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The shadow and quasinormal modes of the asymptotically flat hairy black holes with a dilaton potential

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Pith's one-line read For this exact family of hairy charged black holes, the dilaton coupling visibly alters shadows and quasinormal ringing only near extremal charge, and scalar high-frequency modes are governed by the photon-sphere orbit.

desk verdict Routine but legitimate parameter study of AAM hairy black hole shadows and QNMs; numerics look sound, but a wrong Lyapunov formula and unspecified α make the paper non-reproducible as written. read the letter →

arxiv 2508.03270 v1 pith:4EA2MZYI submitted 2025-08-05 gr-qc

Sheng-Han Xiong , Yong-Zhuang Li , Xiao-Mei Kuang , Jerzy Matyjasek This is my paper
classification gr-qc MSC 83C57
keywords blackholeshadowquasinormalmodesdilatonholeshairyphotonsphereeikonallimitLyapunovexponentEinstein-Maxwell-dilatongravity
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 studies an exact, asymptotically flat, electrically charged hairy black hole with a non-trivial dilaton potential---the AAM family---and asks how the dilaton coupling $\nu$ affects the spacetime's observable signatures. The central claim is that the shadow radius, the Lyapunov exponent, and the coordinate angular velocity of null geodesics are nearly independent of $\nu$, except when the charge $Q$ approaches its extremal value, where the coupling's effect becomes significant, especially as $\nu\to 1$. The paper also computes scalar quasinormal mode frequencies with two independent semi-analytic methods and finds that, as the parameter $\eta\to\infty$, the spectrum approaches the low-energy string-theory limit of the model. In the eikonal limit $\ell\gg 1$, it proves that the real and imaginary parts of the quasinormal frequencies are given by the photon-sphere frequency $\Omega_c$ and the Lyapunov exponent $\lambda$. The interest is that this connects shadow and ringdown observables to the scalar-hair parameters in a model where the Einstein-Maxwell and string-theory regimes are continuously connected.

What carries the argument

The argument is carried by the exact static, spherically symmetric AAM solution, $ds^2 = \Psi(x)[-f(x)dt^2 + \eta^2 dx^2/f(x) + d\Omega_2^2]$, together with the constraint $\nu = 1 + 24Q^2\eta^4/(\alpha + 3\eta^2 + 6M\eta^3)$ linking the hair parameter $\eta$, mass $M$, charge $Q$, and dilaton coupling $\nu$. This constraint turns $\eta$ into the control parameter that interpolates between a hairy Reissner-Nordström-like regime and the low-energy string-theory regime. The shadow and geodesic quantities follow from the null orbital equation, with the photon sphere fixed by $f'(x_{\rm ph})=0$, the shadow radius $x_{\rm sh}=x_{\rm ph}/\sqrt{\Psi(x_{\rm ph})f(x_{\rm ph})}$, and the Lyapunov exponent and angular velocity defined from the second derivative of the geodesic potential. For perturbations, a massless scalar reduces to a Schrödinger-like equation whose effective potential is $V(x) = \frac{f(x)}{\eta^2}\frac{1}{\sqrt{\Psi(x)}}(f(x)\sqrt{\Psi(x)})' + \ell(\ell+1)f(x)$, and the quasinormal frequencies are computed with a Borel-summed high-order WKB method and cross-checked with higher-order WKB with Padé summation.

What would settle it

One decisive check is to substitute the metric (9)--(12) into the equations of motion (2)--(4) for representative parameters and verify the constraint (17) and the horizon-existence condition (26); a violation in any allowed region would invalidate the background. A second check targets the eikonal claim: compute scalar quasinormal modes for $\ell=200$, $n=2$, $Q=0.7$, and $\eta=70$ by direct numerical integration and compare with Fig. 6; if the deviations $\Delta\omega_R$ and $\Delta\omega_I$ are not of those sizes, the $\Omega_c$-$\lambda$ correspondence is refuted.

