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arxiv: 2605.14528 · v2 · pith:JYFDCXF2new · submitted 2026-05-14 · 🌀 gr-qc · hep-th

Quasinormal modes of massless scalar and electromagnetic perturbations for Euler-Heisenberg black holes surrounded by perfect fluid dark matter

Chengfu Feng , Sheng-Yuan Li , Xufen Zhang , Ming Zhang , De-Cheng Zou , Rui-Hong Yue This is my paper

Pith reviewed 2026-05-20 21:14 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords quasinormal modesEuler-Heisenberg black holesperfect fluid dark matterscalar perturbationselectromagnetic perturbationsgreybody factorsasymptotic iteration methodWKB approximation
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The pith

Charge, nonlinear electrodynamics, and dark matter parameters alter quasinormal frequencies and greybody factors for scalar and electromagnetic perturbations around Euler-Heisenberg black holes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper computes the quasinormal modes of massless scalar and electromagnetic fields on the fixed background of a charged Euler-Heisenberg black hole surrounded by perfect fluid dark matter. Frequencies are obtained via the asymptotic iteration method and sixth-order WKB approximation, with their agreement checked quantitatively, while greybody factors are extracted from the WKB transmission coefficients. The work traces how the charge Q, nonlinear parameter a, dark-matter parameter λ, and multipole number l reshape the effective potential barrier and thereby shift the real and imaginary parts of the frequencies together with the reflection and transmission properties.

Core claim

The background metric is taken as given, the wave equations are reduced to Schrödinger-like form, and the resulting effective potentials are shown to depend explicitly on Q, a, and λ; numerical evaluation then demonstrates that larger values of these parameters raise barrier heights or shift peak locations, producing measurable changes in oscillation frequencies, damping times, and greybody factors for both perturbation types.

What carries the argument

The effective potential barrier obtained by separating the wave equation for scalar and electromagnetic perturbations in the Euler-Heisenberg-plus-dark-matter metric.

If this is right

  • Increasing Q raises the real frequency while typically shortening the damping time for both scalar and electromagnetic modes.
  • Nonzero a modifies barrier width and thereby changes the low-frequency transmission coefficient.
  • Larger λ shifts the potential peak outward, reducing the greybody factor at fixed frequency.
  • Higher l increases the real part of the frequency roughly as in Schwarzschild while the imaginary part remains sensitive to a and λ.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same parameter dependence may appear in other nonlinear-electrodynamics black-hole solutions once an analogous dark-matter halo is added.
  • If such black holes form in nature, the altered spectra could produce distinguishable features in the ringdown portion of gravitational-wave signals.
  • Greybody-factor changes imply that the fraction of energy radiated to infinity versus absorbed by the horizon is tunable by the dark-matter density parameter.

Load-bearing premise

The given metric is accepted as an exact solution of the coupled Einstein-nonlinear-electrodynamics equations and is held fixed while linear perturbations are analyzed on top of it.

What would settle it

A calculation of the fundamental quasinormal frequency for a specific choice of Q, a, and λ using an independent high-precision method such as Leaver’s continued-fraction approach that yields a statistically significant mismatch with the reported AIM and WKB values.

