Quasinormal modes response to thermodynamic phase transitions in the charged AdS black hole surrounded by perfect fluid dark matter

Long-Xiang Li; Zhong-Wen Feng

REVIEW 2 major objections 5 minor 55 references

Quasinormal modes response to thermodynamic phase transitions in the charged AdS black hole surrounded by perfect fluid dark matter

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Black hole ringdown jumps at the small-large phase transition

desk verdict A competent, incremental QNM extension to PFDM-dressed charged AdS black holes with a genuinely useful isothermal decomposition, but the novelty claim needs to be squared with ref [47] and the quantitative QNM results would benefit from an independent method check. read the letter →

arxiv 2608.01200 v1 pith:Q5MDLYT3 submitted 2026-08-02 gr-qc

Long-Xiang Li , Zhong-Wen Feng This is my paper · ORCID

classification gr-qc

keywords quasinormalmodesperfectfluiddarkmatterchargedAdSblackholethermodynamicphasetransitionextendedspaceVanderWaals-likeChebyshevpseudospectralmethodsmall/large

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 asks whether a black hole's characteristic ringdown spectrum knows about its thermodynamic phase transitions, and whether an ambient perfect-fluid dark matter environment shifts or destroys that relationship. The authors argue that for a charged anti-de Sitter black hole, the fundamental massless-scalar quasinormal frequencies trace two separated curves below the critical point: one for small black holes and one for large black holes, with the frequency jumping discontinuously when the thermodynamically preferred branch changes. At the critical point the jump disappears and the frequencies vary smoothly, so the fundamental scalar mode carries no sharp second-order signature. The paper presents quasinormal modes as a dynamical complement to the Gibbs free-energy analysis: larger dark-matter intensity raises the critical temperature and pressure, lowers the critical horizon radius, and shifts the quasinormal frequencies to larger values, while the qualitative first-order signature persists.

What carries the argument

The machinery is the scalar-field effective potential on the PFDM background, $V(r)=f(r)[l(l+1)/r^2+f'(r)/r]$, built from the metric function $f(r)=1-2M/r+Q^2/r^2+8\pi P r^2/3+(\alpha/r)\log(r/|\alpha|)$, together with a Chebyshev pseudospectral solution of the resulting radial eigenvalue problem for the complex frequency $\omega$. The same potential drives the qualitative behavior: at fixed pressure, varying $r_h$ reshapes the potential and changes the damping; at fixed $r_h$, varying $P$ (equivalently the AdS radius $L$) rescales the potential and shifts both real and imaginary parts. A first-order Taylor expansion $\omega(r_h+\delta r_h,P+\delta P)$ then separates the two contributions and identifies which thermodynamic variable controls each branch. The thermodynamic side is carried by the equation of state $P=T/(2r_h)-1/(8\pi r_h^2)+Q^2/(8\pi r_h^4)-\alpha/(8\pi r_h^3)$, whose critical-point conditions determine $r_c$, $T_c$, and $P_c$, together with the Gibbs free energy swallowtail that marks the coexistence curve.

What would settle it

Compute the time-domain ringdown of the same charged-AdS-PFDM background through the first-order transition, allowing the background to evolve or including the gravitational perturbation sector, and check whether the emitted frequency actually jumps between the small- and large-branch quasinormal frequencies at the coexistence crossing; if the time-domain signal interpolates smoothly or selects a different frequency, the claimed branch-jump signature fails. Alternatively, compute higher scalar overtones at $P=P_c$, since the paper predicts no sharp critical signature for the fundamental mode and a sharp feature there would refute the universality of that statement.

Watch

Extended reading notes

Core claim

The central claim is that the fundamental scalar quasinormal-mode spectrum of the charged AdS black hole is sensitive to the first-order small/large black hole phase transition, and that this sensitivity survives in a PFDM background. Concretely, for fixed pressure or fixed temperature below the critical point, the small- and large-black-hole branches occupy clearly separated curves in the complex-frequency plane with different slopes; crossing the coexistence curve changes the preferred branch, so the quasinormal frequency jumps discontinuously. Along an isotherm the evolution is not controlled by the horizon radius alone: a first-order expansion in $\delta r_h$ and $\delta P$ shows the small branch is pressure-dominated, while the large branch involves near cancellation in the real part and a horizon-radius-driven increase in damping. At the critical point the same frequencies vary smoothly and monotonically, showing no sharp second-order signal, and along the coexistence curve the two branches' frequencies approach each other as the swallowtail shrinks, merging at criticality. The paper presents this correspondence as the dynamical signature of the phase transition.

