Ay\'on--Beato--Garc\'ia black hole coupled with a cloud of strings: thermodynamics, shadows and quasinormal modes

Amit Kumar; Dharm Veer Singh; Sudhaker Upadhyay

REVIEW 3 major objections 4 minor 1 cited by

Ay\'on--Beato--Garc\'ia black hole coupled with a cloud of strings: thermodynamics, shadows and quasinormal modes

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims an exact black hole solution that couples a cloud of strings to Ayón–Beato–García nonlinear electrodynamics, and it works out the thermodynamics, shadow, and quasinormal modes of that solution.

desk verdict The metric (20) is the expected ABG-plus-string-cloud superposition, but the claimed stress tensor (12)-(13) fails a trace check against the metric, so the exact-solution derivation does not close as written. read the letter →

arxiv 2412.14230 v1 pith:YJV4462E submitted 2024-12-18 gr-qc hep-th

Amit Kumar , Dharm Veer Singh , Sudhaker Upadhyay This is my paper · ORCID

classification gr-qchep-th MSC 83C5783C15

keywords blackholesolutionAyón–Beato–Garcíanonlinearelectrodynamicscloudofstringsthermodynamicsmodifiedfirstlawshadowquasinormalmodes

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 an exact static, spherically symmetric black hole solution to Einstein gravity whose source is the Ayón–Beato–García (ABG) nonlinear electrodynamic field together with a cloud of strings. The metric function is $f(r)=1-\frac{2Mr^2}{(r^2+g^2)^{3/2}}+\frac{g^2r^2}{(r^2+g^2)^2}-a$, with $M$ the mass, $g$ the magnetic-monopole charge, and $a$ the string-cloud parameter. The paper shows that this single function interpolates between the ABG black hole (when $a=0$), the Letelier cloud-of-strings black hole (when $g=0$), and Schwarzschild (when both vanish), and that the combined solution is singular even though the ABG-only solution is regular. It further derives a modified first law of thermodynamics whose correction factor restores the area-law entropy $S=\pi r_+^2$, and it computes photon orbits, shadow radii, and scalar quasinormal modes. This gives a concrete arena in which a string-cloud environment and nonlinear electrodynamics both leave observable traces in black hole shadows and ringdown frequencies.

What carries the argument

The load-bearing object is a single metric function, $f(r)=1-\frac{2Mr^2}{(r^2+g^2)^{3/2}}+\frac{g^2r^2}{(r^2+g^2)^2}-a$, obtained from the mass function $m(r)=\frac{Mr^3}{(r^2+g^2)^{3/2}}-\frac{g^2r^3}{2(r^2+g^2)^2}+\frac{a}{2}r$ through $f(r)=1-2m(r)/r$. That function is the integrated output of the summed sources: the ABG electromagnetic stress tensor (12)–(13) plus the cloud-of-strings stress tensor $a/r^2$, so the solution's horizon structure, thermodynamics, photon sphere, shadow, and quasinormal modes are all read off from this one function. Its role in the argument is to carry every claimed limit: setting $a=0$ recovers the ABG black hole, $g=0$ recovers the Letelier black hole, and both zero recovers Schwarzschild, while the $a$-term is what reintroduces the curvature singularity at $r=0$.

What would settle it

Substitute the metric (20) into the $\theta\theta$ and $\phi\phi$ components of the Einstein equations (9) with the full stress tensor (12)–(13) and check for identity at all $r$; equivalently, compute $\nabla_\mu T^{\mu\nu}$ for the summed stress tensor in this background. If either check fails at any radius, the claimed exact solution is not actually a solution of the original action.