Figures

Figures reproduced from arXiv: 2605.14528 by Chengfu Feng, De-Cheng Zou, Ming Zhang, Rui-Hong Yue, Sheng-Yuan Li, Xufen Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1: the variation of the metric function [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Variation of scalar and electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Variation of scalar and electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Variation of scalar and electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Variation of scalar fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Variation of scalar fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Variation of electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Variation of electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Variation of scalar fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Variation of scalar and electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Variation of electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Variation of scalar and electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Variation of scalar and electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Variation of scalar( [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Variation of scalar fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Variation of fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Variation of electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Variation of scalar fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Variation of scalar fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Variation of electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Variation of electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Variation of scalar ( [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Variation of fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Variation of fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Variation of scalar fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: The scalar field greybody factor [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Variation of electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p024_16.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: The electromagnetic field greybody factor [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Variation of scalar fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p025_17.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: The scalar field greybody factor [PITH_FULL_IMAGE:figures/full_fig_p019_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Variation of electromagnetic fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p025_18.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: The electromagnetic field greybody factor [PITH_FULL_IMAGE:figures/full_fig_p019_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: Variation of fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p026_19.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: The scalar field greybody factor [PITH_FULL_IMAGE:figures/full_fig_p020_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: The scalar field greybody factor [PITH_FULL_IMAGE:figures/full_fig_p028_20.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: The electromagnetic field greybody factor [PITH_FULL_IMAGE:figures/full_fig_p020_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21: The electromagnetic field greybody factor [PITH_FULL_IMAGE:figures/full_fig_p029_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22: The scalar field greybody factor [PITH_FULL_IMAGE:figures/full_fig_p029_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23: The electromagnetic field greybody factor [PITH_FULL_IMAGE:figures/full_fig_p030_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24: The scalar field greybody factor [PITH_FULL_IMAGE:figures/full_fig_p030_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25: The electromagnetic field greybody factor [PITH_FULL_IMAGE:figures/full_fig_p030_25.png] view at source ↗
read the original abstract

We investigate the quasinormal modes of massless scalar and electromagnetic perturbations in charged Euler--Heisenberg black holes surrounded by perfect fluid dark matter. The quasinormal frequencies are calculated using the asymptotic iteration method and the sixth-order WKB approximation, and the relative deviation between the two methods is quantitatively analyzed to verify the reliability of results. The greybody factors for both perturbations are also evaluated within the sixth-order WKB framework. We systematically examine the effects of the black hole charge $Q$, nonlinear electrodynamic parameter $a$, dark matter parameter $\lambda$, and angular quantum number $l$ on the quasinormal frequencies and greybody factors. We find that these parameters significantly modify the structure of the effective potential barriers, and thus affect the oscillation frequencies, damping rates, and wave transmission and reflection properties of the perturbed fields.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the quasinormal modes of massless scalar and electromagnetic perturbations for charged Euler-Heisenberg black holes surrounded by perfect fluid dark matter. It employs the asymptotic iteration method and the sixth-order WKB approximation to compute the quasinormal frequencies, analyzes the relative deviations between these methods, and evaluates greybody factors using the WKB approach. The effects of the charge Q, nonlinear parameter a, dark matter parameter λ, and angular momentum l on the frequencies, damping rates, and transmission properties are systematically studied.

Significance. If the background geometry is confirmed to be an exact solution and the numerical computations include proper error controls, the results would demonstrate how nonlinear electrodynamics and dark matter parameters alter the effective potential barriers, thereby influencing the ringdown signals and scattering of fields around these black holes. This could be relevant for future gravitational wave astronomy in testing extensions of general relativity and dark matter models.

major comments (2)
  1. [Section II] Section II: The line element for the Euler-Heisenberg black hole surrounded by perfect fluid dark matter is presented with free parameters Q, a, and λ, but the manuscript does not explicitly verify that this metric satisfies the Einstein equations coupled to the nonlinear electromagnetic stress-energy tensor and the perfect fluid dark matter energy-momentum tensor. Without this check, the derived effective potentials in the subsequent sections rest on an unconfirmed foundation.
  2. [Results section] Results section: While relative deviations between the asymptotic iteration method and sixth-order WKB are reported, the manuscript lacks explicit error bars, convergence tests with respect to WKB order or AIM iterations, and full details on the derivation of the effective potentials for both scalar and electromagnetic perturbations, which limits the verifiability of the central numerical claims on frequencies and greybody factors.
minor comments (2)
  1. [Abstract] The abstract mentions 'quantitatively analyzed' relative deviations but does not specify the magnitude of these deviations or the range of parameters studied.
  2. [Notation] Ensure consistent use of symbols for the dark matter parameter λ and the nonlinear parameter a throughout the text and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and will revise the manuscript to incorporate the requested clarifications and additional details.

read point-by-point responses

  1. Referee: [Section II] Section II: The line element for the Euler-Heisenberg black hole surrounded by perfect fluid dark matter is presented with free parameters Q, a, and λ, but the manuscript does not explicitly verify that this metric satisfies the Einstein equations coupled to the nonlinear electromagnetic stress-energy tensor and the perfect fluid dark matter energy-momentum tensor. Without this check, the derived effective potentials in the subsequent sections rest on an unconfirmed foundation.

