Scattering of massless scalar waves by higher-dimensional regular black holes in general relativity coupled to nonlinear electrodynamics

Askar Abdikamalov; Bobir Toshmatov; Sardor Tojiev

REVIEW

Scattering of massless scalar waves by higher-dimensional regular black holes in general relativity coupled to nonlinear electrodynamics

Bobir Toshmatov, Askar Abdikamalov, Sardor Tojiev

General Relativity and Quantum Cosmology · Submitted 30 Oct 2026

Pith's verdict: A genuinely new fixed-frequency forward law and D-dimensional glory formula, built on an Abel summation whose regulator independence is asserted rather than shown; worth refereeing with one focused demand.

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arXiv ID2610.41972 (version 2)

Pith ID7YFSDGBO, Pith's permanent record of this paper and its review

CitationAskar Abdikamalov, Bobir Toshmatov, Sardor Tojiev. "Scattering of massless scalar waves by higher-dimensional regular black holes in general relativity coupled to nonlinear electrodynamics." arXiv:2610.41972 (2026). https://pith.science/paper/7YFSDGBO

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Editor's note

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Two things to know. The main new results are real and clean: at fixed frequency the scalar-wave cross section diverges as θ^{-4} in D≥5 with a mass-fixed, charge-independent coefficient, while the steeper classical law θ^{-(D-2)^2/(D-3)} is only an eikonal limit with a transition scale θ* ∝ ω^{-(D-3)/(D-4)}; and the D-dimensional glory formula (21) is a genuine generalization of the standard 4D one, where the forward law is just the familiar θ^{-4} and the order-of-limits issue never surfaces. The second thing is where the paper is exposed: the θ^{-4} law comes from Abel summation of a partial-wave series that does not converge in the ordinary sense for D=5, and the paper asserts rather than demonstrates that the leading singularity is independent of the summation prescription. That is the claim a referee should push on.

Credit where it is due. Appendix A derives the glory formula carefully, and the partial-wave results agree with it at the few-percent level. Appendix B — subtract the known large-l Born tail, resum it in closed form with Gegenbauer generating functions — is detailed enough to follow, and the numerical checks (convergence in l_c, matching radius, 0.1% backward stability) are reported honestly. There are no fitted parameters. The authors also scope their claims properly: the NED well-posedness question for near-extremal backgrounds is flagged as open, and they state plainly that the results concern the exterior geometry only and that the charge scan compares across the metric family rather than within one theory. The approximate ten-percent degeneracy between regular and RNT exteriors is modest but honestly presented.

The soft spots, in proportion. The Abel-summation issue is load-bearing. The backup check in Appendix B is an Abel-damped evaluation — a second use of the same regulator, not an independent one. The D=7 series is better behaved than D=5 but is still not absolutely convergent, and the numerical evaluation shares the subtraction machinery, so the coefficient check against 'numerical' results does not fully test the prescription. That said, I found no sign of an error: the physics (large-b eikonal phase fixed by the mass term) makes θ^{-4} plausible, and the claim may be regulator-independent. The paper should be asked to show it — either through the eikonal-phase argument or with a genuinely different damping. Minor: no code or data archive ('available upon request'), and the metric family comes from the authors' own preprints, though the metric function is stated in full so the calculation is self-contained. The near-extremal α≈1 configurations fall inside the regime the authors themselves flag; the scoping is correct, but the charge-dependence conclusions do lean on those points.

This is a careful, competent paper for people working on higher-dimensional black-hole scattering and on whether regular exteriors are observable. It deserves a serious referee. Send it out with one focused request: demonstrate the regulator independence of the forward law, and release the code or data behind the main figures.

