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Analyzing Deflection Angles and Photon Sphere Dynamics of Magnetically Charged Black Holes in Nonlinear Electrodynamic

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

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Pith's one-line read Magnetically charged black holes in nonlinear electrodynamics bend light more strongly and cast smaller shadows than their classical counterparts, and observations can bound the coupling.

desk verdict A novel weak-deflection calculation for a NED black hole is undermined by a strong-deflection section that accidentally reproduces pure Schwarzschild, contradicting the paper's central claim. read the letter →

arxiv 2502.04044 v2 pith:662FSCSS submitted 2025-02-06 gr-qc

Hira Waseem , Nikko John Leo S. Lobos , Ali Övgün , Reggie C. Pantig This is my paper · ORCID
classification gr-qc PACS 95.30.Sf04.70.-s97.60.Lf04.50.+h
keywords blackholesnonlinearelectrodynamicsmagneticchargeweakdeflectionanglestronglimitphotonsphereholeshadowEventHorizonTelescopeconstraints
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 aims to show that a magnetically charged black hole produced by a nonlinear electrodynamics (NED) model leaves observable signatures in gravitational lensing and shadow images. Starting from an exact solution whose metric approaches Schwarzschild at infinity, it derives a closed-form weak deflection angle: the standard $4M/b$ term plus power-law and logarithmic corrections involving the magnetic charge $Q$ and the NED coupling $\xi$. It then computes the strong deflection angle and shows that, under a Schwarzschild-like approximation with a renormalized mass, the result reproduces the Schwarzschild coefficients, while numerical work indicates the photon sphere and shadow radii shrink as $\xi$ grows. These formulas are combined with solar-system, M87*, and Sgr A* observations to place bounds on $\xi$, giving a route to identify this NED black hole through future precision lensing and shadow measurements. The results would matter because they show that the nonlinearity of the electromagnetic field can be constrained by observations of how black holes bend light.

What carries the argument

The central objects are the NED metric function $f(r)=1-2M/r+Q^2/r^2-\dots$ and its asymptotic expansion $f(r)=1-2M_{ADM}/r+Q^3/(9\xi^2 r^4)+O(1/r^6)$, with the renormalized mass $M_{ADM}=M+(2\sqrt{2}/3)\xi Q^{3/2}\ln(2\xi/\sqrt{2Q})$. The weak deflection is computed with the Gauss–Bonnet theorem applied to the optical metric of the spacetime, which turns the integral of the Gaussian curvature over the photon trajectory into the deflection angle. The strong deflection calculation uses the Tsukamoto/Bozza expansion of the radial orbit equation in the variable $z=1-r_0/r$, with coefficients $\bar a$ and $\bar b$ built from $A(r)$ and its derivatives at the photon sphere. The photon sphere and shadow are obtained from the condition $h'(r)=0$ with $h(r)=C(r)/A(r)$, evaluated numerically because the full metric does not admit an analytic photon-sphere solution.

What would settle it

Recompute the photon sphere radius and the strong-deflection coefficients $\bar a$ and $\bar b$ from the full metric (17) without discarding the $Q^3/(9\xi^2 r^4)$ term; if $r_{ps}\ne 3M_{ADM}$ or the coefficients deviate from $\bar a=1$, then Eq. (39) and the $\xi$ constraints built on it fail.

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Extended reading notes

Core claim

The paper's central claim is that the magnetically charged NED black hole solution of Mazharimousavi imprints measurable deviations on light bending and shadows relative to Schwarzschild and Reissner–Nordström, and that these deviations are controlled by the single coupling parameter $\xi$ together with the magnetic charge $Q$. On the weak-field side, Gauss–Bonnet integration of the optical metric yields $\alpha = 4M/b - 5\pi Q^3/(48 b^4 \xi^2) + \dots$, where the extra terms are proportional to $\xi$, $\xi^2$, and logarithms of $Q$ and $\xi$, so the bending is stronger than the classical RN prediction. In the strong-deflection regime the paper reports that the two coefficients $\bar a$ and $\bar b$ reduce to the Schwarzschild values once the metric function is approximated as $A(r)=1-2M_{ADM}/r$, making the strong deflection angle exactly Schwarzschild-like with $\bar a=1$ and $b_{\rm crit}=3\sqrt{3}M_{ADM}$. Numerically, the photon sphere radius and the shadow radius both decrease as $\xi$ increases. The paper then translates solar-system, M87*, and Sgr A* observations into constraints on $\xi$, finding $0<\xi<4.7\times10^{-2}$ from the PPN deflection bound, $\xi\sim10^{-21}$ from photon ring size estimates, and upper bounds from shadow radii, and it argues the combined effects of $Q$ and $\xi$ produce a more pronounced strong lensing effect than the classical Reissner–Nordström solution.