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Extended reading notes

Core claim

The central claim is that Eq. (20) of the paper, $f(r)=1-\frac{2Mr^2}{(r^2+g^2)^{3/2}}+\frac{g^2r^2}{(r^2+g^2)^2}-a$, is an exact solution of the Einstein equations (9) with the total stress-energy tensor (12)–(13), obtained by linearly adding the ABG nonlinear-electrodynamics stress tensor and the cloud-of-strings stress tensor $T^t_t=T^r_r=a/r^2$ and integrating $m'(r)$ in Eq. (16). The authors verify the interpolating limits, show the curvature invariants (55)–(57) diverge as $r\to 0$ so the string cloud spoils the regularity the ABG field alone would supply, and analyze horizons, showing critical values of $a$ and $g$ where the Cauchy and event horizons merge into an extremal black hole. On the thermodynamics side, the paper argues that because the mass parameter enters the stress tensor, the first law takes the modified form $C(M_+,g,r_+)\,dM_+=T_+\,dS_+$ with correction factor (32), and with this modification the entropy reduces to the area law $S_+=\pi r_+^2$ rather than the NLED-corrected expression (29). Finally, it computes the photon-sphere radius, the shadow radius $r_s=r/\sqrt{f(r)}$, and WKB quasinormal frequencies, finding opposite responses to the two parameters: the shadow and photon radius grow with $a$ and shrink with $g$.

Load-bearing premise

The load-bearing premise is that the cloud of strings and the ABG electromagnetic field do not interact, so their stress tensors can be added line by line and the static spherically symmetric ansatz stays valid; if energy flows between the two sectors or the summed stress tensor fails to conserve, the metric (20) is not an exact solution.

Editorial extensions

If this is right

When the string-cloud parameter $a$ vanishes, every formula in the paper (horizon radii, temperature, entropy, shadow, quasinormal frequencies) reduces to the corresponding ABG-black-hole quantities; when $g$ vanishes it reduces to the Letelier family, so the new solution contains both prior geometries as parameter slices.
Shadow radii computed from Eq. (49) grow with $a$ and shrink with $g$; if a future black-hole image resolves a shadow of the size predicted for some $(M,a,g)$, it can bound the string-cloud parameter once the mass is known from other observations.
The quasinormal-mode calculation gives negative imaginary parts over most of the parameter space, meaning the solution is dynamically stable against scalar perturbations; at large magnetic charge the imaginary part changes behaviour, a feature that could be looked for in ringdown signals.
The heat capacity changes sign at a critical horizon radius where the temperature is maximal and the Gibbs free energy minimal, indicating a second-order phase transition from small to large black holes, with the transition point controlled by both $a$ and $g$.

Reading between the lines

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

The same linear-superposition recipe could be applied to other nonlinear electrodynamics Lagrangians (Born–Infeld, Bardeen-like, etc.) together with a cloud of strings; nothing in the construction seems specific to ABG except the explicit integrals, so one would expect a family of interpolating singular black holes with similar shadow and quasinormal-mode phenomenology.
The shadow-radius degeneracy between $M$, $a$, and $g$ means a single shadow-size measurement cannot determine the string-cloud parameter; a joint fit to shadow size and quasinormal frequency, which respond differently to $a$ and $g$, would be the natural observational test.
The extremal configuration with degenerate horizons (Eqs. (39)–(41)) is a candidate black-hole remnant, but the paper only computes its radius and mass; whether it is thermodynamically stable under perturbations and whether it could serve as a dark-energy or information-loss remnant are questions the paper leaves open.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

One line: the metric (20) is the expected ABG-plus-string-cloud superposition, but the paper's own equations don't close. The claimed stress tensor fails a direct substitution test against the metric.

The interesting object is f(r) = 1 − 2Mr^2/(r^2+g^2)^{3/2} + g^2r^2/(r^2+g^2)^2 − a. The limits are right (ABG at a=0, Letelier at g=0, Schwarzschild at both), nothing is fit to data, and because the Einstein tensor of the f = 1 − 2m/r ansatz is linear in m(r), this superposition is the natural construction. If the paper only contained the metric, I would call it conditionally fine.

It doesn't work as written, though. The stress-test note lands. For this ansatz, R = 2m''/r + 4m'/r^2, and the trace identity R = −T has to hold. At M=g=1, r=2, a=0 the metric gives R ≈ 0.058. The EMT (12)-(13) gives T = 2T^t_t + 2T^θ_θ ≈ −0.45. That fails by a factor of almost eight, not by a convention. Pointwise, G^t_t = −2m'/r^2 ≈ −0.115, while (12) gives −0.099; the g^2-only term (and the a-term, when present) has the wrong sign. Eq. (16) has −3Mr^2g^2/(r^2+g^2)^{5/2}, but differentiating their own (19) gives +3Mg^2r^2/(r^2+g^2)^{5/2}. The appendix's Ricci scalar (55) at the same point gives 0.52, not 0.058. So the central derivation is internally contradictory; the fix is mechanical (reset the signs), and the metric will likely survive, but "exact solution" is not supported as submitted.

