minimum orbital periods around black holes. In this work, we explore the minimum orbital periods of objects around Hayward and Bardeen black holes without central singularities. By combining analytical and numerical methods, we show that both Hayward and Bardeen black holes conform to these bounds. Our results imply that such bounds may be connected to the presence of the black hole horizon rather than the singularity.

Figures

Figures reproduced from arXiv: 2505.23000 by the authors.

Figure 1
Figure 1. FIG. 1: The left panel shows [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The left panel shows [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Bounds on the minimum orbital period in the background of 5-dimensional charged black holes

    gr-qc 2025-06 conditional novelty 4.0 of 10

    For 5D charged black holes, the minimum orbital period lies between 6√π√M and 8√(6π)/3√M, with the bounds reached at maximal and zero charge.

Reference graph

Works this paper leans on

30 extracted references · 20 canonical work pages · cited by 1 Pith paper

  1. [1]

    Bardeen, W.H

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

    work page 1972

  2. [2]

    Chandrasekhar, The Mathematical Theory of Black Hole s (Oxford University Press, New York, 1983)

    S. Chandrasekhar, The Mathematical Theory of Black Hole s (Oxford University Press, New York, 1983)

    1983

  3. [3]

    Shapiro, S.A

    S.L. Shapiro, S.A. Teukolsky, Black holes, white dwarfs , and neutron stars: The physics of compact objects (Wiley, New York, 1983)

    work page 1983

  4. [4]

    Hod, Phys

    S. Hod, Phys. Lett. B 727, 345 (2013), arXiv:1701.06587

    arXiv 2013

  5. [5]

    Cunha, C.A.R

    P.V.P. Cunha, C.A.R. Herdeiro, Phys. Rev. Lett. 124, 181 101 (2020)

    work page 2020

  6. [6]

    Mashhoon, Phys

    B. Mashhoon, Phys. Rev. D 31, 290 (1985)

    1985

  7. [7]

    Goebel, Astrophys

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

    work page 1972

  8. [8]

    Black-hole quasinormal resonances: Wave analysis versus a geometric-optics approximation

    S. Hod, Phys. Rev. D 80, 064004 (2009), arXiv:0909.0314

    work page Pith review arXiv 2009

Show all 30 references
  1. [9]

    Hod, Phys

    S. Hod, Phys. Rev. D 78, 084035 (2008), arXiv:0811.3806

    2008 arXiv

  2. [10]

    Hod, Phys

    S. Hod, Phys. Rev. D 75, 064013 (2007), arXiv:gr-qc/0611004

    2007 arXiv

  3. [11]

    Hod, Class

    S. Hod, Class. Quant. Grav. 24, 4235 (2007), arXiv:0705.2306

    2007 arXiv

  4. [12]

    Hod, Phys

    S. Hod, Phys. Lett. B 715, 348 (2012), arXiv:1207.5282

    2012 arXiv

  5. [13]

    De´canini, A

    Y. De´canini, A. Folacci, B. Raffaelli, Phys. Rev. D 81, 104039 (2010)

    2010

  6. [14]

    De´canini, A

    Y. De´canini, A. Folacci, B. Raffaelli, Phys. Rev. D 84, 084035 (2011)

    2011

  7. [15]

    M. A. Podurets, Astr. Zh. 41, 1090 (1964) [English translation in Soviet Astr.-AJ 8, 868 (1965)]

    1964

  8. [16]

    Ames, K.S

    W.L. Ames, K.S. Thorne, Astrophys. J. 151, 659 (1968)

    1968

  9. [17]

    Stefanov, S.S

    V.B. Stefanov, S.S. Yazadjiev, G.G. Gyulchev, Phys. Re v. Lett. 104, 251103 (2010)

    2010

  10. [18]

    Hod, Phys

    S. Hod, Phys. Rev. D 84, 124030 (2011), arXiv:1112.3286

    2011 arXiv

  11. [19]

    Hod, Phys

    S. Hod, Phys. Rev. D 101, 084033 (2020), arXiv:1202.03962

    2020

  12. [20]

    Hod, Fastest way to circle a black hole, Phys

    S. Hod, Fastest way to circle a black hole, Phys. Rev. D 84, 104024 (2011)

    2011

  13. [21]

    Hod, Lower bound on the time-to-mass ratio of closed c ircular motions around spinning and charged naked singularities, Eur

    S. Hod, Lower bound on the time-to-mass ratio of closed c ircular motions around spinning and charged naked singularities, Eur. Phys. J. C 83, 177 (2023)

    2023

  14. [22]

    Yan Peng, Lower bound on the orbital period of Kerr-Newm an black holes, arXiv:2503.21218

    arXiv

  15. [23]

    Yan Peng, An upper bound on the minimum orbital period of black holes, arXiv:2504.09061

    arXiv

  16. [24]

    S. A. Hayward, Phys. Rev. Lett. 96, 031103(2006)[gr-qc /0506126]. 9

    2006

  17. [25]

    Abbas and U

    G. Abbas and U. Sabiullah, Geodesic Study of Regular Hay ward Black Hole, arXiv:1406.0840

    arXiv

  18. [26]

    J. M. Bardeen, in Conference Proceedings of GR5 (Tbilis i,USSR,1968),p.174

    1968

  19. [27]

    Hideo Iguchi, Gravitational entropy of Hayward black h ole, Annals of Physics 453(2023)169331

    2023

  20. [28]

    Ay´ on-Beato and A

    E. Ay´ on-Beato and A. Garcia, Phys. Rev. Lett. 80, 5056 ( 1998)

    1998

  21. [29]

    Ay´ on-Beato and A

    E. Ay´ on-Beato and A. Garcia, Phys. Lett. B 493, 149 (200 0)

  22. [30]

    Vitalii Vertogradov, Gravitational Collapse and Form ation of Regular Black Holes: Dymnikova, Hayward, and Beyond,Vitalii Vertogradov

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.