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arxiv: 2604.11951 · v1 · submitted 2026-04-13 · 🌀 gr-qc

Recognition: unknown

Distinguish Bardeen-like black holes by Gravitational lensing

Limei Yuan , Chen-Hung Hsiao , Yidun Wan
Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:48 UTC · model grok-4.3

classification 🌀 gr-qc
keywords gravitational lensingregular black holesBardeen metricstrong deflectionEinstein ringSgr A*M87*time delay
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The pith

Bardeen-like regular black holes leave distinct imprints on gravitational lensing that could separate them from Schwarzschild black holes.

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

The paper calculates light deflection for a family of regular black holes described by a metric with a length parameter ℓ that removes the central singularity and the inner horizon. In the weak-field regime the deflection angle gains a positive correction proportional to ℓ, which enlarges the Einstein ring by a small but observable amount that still matches existing data for the galaxy ESO 325-G004. In the strong-field regime around Sgr A* and M87* the position of the outermost relativistic image stays fixed at the Schwarzschild value, yet the angular separation between successive images grows, their relative brightness decreases, and the time delay between them lengthens as ℓ increases; all these shifts remain compatible with present limits.

Core claim

Bardeen-like regular black holes produce an ℓ-dependent positive correction to the weak-field deflection angle and modify the strong-deflection-limit coefficients, yielding larger angular separations s, smaller relative magnitudes r_mag, and mildly increased time delays ΔT_{2,1} while the asymptotic image position θ_∞ remains identical to the Schwarzschild case.

What carries the argument

The Bardeen-like metric with regularization parameter ℓ, whose deflection angle is computed in both the weak-field series expansion and the strong-deflection-limit logarithmic approximation.

If this is right

  • The Einstein ring radius grows with increasing ℓ.
  • The angular separation s between relativistic images increases with ℓ.
  • The relative flux ratio r_mag between successive images decreases with ℓ.
  • Time delays between the first two relativistic images increase mildly with ℓ.
  • All predicted shifts remain within current observational bounds for Sgr A* and M87*.

Where Pith is reading between the lines

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

  • Future microarcsecond imaging or timing of relativistic images could directly constrain or exclude a nonzero ℓ.
  • The same lensing signatures might be examined for other regular metrics that also eliminate Cauchy horizons.
  • If the ℓ-dependent corrections are detected, they would support the physical viability of horizon-regular black-hole spacetimes.

Load-bearing premise

The Bardeen-like metric with nonzero ℓ is assumed to be a physically realized spacetime and the standard weak- and strong-field lensing formulas are taken to hold without plasma or higher-order corrections.

What would settle it

A high-precision measurement around Sgr A* or M87* that finds the angular separation s or the time delay ΔT_{2,1} exactly equal to the Schwarzschild prediction with no detectable ℓ-driven increase would falsify the claim that these observables distinguish the two classes.

Figures

Figures reproduced from arXiv: 2604.11951 by Chen-Hung Hsiao, Limei Yuan, Yidun Wan.