The reader's other flags are real too. The eikonal QNM formula has no business at l=1; the potential V0 is missing a factor f on the f'/r term; Table 2 gives δ = 0.590 for r+−r− = 1.993; Tables 5 and 6 have 0.488 and 0.153 typos; the appendix's last line contradicts the abstract's regularity claim. Novelty is under-supported without comparing to overlapping work by the same authors (ref [9] is the obvious one).

Credit: the circularity burden is genuinely zero, and the interpolation limits check out. The paper is a catalog entry in a busy subfield, not a framework.

Who it's for: readers collecting exact NLED+string-cloud models. It deserves referee time only if the referee is told to verify the field equations by substitution first. I'd send it to review with a major-revision expectation, not desk-reject it, since the defect is sign errors rather than a wrong idea.

Referee Report

3 major / 4 minor

Summary. The manuscript constructs a static, spherically symmetric black hole solution by minimally coupling Ayón--Beato--García (ABG) nonlinear electrodynamics with a cloud of strings, giving f(r)=1-2Mr^2/(r^2+g^2)^{3/2}+g^2r^2/(r^2+g^2)^2-a, and then studies the horizon structure, thermodynamics, local and global stability, photon sphere and shadow, and scalar quasinormal modes in the eikonal approximation. It claims the solution interpolates between the ABG, Letelier, and Schwarzschild black holes and that the thermodynamics follows a modified first law leading to an area-law entropy.

Significance. The proposed construction is a natural one and, if correct, would provide a simple exact solution combining two well-studied matter sources; the manuscript also has the merit of checking known limiting cases and of relating the entropy to the modified first law through the factor C in Eq. (32). No fitting or reverse-engineering of the output is apparent. However, the field equations are not satisfied as written, the quasinormal-mode computation is applied outside its regime of validity, and several tables contradict the surrounding text. These are load-bearing problems, so the quantitative conclusions are not currently reliable.

major comments (3)

[§2, Eqs. (12)–(20)] The claimed exact solution is not a solution of the field equations as printed. With the ansatz (14) and f=1-2m/r, the Einstein tensor component is G^t_t=-2m'/r^2 in the paper's convention, so Eq. (19) gives G^t_t=g^2(r^2-3g^2)/(r^2+g^2)^3-6Mg^2/(r^2+g^2)^{5/2}-a/r^2. This matches the ABG part of Eq. (12) but not the cloud term, which is written as +a/r^2. In addition, Eq. (16) contains a minus sign on the 3Mr^2g^2 term, whereas differentiating Eq. (19) gives a plus sign for that term; Eq. (17) also has a plus sign, so the displayed derivation is internally inconsistent. A direct trace check confirms the problem: for M=1, g=1, r=2 and a=0, the metric (20) gives R≈0.058, while the trace of the EMT in Eqs. (12)–(13) is of order -1 in the same units, so R+T does not vanish. The appendix Ricci scalar (55) contains +2a/r^2, which corresponds to a cloud contribution T^t_t=-a/r^2 rather than the printed +a/r^2. The derivation and the sign conventions must be corrected and rechecked before the central exact-solution claim can be accepted.
[§5.2, Eq. (53) and Tables 5–6] The eikonal WKB formula (53) is a large-l result, but the tables are explicitly for l=1. At l=1 the values of ω_R in Table 5 are simply Ω=1/r_s, e.g. 0.164 for a=0.1, g=0.1 with shadow radius 6.068, rather than the l=1 scalar quasinormal frequency obtained from the full l(l+1) potential; for Schwarzschild, for instance, the fundamental l=1 scalar frequency is about 0.293, roughly 50% above Ω=1/(3√3M)=0.192. The imaginary parts are likewise the eikonal damping rates, not the l=1 damping rates. The quasinormal-mode section should be redone with a proper fixed-l WKB or continued-fraction method, or the claims should be explicitly restricted to the eikonal regime l≫1.
[Tables 1–6] The numerical tables are not self-consistent and cannot be trusted as printed. Table 3 shows the photon radius increasing with g for fixed a (e.g. the a=0.1 row reads 3.319, 3.736, ..., 29.992), while §5 states that the photon radius decreases with g. Table 4 lists '...' for some combinations for which Table 3 gives a finite photon radius (e.g. a=0.1, g=0.8), and it gives a shadow radius for other combinations for which Table 3 has no photon radius (e.g. a=0.8, g=0.1). Table 1 repeats a=0.50 in the g=0.95 block, and in Table 2 the δ column does not equal r_+-r_- (a=0.20, g=0.60: δ=0.590 but r_+-r_-=1.993). Tables 5 and 6 contain decimal-point typos (0.488 and 0.153 in the a=0.6, g=0.2 entries). These issues affect the quantitative claims of the paper and require a full audit of the numerical results.

