Shadow images of regular black hole with finite boundary

A. Sulaksono; H. S. Ramadhan; M. F. Fauzi

REVIEW 2 major objections 6 minor 16 references

Shadow images of regular black hole with finite boundary

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read By imposing a sharp surface radius R on the Hayward regular black hole, this paper finds that horizonless configurations acquire distinct inner light-ring structures in their accretion-disk images, a possible observable fingerprint of a…

desk verdict Competent new metric and ray tracing, but the headline claim of distinct inner rings vs Hayward is untested and likely generic. read the letter →

arxiv 2411.16241 v1 pith:2LBYKWYH submitted 2024-11-25 gr-qc

M. F. Fauzi , H. S. Ramadhan , A. Sulaksono This is my paper · ORCID

classification gr-qc MSC 83C5783C10 PACS 04.70.-s

keywords regularblackholesHaywardmetricfiniteboundaryphotonsphereshadowimageshorizonlesscompactobjectsnullgeodesicsthinaccretiondisk

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 proposes a version of the Hayward regular black hole whose energy density is cut off at a finite radius R, giving the object a well-defined surface while keeping the singularity-free core. It then works out what null geodesics and thin-disk shadow images look like in both 'horizonful' and 'horizonless' versions of this spacetime. The central result is that for horizonless configurations the added boundary creates distinctive inner light-ring structures near the center of the image, which the authors argue differ from what horizonless Hayward spacetimes produce. If correct, this gives a concrete way the presence of a surface could be read off from future high-resolution images of ultracompact objects.

What carries the argument

The machinery is the finite-boundary mass function m(r) obtained by integrating the truncated Hayward energy density, which includes a cutoff term (1 − (r/R)^n) and a normalization κ so that m(R) = M. Outside R the metric reduces to Schwarzschild, so for R ≤ 3M the photon sphere stays at 3M; inside R the modified potential V(r) = A(r)/$r^{2}$ develops a local minimum corresponding to a stable photon orbit. The images are produced by backward ray tracing null geodesics, classifying trajectories as direct emission, lensed emission, or photon ring, and weighting them with the Gralla-Lupsasca-Marrone thin-disk intensity profile. With n = 3 the hypergeometric mass profile simplifies to logarithmic functions, making the model computationally tractable.

What would settle it

A direct numerical check: compute the horizonless image intensity profile for the same spacetime but with the cutoff power n changed from 3 to a larger value, or with the sharp cutoff replaced by a smooth transition, and compare the inner ring positions; if the pattern persists unchanged, the claimed distinctness from horizonless Hayward images is not tied to the boundary's sharpness. Alternatively, generate a horizonless Hayward image with the same ray-traced disk and overlay the two intensity cross-sections; the claim stands only if the central ring pattern visibly differs.

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Extended reading notes

Core claim

The authors construct a Hayward-type regular black hole with a finite boundary by cutting off the energy density at radius R. They find that for horizonless configurations (small α = 2M/ℓ) the effective potential develops a positive local minimum inside the boundary, i.e., a stable photon sphere, so light can orbit inside and produce a sequence of ring-shaped secondary images: three wide rings plus thin rings between the first and second, with the innermost ring shifting inward as R/M grows. They claim this inner structure is distinct from what horizonless Hayward spacetimes produce, while for horizonful configurations with R > 3M the shadow is only slightly modified from Schwarzschild and the shadow radius shrinks as R/M increases.

Load-bearing premise

The argument assumes the sharp cutoff at r = R, where the energy density jumps to zero, is a real physical surface; if a smooth or differently shaped boundary changes the inner ring pattern, the claimed signature disappears.

Editorial extensions

If this is right

For horizonful configurations with R > 3M, the shadow radius is slightly smaller than Schwarzschild's, and at R = 6M the direct image of the disk shifts inward, so a finite boundary is imprinted on standard black hole images at large R/M.
In horizonless configurations the image loses the central shadow entirely and instead shows a pattern of three wide rings plus thin photon rings between the first two, with the inner ring moving inward as R/M increases.
The critical ratio αc = 2M/ℓ becomes a function of R/M and has no real value below R/M ≈ 2.07, below which the spacetime always has two horizons; the boundary changes the global horizon structure and not just the image.
Because the metric is exactly Schwarzschild outside R, any observable departure from Schwarzschild lensing in this model is localized inside R, making the boundary radius a parameter that could in principle be extracted from the inner image structure.

