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Quasinormal modes of black holes. II. Pad\'e summation of the higher-order WKB terms

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Pith's one-line read High-order Padé summation matches black-hole frequencies to 24 digits.

desk verdict A serious, well-benchmarked extension of the WKB-Padé program to order 700; the machinery is prior work, but the high-order results and corrections are new, and the paper deserves refereeing despite an unproven convergence claim and no archived code. read the letter →

arxiv 1908.09389 v1 pith:TJOGNRRB submitted 2019-08-25 gr-qc

Jerzy Matyjasek , Malgorzata Telecka This is my paper · ORCID
classification gr-qc PACS 04.70.-s04.30.-w
keywords quasinormalmodesblackholeperturbationtheoryWKBapproximationPadéapproximantsanharmonicoscillatorSchwarzschildReissner-Nordströmovertones
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 claims that the WKB approach to black-hole quasinormal modes—long treated as only approximate—can be promoted to a highly accurate semianalytic method by diagonal Padé summation of a very long perturbation series for $\omega^2$. The authors extend the standard WKB expansion to hundreds of terms (up to $k=700$) by converting the scattering problem into an anharmonic-oscillator bound-state problem, then build diagonal Padé approximants instead of summing the raw series. They show that for Schwarzschild and Reissner-Nordström black holes the resulting frequencies agree with continued-fraction numerical results to many decimal places, 24 for the gravitational $l=2$, $n=0$ mode, and that slow stabilization means low-order WKB results can mislead. The demonstration matters because it offers a black-box, potential-in/frequencies-out route to accurate quasinormal frequencies, including overtones, without a dedicated numerical eigenvalue solver.

What carries the argument

The load-bearing object is the diagonal Padé approximant $P_k^k$ formed from the formal WKB series $\omega^2 = V(x_0) + \sum_j \varepsilon^j \tilde{\Lambda}_j$, where the $\tilde{\Lambda}_j$ are WKB correction terms built from derivatives of the effective potential at its maximum. To supply the hundreds of terms needed, the authors use the reduction of the resonance problem to a one-dimensional anharmonic oscillator and generate high-order Rayleigh-Schrödinger corrections numerically for the chosen multipole and overtone numbers. The diagonal Padé transform replaces direct summing of the series, which is a poor strategy, and an epsilon-acceleration algorithm yields the same diagonal approximants at lower computational cost.

What would settle it

Compute the Schwarzschild scalar $l=0$, $n=0$ mode with an independent high-precision method—say a pseudospectral discretization of the perturbation equation—and compare with the Padé-stabilized value; a disagreement beyond the quoted 16 digits would show the stabilization is spurious. The same check applies to any $(l,n)$ pair: a single mode whose Padé plateau differs from a reliable independent calculation in more than one decimal place would falsify the universal claim.

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Extended reading notes

Core claim

The central discovery is that the formal WKB series for the squared quasinormal frequency, once computed to sufficiently high order, is a good input to Padé summation: the diagonal transforms $P_k^k$ stabilize to the true complex frequency, and do so even for overtones that standard low-order WKB handles poorly. For the Regge-Wheeler potential the calculations cover scalar, electromagnetic, and gravitational perturbations of Schwarzschild, and for Reissner-Nordström the results match both Borel-Padé and continued-fraction computations. The paper also finds that the longstanding benchmark value for the scalar $l=0$, $n=0$ mode is wrong: the Padé and continued-fraction codes agree on a corrected value to 16 decimal places. The convergence behavior is demonstrated empirically, not proved.

Load-bearing premise

The method assumes that the diagonal Padé transforms of the formal high-order WKB series converge to the true quasinormal frequency; the paper demonstrates this empirically for the cases it studies but does not prove it.

Editorial extensions

If this is right

  • If the stabilization is genuine, low-order WKB results cannot be trusted on their own; only high-order Padé sequences that show a clear plateau are reliable.
  • The method extends with little change to Reissner-Nordström and likely to other spherically symmetric potentials, giving a single black-box pipeline for accurate mode frequencies.
  • Accurate overtones up to at least $n=5$ for low $l$ become accessible to semianalytic WKB techniques, relaxing the usual $n \lesssim l$ rule of thumb.
  • The corrected scalar $(0,0)$ and $(1,0)$ frequencies show that high-order Padé plus continued-fraction agreement can serve as a cross-check on published numerical values.

