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An ambient magnetic field, via an effective scalar mass, can suppress the damping of the fundamental quasinormal mode of an Einasto-supported regular black hole by about an order of magnitude, approaching a quasi-resonant regime.

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2026-08-04 05:35 UTC pith:MCHI2NEU

load-bearing objection Competent incremental extension of massive-scalar QNM/GBF work to Einasto-supported regular black holes; the trend toward longer-lived modes with larger μ is plausible, but the headline quasi-resonant endpoint rests on a single 8th-order WKB value with no independent check in exactly the regime the paper admits WKB can fail. the 3 major comments →

arxiv 2603.28415 v2 pith:MCHI2NEU submitted 2026-03-30 gr-qc

Long-lived quasinormal frequencies for regular black hole supported by the Einasto profile in the presence of the magnetic field

Milena Skvortsova This is my paper
classification gr-qc PACS 04.70.Bw95.35.+d98.62.Js
keywords regular black holesEinasto profilequasinormal modesmassive scalar fieldmagnetic-field effective massgrey-body factorsabsorption cross-sectionsquasi-resonances
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that environmental parameters—the Einasto halo shape and the strength of an ambient magnetic field—leave a strong imprint on how a regular black hole vibrates. Treating the magnetic field as giving a scalar perturbation an effective mass μ=2Bm, the author shows that increasing μ systematically lowers the damping rate |Im(ω)| of the fundamental quasinormal mode; in the most extended halo model (ñ=5), the damping at ℓ=2 drops from about 9.1×10⁻² to 8.7×10⁻³, by an order of magnitude, while the oscillation frequency rises. A sympathetic reader should care because this predicts that the ringdown of such an object is environment-dependent and can be much longer-lived than the standard Schwarzschild-like pattern, and that the same mass term shifts grey-body factors and absorption thresholds to higher frequencies. The trend is obtained with high-order WKB-Padé and independently cross-checked with time-domain profiles, although the paper itself flags regimes where the barrier-based approximation becomes questionable.

Core claim

The central claim: for a regular Einasto black hole, an effective scalar mass μ induced by a magnetic field (μ=2Bm) and the halo parameters ñ, h strongly control the quasinormal spectrum. In the most extended halo (ñ=5), raising μ from 0 to 1.0 cuts |Im(ω)| for the fundamental ℓ=2 mode from ≈9.1×10⁻² to ≈8.7×10⁻³ (an order of magnitude) while Re(ω) rises from 0.707 to 0.877—the quasi-resonant regime. The mechanism is geometric: the μ² term lifts the potential at infinity, flattening and eventually erasing the barrier maximum; near that critical μ the damping plummets. The same μ shifts grey-body factors and absorption onset to higher frequencies. The paper interprets all this as environmen

What carries the argument

The central object is the effective potential for a massive scalar perturbation, V(r)=f(r)[ℓ(ℓ+1)/r² + f'(r)/r + μ²], on the Einasto regular black hole. The geometry follows from the Einasto density profile ρ(r)=ρ₀ exp[-(r/h)^{1/ñ}], producing a de Sitter-like core that removes the singularity and a mass function that tends to Schwarzschild at infinity. The argument turns on how the μ² term changes this single-barrier potential: it raises the potential at large r, lowers and flattens the peak, and at a critical μ the maximum disappears. Near that point the higher-order WKB expansion with Padé resummation—and the related WKB transmission formula for grey-body factors—gives a sharp drop in th

Load-bearing premise

The load-bearing premise is that the WKB approximation remains quantitatively reliable for the fundamental mode in the very regime where the damping is small and the barrier is nearly flat, even though the paper itself shows that in a nearby regime (ℓ=0, ñ=1, μ=0.25) two WKB orders disagree by 129%.

