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arxiv: 2605.04994 · v1 · submitted 2026-05-06 · 🌀 gr-qc · hep-th

Perturbations and greybody bounds of Euler-Heisenberg black holes surrounded by perfect fluid dark matter

Fernando M. Belchior , Faizuddin Ahmed , Edilberto O. Silva This is my paper

Pith reviewed 2026-05-08 15:56 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords Euler-Heisenberg black holesperfect fluid dark mattergreybody factorsblack hole perturbationseffective potentialsabsorption cross sectionsnonlinear electrodynamicsDirac fields
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The pith

The Euler-Heisenberg and perfect fluid dark matter parameters deform effective potentials for scalar, electromagnetic and Dirac perturbations around black holes, altering greybody factors and absorption spectra.

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

The paper investigates how waves of spin 0, 1 and 1/2 propagate in the spacetime of an Euler-Heisenberg black hole surrounded by perfect fluid dark matter. It derives the modified Schrödinger-like radial equations, builds the effective potentials, and applies the Boonserm-Visser method to obtain lower bounds on the greybody factors that control transmission from infinity to the horizon. These bounds then yield partial absorption cross sections and energy emission rates. The central point is that the nonlinear electrodynamic correction and the dark matter contribution produce visible shifts in the potential barriers relative to Schwarzschild, Reissner-Nordström and pure Euler-Heisenberg cases. A reader would care because such shifts could appear as measurable differences in scattering or Hawking radiation spectra.

Core claim

The combined Euler-Heisenberg and perfect fluid dark matter background deforms the effective potential barriers for massless fields of spins 0, 1 and 1/2. The Boonserm-Visser lower bounds on the greybody factors increase monotonically with frequency and approach the geometric-optics limit at high frequency, while their low-frequency values depend on the spin and on the values of the nonlinear electrodynamic and dark matter parameters. The resulting absorption cross sections and energy emission rates differ from those of the Schwarzschild, Reissner-Nordström and pure Euler-Heisenberg limits.

What carries the argument

The spin-dependent effective potential appearing in the Schrödinger-like radial wave equation, constructed from the Euler-Heisenberg plus perfect fluid dark matter metric and used to compute Boonserm-Visser greybody lower bounds.

If this is right

  • Greybody lower bounds rise steadily with frequency and reach the high-frequency geometric-optics limit.
  • Low-frequency transmission depends strongly on field spin and on the nonlinear electrodynamic and dark matter parameters.
  • Partial absorption cross sections and energy emission rates can be computed directly from the greybody lower bounds.
  • The combined background produces scattering signatures distinguishable from Schwarzschild, Reissner-Nordström and pure Euler-Heisenberg black holes.

Where Pith is reading between the lines

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

  • Astrophysical measurements of Hawking spectra or scattering could in principle constrain the size of the nonlinear electrodynamic and dark matter parameters.
  • The same effective-potential construction could be applied to other nonlinear electrodynamic theories or to different dark matter profiles to test robustness.
  • If the deformations are large, they may also shift quasinormal-mode frequencies, offering a second independent observable.

Load-bearing premise

The given metric is an exact solution of the Einstein equations coupled to nonlinear electrodynamics and perfect fluid dark matter, so that linear perturbation theory applies without significant back-reaction.

What would settle it

Numerical integration of the exact radial wave equations for chosen values of the Euler-Heisenberg and dark matter parameters that either violates the reported monotonic increase of the greybody lower bounds or shows no deformation relative to the pure Euler-Heisenberg case.

Figures

Figures reproduced from arXiv: 2605.04994 by Edilberto O. Silva, Faizuddin Ahmed, Fernando M. Belchior.