Reading between the lines

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

  • A natural testable extension is to push the same high-order Padé machinery toward the algebraically special modes the paper says WKB cannot reach; whether any plateau appears with even longer series would reveal whether that limitation is quantitative or structural.
  • The paper computes Borel transforms only for selected cases; a systematic comparison of Padé and Borel-Padé stabilization across modes could show whether the cheaper Padé-only workflow is sufficient whenever a plateau appears.
  • The anharmonic-oscillator equivalence suggests the resummation should transfer to other one-dimensional scattering problems with locally harmonic, asymptotically constant potentials, such as effective potentials in modified gravity or wormhole spacetimes, though that transfer is not demonstrated here.
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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

3 major / 6 minor

Summary. This paper extends the WKB-based method for computing quasinormal modes introduced in the authors' previous work, Paper I. The idea is to generate high-order terms in the formal expansion for ω², up to order 700, using the equivalence with an anharmonic-oscillator perturbation problem and the Bender–Wu/Sulejmanpasic–Ünsal toolkit, and then to sum the resulting formal series with diagonal Padé approximants. The authors compare their results for scalar, electromagnetic, and gravitational perturbations of the Schwarzschild and Reissner–Nordström black holes against continued-fraction results, reporting agreement to up to 24 decimal places for some modes and correcting two values from Andersson's tables. They also discuss the use of the Wynn epsilon algorithm and the Borel–Padé method.

Significance. The strength of the paper is its systematic numerical demonstration: Tables II–XIV show that high-order diagonal Padé transforms of WKB series reproduce known continued-fraction quasinormal frequencies, including overtones, with no free parameters fitted to the target frequencies. The method is simple and black-box in structure, and the paper's cross-checks with Borel–Padé summation and continued fractions are valuable. The main gap is that the convergence of the Padé-summed series to the physically correct frequency is not established, and the paper itself flags a case where stabilization agrees with the authors' own code but disagrees with a published value. Because the method is proposed as a tool for modes that may lack independent benchmarks, this gap is load-bearing.

major comments (3)
  1. [Section III.A, Figs. 1–2 and Appendix Eq. (A.4)] The stabilization of the diagonal sequence P_k^k is the central evidence for the method, but the paper reports no neighboring Padé approximants, such as entries with numerator and denominator degrees differing by one. The appendix explicitly notes that the Wynn algorithm yields only one family of approximants and that other entries of the Padé table must be constructed independently. Without checking that neighboring approximants converge to the same limit, the possibility that the diagonal sequence converges to a different analytic continuation of the divergent series remains open. Please add such a check for at least one low-overtone and one high-overtone case.
  2. [Section III.A, scalar (0,0) mode, Table II] The resolution of the discrepancy with Andersson's published value rests on the authors' own continued-fraction code. The agreement between the Padé result and their continued-fraction result to 16 decimal places is impressive, but no implementation details, convergence criteria, or error estimates for that code are provided. To make the correction convincing to readers, an independent high-precision value from a published and independently implemented method, or a reproducible notebook with machine-precision settings, should be supplied.
  3. [Section II, Eq. (3) and Section IV] The black-box claim in the abstract and in the Final Remarks presupposes that the Padé-summed series converges to the true ω². The paper provides no argument from the structure of the series, such as Stieltjes-type properties, and no criterion for detecting when stabilization is trustworthy. Figs. 1–2 show that apparent stabilization can be slow and that low-order results are unreliable, and the paper admits that the behavior for highly damped modes is unknown. The manuscript should either narrow the claimed domain of applicability or provide a practical consistency test, such as agreement between neighboring Padé approximants and between Padé and Borel–Padé sums.
minor comments (6)
  1. [Abstract and Introduction] The word 'quasiormal' appears in the abstract and Introduction; it should be 'quasinormal'.
  2. [Section II, Eq. (3)] In Eq. (3), the summation index is i in one place but the term is written with ε^j and Λ_j; please make the index notation consistent.
  3. [Table IV] In the left column for l = 0, n = 2, the entry '45907867− 1.08026685i' appears to be missing a leading '0'.
  4. [Section II] The sentence 'Sulejmanpasic and and ¨Unsal' contains a duplicated 'and'.
  5. [Footnote 1] 'BenderWu' should be typeset as 'Bender–Wu' for consistency with the reference list.
  6. [Abstract and Section III.B] The abstract's claim that the l = 2, n = 0 gravitational mode agrees with the continued-fraction method to 24 decimal places should identify the comparison source; Table XIV shows agreement between Padé and Borel–Padé values, while the corresponding continued-fraction value is not listed there.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the WKB coefficients are generated from the potential independently of the numerical QNM benchmarks, and the benchmarks are used only for post-hoc comparison.