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Extended reading notes

Core claim

The paper's central assertion is that for the exact asymptotically flat AAM hairy charged black hole, the dilaton coupling $\nu$ leaves the shadow radius, the Lyapunov exponent $\lambda$, and the coordinate angular velocity $\Omega_c$ essentially unchanged except when $Q$ is close to the extremal value, with the largest deviations appearing near $\nu\to 1$. In the eikonal limit of scalar perturbations, the quasinormal frequencies obey $\omega_R \simeq \ell\Omega_c$ and $\omega_I \simeq (n+1/2)\lambda$, and the numerical data support a refined real part $(\ell + 1/2)\sqrt{f(x_{\rm ph})}$ that reduces the discrepancy with the Borel-summed WKB results. The paper further finds that for large $\eta$, where the dilaton coupling $\gamma$ tends to the low-energy string-theory value, the quasinormal spectrum approaches the known results for that string limit. It reports, for example, $Q=0.7$, $\ell=3$, $n=0$, $\eta=400$, a computed mode $\omega=1.451472-0.089668i$, close to the string-limit value obtained by the higher-order WKB with Padé summation.

Load-bearing premise

The load-bearing premise is that the exact AAM background, its asymptotic flatness, and the constraint tying $\eta$, $M$, $Q$, and $\nu$ are all correct, because every shadow and quasinormal result is computed on that background rather than re-derived from it.

Editorial extensions

If this is right

  • Measuring the shadow radius of these black holes constrains the dilaton coupling only when the charge is near extremal; otherwise the shadow is virtually identical to that of an uncharged hairy black hole.
  • The eikonal correspondence means the scalar ringdown spectrum is encoded entirely in the unstable photon-sphere orbit, so high-frequency observations would directly read off $\Omega_c$ and $\lambda$.
  • At large $\eta$, the scalar quasinormal frequencies of the AAM black hole approach those of the low-energy string-theory limit, bridging the Einstein-Maxwell-like and stringy regimes of the model.
  • Using $(\ell + 1/2)\sqrt{f(x_{\rm ph})}$ for the real part instead of $\ell\sqrt{f(x_{\rm ph})}$ gives the more accurate eikonal prediction for this family.

Reading between the lines

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

  • Because the analysis is restricted to test scalar fields, a natural extension is to repeat it for electromagnetic and gravitational perturbations; if the geometric-optics correspondence survives there, the $\Omega_c$ and $\lambda$ relations would govern the full ringdown, not just scalar modes.
  • A direct observational consequence, not drawn by the paper, is that only near-extremal charged black holes could reveal this dilaton hair through shadows; neutral astrophysical black holes would be blind to it.
  • The larger deviation in the real part than in the imaginary part suggests that next-to-leading eikonal corrections are sensitive to the shape of the photon-sphere potential, so a next-order WKB expansion for general $\nu$ would be a concrete quantitative extension.
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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 paper studies the shadow, photon sphere, and scalar quasinormal modes of the exact asymptotically flat hairy charged black hole solution of Anabalón, Astefanesei, and Mann, with a non-trivial dilaton potential. Using the constraint relation among the black hole mass, charge, dilaton coupling, and integration constant η, the authors compute the shadow radius, Lyapunov exponent λ, and coordinate angular velocity Ω_c as functions of η for several charges. They then compute scalar QNM frequencies with the Hatsuda method and cross-check them with higher-order WKB-Padé summation. The central claims are that the dilaton coupling significantly affects the shadow and QNM observables only when the charge is close to extremal, that the large-η limit reproduces the low-energy string theory results, and that in the eikonal limit the QNM real and imaginary parts are controlled by Ω_c and λ.