What would settle it

Compute the fundamental quasinormal frequency for ñ=5, ℓ=2, μ=1.0 using a method that does not assume a single barrier (e.g., direct integration of the radial equation with outgoing-wave boundary conditions, or a full time-domain extraction). If the resulting Im(ω) deviates substantially from −8.7×10⁻³, or if the effective potential V(r*) at those parameters shows no local maximum, then the claimed quasi-resonant suppression is a WKB artifact.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If an ambient magnetic field supplies μ=2Bm, stronger fields should make scalar ringdown longer-lived, with damping vanishing at a critical field strength that depends on ℓ.
  • Low-multipole modes reach the quasi-resonant regime at smaller μ, so the damping suppression should appear first for ℓ=0 and ℓ=1, then for higher ℓ.
  • Grey-body factors shift upward in frequency with μ, so Hawking-radiation spectra and absorption cross-sections of such objects are environment-sensitive.
  • The time-domain profiles in the paper agree with the WKB trend in the cases checked, supporting the physical reality of the effect rather than a pure truncation artifact.
  • At the critical μ the barrier maximum vanishes; this geometric condition can be verified directly from the metric, independently of any mode calculation.

Where Pith is reading between the lines

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

  • Editor's extension: the effective-mass mechanism is generic—any process giving the scalar a mass (couplings, extra dimensions) should produce similar long-lived modes in regular black holes with sufficiently flat barriers, so the result may extend beyond the magnetic-field reading.
  • Editor's extension: a testable prediction is that at the quasi-resonant endpoint the late-time signal transitions from exponential ringdown to the oscillatory power-law tail characteristic of massive fields; a long time-domain evolution would show this.
  • Editor's extension: because the paper's ñ=5 result at μ=1.0 is taken at the edge of WKB reliability, an independent non-barrier spectral computation is the cleanest check of whether Im(ω) really is ≈−8.7×10⁻³ or whether the WKB approximation simply breaks down there.
  • Editor's extension: if μ=2Bm with m the azimuthal number, the effect is m-dependent, which would break the usual m-degeneracy of quasinormal frequencies in a spherically symmetric spacetime; this could be searched for as m-splitting in magnetized environments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies massive scalar perturbations in regular black-hole spacetimes supported by the Einasto density profile, computing quasinormal modes (QNMs), grey-body factors, and absorption cross-sections for three Einasto indices (n~=1/2, 1, 5). The effective scalar mass μ is interpreted as arising from an external magnetic field via μ=2Bm. Quasinormal frequencies are obtained with high-order WKB expansions with Padé resummation and cross-checked by time-domain evolution for two configurations. The central physical claim is that increasing μ suppresses the damping rate |Im(ω)|, eventually producing quasi-resonant long-lived modes; the most dramatic example is Table III for n~=5, ℓ=2, where |Im(ω)| drops from about 9.1×10^-2 at μ=0 to 8.7×10^-3 at μ=1.0. Grey-body factors from direct WKB transmission and from the QNM-based correspondence agree well in the tested parameter range, and the absorption cross-sections show the expected low-frequency suppression and high-frequency transparency.

Significance. If the central trend is quantitatively reliable, the paper would demonstrate a concrete environmental imprint on ringdown and scattering: regular-core geometry plus an effective mass induced by a magnetic field can make the fundamental mode much longer-lived than the Schwarzschild-like expectation. The manuscript has several genuine strengths: analytic metric functions for n~=1/2 and n~=1; systematic WKB-order comparisons in Tables I and II; an independent time-domain code used as a check; and grey-body factors computed by two separate routes. The paper also explicitly acknowledges the regime where the WKB method breaks down (Sec. III). These features make the work a useful contribution, but the central quasi-resonant claim for n~=5 is not yet supported with the precision claimed because the endpoint of Table III rests on a single WKB implementation with no independent confirmation.

