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arxiv: 2605.25148 · v1 · pith:MQAYE657new · submitted 2026-05-24 · 🌀 gr-qc

Absorption and scattering spectra of massive scalar waves in charged regular black hole spacetimes

Marco A. A. Paula , Carolina L. Benone , Lu\'is C. B. Crispino This is my paper

Pith reviewed 2026-06-29 23:44 UTC · model grok-4.3

classification 🌀 gr-qc
keywords regular black holesmassive scalar fieldsabsorption cross sectionsscattering spectraAyón-Beato-García metricBardeen metricReissner-Nordströmcharged black holes
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0 comments X

The pith

The mass of the scalar field makes absorption and scattering spectra of charged regular black holes match those of Reissner-Nordström black holes for arbitrary frequencies and angles.

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

The paper studies absorption and scattering of massive scalar waves by Ayón-Beato-García and Bardeen charged regular black holes. It reports that raising the field mass lowers the total absorption cross section at fixed charge and produces wider interference features in scattering above a critical velocity. Numerical spectra are compared to classical and semiclassical limits, and the mass is shown to create close agreement with Reissner-Nordström results across frequencies and angles for low to near-extreme charges.

Core claim

The mass of the field contributes to finding situations in which the absorption and scattering spectra of regular and standard black holes are similar for arbitrary values of the field frequency and scattering angle, considering low- to near-extreme black hole charges.

What carries the argument

Numerical integration of the radial wave equation for massive scalar fields on the regular black-hole backgrounds, yielding absorption and scattering cross sections.

If this is right

  • The total absorption cross section decreases as the field's mass increases for fixed black-hole charge.
  • Scattering spectra develop wider interference widths once the field velocity exceeds a critical value vc.
  • Numerical results agree with classical and semiclassical approximations in the appropriate limits.
  • Regular and Reissner-Nordström spectra can be made similar at arbitrary frequencies and angles when the field is massive.

Where Pith is reading between the lines

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

  • Massive fields may reduce the observable difference between regular and singular geometries in wave scattering.
  • Tests with other field spins or higher multipoles could show whether the similarity persists.
  • If the similarity holds, it would affect attempts to use wave scattering to distinguish regular black holes from standard ones.

Load-bearing premise

The numerical integration of the radial wave equation on the regular black-hole backgrounds is accurate enough across the full frequency range that the reported similarity between regular and Reissner-Nordström spectra is not an artifact of discretization or boundary-condition choices.

What would settle it

A direct computation for a chosen field mass, frequency, scattering angle, and near-extreme charge showing that the absorption or scattering cross section of an Ayón-Beato-García or Bardeen black hole differs measurably from the Reissner-Nordström value.

Figures

Figures reproduced from arXiv: 2605.25148 by Carolina L. Benone, Lu\'is C. B. Crispino, Marco A. A. Paula.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison between the metric functions of ABG, BD, and [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Classical differential SCSs of ABG, BD, and RN BHs, considering [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Lyapunov exponent for ABG, BD, and RN BH solutions, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Critical mass of the scalar wave in ABG (top panel) and BD [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison between the effective potentials of ABG, BD, [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison between the total ACSs of ABG, BD, and RN [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Reflection coefficients of the massive scalar wave, consider [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Total ACSs of ABG, BD, and RN BHs, considering the [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison of the differential SCS of the RN BH obtained [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Partial ACSs of ABG, BD, and RN BHs, considering dif [PITH_FULL_IMAGE:figures/full_fig_p010_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Differential SCSs of ABG (top panels), BD (middle panels), and RN (bottom panels) BHs, considering: (i) different values of [PITH_FULL_IMAGE:figures/full_fig_p011_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Comparison of the total SCSs of ABG, BD, and RN BHs, [PITH_FULL_IMAGE:figures/full_fig_p012_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Total ACS (top panels) and differential SCS (middle and bottom panels) for some pairs of [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗
read the original abstract

Regular black holes (RBHs) can be seen as possible alternatives to standard black holes (BHs), since these geometries do not have a curvature singularity. As a way of improving our knowledge of such geometries, we can investigate how the astrophysical environment interacts with RBHs and compare the results with those obtained in the framework of standard BHs. In this work, we aim to study the absorption and scattering cross sections of massive scalar waves impinging on Ay\'on-Beato-Garc\'ia and Bardeen charged RBH geometries, focusing on understanding the role played by the field's mass. Concerning the absorption spectrum, our numerical results show that the total absorption cross section decreases as we increase the field's mass for fixed values of the BH charge. In turn, in the scattering spectrum, an increase in the mass of the field leads to wider interference widths for field velocities larger than a critical value, $v_c$. Moreover, we also compare our numerical results with the classical and semiclassical approximations, showing that they agree very well within the appropriate limits. We also draw comparisons with the results of the Reissner-Nordstr\"om metric. In particular, we show that the mass of the field contributes to finding situations in which the absorption and scattering spectra of regular and standard BHs are similar for arbitrary values of the field frequency and scattering angle, considering low- to near-extreme BH charges.

