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Shadow and geometric scattering analysis of a new black hole with magnetic charge

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper constructs a new magnetically charged black hole from an $F^3$ quasi-topological term and uses its multi-branch shadow radii to constrain the magnetic charge against Sgr A* EHT data.

desk verdict A new magnetic-charge black hole solution with a clean derivation, but the headline naked-singularity branches are ruled out by a one-line continuity argument on the horizon polynomial. read the letter →

arxiv 2505.00280 v1 pith:4S4KFNR5 submitted 2025-05-01 gr-qc hep-th

Yun Soo Myung This is my paper · ORCID
classification gr-qchep-th
keywords blackholeshadowmagneticchargenonlinearelectrodynamicsquasi-topologicaltermnakedsingularityphotonspheregeometricscatteringSgrA*EHTconstraints
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 constructs a static black hole with magnetic charge $q$ by adding a quasi-topological $F^3$ term to Einstein-nonlinear electrodynamics, giving the metric function $f(r)=1-\frac{2m}{r}+\frac{q^2}{r^2}-\frac{4\mu q^6}{9r^{10}}$. For a small coupling $\mu=0.01$, the horizon equation has four positive branches, whereas $\mu=0.1$ gives one branch. The author argues that two of those branches, labelled 3 and 4, extend into naked-singularity states for $q\in[1.00245,1.06]$, and he uses the photon-sphere impact parameter as the shadow radius to confront the Sgr A* EHT observations. The outcome is a set of allowed magnetic-charge windows, most notably $1.022\le q\le 1.03$ at $1\sigma$ and $1.02\le q\le 1.038$ at $2\sigma$ for the 3-NS branch, plus a geometric scattering analysis showing two critical impact parameters diverging at $q=0.989$ and $q=1.00245$. A reader should care because shadow measurements are one of the few direct observational handles on charge and on alternative electrodynamic corrections to the standard Reissner-Nordstr\oding picture.

What carries the argument

The load-bearing object is the metric function $f(r)$ from Eq. (10), obtained by integrating $M'(r)=\frac{q^2}{2r^2}-\frac{2\mu q^6}{r^{10}}$. Its zeros define the horizon branches, and the combination $V(r)=f(r)/r^2$ acts as the effective potential for null geodesics. Photon spheres are fixed by the conditions $V=1/(2b^2)$ and $V'=0$, which determine the critical impact parameters $b_i(m,q,\mu)$ interpreted as shadow radii. The peculiar blow-up structure of $b_3$ and $b_{2+}$ follows from the photon-sphere equations losing real roots at $q=0.989$ and $q=1.00245$.

What would settle it

Evaluate $f(r)$ from Eq. (10) at fixed $m=1$, $\mu=0.01$, $q=1.03$: for $r\to0$ the term $-\frac{4\mu q^6}{9r^{10}}$ forces $f\to-\infty$, while $f\to1$ as $r\to\infty$; the intermediate value theorem then guarantees a positive root of $f(r)=0$. Locating that root, which should lie at small $r$, would settle whether the 3-NS state is a naked singularity or a black hole with an inner horizon.

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Extended reading notes

Core claim

Within Einstein-nonlinear electrodynamics inspired by quasi-topological terms, the paper obtains a new magnetically charged black hole whose metric function is $f(r)=1-\frac{2m}{r}+\frac{q^2}{r^2}-\frac{4\mu q^6}{9r^{10}}$. For $\mu=0.01$, the horizon equation $r^{10}f(r)=0$ yields four positive branches $r_{2-}$, $r_3$, $r_4$, and $r_{2+}$; for $\mu=0.1$ only a single branch $r_2$ remains. The paper identifies the 3- and 4-branches as entering naked-singularity regimes for $q\in[1.00245,1.06]$, where the photon-sphere radii $L_3$ and $L_4$ continue to exist but the horizon branches do not. Using the critical impact parameter $b_i$ as the shadow radius and comparing with the Sgr A* EHT bounds, it finds that the $2-$ branch reproduces the Reissner-Nordstr\om charge limits for $q<1$, while the 3-NS branch passes only in a narrow window around $q\approx 1.02-1.04$ and the 4-NS branch is excluded at $2\sigma$. The geometric scattering cross sections $\sigma_{ci}=\pi b_i^2$ are computed, and $\sigma_{c3}$ and $\sigma_{c2+}$ blow up at $q=0.989$ and $q=1.00245$, indicating total capture of particles at those parameter values.

