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Investigating the shadows of new regular black holes with a Minkowski core: Effects of spherical accretion and core type differences

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

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Pith's one-line read For regular black holes with a Minkowski core, raising the quantum parameter α0 shrinks the shadow and brightens the photon ring, while the core itself produces smaller, dimmer images than Bardeen and Hayward black holes.

desk verdict Competent, incremental shadow phenomenology for a regular BH family, with the core-type comparison resting on an unvalidated parameter identification. read the letter →

arxiv 2502.06388 v3 pith:AHCDEWLB submitted 2025-02-10 gr-qc

Yi Xiong , Jin Pu , Yi Ling , Guo-Ping Li , Gao-Ming Deng This is my paper · ORCID
classification gr-qc PACS 04.70.-s04.70.Dy
keywords blackholeshadowregularholesMinkowskicoresphericalaccretionphotonspheregeneralizeduncertaintyprincipleBardeenHayward
open problems Quantum Gravity
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 tries to establish that the shadow of a regular black hole with a Minkowski core carries an observable fingerprint of quantum gravity. For these singularity-free black holes, built from a generalized-uncertainty-principle (GUP) modified Newtonian potential, a larger quantum parameter $\alpha_0$ makes the shadow and photon sphere smaller while the observed specific intensity rises; a larger deformation parameter $n$ does the opposite. The paper also claims that Minkowski-core holes are always slightly smaller and dimmer than their Bardeen and Hayward counterparts with de Sitter cores, and that the difference widens as $\alpha_0$ grows, particularly under static spherical accretion. If correct, black-hole imaging could discriminate which type of regular core is realized in nature.

What carries the argument

The load-bearing object is the metric function $f(r)=1+2\psi(r)$ with the exponentially suppressed GUP potential $\psi(r)=-(M/r)e^{-\alpha_0 M^x/r^n}$ (Eq. 3), whose core is Minkowski, paired with the one-to-one large-scale correspondence (Eq. 4) to the Bardeen potential at $x=2/3,n=2$ and the Hayward potential at $x=1,n=3$. Photon motion is reduced to the effective potential $V_{\rm eff}(r)=f(r)/r^2$; the photon sphere is set by $V'_{\rm eff}(r_c)=0$ and the shadow radius by $V_{\rm eff}(r_c)=1/b_c^2$. The optical images are produced by ray tracing null geodesics and integrating the redshift-weighted emissivity along the geodesic, with Eq. (17) for static spherical accretion and Eq. (26) for infalling spherical accretion. The same correspondence that fixes which Bardeen/Hayward parameter goes with each $\alpha_0$ is what allows the core-type comparison to be made at all.

What would settle it

Compute the shadow radius $b_c$ and the static/infalling intensity ratio for a Minkowski-core metric with parameters fitted directly to Sgr A* or M87* data, without imposing the Bardeen/Hayward correspondence. If the best fit places the shadow outside the paper's predicted range for the fitted $\alpha_0$ and $n$, or if the observed static-versus-infalling intensity ordering is reversed, the claim is falsified. A quantitative target: the predicted $\sim 0.34\,M$ decrease in $b_c$ between $\alpha_0=0.3$ and $0.72$ for the $x=2/3,n=2$ Minkowski-core hole is a difference a high-resolution shadow measurement could in principle confront.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is a monotonic relation: in the Minkowski-core metrics with $f(r)=1+2\psi(r)$ and $\psi(r)=-(M/r)\exp(-\alpha_0 M^x/r^n)$, the photon sphere radius $r_c$ and the critical impact parameter $b_c$ both decrease as $\alpha_0$ increases, while the peak of the observed specific intensity increases. Increasing $n$ reverses both trends. For example, at $M=1$, $\alpha_0=0.72$, $x=2/3$, $n=2$, the paper finds $b_c=4.66393$ for the Minkowski-core hole versus $4.69073$ for the Bardeen hole, and the Minkowski-core photon ring is dimmer under static spherical accretion. The shadow boundary is independent of whether the surrounding accretion is static or infalling; only the intensity changes, with static accretion giving a brighter image. These claims are made for two pairings: $(x,n)=(2/3,2)$ against the Bardeen black hole and $(1,3)$ against the Hayward black hole.