Reading between the lines

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

The sudden cutoff at r = R makes the mass derivative discontinuous, so the model implicitly includes a thin shell at the surface; checking the Israel junction conditions would reveal what surface stress-energy is required and whether such a boundary is physically realizable.
The same construction could be applied to other regular black hole metrics, such as Bardeen or Simpson-Visser, to test whether the inner ring pattern is a generic feature of finite boundaries or specific to the Hayward cutoff shape.
A clean observational extension is to compute the autocorrelation of the image intensity across the inner ring region; the predicted spacing between wide and thin rings would give an observable fingerprint of the boundary radius R.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version: the paper gives a concrete new metric (Hayward plus a cutoff that forces m(R)=M) and does a clean numerical job of computing photon trajectories and disk images. The effective-potential analysis, the alpha_c(R) curve, and the observation that horizonful shadows are nearly Schwarzschild for R>3M are all solid and useful. The horizonless inner rings in Fig. 6 are real and match the expected stable-photon-orbit signature. That part is fine.

The problem is the abstract's last sentence: it claims the inner rings "differ from those observed in horizonless Hayward black holes," but no Hayward image is ever shown. No baseline, no overlay, no cross-section comparison. On top of that, the generic lore (and the cited Olmo et al. work) is that any horizonless ultracompact spacetime with a photon sphere produces these wide/thin inner rings. The finite boundary may shift the ring radii, but nothing in the paper demonstrates a qualitative difference. The authors should either add the Hayward comparison or drop the claim.

Other soft spots: the n=3 cutoff is ad hoc; the density derivative jumps at R, and no microphysical or energy-condition justification is offered. The claim that a stable photon orbit always forms for arbitrarily small alpha is likely true in this model because the boundary forces a Schwarzschild exterior, but the text's justification (flat center, no horizon) is too hand-wavy and needs an explicit argument. There are also typos: the inequality in Eq. (5) reads "0 > r > R", and Fig. 4 is cross-referenced incorrectly.

If this comes to me as referee, I'd ask for a Hayward baseline figure, a softened abstract, and a few lines on the boundary's physical status. The model itself is a legitimate addition to the catalog, and the numerics are reproducible enough to be worth a round of revision.

Referee Report

2 major / 6 minor

Summary. The paper proposes a modification of the Hayward regular black hole metric in which the energy density is cut off at a finite radius R, yielding a mass function that is exactly M outside R (Eqs. 5-7). The authors study the effective potential for null geodesics, compute photon trajectories, and ray-trace images of a thin accretion disk for both horizonful and horizonless configurations. They report that horizonful images differ only slightly from Schwarzschild for R>3M, while horizonless configurations produce multiple inner ring-shaped secondary images. The central claimed novelty is that these inner rings differ from those of horizonless Hayward black holes.

Significance. If fully established, the model would offer a simple example of how an explicit boundary in a regular black hole metric affects strong-field images, and the effective-potential analysis in Fig. 2 is internally consistent. The paper also correctly identifies that the critical parameter αc becomes R-dependent and reproduces the Hayward limit as R/M→∞. However, the main distinguishing claim is not tested, and the stable inner photon orbit that produces the inner rings is a generic feature of horizonless ultracompact spacetimes with a regular center. The manuscript does not supply machine-checked proofs or code, but the numerical pipeline is standard and the qualitative behavior is plausible. The contribution is therefore useful but limited unless the Hayward comparison is supplied and the physical status of the boundary is clarified.

major comments (2)

[Abstract; Sec. 3.2; Conclusion] The central claim that horizonless images show "distinct inner light ring structures ... which differ from those observed in horizonless Hayward black holes" is never tested. Figures 5 and 6 show images only for the new model; no horizonless Hayward image or intensity cross-section is presented, and the Conclusion states only that "a similar ring-shaped secondary image pattern" appears across R values. The distinction is load-bearing because any horizonless ultracompact metric with A(0)=1 has V(r)→∞ as r→0 and V(r)→0 as r→∞, forcing a potential well and a stable circular photon orbit. The inner rings are therefore a generic feature, not a specific effect of the finite boundary. Please add a direct Hayward comparison (images and intensity cross-sections at the same α/αc and R-compatible parameters) or revise the abstract to claim only a quantitative difference.
[Sec. 2, Eqs. (5)-(7)] The finite boundary is introduced through a sharp cutoff in the energy density (Eq. 5), but the paper does not justify the boundary physically. There is no Israel junction analysis at r=R, no check of energy conditions, and no microphysical realization of the density profile; the constant κ in Eq. (7) simply enforces m(R)=M. Since the paper's stated motivation is to cure the absence of a well-defined boundary in horizonless Hayward spacetimes, the model's physical status needs a clear statement (e.g., a phenomenological toy model) and at least a discussion of the dominant-energy condition and the behavior of pressure at the boundary.