full rationale

The derivation chain is self-contained. The formal series for omega-squared is constructed from derivatives of the effective potential at its maximum via Eq. (3), and the higher-order Lambda_k coefficients are generated numerically using the anharmonic-oscillator equivalence of Eqs. (5)-(7) with the Bender-Wu/Sulejmanpasic-Unsal package, not from any numerical quasinormal frequency. The Pade approximants are then evaluated at epsilon = 1. Literature values from Andersson and continued-fraction results enter only after the calculation, as benchmarks for the deviation measures (10)-(11), and where the authors dispute a literature value they recompute it with an independent continued-fraction code. No parameter is fitted to the target frequencies, no uniqueness theorem is imported from the authors' prior work, and no known result is merely relabeled. The only self-citation is to Paper I for the original Pade-WKB proposal and earlier low-order agreement, but the present high-order results are generated independently and agree with external continued-fraction computations (e.g., 24 decimal places for the gravitational l=2, n=0 mode), so the self-citation is not load-bearing. The concern that diagonal Pade summation of the asymptotic WKB series might converge to the wrong analytic continuation is a correctness risk, not a circularity.

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

The central claim rests on no fitted constants but on five structural assumptions: the single-barrier omega-independent potential, the Iyer-Will quantization condition, the anharmonic-oscillator equivalence, Padé summability of the formal series, and the exactness of continued-fraction benchmarks. The first three are standard for this method family; the fourth is the paper's unproved working hypothesis.

assumptions (5)
  • domain assumption The perturbed black hole equation reduces to Eq. (1) with Q = omega^2 - V, V independent of omega, one maximum, constant asymptotic limits.
    Invoked as the starting point of Section II; excludes rotating and omega-dependent effective potentials.
  • domain assumption The Iyer-Will quantization condition, Eq. (2), yields the quasinormal frequencies when Lambda_k are evaluated at the maximum of V.
    The paper imports this result from Refs. [12,13] without rederiving it.
  • domain assumption Taylor-expanding Q(x) about x0 and mapping to the anharmonic oscillator, Eqs. (5)-(7), gives the same Lambda_k as the WKB method.
    Equality checked symbolically only for k<=5 and numerically for 6<=k<=16; for k>16 it is assumed.
  • ad hoc to paper Diagonal Padé approximants of the formal omega^2 series stabilize to the true frequency as k grows.
    This is the paper's central unproved assumption; support is empirical only, e.g., Figs. 1-2 and tables.
  • domain assumption Continued-fraction benchmark frequencies are exact to the quoted precision.
    The paper uses Andersson's table [32] and its own CF code as ground truth, with no independent error analysis.

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Pith. "Pith review of Quasinormal modes of black holes. II. Pad\'e summation of the higher-order WKB terms." pith.science (2026). https://pith.science/paper/TJOGNRRB

@misc{pith2026190809389,
  author       = {Pith},
  title        = {Pith review of: Quasinormal modes of black holes. II. Pad\'e summation of the higher-order WKB terms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TJOGNRRB}},
  note         = {Machine review of arXiv:1908.09389}
}
abstract

In previous work [1] we proposed an improvement of the WKB-based semianalytic technique of Iyer and Will for calculation of the quasiormal modes of black holes by constructing the Pad\'e approximants of the formal series for $\omega^{2}.$ It has been demonstrated that (within the domain of applicability) the diagonal Pad\'e transforms $\mathcal{P}_{6}^{6}$ and $\mathcal{P}_{7}^{6}$ are always in a very good agreement with the numerical results. In this paper we present a further extension of the method. We show that it is possible to reproduce many known numerical results with a great accuracy (or even exactly) if the Pad\'e transforms are constructed from the perturbative series of a really high order. In our calculations the order depends on the problem but it never exceeds 700. For example, the frequencies of the gravitational mode $l=2,$ $n=0$ calculated with the aid of the Pad\'e approximants and within the framework of the continued fractions method agree to 24 decimal places. The use of such a large number of terms is necessary as the stabilization of the quasinormal frequencies can be slow. Our results reveal some unexpected features of the WKB-based approximations and may shed some fresh light on the problem of overtones.

Figures

Figures reproduced from arXiv: 1908.09389 by the authors.

Figure 1
Figure 1. FIG. 1: The real part of the quasinormal frequency of the odd gravitational perturbation with [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The imaginary part of the quasinormal frequency of the odd gravitational perturbation with [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The few last columns of the [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

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Forward citations

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  6. Connection Between the Shadow Radius and Quasinormal Frequencies for Black Holes in STVG with Perfect Fluid Dark Matter

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Reference graph

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