Referee report

Significance

If the central claims hold, the paper establishes a clean, falsifiable universality statement: in D≥5 the fixed-frequency forward divergence of the wave cross section is θ^{-4} in all dimensions, charge-independent, and fixed by the mass, with the geometric-optics law appearing only as the eikonal limit with a dimension-dependent transition angle. This is a useful organizing result for higher-dimensional black-hole scattering, and the D-dimensional glory formula (21)-(22), the phase-shift expansions (B4), and the explicit summation scheme are valuable in themselves. The manuscript is unusually transparent: the appendices give the derivations in detail; the numerical method is described with convergence checks (claimed 0.1% stability of the backward cross section, matching-radius and parity checks); no parameters are fitted to produce the reported cross sections; and the authors state the scope and limitations clearly (neutral test field in the bulk, exterior geometry only, open question on NED well-posedness for near-extremal backgrounds, and neglected winding-ray contributions).

Major objections (2)

[Sec. IV.B and Appendix B, Eqs. (18), (19), (B5)-(B6)] The headline forward law (19) is extracted from a partial-wave series that the authors themselves state does not converge in the ordinary sense for D=5 and converges only conditionally for D=6 (Sec. IV.B). The paper is explicit that the sum is defined by Abel summation with a wave-packet interpretation, and Appendix B supplies a detailed subtraction scheme, so this is not a missing-proof objection; the issue is the regulator dependence of the leading singularity. The reported cross-checks — stability under l_c, its parity, the fitted term, and agreement with the "Abel-damped evaluation" of the backward amplitude — all operate within the same Abel-type regularization family, and the backward amplitude check does not directly constrain the forward singularity. I therefore ask for an independent derivation or check of Eq. (19): the most direct is the eikonal/Fourier argument in which the small-θ amplitude is the (D-2)-dimensional Fourier transform of exp(2iδ(ωb))-1 with δ ∝ b^{-(D-4)}, which should yield θ^{-4} in every dimension with the mass-determined coefficient; alternatively, a hard-cutoff evaluation or a comparison of the Tangherlini α=0 case with existing higher-dimensional results would serve. Because a different physically motivated summation could in principle change the leading singularity or the coefficient, this point is load-bearing for the universality and charge-independence claims.
[Eq. (19)] The displayed coefficient is internally inconsistent. As printed, C_D = 2^{D-2} Γ^2(D/2) Γ^2(D-2) gives C_5 = 8 × (9π/16) × 4 = 18π, whereas the text states C_5 = 9π/8; the stated value corresponds instead to C_D = 2^{D-2} Γ^2(D/2)/Γ^2(D-2). Since Sec. V.B says the coefficient "has been checked against the numerical results," the formula and the numerical check must be reconciled and the correct value confirmed, because a factor-of-16 ambiguity in the magnitude of the central prediction is material.

Minor objections (5)

[Sec. V.B] The statement that "the comparison for several frequencies and charges is available upon a reasonable request" does not meet current reproducibility standards; the forward-coefficient check and the claimed 0.1% backward stability should be supported by ancillary data files or a repository.
[Sec. V.C] The sentence "for D=6 and D=7 at this frequency, the glory formula has no zero at all" should be clarified, since J_μ has infinitely many zeros; the intended statement is that the first zero lies outside the plotted or observable range because ω b_g < j_{μ,1}.
[Sec. V.D and Fig. 13] The claimed few-percent and ten-percent differences between the regular and RNT cross sections would be directly verifiable if a panel showing the relative difference of the two cross sections as a function of θ were added.
[Sec. IV.B] The Abel summation used to define Eq. (18) is described only verbally; one displayed line, such as h(θ) = lim_{ρ→1^-} Σ_l ρ^l (l+λ)(S_l - 1) C_l^λ(cosθ), would remove ambiguity about which Abel average is meant.
[Sec. V.D] The claim that near-extremal regular BHs have no RNT partner with the same b_g deserves one explanatory sentence, since as written the reader must infer that the matching would require α > 1 in the RNT family.

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The paper

The paper claims

The forward scattering of scalar waves off higher-dimensional black holes diverges as $\theta^{-4}$ at fixed frequency, with a mass-only coefficient; the classical law is only an eikonal limit.