Load-bearing premise

The paper's load-bearing premise is that quasinormal frequencies computed on each equilibrium background separately, for a test scalar field with no backreaction and no dynamical path connecting the branches, can be read as a 'dynamical signature' of what a ringing black hole would actually do when it crosses the first-order transition; if the ringdown is governed by a non-equilibrium or coupled process, the branch discontinuity in the spectrum need not be what a real black hole exhibits.

Editorial extensions

If this is right

Below the critical point, the fundamental scalar quasinormal frequency works as a dynamical marker of the small/large black hole transition: an abrupt jump in $\omega$ accompanies the branch switch in both isobaric and isothermal processes.
Along isotherms, the small-black-hole branch is pressure-dominated while the large-black-hole branch is governed by competing $r_h$ and $P$ effects, so quasinormal measurements could in principle distinguish which thermodynamic quantity drives a perturbation process.
Larger PFDM intensity $\alpha$ strengthens the first-order signature by enlarging the branch separation and slope difference in the complex-frequency plane, while leaving the smooth behavior at criticality intact.
Along the coexistence curve, the shrinking quasinormal gap between the two branches mirrors the shrinking Gibbs free-energy swallowtail, so the dynamical and thermodynamic descriptions converge together near the critical point.
Because the fundamental scalar mode shows no sharp critical-point signal, any second-order dynamical signature would have to come from other perturbation types or higher overtones, a question the paper leaves open.

Reading between the lines

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

Editorial inference: if this branch-jump behavior survives for gravitational axial and polar perturbations, quasinormal spectra could become a gravitational-wave-relevant marker of the phase transition, but the paper's test-field scalar computation alone does not establish that.
Editorial inference: the smooth behavior at the critical point suggests the fundamental scalar mode does not directly see the divergence of thermodynamic response functions; computing higher overtones or the pseudospectrum near $P=P_c$ might reveal a distinct imprint of critical slowing down.
Editorial inference: the isothermal branch analysis implies that in a realistic dark-matter halo, frequency shifts from the environment could be mistaken for horizon-radius shifts; disentangling the two requires measuring the pressure/AdS dependence separately, exactly as the paper does analytically.
Editorial inference: the PFDM model is phenomenological, and replacing the logarithmic correction with a particle-physics-based dark matter profile should change both the phase structure and the quasinormal scaling, offering a route to constrain dark matter models through ringdown observations.

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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

Quick take: this is a solid, incremental numerical study of scalar QNMs across the small/large black hole transition in charged AdS with perfect fluid dark matter. The headline result, separated QNM branches below the critical point, a discontinuous jump at the coexistence crossing, and smooth behavior at the critical point, matches the pattern known from RN-AdS and its variants. The genuinely new bits are the quantitative trajectories for alpha = 0.1, 0.3, 0.5, and, more interesting, the isothermal decomposition into horizon-radius and pressure effects. Tables 3-6 isolate the two variables and Tables 7-9 show a first-order Taylor split; that is a clean piece of analysis and worth keeping.

The thermodynamics in Section 2 checks out. The critical quantities in Eqs. (2.10)-(2.12) reduce correctly to RN-AdS when alpha→0, and the tabulated transition temperatures and pressures are consistent with the coexistence curves. The QNM tables are mutually consistent and the pseudospectral implementation is standard, with a reported N=120 vs N=150 agreement at 10^-6. That is a reasonable resolution check.

Soft spots, in proportion. First, the novelty claim is under-hedged. Ref [47], Abbas and Ali, is cited in the reference list but never discussed in the text; by its title it already treats QNMs and phase transitions for the same charged AdS PFDM black hole. The authors should state explicitly what their work adds beyond [47], or the claim that this 'has not been systematically examined' will not survive. This is my main concern and it is fixable.

Second, all quantitative claims rest on one Chebyshev pseudospectral implementation. No independent method, no code, no error budget. The stress-test note that a systematic offset in the large-|Im w| modes could affect the slopes and the branch separation is a real possibility, though not proven. For a paper whose main message is quantitative (distinct slopes, specific QNM frequencies), an independent cross-check on a handful of representative points, say a continued fraction or time-domain integration, would substantially raise confidence. I would not call it fatal, but it is a clear gap.

Third, the 'dynamical signature' phrasing overstates things a little. These are QNMs of static equilibrium backgrounds, not a time-dependent transition. The authors do acknowledge the step to gravitational perturbations and real observations is open, but the correspondence between the plotted branch jump and a physical ringdown crossing is asserted, not derived. That caveat is worth making prominent.

Minor: tables are long and repetitive; a few figures plus a supplementary dataset would do.

Overall: this paper is for people working on black hole chemistry, PFDM backgrounds, and QNM probes of phase structure. It is a competent extension with a useful new decomposition, but it needs a clear comparison with ref [47] and an independent QNM check before I would trust the quantitative details. Send it to a serious referee; with those two additions it would be a publishable contribution.