Load-bearing premise

The strong-deflection results in Section IV rest on replacing the full metric by $A(r)=1-2M_{ADM}/r$, dropping the $Q^3/(9\xi^2 r^4)$ term before computing the photon sphere; if that term is not negligible at the photon sphere, the Schwarzschild-like strong deflection angle and the photon-ring constraints derived from it are unsupported.

Editorial extensions

If this is right

  • A measurement of the weak deflection angle at precision level can directly constrain $\xi$, with the solar-system PPN bound giving $0<\xi<4.7\times10^{-2}$.
  • EHT photon ring size constraints translate into $\xi$ values near $10^{-21}$ for both M87* and Sgr A*, so a sharper ring measurement pins down the coupling.
  • Shadow radius observations place upper bounds ($\xi\lesssim3.78$ for Sgr A* and $\xi\lesssim4.29$ for M87*), meaning the NED correction cannot be arbitrarily large in these astrophysical black holes.
  • Future ngEHT and space-VLBI observations at few-microarcsecond resolution could test the predicted $\xi$-dependent shadow shrinkage and distinguish this NED model from Einstein–Maxwell theory.

Reading between the lines

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

  • The strong-deflection result is exactly Schwarzschild-like: in the approximate metric the only NED trace is the mass shift $M_{ADM}$, so a measurement that separates $M_{ADM}$ from the Einstein-frame mass $M$ is required to claim detection of the nonlinearity.
  • The large gap between the photon-ring bounds ($\xi\sim10^{-21}$) and the shadow-radius bounds ($\xi\lesssim4$) suggests the two observables constrain different physical combinations; reconciling them would require a computation of the photon ring from the full metric rather than the truncated one.
  • A testable extension is to keep the $Q^3/(9\xi^2 r^4)$ term in the strong-deflection analysis; if that term shifts the photon sphere away from $3M_{ADM}$, the strong deflection coefficients will acquire explicit $\xi$ and $Q$ dependence, restoring a genuine NED signature in the strong-field regime.
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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

5 major / 4 minor

Summary. The manuscript analyzes gravitational lensing and black-hole shadows for a magnetically charged black hole in nonlinear electrodynamics, using the exact metric introduced by Mazharimousavi [105]. It derives a weak-deflection angle via the Gauss-Bonnet theorem, a strong-deflection angle via the Tsukamoto formalism, numerical photon-sphere and shadow radii, and compares the results with solar-system PPN data and EHT observations of M87* and Sgr A*. The central advertised result is that the combined effects of the magnetic charge Q and the NED parameter xi enhance the strong deflection angle relative to the Reissner-Nordstrom solution.

Significance. If the calculations were correct, the paper would provide closed-form lensing formulas and EHT-based constraints for a specific NED black-hole model. The paper has some strengths: it starts from an exact metric, applies the Gauss-Bonnet method in a standard manner, and presents numerical shadow plots. However, the main strong-deflection result is exactly the Schwarzschild formula with a redefined mass, the weak-deflection expression is divergent in the physically relevant xi-to-0 limit, and the reported observational constraints are internally inconsistent. The paper therefore does not establish its claimed NED signatures, and the central comparison with Reissner-Nordstrom is not supported by the derivation.