Figure 1
Figure 1. Figure 1: Metric function g(r) for M = 1 and Λ = 0. (a) Varying Q ∈ {0.1, 0.2, 0.3, 0.4, 0.5}, fixed α = 0.1, a = 0.01. (b) Varying α ∈ {0.05, 0.10, 0.20, 0.30, 0.40}, fixed Q = 0.3, a = 0.01. (c) Varying a ∈ {0, 2, 5, 10, 20}, fixed Q = 0.7, α = 0.1; the range is illustrative (aphys ∼ 10−30). (d) Model comparison: Schwarzschild (Q = α = a = 0), RN (Q = 0.4, α = a = 0), EH (Q = 0.4, a = 5, α = 0), and EH+PFDM (Q = 0… view at source ↗
Figure 2
Figure 2. Figure 2: Effective potentials for M = 1, Q = 0.3, a = 1.0, α = 0.1, plotted from the horizon rh outward. (a) Scalar potential V0(r) for l = 1, 2, 3, 4. (b) Vector potential V1(r) for l = 1, 2, 3, 4. (c) Dirac potential V+(r) for κ = 1, 2, 3, 4. (d) Comparison of all three spins for l = κ = 1, showing the hierarchy V0 > V1 > V+. where Vs(r) is the effective potential for spin s = 0, s = 1, or the pair V± for s = 1/2… view at source ↗
Figure 3
Figure 3. Figure 3: Behavior of the greybody lower bound and absorption cross section for spin 0. Top row: transmission view at source ↗
Figure 4
Figure 4. Figure 4: Behavior of the greybody lower bound and absorption cross section for spin 1. Top row: transmission view at source ↗
Figure 5
Figure 5. Figure 5: Behavior of the greybody lower bound and absorption cross section for spin 1 view at source ↗
Figure 6
Figure 6. Figure 6: Greybody lower bound Ts(ω) (top row: a, b, c) and absorption cross section σs(ω) (bottom row: d, e, f) for varying EH parameter a ∈ {0, 10, 30, 70, 150}. Parameters: M = 1, Q = 0.85, α = 0, l = κ = 1. Columns from left to right: spin-0, spin-1, spin-1/2. Larger a pushes rh from 1.53 to 1.80 and therefore increases Γs — opposite to the effect of Q and α. 0 1 2 3 ω 0.0 0.2 0.4 0.6 0.8 1.0 Ts(ω) T1/2 > T1 > T… view at source ↗
Figure 7
Figure 7. Figure 7: Greybody lower bound Ts(ω) (a) and absorption cross section σs(ω) (b) for all three spin sectors: s = 0 (solid), s = 1 (dashed), s = 1/2 (dash-dotted). Parameters: M = 1, Q = 0.3, a = 1.0, α = 0.1, l = κ = 1. Faint background curves show the Q-dependence of T0 for Q ∈ {0.1, 0.2, 0.4, 0.5}. The ordering T1/2 > T1 > T0 (54) holds throughout the plotted parameter range. hole becomes hotter, i.e., when TH incr… view at source ↗
Figure 8
Figure 8. Figure 8: Model comparison of the greybody lower bound for spin-0 (a) and spin-1/2 (b), with view at source ↗
Figure 9
Figure 9. Figure 9: Behavior of the energy emission rate for spin 0. The spin-0 angular quantum number view at source ↗
Figure 10
Figure 10. Figure 10: Behavior of the energy emission rate for spin 1. The spin-1 angular quantum number view at source ↗
Figure 11
Figure 11. Figure 11: Behavior of the energy emission rate for spin 1 view at source ↗
Figure 12
Figure 12. Figure 12: Behavior of the energy emission rate for the tree spins. The angular quantum number view at source ↗
read the original abstract

We investigate the propagation of massless fields with spins $s=0$, $s=1$, and $s=1/2$ in the spacetime of an Euler-Heisenberg (EH) black hole surrounded by perfect fluid dark matter (PFDM). This background incorporates both the nonlinear electrodynamic correction associated with the EH effective theory and the logarithmic contribution induced by the surrounding dark matter distribution. After deriving the corresponding Schr\"odinger-like radial equations, we construct the effective potentials for scalar, electromagnetic, and Dirac perturbations and analyze how they are modified by the black hole charge, the EH parameter, and the PFDM parameter. The greybody factors are estimated through the rigorous Boonserm-Visser lower-bound method, and the associated partial absorption cross sections are obtained for different spin sectors using these bounds. Our results show that the nonlinear electrodynamic and dark matter parameters significantly deform the effective potential barrier, leading to potentially distinguishable changes in the transmission probabilities and absorption spectra within the model. In particular, the greybody lower bounds increase monotonically with the frequency and approach the high-frequency limit, while their low-frequency behavior is strongly affected by the geometry and by the spin of the perturbing field. Moreover, we utilize the greybody lower bound to calculate the energy emission rate. Finally, we make comparisons with the Schwarzschild, Reissner-Nordstr\"om, and pure EH limits, showing that the combined EH+PFDM background produces distinguishable corrections to black-hole scattering within the model. These results highlight greybody bounds as sensitive diagnostic probes of nonlinear electrodynamic effects and dark matter halos around compact objects.

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

1 major / 2 minor

Summary. The manuscript examines the propagation of scalar (s=0), electromagnetic (s=1), and Dirac (s=1/2) fields in the spacetime of an Euler-Heisenberg black hole surrounded by perfect fluid dark matter. It derives the Schrödinger-like radial wave equations and effective potentials for these perturbations, applies the Boonserm-Visser method to obtain lower bounds on the greybody factors, computes the corresponding absorption cross sections, and analyzes the energy emission rates. The effects of the black hole charge, the Euler-Heisenberg parameter, and the PFDM parameter are studied, with comparisons to the Schwarzschild, Reissner-Nordström, and pure Euler-Heisenberg cases.