Referee Report

2 major / 5 minor

Summary. This paper studies the relationship between thermodynamic phase transitions and fundamental scalar quasinormal mode (QNM) spectra for charged AdS black holes surrounded by perfect fluid dark matter (PFDM). The authors work in the extended phase space, treating the cosmological constant as pressure and the PFDM parameter as an additional thermodynamic variable, and derive the equation of state, critical point, and Gibbs free energy. Using a Chebyshev pseudospectral method, they compute the fundamental l=0 scalar QNMs on equilibrium backgrounds. Below the critical point, along isobaric and isothermal processes, the QNM frequencies of the small and large black hole branches are claimed to form separated trajectories with distinct slopes, with a discontinuous jump at the first-order transition. At the critical point, the QNMs vary smoothly and show no sharp signature. Along the coexistence curve, the separation between branch QNMs shrinks and vanishes at criticality. The paper concludes that the fundamental scalar QNM spectrum is sensitive to the first-order small/large black hole transition and that the PFDM parameter shifts both the thermodynamic and dynamical scales.

Significance. The paper is a straightforward extension of the known QNM-phase transition correspondence to a PFDM background. If the numerical results are reliable, they provide a concrete example of how a dark-matter environment alters both the phase structure and the perturbative response, which could be useful for holographic or observational probes. Strengths include the self-contained thermodynamic derivation (Eqs. (2.8)-(2.13)), the complete data tables for QNM frequencies, and the reported convergence test N=120 vs N=150. However, the central claims are entirely based on a single numerical method with only internal resolution checks, and the 'smoothness' at the critical point is asserted from sparse data. These issues, rather than the physics modeling per se, are the main obstacles to full confidence.

major comments (2)

[§3.1, Tables 1–2, Figs. 2, 4] The central branch-separation, slope, and discontinuity claims rest exclusively on the Chebyshev pseudospectral implementation described in §3.1, with no independent method cross-check. The only validation reported is that N=120 agrees with N=150 to 1e-6 for the fundamental mode. Internal resolution convergence does not exclude spurious eigenvalues or systematic boundary errors, which are known risks for pseudospectral QNM codes, especially for large-|Im ω| modes. I request an independent validation: (i) reproduce the α=0 RN-AdS limit and compare with published results (e.g., Ref. [37]); (ii) compute at least a representative subset of the entries in Tables 1 and 2 with a second method (continued fraction or time-domain integration); and (iii) report the numerical uncertainty for the large-black-hole-branch modes. Without this, the claimed discontinuity and distinct slopes could be partly numerical in origin.
[§4, Tables 10–11, Figs. 7–8] The claim that QNM frequencies vary smoothly at the critical point is based on only a few sampled points, with no analysis of derivatives. For example, in Table 10(a) the points near the critical radius r_c ≈ 1.69 are T=0.0654 (r_h=1.623) and T=0.0655 (r_h=1.913), a gap of 0.29 in r_h; a non-analytic kink in ω(r_h) or dω/dT between these points would go unnoticed. To support the 'no sharp signature' conclusion, please compute a denser grid around r_c and explicitly test continuity of dω/dr_h (or dω/dT) across the critical point, or alternatively soften the claim to state that no discontinuous jump was found rather than that the frequencies vary smoothly.

minor comments (5)

[Abstract and §6] The phrase 'dynamical signature' may overstate the results; the QNMs are computed for a test scalar field on static equilibrium backgrounds, not through a dynamical transition. Please clarify that the observed discontinuity is a spectral branch-structure feature.
[§3.1] The convergence test statement could be strengthened by explicitly stating the comparison for the worst-converging mode (e.g., the large-|Im ω| modes) rather than only the fundamental mode.
[Tables 1–6] The number of decimal places is inconsistent across tables (e.g., 6 decimals in Table 3 vs 5 in Table 1). Please standardize.
[Eqs. (2.10)–(2.12)] It would be helpful to show the α→0 limit explicitly for these expressions (the text states it but does not display the reduction).
[Figure 11] The two branches for Re ω appear to overlap over a wide range of T/T_c; the legend and line styles should be made more distinguishable, and the text should quantify the 'separation' rather than relying on visual impression.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QNM frequencies are obtained by an independent pseudospectral solution of the scalar perturbation equation on the PFDM background, with no parameter fitted to the phase-transition claim.