major comments (3)
  1. [V, Table III] The central quantitative claim—order-of-magnitude suppression of |Im(ω)| at ℓ=2, μ=1.0 for n~=5—rests on one 8th-order WKB-Padé value (m~=4) for a numerically constructed metric. No second WKB order and no time-domain value are reported for any Table III entry. This is exactly the regime that the paper itself identifies as unreliable: Sec. III states that the standard WKB approach 'ceases to be applicable' when the potential barrier loses its maximum, and Sec. V notes that μ=1.0 is the last point before the peak disappears. Tables I–II demonstrate the failure mode: at ℓ=0, μ=0.25 (Table II), the 16th- and 14th-order values differ by 129% and even flip the sign of Im(ω). The statement in Sec. V that the trend is 'physical and not merely a truncation artifact' is therefore not demonstrated at the endpoint. Please provide an independent high-precision computation (for instance, time-domain
  2. [V, Fig. 3] The only time-domain check for the n~=1/2 model uses h=1 (Fig. 3, left), but the WKB values quoted from Table I are for h=1.05. The text calls these 'corresponding' values. Since h shifts the frequency (as the paper itself discusses), the sub-percent agreement in Re(ω) and the few-percent agreement in Im(ω) are not a controlled comparison. Please compute WKB values at the same h as the time-domain run or quantify the sensitivity to h. The n~=1 check (h=0.38) does match Table II, but neither time-domain check covers the quasi-resonant regime used for the main claim.
  3. [II, IV.A] For n~=5, the metric function f(r) is obtained numerically (Sec. II), and Sec. IV.A reports that only 8th-order WKB was used because of computational cost. WKB of this order requires many derivatives of the potential at its maximum; for a numerically constructed f(r) with h=1.5×10^-6, numerical differentiation can introduce uncontrolled errors. The manuscript gives no convergence test for the numerical metric or its derivatives. Please provide evidence that the n~=5 frequencies are stable with respect to grid resolution and to the numerical differentiation scheme, or use a method that does not require high derivatives.
minor comments (4)
  1. [Eq. (27)] The identification μ=2Bm is central to the physical interpretation but is presented without derivation. Since m is the azimuthal quantum number, and the background is spherically symmetric, the QNMs of the massive scalar do not depend on m in the absence of the magnetic field; a brief explanation of how the effective mass arises from the magnetic background would help the reader assess the physical claim.
  2. [Table III] The dashes in Table III are unexplained. State the critical μ for each ℓ at which the potential barrier loses its maximum, and clarify why entries are omitted.
  3. [Eq. (29)] The formula for K is missing parentheses in the first term; as written, the notation may be misread as Ω² minus V0 divided by sqrt(-2V0''). Please clarify the standard form.
  4. [Fig. 4] The quadratic extrapolation that yields μ_c≈0.554 and 0.545 is presented without uncertainties or fit residuals. State the fit errors or at least the number of points used, so the extrapolation can be assessed.

Circularity Check

0 steps flagged

No significant circularity: quasinormal frequencies are computed from the potential with independent time-domain checks; the magnetic-field interpretation is a parameter mapping, not a circular derivation.

full rationale

The paper's central objects are quasinormal frequencies obtained by solving the radial wave equation with the explicit potential (Eq. 16) using high-order WKB/Padé and time-domain integration. No parameter is fitted to the target QNM data: the only fit is an extrapolation of |Im(omega)| to estimate the critical mu, and it is labelled as extrapolated. The metric is taken from a cited construction (Ref. [49]), but it is the input background, not a conclusion derived from the QNM results. The identification mu = 2Bm (Eq. 27) is an interpretive mapping that translates the mass parameter into a magnetic-field language; the computed mu-dependence is nontrivial and is not an identity. The grey-body comparison between direct WKB transmission (Eq. 28) and the QNM-based relation (Eqs. 30-31) is a consistency check between two WKB-level approximations using the same potential, not a prediction from fitted parameters; it is not load-bearing for the main ringdown claim. The paper explicitly flags the regime where WKB is unreliable (Section III) and Tables I-II show large order discrepancies near critical masses; these are correctness/robustness concerns, not circularity. Self-citations are used for standard numerical methods and known massive-field phenomena, but no load-bearing argument reduces to an unverified self-citation. The derivation chain is therefore self-contained, and the central claim is an independent numerical result, so the circularity score is 0.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 1 invented entities

The paper introduces no new free parameters beyond the background model (n-tilde, h) and the effective mass μ. The central numerical results are deterministic computations from the potential; their main inputs are the chosen parameter values. The liabilities are the methodological assumptions (WKB reliability, QNM-GBF correspondence at moderate ℓ) rather than invented entities or fitted constants.

