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Deflection Angle of Regular Black Holes in Nonlinear Electrodynamics: Gauss-Bonnet Theorem, Time Delay, Shadow, and Greybody Bound

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper derives weak-lensing deflection angles, time delay, shadow radius, and greybody bound for a regular charged black hole, and shows how each observable depends on mass and charge.

desk verdict Routine GBT application with a load-bearing error: the deflection series misses the Schwarzschild second-order term when q=0, so the higher-order coefficients are not justified. read the letter →

arxiv 2509.02633 v1 pith:EKF2OS63 submitted 2025-09-01 gr-qc

Susmita Sarkar , Nayan Sarkar , Hasrat Hussian Shah , Pankaj Balo , Farook Rahaman This is my paper · ORCID
classification gr-qc
keywords weakgravitationallensingregularblackholesnonlinearelectrodynamicsGauss-Bonnettheoremdeflectionangleplasmamediumholeshadowgreybodyfactor
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 studies a singularity-free, spherically symmetric black hole spacetime obtained from general relativity coupled to nonlinear electrodynamics, characterized by mass m and electric charge q. Its central contribution is a weak-field deflection angle for light, derived with the Gauss-Bonnet theorem both in vacuum and in a plasma, plus companion results for time delay, shadow radius, and greybody-factor bound. The authors argue that the deflection angle decreases with impact parameter and charge, increases with mass, and is larger in plasma than in vacuum. If correct, these formulas give concrete signatures that could distinguish this regular black hole from the Schwarzschild black hole and other charged spacetimes. The time delay vanishes when m=q=0, the shadow shrinks with q and grows with m, and the greybody bound drops with q and rises with m.

What carries the argument

The argument runs through the Gauss-Bonnet theorem, which relates the Gaussian curvature of an optical metric to the deflection angle. For the optical metric built from the null geodesic condition of the spacetime, the Gaussian curvature G is expanded to high order in 1/r, and the deflection angle is computed as θ = −∫∫ G dA over the region outside a straight-line photon path r(φ)=b/sinφ. The same optical-metric machinery is extended to a plasma by including a refractive index n(r). The shadow and greybody results come from the effective potential of the null geodesic equations and from a rigorous bound on the one-dimensional scattering potential.

What would settle it

Numerically integrate the null geodesic equations for metric (13) to second order in m and q for several impact parameters, and compare the resulting deflection angles with the series in Eq. (41); if the 1/b³ and 1/b⁴ coefficients differ beyond numerical error, the straight-line approximation used in Eq. (33) is the reason.

Watch

Extended reading notes

Core claim

For the regular black hole metric (13), the weak deflection angle in vacuum is claimed to be θ_A ≈ 4m/b − 3πq²/(4b²) − 16mq²/(3b³) + 9πq²(11m²+6q²)/(32b⁴) + 12mq⁴/(25b⁵), with an additional plasma term proportional to ω_e²/ω_∞². The authors show this reduces to 4m/b for q=0, recovering Schwarzschild, and that plasma increases the angle relative to vacuum. They also derive a time-delay expression that vanishes for m=q=0, compute photon-sphere and shadow radii, finding that the shadow shrinks as q increases and expands as m increases, and obtain a rigorous lower bound on the greybody factor that decreases with q and increases with m.

Load-bearing premise

The deflection angle is computed using a straight-line photon path, which is reliable only to leading order, even though the final formula keeps correction terms of order 1/b³ and smaller.

Editorial extensions

If this is right

  • Observers can use the deflection-angle series to estimate m and q from lensing measurements of stars or quasars passing behind a candidate regular black hole.
  • Plasma corrections matter: for a given impact parameter, the bending angle in a plasma is larger than in vacuum, so interpreting lensing observations requires knowing the ambient plasma density.
  • The shadow-radius behavior gives a clear contrast: increasing charge shrinks the shadow and increasing mass expands it, which can be compared with horizon-scale images.
  • The time-delay formula provides a second observational channel, vanishing only when the black hole is absent, useful for testing the spacetime against Schwarzschild.
  • The greybody bound implies that the black hole's emission, as measured by the bound, is suppressed by charge and enhanced by mass.

Reading between the lines

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

  • A natural extension, not pursued in the paper, is to recompute the deflection angle with a perturbed photon geodesic to second order; if the higher-order coefficients change, the b⁻³ and b⁻⁴ terms should be revised.
  • The same Gauss-Bonnet pipeline could be applied to rotating or magnetized regular black holes to see whether the charge-dependent shadow shrinkage survives with spin.
  • The deflection formula could be tested against numerical null-geodesic integrations for moderate impact parameters, providing a direct check of the straight-line approximation at finite b.
  • For plasma, the dependence on ω_e/ω_∞ suggests a frequency-dependent lensing signal: multi-frequency observations of a single source could separate plasma and gravitational contributions.
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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

3 major / 5 minor

Summary. The paper studies the Ayon-Beato-Garcia regular black hole metric (13), which reduces to Schwarzschild for q=0. The main results are the weak vacuum deflection angle (Eq. (41)/(80)) and plasma deflection angle (Eq. (46)/(81)) obtained via the Gauss-Bonnet theorem using a straight-line path r=b/sin(φ), the time delay (Eq. (52)), the shadow radius (Eq. (73)), and the greybody bound (Eq. (79)). The paper claims that both deflection angles decrease with impact parameter b and charge q, increase with mass m, and that plasma increases the deflection relative to vacuum.

