arxiv: 2604.16628 · v1 · submitted 2026-04-17 · 🌀 gr-qc · hep-th

Recognition: unknown

Shadow, Quasinormal Modes, Sparsity, and Energy Emission Rate of Euler-Heisenberg Black Hole Surrounded by Perfect Fluid Dark Matter

Edilberto O. Silva , Faizuddin Ahmed

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Pith reviewed 2026-05-10 07:32 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords black hole shadowquasinormal modesEuler-Heisenbergperfect fluid dark matterHawking radiationgrey-body factorenergy emission

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The pith

The perfect fluid dark matter parameter and black-hole charge strongly affect the photon sphere, shadow size, quasinormal frequencies, and energy emission rate of an Euler-Heisenberg black hole, while the nonlinear correction remains suble

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

The paper studies the combined effects of nonlinear electrodynamics and a perfect fluid dark matter halo on a static black hole. It computes the unstable photon sphere radius that sets the shadow size, the scalar quasinormal frequencies in the eikonal limit, the associated grey-body factors, and the resulting Hawking radiation sparsity and energy emission profile. The calculations show that both the black-hole charge and the dark-matter density parameter produce clear shifts in all these quantities, whereas the Euler-Heisenberg nonlinear term contributes only weakly except possibly at very large charges. The authors conclude that the dark-matter background leaves the dominant observable signature.

Core claim

The central claim is that the perfect fluid dark matter environment imprints dominantly on the photon sphere, shadow size, eikonal quasinormal frequencies, Hawking temperature, and energy emission rate of the Euler-Heisenberg black hole, with the black-hole charge also playing a significant role, while the Euler-Heisenberg nonlinear correction is typically subleading within the explored parameter range but may become relevant in strong-charge regimes.

What carries the argument

The static spherically symmetric metric that combines the Euler-Heisenberg nonlinear electrodynamics Lagrangian with a perfect fluid dark matter density term, together with the eikonal correspondence that relates quasinormal modes and grey-body factors to the unstable photon sphere.

Load-bearing premise

The metric ansatz is an exact solution to the Einstein equations with the chosen matter sources and the eikonal approximation accurately describes the quasinormal modes and grey-body factors throughout the parameter space examined.

What would settle it

An observed shadow radius or ringdown frequency that scales with the Euler-Heisenberg parameter rather than with the perfect fluid dark matter density, or a measured energy emission rate that deviates from the predicted PFDM-dominated profile.

Figures

Figures reproduced from arXiv: 2604.16628 by Edilberto O. Silva, Faizuddin Ahmed.

**Figure 3.** Figure 3: FIG. 3. Three-dimensional representation of the shadow ra [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Shadow silhouettes of the EH–PFDM black hole in the observer’s celestial plane. (a) Varying [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: (d) varies the EH parameter α at Q/M = 0.3. The curves are barely distinguishable, confirming once again that the leading-order QED correction has a negligible [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Eikonal QNM quantities at [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Eikonal grey-body factor Γ [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Hawking temperature [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Dimensionless sparsity parameter [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Spectral energy emission rate [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗

read the original abstract

In this work, we investigate the optical, dynamical, and radiative properties of an Euler--Heisenberg black hole immersed in a perfect fluid dark matter (PFDM) background. We analyze the photon sphere and shadow, the scalar quasinormal-mode spectrum in the eikonal regime, the grey-body factor through the eikonal QNM correspondence, the sparsity of Hawking radiation, and the corresponding energy emission rate. Our results show that both the black-hole charge and the PFDM parameter significantly affect the photon sphere, shadow size, quasinormal frequencies, Hawking temperature, and emission profile, whereas the Euler--Heisenberg correction is typically subleading in the parameter range explored, although it may become more visible in strong-charge regimes for selected observables. Overall, the dark-matter environment provides the dominant imprint on the phenomenology of the system, indicating that shadow and ringdown-related quantities may serve as useful probes of PFDM effects within the approximations considered.

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.

Desk Editor's Note private letter to a colleague

This is a standard application of known techniques to a combined black hole model, with dark matter effects outweighing the nonlinear electrodynamics correction.

read the letter

The main takeaway is that this paper performs standard phenomenological calculations on the shadow, quasinormal modes in the eikonal limit, and Hawking radiation properties for an Euler-Heisenberg black hole immersed in perfect fluid dark matter. The key result is that the perfect fluid dark matter parameter tends to have a larger impact on these observables than the Euler-Heisenberg nonlinear correction, except possibly at high charges. The work does a decent job of mapping out the dependence on the three free parameters: the black hole charge, the Euler-Heisenberg parameter, and the dark matter density parameter. They compute the photon sphere and shadow radius, the quasinormal frequencies and damping times in the eikonal regime, the grey-body factors via the correspondence, the sparsity of the radiation, and the energy emission rate. This gives a broad picture of how the model behaves across different regimes. What the paper does well is presenting these results in a clear way, showing that dark matter effects are dominant in the phenomenology. This aligns with expectations since the dark matter halo affects the spacetime at larger distances while the nonlinear electrodynamics is a local correction near the horizon. Soft spots include the lack of any discussion on the validity of superposing the two metrics or potential issues with the resulting spacetime. The eikonal approximation for quasinormal modes is used without quantifying how good it is for the parameter values chosen, and there are no comparisons to full numerical solutions or error estimates provided in the abstract. The conclusions are qualitative, stating that certain parameters affect things significantly, but without specific numbers or plots described here it's hard to gauge the magnitude. This paper is aimed at the community working on black hole observables in modified gravity or with dark matter components. It could be of interest to those studying potential observational signatures in shadows or ringdown signals. A reader seeking groundbreaking new physics or rigorous analytical results will be disappointed, but it provides concrete computations for this particular setup. Overall, the paper shows clear thinking in applying known tools to a new combination, with no obvious internal contradictions. It deserves to go to peer review so that the numerical methods and parameter selections can be vetted by experts in the field.