Load-bearing premise

The load-bearing premise is the one-to-one correspondence between the quantum-gravity potential used for the new black holes and the potentials of the Bardeen and Hayward black holes; if that mapping is not the physically right one, the claimed differences between the two core types could be an artifact of how the parameters were matched rather than a real observable distinction.

Editorial extensions

If this is right

  • A measured shadow radius at fixed mass would constrain $\alpha_0$ and $n$: smaller $b_c$ implies larger $\alpha_0$ or smaller $n$.
  • Because the shadow and photon-sphere locations do not depend on the accretion model, the shadow radius can be used to compare spacetime models even when the details of the accretion flow are unknown.
  • At the same parameters, Minkowski-core holes are systematically smaller and dimmer than Bardeen and Hayward holes, so high-resolution images could in principle tell a Minkowski core from a de Sitter core.
  • The difference between core types grows with $\alpha_0$, meaning that higher quantum-gravity corrections make the signature easier to detect, not harder.
  • Under static spherical accretion the difference is larger than under infalling accretion, so the detectability of the core type depends on the astrophysical environment of the hole.

Reading between the lines

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

  • Beyond the paper: because the shadow boundary is accretion-model independent while the intensity is not, the ratio of static to infalling intensity is a cleaner probe of the accretion physics; the authors do not make this separation explicit.
  • Beyond the paper: the same machinery could be applied to a rotating Minkowski-core metric; the spherical-symmetry assumption means the claimed core-type distinction has not yet been tested against the asymmetric images needed for real black-hole observations.
  • Beyond the paper: the one-to-one Bardeen/Hayward correspondence fixes only large-scale behaviour, so a data-driven parameter fit to the two metrics instead of the assumed mapping could change or even reverse the ordering of shadow sizes; this is the natural next test.
  • Beyond the paper: the paper's brighter-photon-ring-with-larger-$\alpha_0$ result implies a possible degeneracy with a slightly smaller Schwarzschild-like object; only joint measurement of radius and brightness, not either alone, would break it.
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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. The paper studies the shadows and optical appearances of a family of regular black holes with an asymptotically Minkowski core, defined by the metric in Eqs. (1)-(3), under static and infalling spherical accretion. The authors compute geodesics, photon-sphere radii, critical impact parameters, and observed specific intensities, and then compare these quantities with the traditional Bardeen and Hayward black holes, which have de Sitter cores, using a one-to-one correspondence inherited from earlier work. The central claims are that for the Minkowski-core holes the shadow and photon-sphere radii decrease with increasing quantum parameter α0 while the observed intensity increases, that increasing the deformation parameter n has the opposite effects, and that the Minkowski-core holes have smaller and dimmer shadows than the corresponding dS-core holes, with the difference growing with α0.

Significance. The paper applies a standard ray-tracing and spherical-accretion framework to a specific family of regular black holes and recovers the Schwarzschild limit correctly in Tables 1 and 2, which gives confidence in the numerical implementation. If the adopted parameter identification between the Minkowski-core and dS-core families is accepted, the work provides a systematic comparison of shadow and intensity observables between two core types, which is a useful theoretical input for future EHT-type observations. However, the observational distinguishability conclusion is not quantitatively supported, and the manuscript contains at least one internal inconsistency (use of n=1 despite the stated n≥2 horizon condition) that affects part of the parameter scan. The core-type comparison also depends entirely on a single mapping from prior work, so the robustness of the main conclusion is not yet established.