    Authors: We appreciate the referee highlighting the need for explicit verification. The metric is an exact solution obtained by solving the Einstein equations with the Euler-Heisenberg nonlinear electrodynamics and the perfect-fluid dark matter energy-momentum tensor, following the standard procedure in the literature. In the revised manuscript we will add a brief but explicit check (by direct substitution of the metric into the field equations or by outlining the derivation steps) in Section II or a short appendix to confirm that the given line element satisfies the coupled system. revision: yes

  2. Referee: [Results section] Results section: While relative deviations between the asymptotic iteration method and sixth-order WKB are reported, the manuscript lacks explicit error bars, convergence tests with respect to WKB order or AIM iterations, and full details on the derivation of the effective potentials for both scalar and electromagnetic perturbations, which limits the verifiability of the central numerical claims on frequencies and greybody factors.

    Authors: We agree that greater transparency on numerical accuracy and potential derivations will improve verifiability. In the revision we will (i) include estimated uncertainties or error bars on the reported quasinormal frequencies and greybody factors, (ii) present convergence tests by increasing the WKB order beyond sixth order and by varying the number of AIM iterations, and (iii) supply the complete derivation of the effective potentials for both massless scalar and electromagnetic perturbations (including the explicit form of the tortoise coordinate and the resulting Schrödinger-like equations) either in the main text or in a dedicated appendix. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard QNM computation on given metric

full rationale

The paper presents a background metric with parameters Q, a, and λ, derives the effective potentials for massless scalar and electromagnetic perturbations via the standard Regge-Wheeler or Schrödinger-like form, and computes quasinormal frequencies using the asymptotic iteration method and sixth-order WKB approximation along with greybody factors. These steps constitute direct numerical evaluation from the input line element and do not reduce the output frequencies or transmission coefficients to fitted parameters or prior self-citations by construction. The metric is treated as a fixed solution to the coupled field equations without the present work claiming to re-derive or verify it via uniqueness theorems from the same authors, rendering the derivation self-contained and externally falsifiable through independent numerical codes or observations.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 1 invented entities

The central claim rests on the assumed validity of the Euler-Heisenberg-plus-perfect-fluid-dark-matter metric, standard linear perturbation equations on a fixed background, and the accuracy of the asymptotic iteration and WKB approximations for the chosen parameter ranges.

free parameters (3)
  • Q
    Black hole electric charge, scanned as a free parameter rather than derived.
  • a
    Nonlinear electrodynamic coupling constant, scanned as a free parameter.
  • λ
    Perfect fluid dark matter density parameter, scanned as a free parameter.
axioms (1)
  • domain assumption The Euler-Heisenberg black hole surrounded by perfect fluid dark matter is an exact solution of the Einstein equations coupled to nonlinear electrodynamics and a perfect fluid.
    Invoked as the fixed background geometry for all perturbation calculations.
invented entities (1)
  • perfect fluid dark matter no independent evidence
    purpose: To modify the spacetime metric via the parameter λ and thereby alter the effective potential for perturbations.
    Postulated phenomenological model with no independent falsifiable prediction supplied in the abstract.

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Lean theorems connected to this paper

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  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The spacetime geometry ... is described by the static and spherically symmetric line element ds² = −f(r)dt² + f(r)⁻¹ dr² + r²(dθ² + sin²θ dϕ²) with f(r) = 1 − 2M/r + Q²/r² − a Q⁴/(20 r⁶) + λ/r ln|r/λ|.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    We systematically examine the effects of the black hole charge Q, nonlinear electrodynamic parameter a, dark matter parameter λ, and angular quantum number l on the quasinormal frequencies and greybody factors.

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