minor comments (6)

[Eq. (5)] The inequality in Eq. (5) is printed as "0 > r > R", which is impossible; it should be "0 ≤ r ≤ R".
[Eq. (6)] The placement of the term "l3(3+n)" in the denominator of the hypergeometric expression is hard to parse. Since the paper sets n=3, displaying the explicit logarithmic form of the mass function would make the model more transparent.
[Fig. 4 caption] The caption says "Same description as Fig. 4" but should refer to Fig. 3.
[Sec. 3.2] The word "chaotic" describing the photon ring trajectories in Fig. 4 is inappropriate: geodesics in a static, spherically symmetric spacetime are integrable. "Highly winding" would be more accurate.
[Sec. 3.1] The ray-tracing procedure should specify the observer distance, image-plane resolution, and numerical integration scheme so that the images in Figs. 5 and 6 are reproducible.
[Throughout] The spelling "horizonfull" in the Abstract and Conclusion should be unified with "horizonful" used in the main text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the shadow images follow from the proposed metric by direct geodesic integration, with no fitted target observables and no load-bearing self-citation.

full rationale

The paper's derivation chain is self-contained. A new mass function is defined in Eqs. (5)-(7) by modifying the Hayward energy density with a cutoff at r = R; the constant kappa is fixed by the boundary condition m(R) = M, which is a self-consistency normalization, not a fit to a target observable. The photon effective potential (Eq. 9), geodesic equation (Eq. 10), and GLM accretion-disk profile (Eq. 11) are standard ingredients. The shadow images in Figs. 5 and 6 are computed consequences of these inputs, not quantities built from the outputs. There is no fitting of model parameters to observational data, and no cited result is used as load-bearing support in a way that reduces the argument to a self-citation. The abstract's claim that the inner ring structures 'differ from those observed in horizonless Hayward black holes' is not actually demonstrated by a comparative horizonless-Hayward ray-tracing image, but that is a completeness/correctness limitation, not circularity: the absence of a comparison does not make the finite-boundary prediction equivalent to its inputs by construction. The stable-photon-orbit inner rings are genuine, if perhaps generic, consequences of the horizonless ultracompact geometry presented.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The central model rests on five explicit choices: the length scale ℓ, the boundary exponent n, the boundary radius R, the mass ratio α, and the disk intensity parameters. In addition, the analysis assumes standard general relativity, a static spherically symmetric metric, a de Sitter perfect fluid interior, a Schwarzschild exterior, and a phenomenological accretion disk. The only invented entity is the sharp boundary surface, which has no independent physical support.

free parameters (6)

Length scale ℓ = ℓ = 1
Sets the scale of the regular core; set to 1 in all simulations, described as 'without loss of generality'.
Boundary exponent n = n = 3
Controls the shape of the density cutoff; chosen 'for the sake of simplicity' so the hypergeometric function reduces to logarithms.
Boundary radius R = R = 2.5M, 4M, 6M
The new finite-boundary scale, varied across the study.
Mass-to-core ratio α = 1.1 α_c and 0.9 α_c
Sets horizonful and horizonless configurations, respectively.
Normalization constant κ = Determined by R, ℓ, n via Eq. (7)
Chosen so that the mass function satisfies m(R) = M, enforcing continuity of A(r).
GLM disk parameters γ, µ, σ = γ = -2, µ = 6M, σ = M/4
Parameters of the assumed thin accretion disk intensity profile, taken from common literature choices.

assumptions (5)

standard math General relativity and the geodesic equation govern photon motion
Used without proof to define null geodesics (Eq. 8) and effective potential (Eq. 9).
standard math Spacetime is static and spherically symmetric with the line element (2)
The entire analysis is built on this metric ansatz.
domain assumption The interior is sourced by a perfect fluid with the de Sitter equation of state p = -ε
Motivates the Hayward-inspired form of the energy density in Eqs. (3)-(5).
domain assumption The mass function is exactly M for r > R, so the exterior is Schwarzschild
The model is designed so that the finite boundary isolates the modified region; this is imposed via Eq. (6).
domain assumption The thin accretion disk radiates according to the Gralla-Lupsasca-Marrone intensity profile (11)
A phenomenological prescription adopted from Eq. (11) with the stated parameters.

invented entities (1)

Finite boundary surface at radius R
purpose: Provides a sharp outer edge for the regular black hole density profile and defines where spacetime becomes Schwarzschild.
Introduced by hand in Eq. (5) as a step-function cutoff; no physical mechanism, equation of state, or observational constraint is offered to justify a sharp surface.