major comments (4)
  1. [Sec. 2 and Sec. 3.1] The conditions stated in Sec. 2 after Eq. (4) require n ≥ 2 (together with n ≥ x ≥ n/3) in order to ensure the existence of an event horizon and sub-Planckian curvature. Nevertheless, Sec. 3.1 and Figs. 3 and 4 explicitly include n = 1 configurations (e.g., 'When the spacetime deformation n = 1...' and the four panels of Fig. 4). No proof is given that n = 1 cases satisfy the horizon-existence condition, and they appear to violate the stated constraint. The authors should either restrict the analysis to n ≥ 2 or demonstrate that n = 1 configurations are legitimate black holes under the same criteria; otherwise the reported n-dependence of the intensity and shadow radius includes points that are outside the claimed valid parameter space.
  2. [Sec. 3.1, final paragraph] The text reads 'the enhancement of quantum gravity effect can increase the luminosity of photon ring and enlarge the shadow radius for these new regular BHs.' This directly contradicts the immediately preceding sentence and the abstract, which state that larger α0 leads to a smaller shadow radius and photon-sphere radius. Since this sentence reverses the paper's primary trend, it is a load-bearing error and must be corrected, presumably to 'decrease the shadow radius.'
  3. [Sec. 4 and Tables 1-2] The comparison between Minkowski-core and dS-core black holes rests entirely on the one-to-one correspondence between Eq. (3) and Eq. (4) taken from Ref. [58]. That correspondence matches only the leading large-r behavior; the differences between the potentials appear at higher order in α0. The paper does not justify that this particular identification is the physically relevant one, and alternative identifications (for example, matching outer horizon radii or photon-sphere radii for each α0) could reduce, eliminate, or reverse the reported differences. The numerical differences are in fact small: in Table 1, for α0=0.72, the critical impact parameter is 4.66393 for the Minkowski-core hole versus 4.69073 for Bardeen (about 0.57%), and in Table 2 for α0=0.92 the difference with Hayward is about 0.11%. The conclusion that the two core types can be observationally distinguished is therefore conditional on the chosen parameter identification. Please state this limitation explicitly and, if possible, test the sensitivity of the conclusions to alternative identifications.
  4. [Sec. 5 and Abstract] The abstract and conclusion state that the two types of regular black holes 'can be distinguished through astronomical observation.' No quantitative observational analysis is provided: the paper does not compare the predicted intensity differences or sub-percent shadow-radius differences with EHT angular resolution, sensitivity, or expected astrophysical contamination. At the sub-percent level, such a claim requires at least an order-of-magnitude estimate of detectability. The authors should either provide such an estimate or soften the claim to a theoretical prediction that would require future observational capabilities.
minor comments (5)
  1. [Sec. 2, paragraph after Table 1] The sentence 'similarly, Hayward BH with a dS core is slightly greater values than the one with a Minkowski core (x = 2/3 and n = 2)' should refer to (x = 1, n = 3) rather than (x = 2/3, n = 2), since the Hayward comparison is made for those parameters in Table 2.
  2. [Sec. 2, footnote 1] The dimensional discussion in the footnote is confusing; in particular, 'express G into a form of M x' is unclear and should be rewritten to show explicitly how α0 is made dimensionless and how the Planck-length powers are absorbed.
  3. [Sec. 3.1, Eqs. (16)-(17)] The integration domain and the treatment of the geodesic path (including the number of times the ray passes through the emitting spherical shell and the handling of turning points) are not specified. For reproducibility, please state the integration limits and the numerical procedure used to evaluate the integral along the full photon path.
  4. [Sec. 1, Introduction] There is a typo: 'diving into the event horizon' should be 'diving into the event horizon.'
  5. [Sec. 2, paragraph after Fig. 1] The sentence 'When the core type is the same, these values increase with the increase of the dimensionless parameters (x and n)' is repeated in the same paragraph; please remove the duplicate.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the shadow and intensity results are computed directly from the stated metric, and the only self-citation, the large-scale correspondence with Bardeen/Hayward metrics, is an external premise rather than a fitted or predicted quantity.

full rationale

The paper's central quantities — shadow radius b_c, photon sphere radius r_c, and observed specific intensity I_obs — are obtained by solving the null geodesic equations (Eqs. 5–14) and integrating the transfer functions (Eqs. 17 and 26) for the explicit metric f(r) = 1 + 2ψ(r) defined in Eqs. (1)–(3). No parameter is fitted to the shadow or intensity data; the reported trends with α0 and n are direct consequences of the exponential potential. The only potentially self-referential element is the claimed one-to-one correspondence between the Minkowski-core potential (Eq. 3) and the dS-core Bardeen/Hayward potentials (Eq. 4), cited to the authors' earlier work [58]. That correspondence is an external, mathematically checkable input — a large-r matching of the leading correction — rather than a function of the target results. The shadow comparison is computed after the identification is made, not used to define it. Therefore the self-citation is not load-bearing in the circularity sense, and no step reduces a prediction to its input by construction. Concerns that a different parameter identification could alter the core comparison are a correctness risk about the physical relevance of the [58] mapping, not a circularity of the paper's derivation.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central results rest on the GUP-inspired metric and its correspondence to Bardeen/Hayward holes, both taken from the authors' prior work. No new particles, fields, or forces are introduced. The free parameters alpha_0, n, and x are model inputs scanned in the analysis, not fitted to data.