major comments (5)
  1. [IV, Eq. (35)] The strong-deflection computation begins by dropping the Q^3/(9 xi^2 r^4) term from Eq. (17) and setting A(r) = 1 - 2 M_ADM / r. As a result, Eq. (36) gives r_ph = 3 M_ADM, Eq. (37) gives a_bar = 1, and Eq. (39) is exactly the Schwarzschild strong-deflection formula with mass M_ADM; no dependence on Q or xi survives except through the mass redefinition. The abstract's claim of enhanced bending relative to Reissner-Nordstrom, and the similar statement in the Conclusion, are therefore not consequences of the paper's own derivation. The text itself acknowledges that the strong lensing is 'analogous to the Schwarzschild black hole', in tension with the abstract. The discarded term is not always negligible at the photon sphere: for Q = 0.5 M and xi = 0.05, the term Q^3/(9 xi^2 r^4) at r ~ 3 M_ADM is roughly 10% of the leading 2 M_ADM / r term, so the strong-deflection coefficients of the full metric remain untested.
  2. [III, Eq. (26)] The weak-deflection angle contains the term -5 pi Q^3 / (48 b^4 xi^2), which diverges as xi tends to zero. Since the metric function Eq. (11) reduces to the Reissner-Nordstrom metric in the limit xi -> 0, a physically meaningful deflection angle should tend to the RN value in that limit. The presence of this divergent term means the expansion in Eq. (26) is not valid for small xi, and it cannot be used to derive a PPN constraint on xi.
  3. [III, Eq. (28)] Equating Eq. (26) to Eq. (27) cannot yield a unique bound 0 < xi < 4.7 x 10^-2 unless the values of the impact parameter b, the mass M, and the charge Q are specified. Equation (27) is a fixed numerical value for solar-system light deflection, while Eq. (26) depends nontrivially on b, Q, and xi; the manuscript does not state which b and Q are used. Moreover, in the Q -> 0 limit Eq. (26) reduces to the GR deflection 4M/b and provides no constraint on xi, so the quoted bound is not a robust result of the calculation.
  4. [IV, photon-ring constraints] The reported ranges 3.8 x 10^-21 <= xi <= 4.8 x 10^-21 (M87*) and 1.2 x 10^-20 <= xi <= 5.7 x 10^-21 (Sgr A*) are internally inconsistent, since the lower bound for Sgr A* exceeds the upper bound. Furthermore, these constraints are derived from Eq. (39), which is the Schwarzschild strong-deflection formula; through M_ADM it depends on xi only via a mass redefinition, so it cannot provide a meaningful direct constraint on the NED parameter xi as advertised.
  5. [V and Conclusion] The paper gives contradictory statements about the effect of xi on the shadow size. Section V says 'as xi increases, the shadow size increases, especially at larger values of xi', while the Conclusion says 'increasing the coupling parameter xi or the charge Q leads to a reduction in the radius of the photon sphere, thereby producing a smaller shadow.' These two statements cannot both describe the same model, and the contradiction affects the main observational interpretation of the results.
minor comments (4)
  1. [Throughout] The manuscript uses both NED and NLED for nonlinear electrodynamics; please standardize the terminology.
  2. [Introduction] The name Soldner is misspelled as 'Solder' in the historical discussion of light deflection.
  3. [References] Reference [22] appears to be about an extension of the Higgs sector and seems unrelated to gravitational lensing; please check the citation.
  4. [II, Eq. (18)] The logarithm in Eq. (18) contains a dimensionful argument, 2 xi / sqrt(2 Q); the units or implicit scale of xi should be stated explicitly, since this affects the interpretation of all numerical bounds on xi.

Circularity Check

1 steps flagged · score 6.0 of 10

Strong-deflection section sets A(r) to Schwarzschild, so Eq. (39) is the known Schwarzschild formula; the advertised NED/RN enhancement is an input-output identity, not a derivation.

  1. renaming known result [Section IV, after Eq. (17), Eq. (35) through Eq. (39)]
    "Additionally, the contribution of the third term in Eq. (17) is negligible because Q ≪ M, and its impact is proportional to the fourth power of the radial position. However, the effect of the charge on the strong deflection angle (SDA) can still be analyzed by considering only the first two terms of the metric function A(r). Now, we have an approximate expression of the metric function given as, A(r) = 1 − 2MADM/r, B(r) = 1/A(r), where it resembles a Schwarzschild metric. ..."

    This step replaces the asymptotic NED metric of Eq. (17) by the exact Schwarzschild metric with a redefined mass M_ADM. The photon sphere then solves r_ph = 3M_ADM (Eq. 36), the strong-deflection coefficients take their Schwarzschild values (a = 1, b from Bozza's result, Eqs. 37-38), and Eq. (39) is exactly Bozza's Schwarzschild strong-deflection formula. No dependence on Q or ξ survives except inside M_ADM, i.e., by mass redefinition. The abstract's conclusion that Q and ξ enhance the strong deflection angle relative to RN is therefore not derived from the NED metric; the Schwarzschild result is inserted as the input and then reported as the NED strong-deflection prediction. The claimed enhancement over RN is a re-labeling, not a computed effect.

full rationale

The weak-deflection calculation (Section III) and the shadow analysis (Section V) use the full/expanded metric and are compared with independent PPN and EHT data, so those branches are not circular; citations to Tsukamoto, Bozza, Gibbons-Werner, and EHT papers provide external, non-self-referential machinery. The circularity is confined to the strong-deflection section: after quoting the asymptotic metric with the Q^3/(9ξ^2 r^4) term, the authors explicitly drop that term and set A(r)=1−2M_ADM/r, which is Schwarzschild by construction. Every subsequent strong-deflection quantity—photon sphere, a, b, and Eq. (39)—is the standard Schwarzschild result, so the paper's advertised 'more pronounced lensing than RN' claim is not an output of the derivation but the input metric renamed. This is a partial circularity (one of the paper's main claims reduces by construction), while other independent results keep the overall score below 8.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central results are carried by the NED parameter xi and magnetic charge Q inherited from the Mazharimousavi solution, plus standard lensing formulas. The paper introduces no new particles or forces. The xi parameter is constrained, not fixed a priori, and the strong-deflection result additionally assumes away the only term that would distinguish NED from Schwarzschild.