free parameters (3)
  • Einasto index n-tilde = 1/2, 1, 5
    Chosen as representative values; the paper does not fit n-tilde to data, but it is a free parameter of the model that controls the density profile and is varied across the paper.
  • halo scale h = 1.05 (n-tilde=1/2), 0.38 (n-tilde=1), 1.5e-6 (n-tilde=5)
    The central results depend on h; these values are chosen by hand (not fitted), but are free parameters that determine the horizon structure and potential shape. The n-tilde=5 case is an extreme choice (h=1.5e-6) that produces a Schwarzschild-like geometry except for a tiny regular core.
  • effective scalar mass μ = varied 0 to 1.0
    The free parameter producing the long-lived modes. The paper interprets it as μ=2Bm, but does not fit B; it is a free knob whose increase drives the quasi-resonant trend.
axioms (4)
  • domain assumption The Einstein equations with the anisotropic-fluid condition P_r = -ρ produce the metric (1)-(4) and that the geometry is regular and asymptotically flat.
    The background construction is taken from Ref. [49]; the paper does not re-derive the existence or stability of such a solution. If this matter model is inconsistent or unstable, the central results would not apply.
  • domain assumption The scalar field is a test field that does not backreact on the geometry.
    Implicit in 'massive test scalar field' (Sec. III); all QNM and grey-body computations treat the field as a perturbation, so the background is fixed.
  • domain assumption The WKB expansion with Padé resummation provides a valid approximation for the fundamental QNM in the parameter ranges used.
    A methodological assumption; the authors themselves note (Sec. IV/V) that WKB fails when the potential has no barrier. Tables I and II show large order-dependence for ℓ=0, so the reliability of WKB is the weakest methodological premise.
  • domain assumption The QNM-based grey-body factor formula (Eqs. 30-31) is applicable at moderate ℓ (1,2) for this potential.
    The paper cites [130,131] for this correspondence; it is exact in the eikonal limit and approximate at moderate ℓ. The paper's conclusion that the two methods agree is based on this assumption.
invented entities (1)
  • Effective mass μ interpreted as due to magnetic field through μ = 2Bm no independent evidence
    purpose: To connect the mass parameter to the magnetic-field strength and to make the mass dependence physically interpretable as magnetically induced.
    No new physical entity is invented; the identification μ=2Bm is imported from earlier literature (refs [54,55,120,121]). The paper does not model the magnetic field explicitly in the metric or the Klein-Gordon equation—it only re-labels μ. There is no falsifiable handle outside the existing literature.

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read the original abstract

We investigate quasinormal modes, grey-body factors, and absorption cross-sections of a massive scalar field in regular black-hole spacetimes supported by the Einasto density profile. The analysis is performed for $\tilde n=1/2$, $1$, and $5$, where the scalar mass $\mu$ is treated as an effective parameter induced by an external environment. Quasinormal frequencies are computed with high-order WKB expansions and Pad\'e resummation, and are cross-checked by time-domain evolution. We show that increasing the effective mass and varying the Einasto parameters can strongly suppress the damping rate, leading to long-lived modes and clear quasi-resonant behavior. Grey-body factors obtained from direct WKB transmission and from the QNM-based correspondence agree well in the considered regimes, while their differences remain controlled. Using the transmission coefficients, we derive partial and total absorption cross-sections and demonstrate the expected transition from low-frequency suppression to efficient high-frequency absorption. Our results show that regularity of the core together with environmental parameters leaves a noticeable imprint on both ringdown and scattering observables. Within this setup, the magnetic field acts as the physical agent that controls the effective mass scale and therefore governs how close the system can approach the quasi-resonant regime.

Figures

Figures reproduced from arXiv: 2603.28415 by Milena Skvortsova.

Figure 1
Figure 1. Figure 1: FIG. 1. Model with [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Model with [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Left: Semi-logarithmic time-domain profile for [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Extrapolation of the damping rate [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Partial and total absorption cross-sections for the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Grey-body factors for [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Grey-body factors for [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

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