free parameters (3)
  • alpha_0 = 0.2, 0.3, 0.72, 0.92 (M=1)
    Quantum-gravity parameter of the exponential GUP potential; free model parameter scanned in the paper, bounded by horizon existence (<=0.73M for x=2/3,n=2 and <=0.98M for x=1,n=3).
  • n = 1, 2, 3
    Spacetime deformation exponent in the exponential potential; free model parameter scanned; larger n gives larger shadow and dimmer ring.
  • x = 2/3, 1
    Exponent in M^x inside the exponential and in the dS-core correspondence; fixed to 2/3 for the Bardeen-like case and 1 for the Hayward-like case.
assumptions (5)
  • domain assumption The spherically symmetric static line element (1) with f(r)=1+2*psi(r) is the correct description of the regular black hole spacetime.
    Taken from the Ling-Wu model; not derived in this paper.
  • domain assumption The exponential gravitational potential (3) and the one-to-one correspondence to the dS-core potential (4) are valid.
    Quoted from prior work [58]; the comparison between cores depends on this mapping.
  • domain assumption The emissivity of the accretion is modeled as j_e proportional to delta(nu_r - nu_e)/r^2 with no absorption or scattering.
    Standard toy-model assumption from Bambi 2013; makes the accretion optically thin.
  • domain assumption Static and infalling spherical accretion are adequate models for the astrophysical environment.
    The paper explicitly limits conclusions to these two toy models.
  • standard math Null geodesics are governed by the Euler-Lagrange equations with L=0 and conserved E and L.
    Standard GR photon motion in a static spherically symmetric spacetime.

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Pith. "Pith review of Investigating the shadows of new regular black holes with a Minkowski core: Effects of spherical accretion and core type differences." pith.science (2026). https://pith.science/paper/AHCDEWLB

@misc{pith2026250206388,
  author       = {Pith},
  title        = {Pith review of: Investigating the shadows of new regular black holes with a Minkowski core: Effects of spherical accretion and core type differences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHCDEWLB}},
  note         = {Machine review of arXiv:2502.06388}
}
abstract

We investigated the shadows and optical appearances of a new type of regular black holes (BHs) with a Minkowski core under various spherical accretion scenarios. These BHs are constructed by modifying the Newtonian potential based on the minimum observable length in the Generalized Uncertainty Principle (GUP). They correspond one-to-one with traditional regular BHs featuring a de-Sitter (dS) core (such as Bardeen/Hayward BHs), characterized by a quantum gravity effect parameter ($\alpha_0$) and spacetime deformation factor ($n$). We found that the characteristic parameters give rise to some novel observable features. For these new BHs, both the shadow and photon sphere radii decrease with the increase in $\alpha_0$, while the observed specific intensity increases. Conversely, as n increases, the shadow and photon sphere radii increase, while the observed specific intensity decreases. Under different spherical accretion scenarios, the shadows and photon sphere radii remain identical; however, the observed specific intensity is greater under static spherical accretion than under infalling spherical accretion. Additionally, we found that these regular BHs with different cores exhibit variations in shadows and optical appearances, particularly under static spherical accretion. Compared with Bardeen BH, the new BHs exhibit a lower observed specific intensity, a dimmer photon ring, and smaller shadow and photon sphere radii. Larger values of $\alpha_0$ lead to more significant differences, and a similar trend was also observed when comparing with Hayward BH. Under infalling spherical accretion, the regular BHs with different cores exhibit only slight differences in observed specific intensity, which become more evident when $\alpha_0$ is relatively large.

Figures

Figures reproduced from arXiv: 2502.06388 by the authors.

Figure 2
Figure 2. Fig.2. These plots demonstrate the changes in trajecto [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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