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Scattering and absorption of massless scalar waves by a ModMax black hole

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Pith's one-line read Massless scalar waves scatter off a ModMax black hole as off a Reissner-Nordström hole with a screened charge.

desk verdict Routine but solid partial-wave calculation for ModMax black holes, undercut by a dimensionally wrong printed Lyapunov exponent that must be fixed before the high-frequency absorption curves can be trusted. read the letter →

arxiv 2506.17489 v1 pith:2PBGPRYN submitted 2025-06-20 gr-qc

Arturo Quispe Baptista , M. L. Peñafiel This is my paper · ORCID
classification gr-qc
keywords ModMaxelectrodynamicsmasslessscalarwavesblackholescatteringabsorptioncrosssectionpartialwavemethodchargescreeningReissner-Nordströmglory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether scattering and absorption of massless scalar waves can distinguish a ModMax black hole from an ordinary Reissner-Nordström black hole. The ModMax metric is the Reissner-Nordström form with the charge term multiplied by $\beta=e^{-\gamma}$, so the partial-wave computation becomes the known charged-black-hole calculation with an effective charge $Q_{\mathrm{eff}}^2=\beta Q_e^2$ (or $\beta(Q_e^2+Q_m^2)$ in the dyonic case). The central claim is that this screened charge is visible in the cross sections: for a fixed charge, smaller $\beta$ pushes the scattering and absorption curves toward the Schwarzschild limit, while $\beta\to 1$ recovers Maxwell electrodynamics. The paper checks the numerics against the classical geodesic, glory, low-frequency horizon-area, and high-frequency sinc approximations and reports agreement in every regime.

What carries the argument

The load-bearing object is the radial metric function $f(r)=1-2M/r+\beta(Q_e^2+Q_m^2)/r^2$, with $\beta=e^{-\gamma}$, which turns the electromagnetic charge into a screened Coulomb-type term. On this background the scalar Klein-Gordon equation is separated with spherical harmonics, transformed to tortoise coordinate $r_*$, and solved numerically from near the horizon to large radius; the asymptotic coefficients $A_{\mathrm{in}}$ and $A_{\mathrm{out}}$ define the phase shifts $e^{2i\delta_\ell}=(-1)^{\ell+1}A_{\mathrm{out}}/A_{\mathrm{in}}$. The partial-wave sums for the differential scattering cross section are regularized by an iterative series-regularization method, and the absorption cross section is summed from the ratio $|A_{\mathrm{out}}/A_{\mathrm{in}}|^2$. Analytical anchors are the classical deflection formula, the glory approximation with impact parameter $b_g$, the low-frequency horizon-area formula, and the high-frequency sinc formula with the null-geodesic instability exponent $\lambda$ and orbital frequency $\Omega_c$; these are what turn the screened metric into concrete, checkable cross sections.

What would settle it

Recompute any single partial-wave phase shift, say $\delta_0(\omega)$, directly from the full Einstein-ModMax field equations rather than from the screened metric; if it deviates from the Reissner-Nordström-with-effective-charge prediction beyond numerical error, the central claim fails. A concrete observational check is to measure the weak-field deflection angle $\Theta\approx 4M/b+3\pi(5M^2-\beta Q_e^2)/(4b^2)$ or the low-frequency absorption cross section for a black hole with known mass and charge, and test whether the inferred $\beta Q_e^2$ combination is consistent with the screening model.

Watch

Extended reading notes

Core claim

The paper's central claim is that the full partial-wave scattering and absorption cross sections for massless scalar waves on the ModMax black hole have the same structure as those of Reissner-Nordström with the charge replaced by a screened value, $Q^2\to\beta Q^2$. Consequently, for any fixed charge-to-mass ratio $q$, decreasing $\beta=e^{-\gamma}$ (increasing the ModMax nonlinearity) moves the scattering pattern, glory oscillations, and absorption curve toward the Schwarzschild baseline, and different $(\beta,q)$ pairs can produce the same scattering features. The paper demonstrates this by solving the radial Klein-Gordon equation, extracting the phase shifts $e^{2i\delta_\ell}$ from asymptotic matching, and comparing with analytical approximations. In the $\beta\to 1$ limit the results match the existing Reissner-Nordström scalar-wave results, which the paper treats as validation of the method.

Load-bearing premise

The computation assumes, without re-derivation, that $f(r)=1-2M/r+\beta Q_e^2/r^2$ is the true static black hole solution of Einstein-ModMax theory and that a massless scalar field minimally coupled to gravity is the right probe; if either premise fails, every phase shift, cross section, and approximation in the paper changes.