Significance. If the results hold, the paper provides a useful comparative framework for hairy black hole observables and a further test of the geometric-optical correspondence between null geodesics and quasinormal modes. The numerical cross-checks are a genuine strength: the Schwarzschild and Reissner-Nordström limits are checked, the large-η string limit is compared with Ref. [129], and two independent numerical methods (Hatsuda and higher-order WKB with Padé summation) are used. No parameter fitting is performed, so the geodesic/QNM comparison is not circular. The main obstacles are a displayed formula for λ that is inconsistent with the Schwarzschild limit and with Eq. (33), and an unspecified convention for the potential parameter α; both are correctable within the manuscript's scope.

major comments (2)
  1. [III, Eq. (23)] The displayed formula for the Lyapunov exponent is inconsistent with the Schwarzschild limit and with the paper's own eikonal result. Taking Q→0, ν→1, with Ψ=1/[η²(x-1)²], f=η²(x-1)²(1-2η(x-1)) and x_ph=1+1/(3η), Eq. (23) gives λ² = -1/27, whereas the correct Cardoso-type result, and the result quoted in Eq. (33), give λ² = +1/27. The corrected expression should carry an overall minus sign, equivalently λ² = -(Ψ²f²/2) d²/dx²(1 - b²f/(η²Ψ²)) at x_ph, which reduces to λ² = -f(x_ph)f''(x_ph)/(2η²). Because λ is one of the three headline observables and is used to verify the geometric-optical correspondence in Fig. 6, please correct Eq. (23) and confirm that all plotted λ(η) values were computed with the corrected formula.
  2. [II, Eqs. (14)-(17)] The parameter α is never assigned a numerical value, and the rescaling sentence after Eq. (16) does not make the convention transparent. Equation (17) still contains α explicitly, so the relation ν(Q,η) used in Fig. 1 and in every subsequent QNM calculation depends on α; the proposed transformations η→√α η, M→√α M, Q→√α Q do not by themselves remove α once M=1 is imposed. Please state explicitly the value or convention used in the numerics (for example, 'we set α=1 in units with M=1'), or demonstrate the α-independence of all reported observables by rewriting the metric and the constraint equations in explicitly rescaled variables.
minor comments (5)
  1. [Abstract and Sec. IV.B] The abstract says the real and imaginary parts of the eikonal QNMs are 'proved to be given by Ω_c and λ', but Sec. IV.B immediately finds that a (l+1/2)√f correction is needed to reduce Δω_R. The statement is acceptable as an asymptotic statement for l≫1, but the wording 'proved' should be qualified to avoid overstating the precision of the eikonal formula.
  2. [Throughout] There are several typos: 'Chian' should be 'China' in the author affiliation; the phrase 'the the dilaton field' appears twice in the introduction; and 'diatonic Reissner-Nordström' in Sec. V should presumably be 'dilatonic Reissner-Nordström'.
  3. [IV.B, Eq. (32)] The phrase 'expanding Eq. 32 to infinity about l' is unclear; it should read 'expanding Eq. (32) in powers of 1/l to high order' or similar.
  4. [Ref. [132]] The footnote 'the expression in [130] should be slightly modified' is too vague for a reader who wants to reproduce the third-order WKB calculation; please specify the modification or give a precise reference for the corrected term.
  5. [Fig. 1] The caption says 'for all subfigures, the horizontal coordinate is η', but the top-left panel plots ν on the vertical axis while the other panels plot x_ph, Ω_c, and λ; please make the panel labels more explicit so the reader can identify which curve corresponds to which quantity in each panel.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the geodesic observables and QNMs are computed independently from an externally adopted exact solution, and the eikonal relation is a mathematical consistency check rather than an input.