full rationale

The derivation chain is self-contained. Section 3.1 fixes the background from (r_h, P, Q, alpha) via the horizon condition, and then solves the Klein-Gordon equation (3.1) as a matrix eigenvalue problem (3.7) with Chebyshev pseudospectral discretization. The QNM frequencies are numerical outputs of this eigenvalue problem and are not fitted to, or defined in terms of, the thermodynamic phase-transition quantities. The thermodynamic input is limited to selecting which background (small/large branch, isobar, isotherm, coexistence state) to evaluate; this is the intended physical setup, not a circular reduction. The critical point formulas (2.10)-(2.12) follow analytically from the equation of state (2.8), and the alpha-dependence reported is a direct consequence of those formulas. No claim in the paper reduces an equation to itself by construction: for example, the smooth behavior at the critical point is read off from separately computed QNM eigenvalues in Tables 10-11, not derived from the smoothness of the thermodynamics. The coexistence-curve comparison in Section 5 correlates two independently computed quantities (Gibbs free energy and QNM frequencies) that share the same underlying branch structure, but the QNM values are not used to construct the swallowtail or vice versa. There are no relevant self-citations: the cited metric, first law, and PFDM action come from external prior work [43,44,52-54], and the representative alpha values are taken from [50], none of which are authored by the present authors. The only methodological caveat is that the QNM computation relies on a single pseudospectral implementation with internal convergence checks but no independent method cross-check; that is a numerical robustness concern, not a circularity of the argument. Accordingly, no load-bearing step reduces to its own input.

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

The central claim rests on the PFDM metric and extended phase space from prior literature, two hand-chosen parameters, the test-field approximation, and the numerical method. No new physical entity is postulated.

free parameters (2)

alpha (PFDM intensity) = 0.1, 0.3, 0.5
Chosen by hand as representative positive values (Section 2). All critical quantities and QNM frequencies depend on it; no fitting to target data.
Q (electric charge) = 0.75
Fixed by hand. The QNM frequencies and critical quantities are evaluated at this one charge, so quantitative conclusions are not shown to be charge-independent.

assumptions (5)

domain assumption The PFDM metric ansatz with the alpha/r log(r/|alpha|) term (Eq. 2.3) correctly describes a charged AdS black hole surrounded by perfect fluid dark matter.
Imported from refs [42-44,52]. The entire thermodynamic and QNM analysis rests on this background. The paper restricts to positive alpha and does not test negative alpha.
domain assumption Extended phase space identifications P = -Lambda/8pi and the first law dM = T dS + V dP + Phi dQ + Psi dalpha (Eq. 2.7).
Standard black hole chemistry input, cited from refs [5,44,50]. Used to define the coexistence curve and critical point.
domain assumption The massless scalar perturbation is a test field with no backreaction on the spacetime, electromagnetic field, or PFDM background.
Stated in Section 3.1. This is the assumption that lets the authors compute QNMs on fixed backgrounds and compare them with equilibrium phase structure.
standard math Maxwell's equal-area law, equivalently equality of Gibbs free energy, determines the first-order coexistence curve.
Used for the coexistence lines in Figs. 1, 3, and 9. Standard thermodynamics, not re-derived in the paper.
ad hoc to paper The Chebyshev pseudospectral discretization with N = 120 converges to the true QNM frequencies.
Convergence is tested against N = 150 to 1e-6, but no independent method such as continued fraction or WKB is used for cross-checking.

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Cite this review

Pith. "Pith review of Quasinormal modes response to thermodynamic phase transitions in the charged AdS black hole surrounded by perfect fluid dark matter." pith.science (2026). https://pith.science/paper/Q5MDLYT3

@misc{pith2026260801200,
  author       = {Pith},
  title        = {Pith review of: Quasinormal modes response to thermodynamic phase transitions in the charged AdS black hole surrounded by perfect fluid dark matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q5MDLYT3}},
  note         = {Machine review of arXiv:2608.01200}
}

read the original abstract

We investigate how thermodynamic phase transitions are reflected in the quasinormal modes (QNMs) of charged anti-de Sitter (AdS) black holes surrounded by perfect fluid dark matter (PFDM). In the extended phase space, increasing the positive PFDM parameter raises the critical temperature and pressure while reducing the critical horizon radius. We compute the fundamental QNMs of a massless scalar perturbation using a Chebyshev pseudospectral method and analyze their evolution along isobaric and isothermal processes below the critical point. The small and large black hole branches trace clearly separated QNM trajectories and display sharply different slopes near the first-order transition, providing a dynamical signature of the branch change. Along isotherms, this evolution results from the competing effects of the horizon radius and pressure, or equivalently the AdS radius, rather than from the horizon radius alone. At the critical point, however, the QNM frequencies vary smoothly with the horizon radius and show no sharp signature of the second-order transition. Along the coexistence curve, the separation between the small and large black hole QNMs decreases as the Gibbs free energy swallowtail shrinks and vanishes at criticality. These results show that PFDM shifts both the thermodynamic phase structure and the associated QNM response, while the fundamental scalar QNM spectrum remains sensitive to the first-order small/large black hole transition.

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