Significance. The manuscript has the virtue of being parameter-free: all predictions follow from the known ABG metric (13) without free parameters, and several q=0 limits are correctly recovered (leading deflection 4m/b, shadow radius 3√3 m, Kretschmann scalar 48m²/r⁶). The shadow and greybody sections are explicit and could be of interest. However, the central quantitative claim is the deflection series, and that series is not derived consistently beyond leading order: the q=0 limit misses the known Schwarzschild second-order term 15πm²/(4b²). If the deflection series were corrected, the paper could be a useful reference for regular-black-hole lensing, but as it stands the main novelty is not established.

major comments (3)
  1. [Secs. III-IV, Eqs. (33), (39), (41)] The deflection angle is evaluated by integrating the Gaussian curvature over the region bounded by the straight-line path r=b/sin(φ). This path is not a null geodesic of the optical metric, so the simple formula θ_A = -∫∫ G dA is valid only to leading order in the weak-field expansion. The paper itself notes at the end of Sec. III that b is only a first-order approximation. Despite this, Eq. (41) retains terms through O(b^{-5}). The inconsistency is exposed by the q=0 limit: Eq. (39) contains the +3m²/r⁴ term, whose straight-line area integral gives a nonzero O(m²/b²) contribution, yet Eq. (41) reduces to θ_A=4m/b. The known Schwarzschild weak-deflection series is 4m/b + 15πm²/(4b²)+…, so Eq. (41) is missing an entire O(m²/b²) term. Consequently the b^{-3}, b^{-4}, and b^{-5} coefficients in Eq. (41) are not justified.
  2. [Sec. VI, Eqs. (46), (81)] The plasma deflection angle uses the same straight-line integration domain, so it inherits the same leading-order limitation. In the q=0 limit Eq. (46)/(81) reduces to 4m/b + πm²ω_e²/(4b²ω_∞²), but the same-order vacuum term 15πm²/(4b²) is absent. The mixed terms involving m³q² and m q⁴ at O(b^{-5}) are therefore unsupported. A valid second-order deflection calculation in plasma must use the perturbed photon trajectory, not the undeflected straight line.
  3. [Sec. XI, after Eq. (81)] The concluding statement that for q=0 the results reduce to the Schwarzschild deflection 'up to the first order term' is an acknowledgment that the calculation is only first order in m/b. However, Eqs. (80)-(81) are presented, plotted, and interpreted as higher-order results. The manuscript must either provide a genuine second-order GBT calculation that includes the geodesic curvature of the perturbed trajectory, or explicitly restrict all claims to leading order and remove the higher-order terms and all figures and conclusions based on them.
minor comments (5)
  1. [Sec. II, Eq. (2)] Typo: 'funstion' should be 'function'.
  2. [Sec. V.A] The text lists q = {0, 0.4, 0.8, 0.11}, while the Fig. 3 legend shows q=1.1; these should be reconciled.
  3. [Sec. VIII, Eq. (48)] The first integral is displayed with both limits as r_c; presumably the upper limit should be r_v (or r_s as appropriate).
  4. [Ref. [42]] The author name is mangled ('Azreg-A /dieresis.ts1ınou'); it should be corrected.
  5. [Throughout] Section headers appear as 'See.- IV', 'See.- V', etc.; please standardize the formatting.

Circularity Check

0 steps flagged · score 1.0 of 10

No substantive circularity: all reported observables are computed from the cited ABG metric through standard, parameter-free methods; self-citations are contextual and not load-bearing.