minor comments (4)

[§5.2] The stability criterion stated just after Eq. (53) refers to 'ω>0' versus 'ω<0'; it should refer to the sign of the imaginary part of the quasinormal frequency, since the real part is positive for stable modes.
[Appendix A] The last sentence of Appendix A, 'In the absence of a CS parameter, these invariants become singular,' contradicts the abstract and the earlier discussion, where the a=0, g≠0 ABG solution is regular; the intended condition is presumably g=0.
[§2] The sentence 'The BH horizon decreases with the growing CS parameter, a, and increases with the growing MM charge, g' is opposite to the values in Tables 1 and 2 and to the later conclusions; the text and tables should be reconciled.
[Tables 3–4] The entries marked '...' in the photon-radius and shadow-radius tables should be explicitly explained; if they denote the absence of a photon sphere, the corresponding entries in the companion tables should be removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained from the stated field equations.

full rationale

The paper's central derivation is self-contained. Starting from the stated action (1), the ABG Lagrangian (2), and the cloud-of-strings energy-momentum tensor (8), the authors derive the total EMT components (12)-(13), impose the static spherically symmetric ansatz (14)-(15), and integrate the field equation m'(r) = ... (16) to obtain m(r) (19) and the metric function f(r) (20). The shadow radius and quasinormal-mode frequencies are then computed from this f(r) through the standard formulas (49) and (53)-(54); no parameter is fitted to any target output, and no predicted quantity is used as an input to determine the metric. The thermodynamic entropy result (34) follows from the cited Ma-Zhao modified first-law correction (32), which is an externally cited general result, and it is not used to construct the metric or the shadow/QNM predictions. The self-citations present in the paper, e.g., refs. [24], [36], and [70], serve as context or limit-check references and are not load-bearing for the derivation. No reduction of a claimed prediction to a fitted input, to a definitional identity, or to a self-citation chain was found. Any inconsistency between the Ricci invariant (55) and the claimed EMT is a correctness issue, not a circularity issue.

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

The paper's central result rests on standard inputs from prior literature (ABG action, Letelier cloud of strings) combined by a linear superposition of stress tensors. No new degrees of freedom are introduced. The only ad hoc assumption is applying a large-l WKB formula at l=1. M, a, and g are model parameters, not fitted constants.

free parameters (3)

M (mass parameter) = Set to 1 in all numerical tables
ADM mass integration constant; normalized to unity in numerical work, no fit to data.
a (cloud of strings parameter) = Varied from 0.1 to 0.9
Cloud-of-strings parameter from Eq. (8); treated as a free model parameter in numerical exploration.
g (magnetic monopole charge) = Varied from 0.1 to 0.9
Magnetic charge parameter from ABG NLED; treated as a free model parameter, with existence bounds from horizon analysis.

assumptions (4)

domain assumption ABG NLED Lagrangian L_ABG(F) (Eq. 2) correctly describes the electromagnetic source
Taken from Ayon-Beato-Garcia refs [14-17]; not derived in this paper.
domain assumption Cloud of strings contributes T^t_t = T^r_r = a/r^2 (Eq. 8) and superposes linearly with the NLED stress tensor (Eq. 11)
This is the Letelier model [25,27]; the minimal-coupling superposition is assumed without interaction terms.
domain assumption Static spherically symmetric metric ansatz (14) is sufficient
The solution is sought within this symmetry class; no uniqueness theorem is invoked.
ad hoc to paper Eikonal WKB formula (53) gives the l=1 quasinormal frequencies
Eq. (53) is a large-l eikonal expression; the paper applies it to l=1 without verifying its accuracy.

