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REVIEW 2 major objections 6 minor 67 references

Strong gravitational lensing effects around rotating regular black holes

T0 review · 2 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For two rotating regular black holes with a Minkowski core, the time delays between relativistic images deviate from Kerr by up to tens of hours for M87* parameters, while image-position deviations remain below 10 microarcseconds.

desk verdict A competent application of Bozza's strong-lensing formalism to a specific rotating regular black hole family; the new numerical time-delay predictions are interesting, but the physical status of the rotating metric is assumed rather than demonstrated. read the letter →

arxiv 2501.00292 v2 pith:B2R43AAK submitted 2024-12-31 gr-qc

Mingyu Guo , Menghe Wu , Hong Guo , Xiaomei Kuang , Fuyao Liu This is my paper · ORCID
classification gr-qc MSC 83C5783C10
keywords stronggravitationallensingregularblackholesMinkowskicoreKerrholerelativisticimagestimedelayM87*Newman-Janismetric
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

This paper asks whether two families of rotating regular black holes with a non-singular Minkowski core can be told apart from a Kerr black hole by the way they bend light. Using a strong-field deflection-angle expansion, it computes the relativistic image position, separation, magnification, and time delays for a Bardeen-type and a Hayward-type regular black hole, scaled to the masses and distances of M87* and SgrA*. It finds that the image-position deviations from Kerr stay below 10 microarcseconds, below current resolution, but the time delays between relativistic images deviate from Kerr by up to tens of hours for M87*, which could be measurable. The authors conclude that time-delay observations, not image positions, are the promising route for testing whether a supermassive black hole is regular.

What carries the argument

The machinery is the Bozza strong-deflection-limit expansion for the deflection angle, $\alpha_D(u) = -\bar a \log(u/u_m - 1) + \bar b$, with the strong-deflection coefficients $\bar a$ and $\bar b$ extracted from the metric functions $A, B, C, D$ of the rotating regular spacetime in the equatorial plane. The same expansion, augmented by the time-delay coefficients $\tilde a$ and $\tilde b$, yields the image position $\theta_\infty = u_m/D_{OL}$, the separation $s$, the magnification ratio $r_{\rm mag}$, and the inter-image time delays. These coefficients carry the entire dependence on the regularity parameter $g$ and the spin $a$, and the paper evaluates them numerically for the two regular models to predict observable shifts from Kerr.

What would settle it

Compute the effective energy\textendash momentum tensor of the rotating metric in Eq. (4) and check whether it satisfies standard energy conditions; if no reasonable matter source exists, the lensing predictions lack an astrophysical basis. Observationally, a next-generation Event Horizon Telescope measurement of the time delay between the first and second relativistic images of M87* that matches the Kerr value to better than the predicted tens-of-hours deviation would rule out these two regular models.

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Extended reading notes

Core claim

The paper's central claim is that strong gravitational lensing by the two rotating regular black holes differs from Kerr in a characteristic, g-dependent way, and that the difference is concentrated in the time-delay channel rather than the image geometry. For both parameter families, increasing the regularity parameter g reduces the critical impact parameter, the asymptotic image radius $\theta_\infty$, and the relative magnification $r_{\rm mag}$, while increasing the angular separation $s$; the same-side time delay $\Delta T_{2,1}$ between the first and second images is shorter than in Kerr, and the prograde\textendash retrograde delay $\Delta\tilde{T}_{1,1}$ is longer. For M87* parameters the time-delay deviations reach tens of hours (up to about 44 hours in the Bardeen-type case at $g=0.48$, $a=0.2$), whereas the deviations in $\theta_\infty$ and $s$ are at most a few microarcseconds. Hence the paper argues that current Event Horizon Telescope observations cannot resolve the regularity through image position, but next-generation instruments may test it through time delays.

Load-bearing premise

The entire calculation rests on the assumption that the rotating metric obtained by the Newman\textendash Janis algorithm actually describes a real black hole in some theory of gravity; the paper never verifies that it solves any field equations or obeys energy conditions.

Editorial extensions

If this is right

  • If the paper is right, the time delay between the first and second relativistic images of M87* is the most promising strong-lensing observable for distinguishing these regular black holes from Kerr, with deviations of up to tens of hours.
  • The regularity parameter $g$ shrinks the critical impact parameter and hence the apparent image radius, but the effect stays below about 10 microarcseconds, below current Event Horizon Telescope resolution.
  • The separation $s$ between the outermost and asymptotic images grows with $g$, so higher-resolution observations of image pairs could complement time-delay measurements.
  • For SgrA* the same trends hold but the time delays are only minutes and the deviations are correspondingly smaller, making M87* the better target.