The reading

At a fixed frequency, the forward scattering cross section of a massless scalar wave off any member of this family of higher-dimensional regular black holes diverges as $\theta^{-4}$ in every dimension $D\geq5$, with a coefficient $C_D M^2\omega^{D-4}$ fixed by the mass alone, independent of charge and of the Bardeen/Hayward model parameter. The steeper geometric-optics law $\theta^{-(D-2)^2/(D-3)}$ is not the forward asymptotics at fixed frequency; it is the eikonal limit, reached only at angles $\theta\gg\theta_*$ with $\theta_*\propto\omega^{-(D-3)/(D-4)}$. The paper combines null-geodesic scattering, a glory approximation generalized to $D$ dimensions, and numerical partial-wave analysis in $D=5,6,7$, regularizing the poorly convergent partial-wave series by Abel summation plus an analytic large-$l$ Born subtraction. It further claims that a regular (Bardeen-like) black hole and a Reissner-Nordström-Tangherlini black hole with the same glory impact parameter scatter scalar waves within about ten percent of each other, with the clearest differences in backward intensity and low-frequency absorption; the authors stress that the scattering probes only the exterior geometry, since the horizon boundary condition never reaches the regular core, and that near-extremal backgrounds may not be acceptable nonlinear-electrodynamics solutions. If right, this makes the forward law a universal, mass-only observable, while the frequency at which the classical law sets in carries the charge and geometry information.

What carries the argument

The load-bearing objects are the unified metric family $f(r)=1-Mr^2/(r^\gamma+Q^\gamma)^{(D-1)/\gamma}$, the unstable circular null geodesic with critical impact parameter $b_c$ and Lyapunov parameter $\eta_c$ (which controls the logarithmic divergence of the deflection angle and the $e^{-2\pi k\eta_c}$ suppression of the winding rays), and the partial-wave amplitude expanded in Gegenbauer polynomials $C_l^{\lambda}(\cos\theta)$ with $\lambda=(D-3)/2$ on the $(D-2)$-sphere. The series converges poorly for $D\geq5$ (in $D=5$ it does not converge in the ordinary sense), so the paper defines it by Abel summation: the analytically known large-$l$ Born expansion of the coefficients, obtained from Bessel-function integrals, is subtracted and then added back in closed form via the generating function of the Gegenbauer polynomials, leaving a remainder that decays fast enough even at $\theta=\pi$. The forward $\theta^{-4}$ law follows from the leading Born term together with $\Sigma_{D-5}(x)\simeq\theta^{-2}/(D-5)!$ near $x=1$. The $D$-dimensional glory formula follows from stationary phase applied to the Gegenbauer (Hilb-type) asymptotics, with the glory ray $\Theta(b_g)=\pi$ as the stationary point; numerical radial integration with ingoing conditions at the horizon supplies the phase shifts and the absorption cross section, whose low-frequency limit is the horizon area.

Extended reading notes

Core claim

The paper's central claim is small-angle universality of the wave cross section: $d\sigma/d\Omega_{D-2}\simeq C_D M^2\omega^{D-4}\theta^{-4}$ as $\theta\to0$ at fixed $\omega$, with $C_D=2^{D-2}\Gamma^2(D/2)\Gamma^2(D-2)$ and $C_5=9\pi/8$, for the unified metric family $f(r)=1-Mr^2/(r^\gamma+Q^\gamma)^{(D-1)/\gamma}$ in any $D\geq5$. The physical mechanism is the universality of the high-multipole phase shifts, $\delta(\nu)\simeq\sqrt{\pi}\,\Gamma(D/2)\,M\omega^{D-3}/[2(D-4)\Gamma((D-1)/2)\,\nu^{D-4}]$, which depend on the mass only, so the forward amplitude forgets the charge; charge enters only through subleading corrections to the phase shifts and through the low multipoles. Because the scalar field is neutral and propagates in the bulk, the scattering is governed by the unstable circular null geodesic of the spacetime metric, not by the effective-geometry photonsphere that would control electromagnetic rays. Derived alongside is the $D$-dimensional glory cross section, $d\sigma/d\Omega\simeq2\pi\omega b_g^{D-2}|db/d\theta|_{\theta=\pi}J_{(D-4)/2}^2(\omega b_g\sin\theta)/\sin^{D-4}\theta$, which is finite at $\theta=\pi$, grows as $\omega^{D-3}$, and widens its fringes as the charge increases; in higher dimensions the pattern has fewer, wider fringes and weaker backward intensity, and the first minimum may disappear. Finally, for matched glory impact parameters, a five-dimensional regular black hole and an RNT black hole differ by less than ten percent in the scalar cross section, with low-frequency absorption (set by the horizon area) and backward intensity as the discriminators, an approximate rather than exact degeneracy that the paper presents as the $D$-dimensional counterpart of earlier four-dimensional results.