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Cite this review

Pith. "Pith review of Ay\'on--Beato--Garc\'ia black hole coupled with a cloud of strings: thermodynamics, shadows and quasinormal modes." pith.science (2026). https://pith.science/paper/YJV4462E

@misc{pith2026241214230,
  author       = {Pith},
  title        = {Pith review of: Ay\'on--Beato--Garc\'ia black hole coupled with a cloud of strings: thermodynamics, shadows and quasinormal modes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJV4462E}},
  note         = {Machine review of arXiv:2412.14230}
}

read the original abstract

We find an exact black hole solution for the Einstein gravity in the presence of Ay\'on--Beato--Garc\'ia non-linear electrodynamics and a cloud of strings. The resulting black hole solution is singular, and the solution becomes non-singular when gravity is coupled with Ay\'on--Beato--Garc\'ia non-linear electrodynamics only. This solution interpolates between Ay\'on--Beato--Garc\'ia black hole, Letelier black hole and Schwarzschild black hole { in the absence of cloud of strings parameter, magnetic monopole charge and both of them, respectively}. We also discuss the thermal properties of this black hole and find that the solution follows the modified first law of black hole thermodynamics. Furthermore, we estimate the solution's black hole shadow and quasinormal modes.

Figures

Figures reproduced from arXiv: 2412.14230 by the authors.

**Figure 1.** The plot of the metric function f(r) versus r. Upper Panel: For various values of the CS parameter but fixed MM charge (g = 0.75 and g = 0.95). Lower Panel: For various values of MM charge but fixed CS parameter (a = 0.1 and a = 0.3) in a unit of M, where M = 1. g = 0.75 g = 0.95 a r− r+ δ a r− r+ δ 0.183 1.21 1.21 0 0.395 1.615 1.615 0 0.30 0.761 2.07 1.309 0.50 1.749 1.509 0.240 0.40 0.627 2.67 0.943 0.50 2.945 0.… view at source ↗

**Figure 2.** The plot of temperature T+ vs horizon radius r+. Upper Panel: For various CS parameter values with a fixed MM charge value (g = 0.75 and g = 0.95). Lower Panel: For various values of MM charge with a fixed value of CS parameter (a = 0.1 and a = 0.2) in the unit of M, where M = 1. However, these quantities coincide with Schwarzschild BH when both the parameters g = a = 0. The entropy of the obtained BH solution can b… view at source ↗

**Figure 3.** The plot of heat capacity C+ vs horizon radius r+ Upper Panel: For various values of CS parameter with a fixed value of MM charge (g = 0.75) and (g = 0.95). Lower Panel: For various values of MM charge with a fixed value of CS parameter (a = 0.1) and (a = 0.3). The BH remnant is a well-merited entity in theoretical astrophysics that can be a source of dark energy [74] and is one of the candidates to resolve the info… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The plot of heat capacity C+ vs horizon radius r+ Upper Panel: For various values of CS parameter with a fixed value of MM charge (g = 0.75) and (g = 0.95). Lower Panel: For various values of MM charge with a fixed value of CS parameter (a = 0.1) and (a = 0.3). The can…

**Figure 5.** Figure 5: The BH shadow for different CS parameter ( [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Left panel: The real part of QNFs for various values [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Thermodynamics and Shadows of Kerr black holes endowed with a global monopole charge
gr-qc 2025-06 reject novelty 4.0 of 10

The authors derive shadow and thermodynamic quantities for a Kerr-like metric with a global monopole term in Δ, and claim EHT observations constrain the monopole charge α.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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