free parameters (2)
  • NED coupling parameter xi = constrained to ranges 0 < xi < 4.7e-2, around 1e-21, and around 3.78 or 4.29 in different sections, with no stated units
    Appears in the metric, deflection angle, photon sphere, shadow, and all constraints; no independent value is fixed before comparison.
  • magnetic charge Q = Q/M = 0.50, 0.30, 0.40, and 0.10 used in figures and EHT constraints
    Chosen by hand; no physical estimate for a particular black hole is given, and the EHT constraint section fixes Q = 0.10 M without motivation.
assumptions (5)
  • domain assumption The metric function f(r) in Eq. (11), taken from Ref. [105], is a valid static spherically symmetric solution of the Einstein-NED equations and is asymptotically flat.
    Used as the starting point of all calculations; the paper does not derive the solution from the Lagrangian.
  • domain assumption The Gauss-Bonnet deflection formula alpha = integral of K sqrt(det g_opt) dr dphi in Eq. (25) applies to this optical metric without additional boundary terms or finite-distance corrections.
    The paper cites the Gibbons-Werner method but does not justify the domain choice or the neglect of boundary contributions for this metric.
  • ad hoc to paper For strong deflection, A(r) = 1 - 2 M_ADM / r with Q much less than M, giving r_ph = 3 M_ADM.
    Eq. (35)-(36) in Section IV; this discards the Q^3 / (9 xi^2 r^4) term and makes the strong deflection result identical to Schwarzschild.
  • ad hoc to paper The EHT Schwarzschild shadow radius bounds for Sgr A* and M87* can be applied directly to this NED model with Q = 0.10 M and robs at infinity.
    Section V uses 4.209M <= R <= 5.560M for Sgr A* and 4.313M <= R <= 6.079M for M87* without discussing whether the NED model's shadow can be compared one-to-one with a Schwarzschild template.
  • ad hoc to paper Equating Eq. (26) to Eq. (27) yields a meaningful PPN constraint on xi.
    Eq. (28) states 0 < xi < 4.7e-2 without specifying the mass, charge, impact parameter, or arcsecond-to-radian conversion used.

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Pith. "Pith review of Analyzing Deflection Angles and Photon Sphere Dynamics of Magnetically Charged Black Holes in Nonlinear Electrodynamic." pith.science (2026). https://pith.science/paper/662FSCSS

@misc{pith2026250204044,
  author       = {Pith},
  title        = {Pith review of: Analyzing Deflection Angles and Photon Sphere Dynamics of Magnetically Charged Black Holes in Nonlinear Electrodynamic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/662FSCSS}},
  note         = {Machine review of arXiv:2502.04044}
}
abstract

In this paper, we investigate the gravitational lensing properties of magnetically charged black holes within the framework of nonlinear electrodynamics. We derive the deflection angle and examine the influence of the nonlinear electrodynamics parameter $\xi$ on light bending. We initially employ a geometric approach based on the Gauss-Bonnet theorem to analyze the gravitational deflection of null and timelike particles. This method encapsulates the global characteristics of the lensing effect in an elegant manner. In the subsequent part of the work, we explore the impact of nonlinear electromagnetic corrections on the black hole shadow. Using numerical techniques, we study the behavior of the photon sphere and demonstrate that a reduction in the photon sphere radius leads to a correspondingly smaller shadow. We compare these results with those for the Schwarzschild and Reissner-Nordstr\"om black holes, highlighting the distinctive features introduced by nonlinear electrodynamics. Furthermore, we examine the strong deflection limit for light trajectories near these black holes, focusing on the roles of both the magnetic charge $Q$ and the nonlinear parameter $\xi$. Our analysis reveals that the combined effects of $Q$ and $\xi$ enhance the strong deflection angle, resulting in a more pronounced lensing effect than that predicted by the classical Reissner-Nordstr\"om solution. These findings suggest that the nonlinear interactions may provide a potential observational signature for identifying NED black holes.

Figures

Figures reproduced from arXiv: 2502.04044 by the authors.

Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Behavior of the weak deflection angle [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Strong deflection angle with varying parameters [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Behavior of the photon sphere. Here, we have chosen an [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Left: Behavior of the shadow cast. Here, we have chosen [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Reviewed August 8, 2026 · model on record in the stance chip above.