Reading between the lines

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

  • The results suggest that time-delay measurements are a more sensitive probe of near-horizon regularity than shadow-size or image-position measurements; if so, future lensing campaigns should prioritize timing observations.
  • The calculation assumes equatorial prograde and retrograde photon orbits; off-equatorial or photon-ring autocorrelation observations could expose additional $g$-dependent signatures not captured here.
  • If the Newman\textendash Janis metric does not correspond to a known matter source, the predicted time-delay deviations would still serve as a bound on what lensing can say about regularity; conversely, a positive detection would motivate finding a theory that produces this metric.
  • The trend that the Bardeen-type model ($\gamma = 2/3, n=2$) shows larger deviations than the Hayward-type model ($\gamma = 1, n=3$) suggests that the specific form of the mass function matters, and probing multiple regular models may be needed to avoid degeneracy with spin.
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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

2 major / 6 minor

Summary. The paper studies strong gravitational lensing by two classes of rotating regular black holes with a Minkowski core, obtained by applying the Newman-Janis algorithm to the static regular black hole of Ref. [53]. The authors derive the strong-field-limit deflection angle for equatorial photons, then compute the lensing observables (asymptotic image position, angular separation, relative magnification, and time delays between relativistic images) under the assumption that the objects are M87* and SgrA*. The central claims are that the time-delay differences between these regular black holes and Kerr are potentially measurable, especially for M87*, while the deviations in image position and separation are below 10 microarcseconds and hence require next-generation Event Horizon Telescope capabilities.

Significance. If the rotating metric is taken as physically viable, the paper provides a concrete and falsifiable prediction: strong-lensing time delays can discriminate between Kerr and these regular black holes even when image-position differences are too small for current observations. The analytical derivation follows the standard Bozza strong-field-limit formalism, and the numerical values are internally plausible and consistent with the mass-scaling between M87* and SgrA*. The paper does not, however, ship machine-checked proofs or reproducible code, and no independent numerical verification or uncertainty estimates are presented. The main value is therefore as a model-dependent theoretical estimate rather than a demonstrated observational test.

major comments (2)
  1. [Section II, Eq. (4)] The rotating metric in Eq. (4) is introduced as the Newman-Janis rotating counterpart of the static regular black hole, but the paper never checks whether this line element is a solution of any concrete gravitational field equations with a reasonable matter source, nor whether the regularity and Minkowski-core properties of the static seed survive rotation (e.g., by examining G_mu_nu, energy conditions, or curvature invariants at Sigma=0). Because every lensing observable in Sections III and IV is computed from Eq. (4), the astrophysical claims about M87* and SgrA* are conditional on this unverified premise. Please either supply the missing verification (or a reference that provides it) and discuss the resulting validity region, or explicitly reframe the conclusions as predictions of the metric ansatz (4).
  2. [Section IV.B, Tables I-VIII] The paper claims that the time-delay differences between the regular black holes and Kerr are 'measurable,' especially for M87*, but it does not provide a quantitative comparison with any observational uncertainty, such as expected EHT timing precision or mass and distance errors. The tables report predicted delays but no error bars, and the sub-10 microarcsecond image-position deviations are not translated into a signal-to-noise or detection-threshold statement. Please add a quantitative detectability discussion or temper the 'measurable' wording in the abstract and conclusion.
minor comments (6)
  1. [Section IV.B, Table IV] In the a=-0.1, g=0.4 entry, the deviation delta Delta T_2,1 is printed as -0.845 min, while the difference of the preceding columns is -0.085 min; as printed this breaks the monotonicity of the g=0.4 column and should be corrected.
  2. [Section III.A.2] The last sentence refers to 'gamma=1,n=2', but the models studied in the paper are gamma=1,n=3; please correct the typo to match Eq. (2) and Fig. 2.
  3. [Eq. (9)] The displayed expression for x-dot appears corrupted in the version under review, containing non-mathematical characters; please ensure the formula is typeset correctly.
  4. [Section IV.A, Eqs. (30)-(36)] The conversion from the dimensionless rescaling x=r/(2M) used in Section III.A.1 to the physical hours and minutes reported in Tables I-VIII is not shown; please state the overall mass/distance prefactor explicitly.
  5. [Section II, Eq. (2)] The dimension of the regularity parameter g is not stated; for gamma=2/3,n=2 the exponent g^2 M^{2/3}/r^2 implies an unusual dimension that should be specified to avoid confusion about the plotted g/M values.
  6. [Section V] In the conclusion, 'Delta-tilde-T_2,1' should read 'Delta T_2,1' to match the notation in Tables I-IV; the two different time-delay quantities are otherwise easy to confuse.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the lensing observables are computed from an assumed metric, and the only self-citation (the seed metric of Ref. [53]) is not load-bearing.