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Cite this review

Pith. "Pith review of Shadow images of regular black hole with finite boundary." pith.science (2026). https://pith.science/paper/2LBYKWYH

@misc{pith2026241116241,
  author       = {Pith},
  title        = {Pith review of: Shadow images of regular black hole with finite boundary},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2LBYKWYH}},
  note         = {Machine review of arXiv:2411.16241}
}

abstract

Regular black hole is one of the bottom-up solutions designed to eliminate the singularity at the center of black holes. Its horizonless solution has gained interest recently to model ultracompact star. Despite interesting, this proposal is problematic due to the absence of a well-defined boundary. In this work, we introduce a novel regular black hole model inspired by the Hayward black hole, incorporating additional terms to define a clear and well-defined `surface' radius $R$. We analyze the null geodesics around the object, both horizonful and horizonless configurations, by studying the photon effective potential. We further simulate the shadow images of the object surrounded by a thin accretion disk. Our results indicate that for $R > 3M$ the horizonfull shadow differs slightly from that of a Schwarzschild black hole. In the horizonless configuration, we identify distinct inner light ring structures near the central region of the shadow image, which differ from those observed in horizonless Hayward black holes.

Figures

Figures reproduced from arXiv: 2411.16241 by the authors.

**Figure 1.** Critical value of α for different values of R/M. 3 Photon effective potential and shadow images Photon geodesics on a spacetime manifold can be calculated by solving the geodesic equation [1], duµ dτ + Γµ αβ dxα dτ dxβ dτ = 0 (8) where x µ = [t, r, θ, ϕ] is the photon four-vector, u µ = dxµ dτ is the photon four-velocity, and τ is an affine parameter. In a static and spherically symmetric spacetime of the form given… view at source ↗

**Figure 2.** Photon effective potential for several values of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Photon trajectories around the horizonful configuration. The direct emission, lensed emission, [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Same description as Fig. 4 for the horizonless configuration. The black solid circle is absent [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: (Top row) Shadow images of the horizonful configuration for several values of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: (Top row) Optical images of the horizonless configuration for several values of [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

General relativity: An introduction for physi- cists,

M. P. Hobson, G. P. Efstathiou and A. N. Lasenby, “General relativity: An introduction for physi- cists,”

work page

[2] [2]

Effective models of nonsingular quantum black holes,

M. Cadoni, M. Oi and A. P. Sanna, “Effective models of nonsingular quantum black holes,” Phys. Rev. D 106 (2022) no.2, 024030

work page 2022

[3] [3]

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C. Lan, H. Yang, Y. Guo and Y. G. Miao, “Regular Black Holes: A Short Topic Review,” Int. J. Theor. Phys. 62 (2023) no.9, 202

work page 2023

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R. Carballo-Rubio, F. Di Filippo, S. Liberati and M. Visser, “A connection between regular black holes and horizonless ultracompact stars,” JHEP 08 (2023), 046

work page 2023

[5] [5]

First Very Long Baseline Interferometry Detec- tions at 870 µm,

A. W. Raymond et al. [Event Horizon Telescope], “First Very Long Baseline Interferometry Detec- tions at 870 µm,” Astron. J. 168 (2024) no.3, 130

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[6] [6]

Toward very large baseline interferometry observa- tions of black hole structure,

R. Carballo-Rubio, V. Cardoso and Z. Younsi, “Toward very large baseline interferometry observa- tions of black hole structure,” Phys. Rev. D 106 (2022) no.8, 084038

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[7] [7]

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J. L. Rosa, D. S. J. Cordeiro, C. F. B. Macedo and F. S. N. Lobo, “Observational imprints of gravastars from accretion disks and hot spots,” Phys. Rev. D 109 (2024) no.8, 084002

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Images from disk and spherical accretions of hairy Schwarzschild black holes,

Y. Meng, X. M. Kuang, X. J. Wang, B. Wang and J. P. Wu, “Images from disk and spherical accretions of hairy Schwarzschild black holes,” Phys. Rev. D 108 (2023) no.6, 064013

work page 2023