Editorial extensions

If this is right

  • In the low-frequency limit the absorption cross section is set by the horizon area, which grows as $\beta$ decreases; charge screening therefore makes low-energy absorption larger for a fixed charge.
  • The glory peak amplitude has a local minimum at the same value for every $\beta$, reached at different charge configurations, so glory measurements can fix the screening strength only if the charge is known independently.
  • The classical and glory approximations bracket the full numerical scattering cross section in their respective angular regimes for every $\beta$ tested, so the semiclassical picture is reliable for this spacetime.
  • Because the scalar field couples only through the metric, these results carry no imprint of ModMax vacuum birefringence; the paper explicitly expects photon and electromagnetic-wave scattering to show such departures, an extension left for future work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Not stated in the paper, but because the only change is $Q^2\to\beta Q^2$, the whole computation is a one-parameter map of the Reissner-Nordström calculation; any existing RN phase-shift code can be reused by inputting an effective charge, making the numerical result straightforward to cross-check.
  • A direct consequence not pursued here is a degeneracy: for fixed mass, $\beta$ and $q$ enter only through the combination $\beta q^2$, so a single scalar-wave scattering observation cannot separately determine the charge and the ModMax parameter; independent measurements or multi-channel probes would be needed.
  • A concrete extension would be to compute photon scattering with the full ModMax field equations: vacuum birefringence should split the phase shifts by polarization, giving a two-channel signature that scalar waves cannot provide.
  • The weak-field deflection formula could be confronted with precision lensing observations; if mass and charge were measured independently, the term $\beta Q_e^2$ would place an observational bound on the ModMax parameter $\gamma$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper studies the scattering and absorption of massless scalar waves by the static, spherically symmetric ModMax black hole. It computes null geodesics, the classical deflection angle, the glory approximation, and full partial-wave numerical cross sections, comparing them with the Reissner-Nordström limit and with analytic approximations. The central physical message is that the ModMax parameter β=e^{-γ} acts as an effective screening of the electromagnetic charge, making the cross sections interpolate between Reissner-Nordström and Schwarzschild behavior.

Significance. If the results stand after correction, this is a useful first partial-wave treatment of scalar wave scattering by ModMax black holes. The paper follows standard and well-tested methods, contains no free parameters, and explicitly validates the numerical scheme by recovering Reissner-Nordström results (Fig. 7) and the known Schwarzschild weak-field deflection limit (Eq. (17)). The screening interpretation is physically suggestive. However, the high-frequency analytical approximation contains a dimensional error that currently makes the agreement claimed in Fig. 10 non-reproducible, so the present version cannot be accepted as is.

major comments (1)
  1. [Sec. III B, Eqs. (30)-(32)] The printed Lyapunov exponent in Eq. (32) is dimensionally inconsistent. Since f_c is dimensionless, b_c has units of length, and f''_c has units of inverse length squared, the right-hand side of Eq. (32) scales as L^{-1/2}, whereas a Lyapunov exponent must scale as L^{-1} in geometrized units. In the Schwarzschild limit (β=1, Q=0, M=1, r_c=3, b_c=3√3), the printed formula gives λ≈1.075, while the standard value from Ref. [57] is λ=1/(3√3)≈0.192. The correct expression is λ = sqrt{ f_c (2 - b_c² f''_c)/(2 b_c²) } = sqrt{ (f_c/2)(2 f_c/r_c² - f''_c) }. Because Eq. (31) uses λ both in the prefactor λ/Ω_c³ and in the damping exponent e^{-πλ/Ω_c}, the dot-dashed curves in Fig. 10 and the claim that the high-frequency approximation agrees with the partial-wave results depend on this quantity. Please correct Eq. (32) and either confirm that Fig. 10 was generated with the corrected expression or, if it used the printed formula, recompute the figure and the related text.
minor comments (5)
  1. [Sec. II, Eq. (6)] The reduction to G=0 appears to have the wrong sign for a standard electric configuration. With the convention F<0 for a purely electric field, |F|=-F in Eq. (2) gives L=-F e^{γ}/2, not -F e^{-γ}/2. Additionally, the text defines L_Max=-F/8π just above, while Eq. (6) identifies L_Max with -F/2. Please clarify the sign and normalization conventions.
  2. [Sec. II B, after Eq. (17)] The sentence 'Since β ≤ 1, for a given Q_e, the deflection angle for the ModMax BH is smaller than the case of RN' is opposite to what Eq. (17) shows: smaller β makes the charge term less negative, so the deflection is larger, not smaller, than in the RN case. Please correct this sentence.
  3. [Fig. 8 caption] The caption of Fig. 8 appears to be a copy of the Fig. 7 caption ('Differential scattering cross section for RN spacetime'), although the text describes Fig. 8 as displaying ModMax results. Please replace the caption.
  4. [Fig. 10 caption] The caption of Fig. 10 does not identify which β and q values correspond to each panel or curve. Please specify the parameters and clearly describe the solid, dashed, and dot-dashed curves.
  5. [Eq. (17)] The notation O(1/b)^3 should be O(b^{-3}), and the same convention should be used consistently throughout the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the scalar cross sections are computed from the cited ModMax metric and standard partial-wave methods, with no fit or self-citation chain feeding the output back into the input.