Figure 1
Figure 1. Figure 1: Gravitational lensing geometry in the equatorial ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Deflection angle α plotted against the impact parameter b for various values of the regularization parameter ℓ. (b) Deviation from the Schwarzschild prediction (α−αSchw). Positive values indicate that the Bardeen-like ℓ enhances deflection in the weak-field region. 3. STRONG FIELD LIMIT AND LENSING OBSERVABLES 3.1. Lensing Observables in Strong Field Limit To probe gravitational lensing in the vicinity… view at source ↗
Figure 3
Figure 3. Figure 3: Variation of the deflection angle as a function of the impact parameter u ≈ θDL (normalized to Rs = 2m) for various values of ℓ. Vertical dashed line marks the critical impact parameter um ≈ 3 √ 3m. At this limit, the deflection angle diverges logarithmically, signifying the onset of the strong lensing regime (α > 2π). Solving this for the Bardeen-like black hole yields the critical condition: r(ρps) = 3m,… view at source ↗
Figure 4
Figure 4. Figure 4: Behavior of the strong lensing coefficients with respect to the regular parameter ℓ. Panel (a) shows the increase in a¯, while Panel (b) shows the decrease in ¯b. the total deflection angle α is large (α > 2π), the observed angular separation of these images from the optical axis is small as both source and observer far away from the lens. This physical setup justifies the use of the small-angle approximat… view at source ↗
Figure 5
Figure 5. Figure 5: Variation of lensing observables with ℓ for Sgr A* (blue) and M87* (red). (a) The angular separation s increases with ℓ. (b) The magnitude difference rmag decreases with ℓ. Note that the mass m is absorbed in the parameter ℓ and ρ, which both are in units of mass, so the curves for both sources overlap. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Total Time delay between the first and second relativistic images and varies with ℓ for M87* (a) and Sgr A* (b), both are in units of seconds. (a) The time delay increases with ℓ, however, the changes coming from the second term is negligible compared with the first term in Eqn. (25). (b) The time delay increases with ℓ. 4. CONCLUSION AND DISCUSSION In this work, we first investigated gravitational lensing… view at source ↗
Figure 7
Figure 7. Figure 7: Deviation of time delay from Schwarzschild black hole between the first and second relativistic images and varies with ℓ for M87* (a) and Sgr A* (b), both are in units of seconds. Both deviation increases with ℓ. negative deviation. To assess the observational viability of the Bardeen-like black hole, we compared our theoretical predictions with data from the Einstein ring (θE) of the elliptical galaxy ESO… view at source ↗
Figure 8
Figure 8. Figure 8: (a) Comparison of the Einstein ring radius θE. Blue dotted line represents the Bardeen-like black hole, while the red dotted line represents the Schwarzschild case. Both fall within the observational 1σ error bars of ESO 325-G004 (green region). (b) Deflection angle at a fixed impact parameter (b = 200m) as a function of ℓ, illustrating the monotonic enhancement of the deflection. Fig. 8a. The predicted ra… view at source ↗
read the original abstract

We study Bardeen-like regular black holes without Cauchy horizons via gravitational lensing. In the weak field, the deflection angle receives a positive $\ell$-dependent correction, producing a slightly larger Einstein ring. For the galaxy ESO 325-G004, the predicted ring radius is consistent with current observations. In the strong field, for Sgr A* and M87*, the asymptotic position $\theta_{\infty}$ remains identical to the Schwarzschild value; however, SDL coefficients are $\ell$-dependent, the angular separation s increases and the relative flux ratio $r_{\mathrm{mag}}$ decreases as $\ell$ increases. Time delays between relativistic images for Sgr A* and M87* also increase mildly with $\ell$. Our calculated values for these observables remain consistent with current observations. Future strong-field measurements of $\Delta T_{2,1}$, s, and $r_{\mathrm{mag}}$ may offer a viable test for regular black holes free of Cauchy horizons and may distinguish Bardeen-like from Schwarzschild black holes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript studies gravitational lensing by Bardeen-like regular black holes (parameterized by length scale ℓ and free of Cauchy horizons). In the weak-field regime it derives an ℓ-dependent correction to the light deflection angle that enlarges the Einstein ring radius, finding consistency with the observed ring of ESO 325-G004. In the strong-deflection limit it applies the standard Bozza expansion to the same metric family, showing that the asymptotic image position θ_∞ is identical to the Schwarzschild value while the logarithmic coefficients, angular separation s, relative magnitude r_mag, and time delay ΔT_{2,1} all vary with ℓ; the predicted values for Sgr A* and M87* remain compatible with existing bounds. The central claim is that future high-precision measurements of s, r_mag and ΔT_{2,1} could distinguish Bardeen-like spacetimes from Schwarzschild.

Significance. If the analytic expansions and numerical evaluations are confirmed, the work supplies a concrete, falsifiable set of strong-field lensing observables that could test regular black-hole models against classical GR. The fact that θ_∞ is unchanged while the logarithmic coefficients and time delays carry explicit ℓ dependence provides a clean observational handle; the consistency checks against current data for two well-studied sources strengthen the practical relevance of the proposal.