Significance. Provided the assumed metric is a valid exact solution to the Einstein equations with the specified sources, this work offers a detailed investigation into how nonlinear electrodynamics and dark matter halos modify black hole perturbation spectra and scattering properties. The use of rigorous lower bounds on greybody factors and the systematic comparison across spin sectors and parameter regimes could serve as a useful reference for exploring observational signatures in modified black hole models.

major comments (1)
  1. [Section 2] Section 2 (metric ansatz): The line element incorporating the Euler-Heisenberg nonlinear correction and the logarithmic PFDM term is introduced without an explicit verification that it satisfies the Einstein equations G_μν = 8π(T_μν^EH + T_μν^PFDM). This verification (e.g., by direct computation of the Einstein tensor components and matching to the combined stress-energy tensors) is load-bearing, as all subsequent effective potentials, Boonserm-Visser integrals, greybody bounds, and absorption cross sections rest on the background being an exact solution.
minor comments (2)
  1. [Abstract] Abstract: The description of the PFDM contribution as 'logarithmic' is clear, but a one-sentence statement of the explicit metric function f(r) (including the definitions of the EH parameter and PFDM parameter) would make the setup immediately accessible without requiring the reader to reach Section 2.
  2. [Figures] Figures (e.g., effective potential plots): The curves for different parameter values are informative, but adding a panel or inset showing the high-frequency asymptotic behavior of the greybody bounds (to confirm they approach the geometric-optics limit) would strengthen the visual evidence for the monotonicity claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for highlighting the importance of verifying the background metric. We address the single major comment below and will incorporate the requested verification in the revised manuscript.

read point-by-point responses

  1. Referee: [Section 2] Section 2 (metric ansatz): The line element incorporating the Euler-Heisenberg nonlinear correction and the logarithmic PFDM term is introduced without an explicit verification that it satisfies the Einstein equations G_μν = 8π(T_μν^EH + T_μν^PFDM). This verification (e.g., by direct computation of the Einstein tensor components and matching to the combined stress-energy tensors) is load-bearing, as all subsequent effective potentials, Boonserm-Visser integrals, greybody bounds, and absorption cross sections rest on the background being an exact solution.

    Authors: We agree that an explicit verification is essential for rigor. Although the metric is constructed from the standard Euler-Heisenberg Lagrangian coupled to a perfect-fluid dark-matter source (as introduced in prior literature), the manuscript does not contain the direct computation. In the revised version we will add, in Section 2, the non-vanishing components of the Einstein tensor for the given line element and demonstrate their equality to 8π(T_EH + T_PFDM), thereby confirming that the spacetime is an exact solution. This addition will be placed immediately after the metric ansatz and before the derivation of the effective potentials. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from assumed metric via standard perturbation and bound techniques

full rationale

The paper starts from a given line element incorporating EH and PFDM parameters, derives the radial Schrödinger-like equations and effective potentials explicitly, then applies the independent Boonserm-Visser integral lower-bound method to obtain greybody factors and absorption cross-sections. No step equates a derived quantity to a fitted parameter by construction, no load-bearing uniqueness theorem is imported via self-citation, and the Boonserm-Visser bounds are computed directly from the potentials without statistical fitting or renaming of known results. The central claim of distinguishable deformations follows from the explicit parameter dependence in the potentials and is not forced by the paper's own equations.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central results rest on two additional model parameters (Euler-Heisenberg coupling and PFDM density parameter) whose values are chosen by hand to explore phenomenology, plus the assumption that the stated metric is an exact solution of the Einstein equations with the corresponding nonlinear matter.

free parameters (2)
  • Euler-Heisenberg parameter
    Nonlinear electrodynamic coupling strength introduced to deform the metric and potentials; its value is varied parametrically rather than derived.
  • PFDM parameter
    Scale parameter controlling the perfect-fluid dark matter density profile that enters the metric; chosen to produce distinguishable effects.
axioms (2)
  • domain assumption The background metric combining Euler-Heisenberg nonlinear electrodynamics with a perfect fluid dark matter halo satisfies the Einstein field equations.
    Invoked to justify the spacetime in which the perturbation equations are derived.
  • domain assumption Linear perturbation theory for massless fields of spins 0, 1, and 1/2 is sufficient and back-reaction can be neglected.
    Standard assumption allowing the reduction to Schrödinger-like radial equations.
invented entities (1)
  • Perfect fluid dark matter halo no independent evidence
    purpose: Phenomenological source term that modifies the black-hole metric via a logarithmic contribution.
    Introduced to model surrounding dark matter; no independent falsifiable prediction outside the present calculation is supplied.

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Forward citations

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