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arxiv: 2605.14528 · v1 · submitted 2026-05-14 · 🌀 gr-qc · hep-th

Recognition: 1 theorem link

· Lean Theorem

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
Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:39 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords quasinormal modesEuler-Heisenberg black holesperfect fluid dark mattergreybody factorsscalar perturbationselectromagnetic perturbationsWKB approximationasymptotic iteration method
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The pith

Charge, nonlinear parameter, and dark matter density modify quasinormal frequencies and greybody factors in Euler-Heisenberg black holes.

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

The paper computes quasinormal modes for massless scalar and electromagnetic perturbations on the background of charged Euler-Heisenberg black holes surrounded by perfect fluid dark matter. It applies the asymptotic iteration method and sixth-order WKB approximation, then quantifies the relative deviation between the two techniques to check consistency. The central result is that the black hole charge Q, nonlinear electrodynamics parameter a, dark matter parameter λ, and angular index l reshape the effective potential barrier, which changes the real and imaginary parts of the frequencies together with the greybody factors that govern transmission and reflection.

Core claim

The parameters Q, a, λ, and l 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.

What carries the argument

The effective potential barrier in the Schrödinger-like wave equation for scalar and electromagnetic perturbations on the Euler-Heisenberg metric with perfect fluid dark matter.

Load-bearing premise

The given metric is an exact solution of the Einstein equations with the chosen sources, and the asymptotic iteration and sixth-order WKB methods capture the quasinormal modes accurately without large systematic errors over the parameter ranges studied.

What would settle it

A high-resolution numerical integration of the perturbation wave equation that produces frequencies differing by more than the reported method-to-method deviation for any chosen set of Q, a, λ, and l 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 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 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 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 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 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 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 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 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 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 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 14
Figure 14. Figure 14: FIG. 14: Variation of fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p021_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 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 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 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 19
Figure 19. Figure 19: FIG. 19: Variation of fundamental QNFs ( [PITH_FULL_IMAGE:figures/full_fig_p026_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 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 quasinormal modes of massless scalar and electromagnetic perturbations around charged Euler-Heisenberg black holes surrounded by perfect fluid dark matter. It computes the frequencies via the asymptotic iteration method and sixth-order WKB approximation, quantifies their relative deviations, evaluates greybody factors within the WKB framework, and examines the dependence on parameters Q, a, λ, and l, concluding that these parameters modify the effective potential barriers and thereby alter oscillation frequencies, damping rates, and transmission/reflection properties.

Significance. If the numerical results hold, the work adds a systematic parameter study of QNMs and greybody factors in a nonlinear electrodynamics plus dark-matter background. The explicit cross-check between AIM and WKB with relative-deviation analysis is a clear methodological strength that increases in the reported frequencies. The findings are incremental but useful for black-hole spectroscopy and greybody calculations in modified spacetimes.

major comments (2)
  1. [Greybody factor computation] Greybody-factor section: the sixth-order WKB transmission formula is applied directly to the effective potential modified by the λ term, yet the paper provides no independent check (e.g., numerical integration of the wave equation or higher-order WKB) on the accuracy of the greybody values once λ alters the large-r tail. This is load-bearing for the claim that λ significantly affects wave transmission and reflection.
  2. [Effective potential and wave equation] Effective-potential derivation: the explicit form of V(r) for both scalar and electromagnetic perturbations (including the Euler-Heisenberg and perfect-fluid-dark-matter contributions) is not cross-verified against the background metric solution; any inconsistency here would propagate into all reported frequencies and greybody factors.
minor comments (2)
  1. [Numerical results] The relative-deviation plots between AIM and WKB would benefit from an additional panel or table showing absolute errors as a function of λ to make the convergence assessment more quantitative.
  2. [Methods] A brief statement on the range of validity of the sixth-order WKB formula for the chosen parameter intervals (especially large λ) would clarify the domain of applicability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the positive assessment and the insightful comments that have helped improve the manuscript. Below we provide point-by-point responses to the major comments.

read point-by-point responses

  1. Referee: [Greybody factor computation] Greybody-factor section: the sixth-order WKB transmission formula is applied directly to the effective potential modified by the λ term, yet the paper provides no independent check (e.g., numerical integration of the wave equation or higher-order WKB) on the accuracy of the greybody values once λ alters the large-r tail. This is load-bearing for the claim that λ significantly affects wave transmission and reflection.

    Authors: We thank the referee for this valuable comment. The sixth-order WKB approximation is widely used for computing greybody factors in asymptotically flat spacetimes, and we have already demonstrated its reliability through cross-validation with the AIM method for the quasinormal frequencies. Regarding the large-r tail modified by λ, we note that the perfect fluid dark matter term affects the metric at large distances, but the WKB method remains applicable as the potential still decays sufficiently. In the revised manuscript, we have added a paragraph discussing the convergence of the WKB series for the greybody factors by comparing sixth-order results with fourth-order ones, showing small deviations. A full numerical integration of the wave equation for greybody factors is computationally intensive and was not performed in this study, but we agree it would be a useful extension. revision: partial

  2. Referee: [Effective potential and wave equation] Effective-potential derivation: the explicit form of V(r) for both scalar and electromagnetic perturbations (including the Euler-Heisenberg and perfect-fluid-dark-matter contributions) is not cross-verified against the background metric solution; any inconsistency here would propagate into all reported frequencies and greybody factors.

    Authors: We appreciate this suggestion for ensuring rigor. The effective potentials are derived from the Klein-Gordon equation for scalar fields and the Maxwell equations for electromagnetic perturbations using the standard tortoise coordinate transformation and separation of variables on the given metric. To address the concern, we have now explicitly included the derivation steps in the revised manuscript, showing how the metric components (f(r) from the Euler-Heisenberg plus dark matter solution) enter into V(r) for each case. We have cross-checked that in the limit λ=0 and a=0, our V(r) reduces to the known Reissner-Nordström form, confirming consistency. revision: yes

Circularity Check

0 steps flagged

Direct numerical solution of wave equation on assumed exact metric; minor self-citation not load-bearing

full rationale

The paper computes quasinormal frequencies via asymptotic iteration method and sixth-order WKB on the provided background metric, then evaluates greybody factors within the same WKB framework. No quantity is defined in terms of a fitted parameter from the output data, and no derivation step reduces by construction to its own inputs. The metric is adopted as an exact solution from prior literature (not self-citation load-bearing for the present results), and relative deviations between AIM and WKB are reported only for frequencies. This is standard independent computation, yielding only a minor score for routine citation of the background.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 0 invented entities

The central results rest on the assumed validity of the Euler-Heisenberg-plus-dark-matter metric and on the accuracy of standard perturbation techniques; no new entities are postulated.

free parameters (3)
  • Q
    Black hole electric charge parameter varied in the study
  • a
    Nonlinear electrodynamic parameter controlling higher-order field corrections
  • λ
    Perfect fluid dark matter parameter
axioms (1)
  • domain assumption The line element for the charged Euler-Heisenberg black hole surrounded by perfect fluid dark matter is a valid exact solution of the field equations
    Invoked as the background spacetime for all perturbation calculations

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