full rationale

The paper's derivation chain is self-contained against external benchmarks. The AAM metric and dilaton potential are adopted from Refs. [71,73], which are not by the present authors, so the background is external input rather than a self-citation. The shadow radii, Lyapunov exponent, and coordinate angular velocity are computed from null geodesics of that metric, while QNMs are obtained separately by solving the scalar perturbation equation with the Hatsuda method and cross-checked with higher-order WKB-Padé approximation. No parameter is fitted to any subset of the data and then renamed as a prediction; the constraint relation (17) merely parametrizes the chosen background. The eikonal identification omega_R = l Omega_c and omega_I = (n+1/2) lambda is derived in Eqs. (32)-(34) from the WKB expansion of the same potential and then verified numerically; this is a standard consistency relation, not a tautology. The only self-citation is to Matyjasek's earlier WKB-Padé papers [120,121], but that method is an independent, widely used computational tool and is not load-bearing for the paper's conclusions. The skeptical observation that Eq. (23) may have a sign error in the Schwarzschild limit concerns correctness of a displayed formula, not circularity, since the plotted results could in principle be checked against a corrected expression and the numerical QNM computation does not rely on Eq. (23). Overall, no step reduces by construction to its own input.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the exactness of the adopted AAM solution, the horizon condition, and the validity of two WKB-based spectral methods. No new particles, fields, or forces are introduced. The main hidden input is the unspecified value of alpha, which a rescaling is said to absorb but which is never fixed in the text.

free parameters (1)
  • alpha (dilaton potential strength) = not stated
    Appears in f(x), the mass formula (14), and the constraint (17). The paper sets M=1 and claims the rescaling (16) removes alpha, but never states the chosen alpha value; the nu(eta) curves and all parameter scans presuppose a definite alpha convention.
assumptions (5)
  • domain assumption The AAM metric (9)-(12) is an exact asymptotically flat solution of action (1) with potential (7).
    Taken without re-derivation from Refs. [71,73]; every shadow and QNM result inherits its validity.
  • domain assumption The horizon-existence condition (26) follows from demanding f'(x to infinity) < 0.
    Stated in Sec. III without a proof; it restricts the (Q, eta) parameter space used in all figures.
  • standard math Hatsuda/Borel-Padé WKB and higher-order WKB-Padé converge to the true quasinormal frequencies for the modes studied.
    Relies on Refs. [117-121]; no independent spectral method or error bound is provided.
  • standard math The eikonal QNM-geodesic correspondence of Cardoso et al. [92] applies to this spherically symmetric asymptotically flat metric.
    Standard result invoked in Sec. IV.B; used to identify Re(omega) and Im(omega) with Omega_c and lambda.
  • ad hoc to paper With M=1, the parameter alpha can be fixed by the rescaling (16) without loss of generality.
    Not stated explicitly; the scanned nu(eta) curves in Fig. 1 presuppose a definite alpha value.

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Cite this review

Pith. "Pith review of The shadow and quasinormal modes of the asymptotically flat hairy black holes with a dilaton potential." pith.science (2026). https://pith.science/paper/4EA2MZYI

@misc{pith2026250803270,
  author       = {Pith},
  title        = {Pith review of: The shadow and quasinormal modes of the asymptotically flat hairy black holes with a dilaton potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4EA2MZYI}},
  note         = {Machine review of arXiv:2508.03270}
}
abstract

In this article, the shadow and the quasinormal modes (QNMs) of an exact asymptotically flat hairy electrically charged black hole solution with a dilaton potential are investigated. Using the {constraint} equation among the integration constant $\eta$ of the gravitational field, the mass $M$, the electric charge $Q$ and the coupling constant $\nu$ between the $U(1)$ field and the dilaton field, we find that the shadow radii, the Lyapunov exponent $\lambda$ and the coordinate angular velocity $\Omega_{c}$ only significantly affected by $\nu$ if the $Q$ is close to the extremal value, especially when $\nu$ approaches to one. Furthermore, the QNMs are numerically computed by using the Hatsuda method and verify with the higher-order WKB approximations with the Pad\'e summation. We find that the QNMs are close to that of the low energy limit of the string theory when $\nu$ is large enough. In the eikonal limit, the real and imaginary parts are proved to be given by $\Omega_{c}$ and $\lambda$, respectively.

Figures

Figures reproduced from arXiv: 2508.03270 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The behaviors of the effective potential [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The maximum [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The behaviors of Re [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The behaviors of Re [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The behaviors of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Forward citations

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