Load-bearing premise

The paper assumes that when the 3 and 4 horizon branches stop having positive roots at $q\approx 1.00245$, no other horizon remains, so the spacetime is a naked singularity rather than a black hole with an extra inner horizon.

Editorial extensions

If this is right

  • For $\mu=0.01$, the $2-$ branch has the same shadow radius as the Reissner-Nordstr\om black hole for $q<1$, so the Sgr A* bounds translate to the same charge limits, $q\lesssim 0.798$ at $1\sigma$ and $q\lesssim 0.939$ at $2\sigma$.
  • The 3-NS branch constrains the magnetic charge to the narrow ranges $1.022\lesssim q\lesssim 1.03$ at $1\sigma$ and $1.02\lesssim q\lesssim 1.038$ at $2\sigma$, while the 4-NS branch is ruled out.
  • For $\mu=0.1$, there is a single horizon branch with no naked-singularity extension, and its shadow radius decreases monotonically with $q$, so no $q>1$ EHT constraint applies.
  • The geometric cross sections $\sigma_{c2-}$ and $\sigma_{c2}$ decrease with $q$ and increase with mass $m$, whereas $\sigma_{c3}$ and $\sigma_{c2+}$ diverge at specific charges and masses, implying total gravitational capture at those points.
  • In the limits $q\to\infty$ and $q\to0$, the cross sections approach finite values, namely $0.54$, $1.72$, and the Schwarzschild value $84.823$, indicating that most particles scatter off a point-like center rather than being captured.

Reading between the lines

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

  • A direct continuity check indicates that $f(r)=0$ should have a positive root for every $q>0$ and $\mu>0$, because $f\to-\infty$ as $r\to0$ and $f\to1$ as $r\to\infty$; if so, the 3-NS and 4-NS states would be covered by an inner horizon rather than being naked singularities, though the photon-sphere and shadow computations for the outer region may still be valid.
  • The same construction with the $F^2$ term gives a different metric, Eq. (11), so comparing the two theories would test whether the four-branch horizon structure is generic to nonlinear electrodynamics corrections or specific to the chosen quasi-topological $F^3$ term.
  • The diverging critical impact parameters suggest a photon sphere merging with the horizon; wave-based scattering calculations, not just geodesic ones, would be needed to decide whether the blow-up has observable absorption consequences.
  • The method could be applied to electrically charged or dyonic solutions of the same theory, where the quasi-topological term acts differently, to see whether similar multi-branch shadow patterns arise.
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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 / 5 minor

Summary. The manuscript constructs a static, spherically symmetric magnetically charged black hole from the Einstein-nonlinear electrodynamics Lagrangian (5), yielding the metric function (10). It then studies the roots of f(r)=0, identifies four horizon branches for μ=0.01 and one for μ=0.1, computes photon-sphere radii and critical impact parameters, compares shadow radii with EHT constraints, and analyzes geometric scattering cross sections. The headline result is that the 3 and 4 branches become naked singularities for q∈[1.00245,1.06], leading to EHT bounds 1.022≤q≤1.03 (1σ) and 1.02≤q≤1.038 (2σ). The derivation from the action to the metric and the photon-sphere equations are standard and self-contained.

Significance. The paper is self-contained in its derivation: the metric follows directly from the action, the shadow radii follow from the null-geodesic equations, and the EHT comparison is an external benchmark rather than a fitted output. If the naked-singularity classification were correct, the q>1 EHT windows would be a new constraint on a magnetically charged black hole. However, that classification is internally inconsistent with the paper's own horizon equation. The residual content is a routine shadow and scattering computation for an exact nonlinear-electrodynamics black hole, and the advertised new phenomenon (naked-singularity branches constrained by EHT) is not supported. The central claim therefore fails, despite the correctness of the basic metric integration and photon-sphere conditions.