full rationale

The paper's central results are not equivalent to its inputs by construction. The metric (13) is imported from Ayon-Beato and Garcia (Ref. [24]), an external prior work, and all subsequent quantities are computed from that metric. The vacuum deflection angle (41) is obtained by inserting the optical Gaussian curvature (39), computed from the metric via Eq. (38), into the Gauss-Bonnet integral (33); no parameter appearing in theta_A is fitted to the deflection itself. The plasma expression (46) similarly follows from the refractive index (42) and the same GBT integral. The time delay (52) comes from integrating the metric components in Eqs. (49)-(51); the shadow radius (73) is the standard null-geodesic critical impact parameter obtained from Eqs. (61)-(69); the greybody bound (79) is Visser's general bound (74) applied to the Regge-Wheeler potential (77) of the same metric. Each result is a first-principles application of a standard theorem to a stated metric, and the q=0 Schwarzschild limits are checked against the known 4m/b leading term. The self-citations (e.g., Refs. [79], [127]) appear only in review/context paragraphs and are not load-bearing for any derivation. The paper's use of the straight-line photon path r=b/sin(phi) at Eq. (33) while retaining terms up to b^{-5} is a possible accuracy/consistency concern about the weak-field expansion, not a circularity: the output is not assumed in the input, and the missing Schwarzschild O(m^2/b^2) term (if confirmed) would be a numerical error rather than a reduction of the prediction to its assumptions. Hence no circular step can be exhibited.

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

The central results rest on a known regular-black-hole metric from nonlinear electrodynamics, the straight-line approximation in the Gauss-Bonnet method, a plasma refractive-index model, and standard theorems for greybody bounds. No free parameters are fitted; m, q, b, omega_e/omega_inf are physical inputs.

assumptions (5)
  • domain assumption The metric (13) is a valid regular, static, spherically symmetric solution of Einstein equations coupled to nonlinear electrodynamics satisfying the weak energy condition.
    Taken from Ayon-Beato-Garcia (Ref [24]); the paper verifies regularity via the Kretschmann scalar (16) and horizon condition q <= 0.6m but does not re-derive the solution from the action principle.
  • ad hoc to paper The Gauss-Bonnet deflection angle can be computed using the straight-line photon path r = b/sin(phi) in the integral (33).
    Invoked in Section III; this approximation is valid only to leading order in m/b, yet the final deflection formulas (41)/(46) retain higher-order terms without a perturbed-trajectory correction.
  • domain assumption Light in a plasma obeys the refractive index n(r) = sqrt(1 - (omega_e^2/omega_inf^2) f(r)) from Ref [113].
    Adopted in Eq (42) to construct the optical metric (43) for the plasma case; standard in the literature.
  • standard math The greybody bound T_b >= sech^2( (1/2omega) integral V dr* ) from Visser (Ref [118]) applies to this potential.
    Used in Eq (74) to derive the bound (79); the theorem is a general scattering bound, not specific to this metric.
  • standard math The photon sphere condition V_eff = V'_eff = 0 yields Eq (69) after squaring, ignoring possible extraneous roots.
    Section IX; the 12th-order polynomial is obtained by squaring 1 - 3m r^4/(r^2+q^2)^{5/2} + 2q^2 r^4/(r^2+q^2)^3 = 0, which may introduce spurious roots not checked against the original equation.

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Pith. "Pith review of Deflection Angle of Regular Black Holes in Nonlinear Electrodynamics: Gauss-Bonnet Theorem, Time Delay, Shadow, and Greybody Bound." pith.science (2026). https://pith.science/paper/EKF2OS63

@misc{pith2026250902633,
  author       = {Pith},
  title        = {Pith review of: Deflection Angle of Regular Black Holes in Nonlinear Electrodynamics: Gauss-Bonnet Theorem, Time Delay, Shadow, and Greybody Bound},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EKF2OS63}},
  note         = {Machine review of arXiv:2509.02633}
}
abstract

In this article, we study the weak gravitational lensing in the background of regular, static, spherically symmetric black hole solutions of Einstein's standard general relativity coupled with nonlinear electrodynamics. The weak deflection angles are estimated in the context of vacuum medium and plasma medium using the Gauss-Bonnet method. The obtained deflection angles decrease as the impact parameter $b$ and the charge parameter $q$ increase, while the deflection angles increase gradually with increasing values of the black hole mass $m$. Moreover, the effect of a plasma medium has increased the deflection angle than the vacuum medium scenario. We also estimate the time delay in the field of the described black holes that vanishes for $m = q =0$, i.e., the absence of the black holes. The shadow cast of the present black holes is also analyzed with respect to the impact $q$ and mass $m$, which ensures that the shadow region shrinks for increasing values of $q$ and expands for increasing values of $m$. In addition, we estimated the rigorous bounds of the greybody factor $\mathcal{T}_b$ for the described black holes and the graphical analysis ensures that the increasing charge parameter $q$ decreases the rigorous bound of $\mathcal{T}_b$ and the increasing mass $m$ increases the rigorous bound of $\mathcal{T}_b$.

Figures

Figures reproduced from arXiv: 2509.02633 by the authors.

Figure 1
Figure 1. FIG. 1: Behavior of - [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: This figure illustrates the chosen region [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Deflection angle [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Deflection angle [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Deflection angle [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Deflection angle [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Deflection angle [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Deflection angle [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Effective potential again the radial coordinate [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Shadows of the black hole corresponding to [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Greybody factor [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

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