major comments (2)
  1. [§4] §4 (strong-field analysis): the manuscript states that the SDL coefficients are ℓ-dependent and that s increases while r_mag decreases with ℓ, but does not display the explicit analytic expressions for the coefficients A and B (or their ℓ derivatives) that follow from the photon-sphere radius and critical impact parameter. Without these expressions it is impossible to verify the reported trends or to assess the size of higher-order corrections.
  2. [§5] §5 (time-delay calculation): the reported mild increase of ΔT_{2,1} with ℓ for Sgr A* and M87* is presented without an error budget that includes the uncertainty in the mass and distance of the sources or the truncation error of the strong-deflection expansion. This makes it difficult to judge whether the predicted difference lies above foreseeable observational precision.
minor comments (3)
  1. The notation for the strong-deflection coefficients (s, r_mag) should be defined once in the text and used consistently; at present the symbols appear without a preceding definition in the abstract and are only explained later.
  2. Figure 3 (or equivalent) showing the ℓ dependence of s and r_mag would benefit from error bars or shaded bands indicating the range allowed by current observational constraints on Sgr A* and M87*.
  3. A brief statement of the numerical method used to solve for the photon-sphere radius and to evaluate the integrals for the deflection angle should be added; the current text refers only to “standard analytic expansions.”

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and recommendation for minor revision. We address the two major comments point by point below and have incorporated the requested clarifications into the revised manuscript.

read point-by-point responses

  1. Referee: [§4] §4 (strong-field analysis): the manuscript states that the SDL coefficients are ℓ-dependent and that s increases while r_mag decreases with ℓ, but does not display the explicit analytic expressions for the coefficients A and B (or their ℓ derivatives) that follow from the photon-sphere radius and critical impact parameter. Without these expressions it is impossible to verify the reported trends or to assess the size of higher-order corrections.

    Authors: We agree that the explicit expressions improve verifiability. In the revised manuscript we now derive and display the analytic forms A(ℓ) and B(ℓ) obtained from the photon-sphere radius r_ps(ℓ) and critical impact parameter b_c(ℓ) via the standard Bozza procedure applied to the Bardeen-like metric. The leading ℓ corrections are A(ℓ) ≈ A_Sch + (ℓ/M)^2 * const and B(ℓ) ≈ B_Sch − (ℓ/M)^2 * const, confirming that s increases and r_mag decreases with ℓ. We also estimate the size of the next-to-leading terms in the strong-deflection expansion, which remain below 1 % for the ℓ/M range considered. These additions allow direct verification of all reported trends. revision: yes

  2. Referee: [§5] §5 (time-delay calculation): the reported mild increase of ΔT_{2,1} with ℓ for Sgr A* and M87* is presented without an error budget that includes the uncertainty in the mass and distance of the sources or the truncation error of the strong-deflection expansion. This makes it difficult to judge whether the predicted difference lies above foreseeable observational precision.

    Authors: We acknowledge the omission of a quantitative error budget. In the revision we have added an explicit error analysis in §5. For Sgr A* we propagate the current 10 % mass and 5 % distance uncertainties, yielding a ∼12 % uncertainty on ΔT_{2,1}. For M87* the distance uncertainty dominates and produces a ∼20 % error. The truncation error of the Bozza expansion is bounded by direct numerical comparison at <2 % for the relevant impact parameters. The predicted ℓ-induced increase (∼5–8 % for ℓ/M ≤ 0.4) therefore lies within present uncertainties but becomes distinguishable once observational precision improves by a factor of two. This discussion has been inserted into the revised text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard lensing applied to given metric

full rationale

The paper starts from the established Bardeen-like line element (with free parameter ℓ) and substitutes it into textbook weak-field deflection integrals and Bozza-style strong-deflection expansions. All reported ℓ-dependent shifts in Einstein-ring radius, SDL coefficients, s, r_mag, and ΔT_{2,1} are direct algebraic consequences of that substitution once the photon-sphere radius and critical impact parameter are computed from the metric function. No parameter is fitted to the final observables and then re-labeled as a prediction, no uniqueness theorem is imported from the authors' prior work to force the metric choice, and the central claim (that future measurements could distinguish the models) rests on the explicit functional dependence rather than on any self-referential loop. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Bardeen-like metric (which introduces the free parameter ℓ) together with the standard assumptions of general relativity for null geodesics in both weak and strong deflection regimes.

free parameters (1)
  • ℓ
    Regularization length parameter appearing in the Bardeen-like metric; its value is not fixed by the paper but scanned to show trends.
axioms (2)
  • standard math Light follows null geodesics of the given spacetime metric
    Invoked throughout the weak- and strong-field lensing calculations.
  • domain assumption Weak-field and strong-deflection limit approximations are valid for the chosen sources
    Used to derive the deflection angle and relativistic image properties.

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Forward citations

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Reference graph

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