major comments (2)
  1. [Sec. 2, Eq. (10); Sec. 3, Figs. 2-3] The central claim that the 3- and 4-branches become naked singularities for q∈[1.00245,1.06] is contradicted by the paper's own metric. The horizon condition f(r)=0 is equivalent to P(r)=r^10-2mr^9+q^2r^8-(4μ/9)q^6=0. For m,q,μ>0, P(0)=-(4μ/9)q^6<0 and P(r)→+∞ as r→∞, so P has a positive root for every parameter choice. Taking the largest such root r_h gives f(r)>0 for all r>r_h, so an event horizon exists for all q, including the shaded region labelled NS. The spacetime is therefore not naked; the r3/r4 roots becoming complex merely reduces the number of positive horizon roots without eliminating the horizon itself. Consequently the abstract's NS claim, the interpretation of the EHT windows as constraints on naked singularities, and the labels '3-NS' and '4-NS' in Figs. 2-3 are unsupported.
  2. [Sec. 5, Fig. 4] The scattering blow-ups are also misattributed. The divergence of b3(1,q,0.01) at q=1.00245 and of b2+(1,q+,0.01) at q=0.989 is the standard critical-impact-parameter divergence that occurs when the photon-sphere radius approaches a horizon (f(L)→0 makes b^2=L^2/f(L) diverge), not capture by a naked singularity. Since a horizon exists at those parameters, the statements in Sec. 5 that all particles 'pass into the 3-NS and 2+' and are 'captured by 3-NS and 2+' do not follow from the geodesic equations. The geometric scattering analysis only reflects the limit where a photon sphere merges with a horizon; the NS interpretation should be removed.
minor comments (5)
  1. [Sec. 2, Eq. (22)] The displayed Reissner-Nordström critical impact parameter contains an apparently garbled factor 'r' and should be checked; as printed it is not dimensionally consistent.
  2. [Sec. 2, Table 1] The text states that the q upper limit for the existence of r3 and r4 is 1.0062, while the table entry for q=1.0063 is 'N.A.'; these should be aligned.
  3. [Sec. 3] The phrase 'equational plane' should read 'equatorial plane'.
  4. [Sec. 4] The statement that the 2-branch shadow radii are 'the same' as the RNBH for q<1 is an approximation; the metric (10) differs from the RN metric by the μ term, so the equality is only numerical to leading order.
  5. [Sec. 5] The sentence 'all particles pass into the 3-NS and 2+' is vague; it should specify the orbital parameters and the sense in which the particles are captured.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new metric is integrated from the field equations and all shadow/scattering outputs are derived from null geodesics, with the EHT comparison serving as an external benchmark.

full rationale

The paper's derivation chain is self-contained. The metric function f(r)=1-2m/r+q^2/r^2-(4mu/9)q^6/r^10 is obtained directly by integrating Eq. (9), M'(r)=q^2/(2r^2)-2mu q^6/r^10, which follows from the Einstein equation for the action (5). No fitted parameter is renamed as a prediction: the coupling mu is chosen as 0.01 and 0.1 to exhibit different horizon branch structures, but it is not tuned to reproduce the EHT shadow data. The shadow radii and critical impact parameters are computed from the standard null-geodesic effective potential, Eqs. (16)-(19), and the EHT intervals in Eqs. (23)-(24) are external observational constraints. The self-citations to Refs. [20] and [33] appear only as contextual references to earlier thermodynamic/shadow and geometric-scattering studies; they are not load-bearing for the derivation of Eq. (10), the branch structure, or the shadow comparison. Even if the paper's naked-singularity classification were questionable on physical grounds, that would be a correctness concern, not a circularity concern: the classification is not assumed as an input but inferred from the roots of the horizon polynomial. Overall, no step reduces to its own input by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

Central derivation is a direct integration of M'(r) from Eq (9) with one new coupling μ; no invented fields. However, the naked-singularity branch classification relies on an assumption, contradicted by the horizon polynomial, that no horizon remains after r3/r4 disappear.

free parameters (3)
  • m = 1
    Mass set to 1 in all numerical branch and shadow plots; all lengths scale with m.
  • q = scanned over [0,2]; constrained intervals q≈1.022-1.030 (1σ) and 1.02-1.038 (2σ)
    Magnetic charge parameter; the EHT comparison selects intervals rather than predicting the charge.
  • μ = 0.01 and 0.1
    Coupling constant of the F^3 term; handpicked to realize four horizon branches versus a single branch, not fitted to data.
assumptions (4)
  • domain assumption The nonlinear Maxwell equation (7) is satisfied by the magnetic ansatz A_phi = -q cosθ with F = 2q^2/r^4.
    The metric (8)-(10) is only a solution if the F^3-modified field equation holds; the paper does not verify this explicitly, though for a purely magnetic ansatz it can be checked by the divergence identity.
  • standard math The photon sphere conditions V(r=L)=1/(2b^2) and V'(r=L)=0 give the critical impact parameter equal to the shadow radius.
    Standard null-geodesic result for static spherically symmetric spacetimes, cited via [24,25].
  • ad hoc to paper When the r3 and r4 horizon roots cease to exist, the spacetime becomes a naked singularity.
    This is the premise for the 3-NS and 4-NS labels in Fig 2 and Fig 3. It is not demonstrated and is contradicted by Eq (10): the horizon polynomial is negative at r=0 and positive at large r, so a positive root exists for all q.
  • domain assumption The EHT Keck/VLTI constraint 4.55 ≤ r_sh ≤ 5.22 (1σ) and 4.21 ≤ r_sh ≤ 5.56 (2σ) applies to the shadow radius b_i of this metric.
    The comparison follows the summary in [10]; the mass normalization m=1 maps the EHT bounds onto the dimensionless b_i.