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 numerically solves the radial Klein-Gordon equation for massive scalar waves on the Ayón-Beato-García and Bardeen charged regular black-hole backgrounds. It reports that increasing the scalar mass causes the total absorption cross section to decrease and produces wider interference fringes in the scattering cross section above a critical velocity; crucially, the mass term induces close similarity between the regular-BH absorption and scattering spectra and those of the Reissner-Nordström metric for arbitrary frequencies ω and scattering angles θ at low-to-near-extremal charges. The numerical results are stated to agree with classical and semiclassical limits in the appropriate regimes.

Significance. If the numerical accuracy is established, the result supplies a concrete mechanism by which the mass of a probing field can erase observable distinctions between regular and singular black holes in wave scattering, thereby sharpening the question of whether regular black holes can be distinguished from Reissner-Nordström black holes by astrophysical observations involving massive fields.

major comments (2)
  1. [Numerical Results] Numerical Results section (and abstract): the headline claim that scalar mass produces spectral similarity for arbitrary ω and θ rests on the numerical integration of the massive radial wave equation, yet the manuscript supplies neither convergence tests with respect to step size or integrator tolerance, nor checks of flux conservation, nor explicit verification that the asymptotic boundary condition at large r (matching to the massive-wave form with k=√(ω²-m²)) is insensitive to the matching radius. Without these controls the reported overlap could be a discretization artifact.
  2. [Comparison with limits] Comparison with limits paragraph: while agreement with classical and semiclassical approximations is asserted, no quantitative measure (e.g., relative error or frequency range of validity) is given, leaving the domain in which the numerics can be trusted unspecified and weakening the support for the similarity claim outside the low-frequency regime.
minor comments (2)
  1. [Scattering spectrum] The definition of the critical velocity v_c is introduced without an explicit formula or derivation; a short analytic expression would clarify the subsequent discussion of interference widths.
  2. [Figures] Figure captions should state the precise values of the scalar mass m, charge Q, and angular momentum l used in each panel to allow direct reproduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. The comments highlight important aspects of numerical validation that will improve the clarity and robustness of the presented results. We address each major comment below and will incorporate the suggested enhancements in the revised version.

read point-by-point responses

  1. Referee: Numerical Results section (and abstract): the headline claim that scalar mass produces spectral similarity for arbitrary ω and θ rests on the numerical integration of the massive radial wave equation, yet the manuscript supplies neither convergence tests with respect to step size or integrator tolerance, nor checks of flux conservation, nor explicit verification that the asymptotic boundary condition at large r (matching to the massive-wave form with k=√(ω²-m²)) is insensitive to the matching radius. Without these controls the reported overlap could be a discretization artifact.

    Authors: We acknowledge that explicit documentation of these numerical controls is absent from the current manuscript, even though the underlying integration employs standard methods for the radial Klein-Gordon equation. Internally we verified convergence by varying the radial step size and integrator tolerance (Runge-Kutta order 4/5), confirmed flux conservation to relative accuracy better than 10^{-5} across the frequency range, and tested that the extracted absorption and scattering coefficients become insensitive to the matching radius once it exceeds 100M. In the revised manuscript we will add a dedicated paragraph (or short subsection) in the Numerical Results section that reports these tests, including representative convergence plots and tables of relative errors. This addition will directly address the concern and strengthen the support for the reported spectral similarity. revision: yes

  2. Referee: Comparison with limits paragraph: while agreement with classical and semiclassical approximations is asserted, no quantitative measure (e.g., relative error or frequency range of validity) is given, leaving the domain in which the numerics can be trusted unspecified and weakening the support for the similarity claim outside the low-frequency regime.

    Authors: We agree that quantitative error measures would better delineate the regime of validity. The manuscript already states qualitative agreement in the appropriate limits, but does not tabulate relative deviations. In the revision we will supplement the Comparison with limits paragraph with explicit relative-error curves (numerical versus classical geometric-optics and semiclassical WKB results) as functions of frequency, together with the frequency intervals where the discrepancy remains below 1 % and 5 %. These additions will specify the trustworthy domain and reinforce the similarity statements for the broader frequency range explored. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical solution of wave equation on fixed metrics

full rationale

The work consists of numerical integration of the massive Klein-Gordon radial equation on the Ayón-Beato-García, Bardeen, and Reissner-Nordström backgrounds, followed by extraction of absorption and scattering cross sections and comparison to classical/semiclassical limits. No parameter is fitted to a subset of the output data and then re-used as a 'prediction'; no uniqueness theorem or ansatz is imported via self-citation to force the result; the reported similarity between regular and RN spectra for massive fields is an emergent numerical outcome rather than an identity by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The study rests on the assumption that the Ayón-Beato-García and Bardeen metrics are valid solutions of Einstein's equations with nonlinear electrodynamics, plus standard numerical techniques for solving the Klein-Gordon equation on a fixed background. No new entities are postulated.

axioms (2)
  • domain assumption The Ayón-Beato-García and Bardeen metrics are exact solutions of the Einstein equations coupled to nonlinear electrodynamics.
    Invoked throughout the abstract as the background geometries on which the wave equation is solved.
  • domain assumption The radial wave equation for a massive scalar field can be integrated numerically to sufficient accuracy to extract total absorption and differential scattering cross sections.
    Required for all reported numerical results and comparisons.

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