Load-bearing premise

The load-bearing premise is that the partial-wave amplitude in $D\geq5$, a series that does not converge in the ordinary sense, is correctly defined by the paper's Abel summation with its large-$l$ Born subtraction, so that the $\theta^{-4}$ forward singularity of that regularized sum is the physical one; the paper cross-checks against Abel-damped evaluations, but the universality claim is not proven as a theorem about the unregularized scattering amplitude.

Editorial extensions

If this is right

At fixed frequency the forward divergence is $\theta^{-4}$ in any dimension $D\geq5$, with the mass-only coefficient $C_D M^2\omega^{D-4}$; charge and the Bardeen/Hayward model parameter cancel out at small angles.
The classical law $\theta^{-(D-2)^2/(D-3)}$ is an eikonal limit, not the fixed-frequency forward asymptotics; the transition angle $\theta_*\propto\omega^{-(D-3)/(D-4)}$ shrinks with frequency, and higher dimensions need higher frequencies before the classical slope appears.
Backward glory scattering is finite in every dimension, grows as $\omega^{D-3}$, and has fringe spacing $\Delta\theta\simeq\pi/(\omega b_g)$; charge widens the fringes, while higher dimensions give fewer, wider fringes and a weaker backward intensity.
A five-dimensional regular (Bardeen-like) black hole and an RNT black hole with the same glory impact parameter scatter massless scalars within about ten percent at all angles, and absorb within one percent at high frequencies; low-frequency absorption and backward intensity are the distinctions.
In the high-frequency limit the wave cross section approaches the classical one outside a forward cone of shrinking angular size and outside a backward region of size $1/(\omega b_g)$, where the wave cross section stays finite.

Reading between the lines

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

The mass-only forward law should extend to any spherically symmetric asymptotically flat black hole in $D\geq5$ with the Schwarzschild-Tangherlini tail, since it follows from the universal high-multipole phase shifts; testing a different family (for example a higher-curvature or dilaton black hole) would show whether the charge-independence found here is truly universal.
The two-parameter structure suggests a practical separation of observables: at fixed frequency the forward slope measures only the mass, while the frequency at which the classical slope sets in measures the critical impact parameter and hence the charge, in principle a way to probe exterior geometry without resolving the photonsphere.
For electromagnetic waves, whose rays follow the effective optical geometry of the nonlinear electrodynamics rather than the spacetime null geodesics, the glory formula should be reapplied with the glory impact parameter of the effective geometry; the authors flag this as a forthcoming problem.
The winding-ray interference, suppressed by $e^{-2\pi k\eta_c}$ and already a few percent for the near-extremal five-dimensional regular black hole, is the natural first candidate for the dominant systematic correction in any precision application of the glory formula.