full rationale

The paper's derivation chain is linear and self-contained. The input is the static spherically symmetric ModMax black hole metric taken from Refs. [37,42] (Eqs. (7)-(9)), and the dynamical equation is the standard minimally coupled Klein-Gordon equation (Eq. (19)) solved by partial waves. The scattering and absorption cross sections (Eqs. (25)-(28)) are direct numerical outputs of this boundary-value problem; nothing in them is fitted to the target result or defined in terms of the cross sections themselves. The analytical approximations (classical geodesic cross section, glory formula, geometric absorption, and sinc oscillation) are standard external results from Refs. [16,52,57], not results derived in this paper and then recycled. The Reissner-Nordström limit β=1 is used as an independent check against the published RN results [11,45], which is a genuine external benchmark. The observation that charge screening makes cross sections approach the Schwarzschild case as β=e^{-γ} decreases is a property already encoded in the metric factor Q^2e^{-γ}; the paper honestly attributes this to the ModMax Lagrangian, rather than claiming to derive the screening from the scattering calculation. There are no self-citations by the present authors that carry a load-bearing argument. The skeptical remarks about Eq. (32) having incorrect dimensions and Eq. (6) having a possible sign error are correctness concerns, not circularity: even if those formulas or typographical conventions were wrong, that would not make the computation feed its own output back as input. Accordingly, no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted constants and no invented entities. The only inputs are the physical parameters M, Q, γ (and the field frequency ω), which are varied to produce plots. The computation rests entirely on the ModMax metric from the cited literature and standard wave-scattering approximations. The absence of fitted parameters means the circularity burden is low.

assumptions (5)
  • domain assumption The ModMax BH line element (Eq. 8 or dyonic Eq. 9) is the correct static, spherically symmetric vacuum solution of Einstein-ModMax theory.
    Adopted from Ref. [37] (and Ref. [42] for the dyonic case); the paper does not rederive the metric. All subsequent wave equations and cross sections inherit this assumption.
  • domain assumption The massless scalar field is minimally coupled and obeys the Klein-Gordon equation (19).
    Introduced in Sec. III; if the scalar field coupled directly to the electromagnetic sector, the radial equation (21) would gain additional terms, changing the phase shifts and cross sections.
  • standard math The Yennie regularization procedure converges to the true partial-wave sum.
    Used in Sec. III.A; the procedure from Refs. [55] and [9] is established, but no convergence proof or convergence test is shown in this paper.
  • standard math The high-frequency absorption is given by the geodesic capture cross section plus the sinc-oscillation term (Eqs. 29-31) with the Lyapunov exponent (32).
    Taken from Refs. [16] and [57]; used as an independent approximation to check the numerical σ_abs in Fig. 10, not part of the primary partial-wave derivation.
  • domain assumption The weak-field deflection expansion (17) is valid for the classical cross section at small angles.
    Used in Sec. II.B to derive the classical differential cross section; the full numerical geodesic calculation in the same section does not rely on this expansion.

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Pith. "Pith review of Scattering and absorption of massless scalar waves by a ModMax black hole." pith.science (2026). https://pith.science/paper/2PBGPRYN

@misc{pith2026250617489,
  author       = {Pith},
  title        = {Pith review of: Scattering and absorption of massless scalar waves by a ModMax black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2PBGPRYN}},
  note         = {Machine review of arXiv:2506.17489}
}
read the original abstract

ModMax electrodynamics is a remarkable example of nonlinear electrodynamics that preserves both conformal and duality invariance. When coupled to General Relativity, the resulting black hole solutions introduce a tunable nonlinearity parameter that effectively screens the electromagnetic charge. We study the scattering and absorption of massless scalar waves by a ModMax black hole by applying the partial waves method. We also compute some well-known analytical approximations for the scattering and absorption cross sections and contrast them with the full numerical solution. Our results adequately reproduce those of Maxwell electrodynamics in the appropriate limit and illustrate the effect of the charge screening in ModMax electrodynamics.

Figures

Figures reproduced from arXiv: 2506.17489 by the authors.

Figure 1
Figure 1. FIG. 1. Radius of the horizons, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Effective potential governing radial motion of massl [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Trajectories for massless particles (other than pho [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (Left panel) Classical differential scattering cros [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: displays the glory parameter bg and b 2 g |db/dθ|θ=π, which is related to the magnitude of the glory peak, for the n = 0 mode in Eq. (18) as a function of the normalized charge for different values of β = e −γ . It can be seen that the parameter b 2 g is always a monot…
Figure 6
Figure 6. Figure 6: displays the effective potential in tortoise coordinates for different values of ℓ and different ModMax scenarios for a fixed charge q = Q/M. It can be seen that the maximum for a given charge corresponds to the RN situation and it decreases as β does, which is consist…
Figure 7
Figure 7. Figure 7: FIG. 7. Differential scattering cross section for RN spaceti [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Differential scattering cross section for RN spaceti [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Horizon area with respect to the normalized charge [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Plot of the different absorption cross sections for d [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Differential scattering cross section for Einstein [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

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