full rationale

The derivation chain is explicit and non-circular: the static metric (1)-(2) is adopted from Ref. [53]; the rotating line element (4) is obtained by applying the Newman-Janis procedure; the null geodesic equations (8)-(11) and the Bozza strong-deflection expansion (17)-(24) produce the deflection angle (21); and the observables (26)-(36) are evaluated using the external EHT mass and distance values quoted for M87* and SgrA*. At no point is an observable fitted to the data that it is then used to predict: the parameter g is scanned over chosen values (0, 0.3, 0.4, 0.48), not calibrated from lensing measurements, and the Kerr comparison is the same computation at g = 0. The only self-citation is the seed metric of Ref. [53], which is coauthored by one of the present authors; that citation supplies an input model for the spacetime, not a theorem that forces the predicted lensing quantities, so it is not load-bearing in the circularity sense. The physical status of Eq. (4) (satisfaction of field equations, energy conditions, and regularity at Sigma = 0) is asserted rather than demonstrated, but that is a model-validity and correctness concern, not a circular reduction of the predictions to their inputs.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

No new particles, fields, or entities are proposed. The regular black hole metric is inherited from prior work; the paper's contribution is the numerical lensing analysis and observational comparison.

free parameters (2)
  • g (regularity parameter) = 0.3, 0.4, 0.48; Kerr limit g=0
    Controls the mass-function suppression m(r)=M exp(-g^n M^gamma / r^n). The entire deviation from Kerr and all tables and figures are scanned over these hand-picked values; no data determine g.
  • a (spin parameter) = -0.2 to 0.2 in tables; -0.5 to 0.2 in figures
    Chosen to display prograde and retrograde photon behavior; not fitted to data. It is a physical parameter of the metric, but the observables depend on it strongly.
assumptions (3)
  • ad hoc to paper Newman-Janis rotated metric Eq. (4) is a legitimate spacetime for a rotating regular black hole.
    Adopted from Ref. [57] without proving it solves gravitational field equations or satisfies energy conditions in the rotating case; all lensing results depend on this metric.
  • domain assumption Bozza strong-field expansion Eq. (21) is accurate for the chosen parameter range.
    The expansion is from Refs. [25,26] and is standard, but the paper does not quantify truncation error for g near 0.48 or for spins near the extremal limit.
  • domain assumption Lensing observables are evaluated in the equatorial plane with both source and observer far away.
    The formulas for θ∞, s, rmag, and time delays assume this geometry, as stated in Section IV.A.

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Pith. "Pith review of Strong gravitational lensing effects around rotating regular black holes." pith.science (2026). https://pith.science/paper/B2R43AAK

@misc{pith2026250100292,
  author       = {Pith},
  title        = {Pith review of: Strong gravitational lensing effects around rotating regular black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B2R43AAK}},
  note         = {Machine review of arXiv:2501.00292}
}
abstract

In this letter, we investigate the strong gravitational lensing effects around two classes of rotating regular black holes, which behave as non-singular Minkowski core at the center. Starting from the null geodesic in the equatorial plane of the regular black holes, we analyze the deflection angle for both prograde photons and retrograde photons, which are found to significantly shift from that for Kerr black hole. Then we suppose the rotating regular black holes as the supermassive M87* and SgrA* black holes, respectively, and evaluate the lensing observables such as the image position, separation, magnification and the time delays between the relativistic images for the black holes. In both cases, the time delay differences between the regular black holes and Kerr black hole seem to be measurable, especially for the M87* black hole, but the deviations of outermost relativistic image and the asymptotic relativistic image for the two rotating regular black holes from those for Kerr black hole are less than 10 $\mu as$. This means that it is difficult to probe the regularity of the black hole via the current observations, but we can expect the next generation Event Horizon Telescope to test more precise properties of strong field regime.

Figures

Figures reproduced from arXiv: 2501.00292 by the authors.

Figure 1
Figure 1. FIG. 1: The figure shows that different spins [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The radius of the photon circle [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The behaviors of the lensing coefficients ¯a [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The deflection angle [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The schematic diagram of the gravitational lensing. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The behaviors of lensing observable [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The behaviors of lensing observables [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The behaviors of lensing observables [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: The differences of [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

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