Original abstract

We study the scattering of planar massless scalar waves by the family of higher-dimensional regular black holes in general relativity coupled to nonlinear electrodynamics, which unifies the Bardeen and Hayward models in any dimension. The scalar field is a neutral test field that propagates in the bulk, and in the geometric-optics limit, its rays are the null geodesics of the spacetime, so that the unstable circular null geodesics of the background geometry control the scattering. We combine the classical scattering of null geodesics, the glory approximation, which we generalize to $D$ dimensions, and the partial-wave analysis in $D=5,6,7$. We show that at a fixed frequency, the wave cross section diverges in the forward direction as $\theta^{-4}$ in any dimension, with a coefficient fixed by the mass, while the steeper classical law $\theta^{-(D-2)^{2}/(D-3)}$ appears only at larger angles and higher frequencies. The charge parameter decreases the critical and the glory impact parameters and widens the glory fringes, and in higher dimensions, the glory pattern has fewer and wider fringes and a weaker backward intensity. For the matched configurations studied here, backward intensity and low-frequency absorption provide the clearest differences, while the intermediate-to-high-frequency spectra are close.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-02: 1 request, 5,234 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.50 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.92 · compares with prior results 0.41 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study the scattering of planar massless scalar waves by the family of higher-dimensional regular black holes in general relativity coupled to nonlinear electrodynamics, which unifies the Bardeen and Hayward models in any dimension. The scalar field is a neutral test field that propagates in the bulk, and in the geometric-optics limit, its rays are the null geodesics of the spacetime, so that the unstable circular null geodesics of the background geometry control the scattering. We combine the classical scattering of null geodesics, the glory approximation, which we generalize to $D$ dimensions, and the partial-wave analysis in $D=5,6,7$. We show that at a fixed frequency, the wave cross section diverges in the forward direction as $\theta^{-4}$ in any dimension, with a coefficient fixed by the mass, while the steeper classical law $\theta^{-(D-2)^{2}/(D-3)}$ appears only at larger angles and higher frequencies. The charge parameter decreases the critical and the glory impact parameters and widens the glory fringes, and in higher dimensions, the glory pattern has fewer and wider fringes and a weaker backward intensity. For the matched configurations studied here, backward intensity and low-frequency absorption provide the clearest differences, while the intermediate-to-high-frequency spectra are close.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the scattering of planar massless scalar waves by the family of higher-dimensional regular black holes in general relativity coupled to nonlinear electrodynamics, which unifies the Bardeen and Hayward models in any dimension.	contribution 0.99	0.86	0.06	asserted 0.96
s1	The scalar field is a neutral test field that propagates in the bulk, and in the geometric-optics limit, its rays are the null geodesics of the spacetime, so that the unstable circular null geodesics of the background geometry control the scattering.	background 0.50	0.15	0.05	asserted 1.00
s2	We combine the classical scattering of null geodesics, the glory approximation, which we generalize to $D$ dimensions, and the partial-wave analysis in $D=5,6,7$.	method 0.95	0.93	0.05	asserted 0.93
s3	We show that at a fixed frequency, the wave cross section diverges in the forward direction as $\theta^{-4}$ in any dimension, with a coefficient fixed by the mass, while the steeper classical law $\theta^{-(D-2)^{2}/(D-3)}$ appears only at larger angles and higher frequencies.	result 1.00	0.94	0.45	asserted 1.00
s4	The charge parameter decreases the critical and the glory impact parameters and widens the glory fringes, and in higher dimensions, the glory pattern has fewer and wider fringes and a weaker backward intensity.	result 1.00	0.92	0.12	asserted 1.00
s5	For the matched configurations studied here, backward intensity and low-frequency absorption provide the clearest differences, while the intermediate-to-high-frequency spectra are close.	result 0.67	0.93	0.33	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.91.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Null geodesics around the five-dimensional Bardeen-like

**Figure 2.** FIG. 2. Critical impact parameter

**Figure 3.** FIG. 3. Deflection angle of null geodesics for

**Figure 4.** FIG. 4. Classical scattering cross section (

**Figure 5.** FIG. 5. Effective potential of the scalar field for the five-dimensional

**Figure 6.** FIG. 6. Effective potential for

**Figure 7.** FIG. 7. Real parts of the phase shifts for

**Figure 8.** FIG. 8. Scattering cross section of the scalar field with

**Figure 9.** FIG. 9. Scattering cross section for

**Figure 11.** FIG. 11. Scattering cross section for

**Figure 12.** FIG. 12. Scattering cross section for the Schwarzschild-Tangherlini

**Figure 13.** FIG. 13. Scattering cross sections of the five-dimensional Bardeen-like BH with

**Figure 14.** FIG. 14. Absorption cross sections of the five-dimensional Bardeen

Showing 13 of the paper's 14 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.41972 by the authors.

Discussion

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