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Pith. "Pith review of Shadow and geometric scattering analysis of a new black hole with magnetic charge." pith.science (2026). https://pith.science/paper/4S4KFNR5

@misc{pith2026250500280,
  author       = {Pith},
  title        = {Pith review of: Shadow and geometric scattering analysis of a new black hole with magnetic charge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4S4KFNR5}},
  note         = {Machine review of arXiv:2505.00280}
}
abstract

We obtain a newly charged black hole with magnetic charge $q$ and coupling constant $\mu$ from the Einstein-nonlinear electrodynamics theory inspired by quasi-topological terms. We perform the shadow and geometric scattering analysis of this black hole. For $\mu=0.01$, we have four solution branches labelled by $2\pm$, 3, and 4 of the horizon, while there exists a single branch labelled by 2 for $\mu=0.1$. There is the naked singularity (NS) arisen from the magnetic charge extension of the photon spheres for the 3 and 4-branches. In case of $q<1$, shadow radii of the $2$-branch with $\mu$=0.01 and 0.1 are the same as that of the Reissner-Nordstr\"om black hole, while the 3-NS branch of $q>1$ is constrained by the EHT observation. The geometric scattering analysis is performed to understand the peculiar forms of the critical impact factors.

Figures

Figures reproduced from arXiv: 2505.00280 by the authors.

Figure 1
Figure 1. Four branches of the horizon r2−(1, q−, µ = 0.01) ≥ r3(1, q, 0.01) ≥ r4(1, q, 0.01) ≥ r2+(1, q+, 0.01) as functions of q with r6/9(1, q, 0) representing outer/inner horizons for the RNBH. We note that r2(1, q, 0.1) is defined as the single horizon without limitation of q. noting that µ does not exist beyond q = 1.0062. For q ∈ [0.989, 1.00245], µ = 0.01 is the upper and lower bounds, guaranteeing the existence of r3… view at source ↗
Figure 2
Figure 2. (Left) Four photon sphere radii Li(m = 1, q, µ = 0.01) for i = 2−, 3, 4, 2+ and L2(1, q, 0.1) are as functions of q. The 3 and 4-branches are extended to include their NS versions defined in the shaded column (q ∈ [1.00245, 1.06]). (Right) Four critical impact parameters bi(m = 1, q, µ = 0.01) for i = 2,3, 4, 2+ and b2(1, q, 0.1) is as functions of q. A shaded column (q ∈ [1.00245, 1.06]) includes their 3, 4-NS bran… view at source ↗
Figure 3
Figure 3. (Left) Five critical impact parameters bi(m = 1, q, µ = 0.01) for i = 2−, 3, 4, 2+ and b2(1, q, 0.1) are as functions of q ∈ [0, 2]. There are two blow-up points (dotted lines) at q = 0.989, 1.00245. Here, we introduce 1σ and 2σ ranges. (Right) Enlarged critical impact parameters bi(1, q, 0.01) for i = 2−, 3, 4, 2+ and b2(1, q, 0.1) are functions of q ∈ [0.9, 1.10]. The shaded column of q ∈ [1.00245, 1.06] includes … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: (Left). Five geometric cross sections σci(m = 1, q, 0.01) for i = 2−, 3, 4, 2+ and σc2(1, q, 0.1) as functions of charge q ∈ [0, 2]. Two dotted lines are blow-up points at q = 0.989, 1.00245. (Right) Five geometric cross sections σc2−(m, 0.5, 0.01), σc2(m, 0.5, 0.1), σ…

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