Referee Report

1 major / 2 minor

Summary. The paper computes the photon sphere radius, shadow size, eikonal-limit scalar quasinormal frequencies, grey-body factors via the eikonal correspondence, Hawking temperature, radiation sparsity, and energy emission rates for the Euler-Heisenberg black hole metric immersed in a perfect-fluid dark-matter halo. It concludes that the PFDM parameter produces the dominant shifts in all these observables while the Euler-Heisenberg correction remains sub-leading except possibly at large charge values.

Significance. If the metric and approximations are accepted, the work supplies a systematic comparison of nonlinear-electrodynamics versus dark-matter effects across several observables, offering concrete predictions that could be tested with shadow imaging and ringdown data. The explicit statement that PFDM dominates is a falsifiable claim within the adopted framework.

major comments (1)

[Sections on metric and QNMs] The central claim that PFDM dominates rests on the validity of the composite metric and the eikonal correspondence for QNMs and grey-body factors. The manuscript should provide a brief justification or reference for the metric construction (likely in the introduction or §2) and a quantitative check of the eikonal approximation's accuracy over the explored parameter ranges, for example by comparing a few low-lying modes to the full wave equation solution.

minor comments (2)

[Abstract] The abstract refers to 'the parameter range explored' without quoting the numerical intervals for the Euler-Heisenberg parameter, PFDM parameter, and charge; these intervals should be stated explicitly.
[Figures] Figures showing shadow radius or emission spectra versus parameters would be clearer if they included a reference curve for the Schwarzschild case and indicated the direction of increasing PFDM parameter.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful reading of the manuscript and the constructive suggestion. We address the major comment below and have revised the manuscript accordingly where feasible.

read point-by-point responses

Referee: The central claim that PFDM dominates rests on the validity of the composite metric and the eikonal correspondence for QNMs and grey-body factors. The manuscript should provide a brief justification or reference for the metric construction (likely in the introduction or §2) and a quantitative check of the eikonal approximation's accuracy over the explored parameter ranges, for example by comparing a few low-lying modes to the full wave equation solution.

Authors: We thank the referee for highlighting these points. We have added a concise justification for the composite metric in the introduction and Section 2, together with references to the standard construction of Euler-Heisenberg black holes immersed in perfect-fluid dark-matter halos. Regarding the eikonal correspondence, we have expanded the discussion in the QNM and grey-body sections to recall the regime of validity of the high-frequency approximation and to cite supporting literature on its accuracy for similar spacetimes. A direct numerical comparison of a few low-lying modes against the full wave equation, however, lies outside the analytical and semi-analytical scope of the present study; such a computation would require a separate numerical campaign that we consider beyond the current work. revision: partial

standing simulated objections not resolved

Performing a quantitative numerical check of the eikonal approximation by solving the full wave equation for low-lying modes across the explored parameter ranges.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper first assembles a composite metric by superposing the Euler-Heisenberg nonlinear-electrodynamics correction onto a perfect-fluid dark-matter halo solution, then computes photon-sphere radius, shadow size, eikonal quasinormal frequencies, grey-body factors, Hawking temperature, sparsity, and energy emission rate directly from the metric functions using standard geodesic and wave-equation techniques. All reported trends (PFDM dominance, sub-leading EH corrections) follow from numerical evaluation over the parameter space; no parameter is fitted to the output observables, no uniqueness theorem is invoked via self-citation, and no ansatz is smuggled in. The central claims therefore reduce to the accepted metric plus textbook GR calculations rather than to any internal redefinition or self-referential loop.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The analysis rests on the assumed exact metric solution for the Euler-Heisenberg black hole in PFDM, standard GR field equations, and the validity of the eikonal limit; several model parameters are free inputs with no independent derivation.

free parameters (3)

Euler-Heisenberg parameter
Controls the strength of nonlinear electrodynamics correction; value chosen within explored range.
PFDM parameter
Sets the density or coupling strength of the perfect fluid dark matter; dominant effect claimed.
Black hole charge
Electric charge parameter appearing in the metric; affects photon sphere and temperature.

axioms (2)

domain assumption The spacetime metric is the standard Euler-Heisenberg solution immersed in perfect fluid dark matter as given in the literature.
Invoked to define the background geometry for all subsequent calculations.
domain assumption Eikonal (high-frequency) approximation accurately describes the quasinormal modes and grey-body factors.
Used to link QNMs to shadow and emission properties.

pith-pipeline@v0.9.0 · 5476 in / 1533 out tokens · 55198 ms · 2026-05-10T07:32:25.596378+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

Quasinormal modes of massless scalar and electromagnetic perturbations for Euler Heisenberg black holes surrounded by perfect fluid dark matter
gr-qc 2026-05 unverdicted novelty 5.0

Quasinormal frequencies and greybody factors for massless scalar and electromagnetic perturbations in Euler-Heisenberg black holes with perfect fluid dark matter are calculated via AIM and sixth-order WKB, showing tha...

Reference graph

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