Exact Cancellation of Horizon Modes in Kerr Ringdown

Daiki Watarai; Hayato Motohashi; Kei-ichiro Kubota; Yusuke Manita

REVIEW

Exact Cancellation of Horizon Modes in Kerr Ringdown

Kei-ichiro Kubota, Yusuke Manita, Hayato Motohashi, Daiki Watarai

General Relativity and Quantum Cosmology · Submitted 7 Sep 2026 · 36 references

Pith's verdict: This paper convincingly shows that the source-induced horizon mode is exactly canceled by an infinite tower of Matsubara zeros in Kerr, undermining the 'direct wave' interpretation—but the argument hinges on the zero structure of 1/B_inc from Ref. [23].

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arXiv ID2609.07326 (version 1)

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CitationDaiki Watarai, Hayato Motohashi, Kei-ichiro Kubota, Yusuke Manita. "Exact Cancellation of Horizon Modes in Kerr Ringdown." arXiv:2609.07326 (2026). https://pith.science/paper/FWGBUWUE

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Editor's note

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This is a clean, well-structured paper. The main result—that every source-induced horizon-mode pole is canceled by an infinite sequence of Matsubara zeros in Kerr—is new and important. The authors give a clear analytic derivation (Eqs. 24–27) that generalizes the Schwarzschild cancellation to spinning black holes, and they back it with numerical integration showing sequential screening through the first four Matsubara frequencies. The failure of the saddle-point approximation in the near-horizon limit is convincingly demonstrated. If the result holds, it kills the 'direct wave' interpretation of post-ringdown oscillations as a surviving horizon-mode signal, which has practical consequences for how people interpret filtered waveforms from LIGO/Virgo.

The soft spot is the dependence on Ref. [23] for the higher Matsubara zeros (j ≥ 3). The paper explicitly leans on the claim that 1/B_inc has zeros exactly at all ω_MM^j. The numerical verification in Fig. 2 builds those zeros into Π(ω) by construction, so it does not test the Ref. [23] claim itself. That said, Ref. [23] is from the same group and has independent verification, and the dependence is openly acknowledged. This is a genuine load-bearing assumption, but not a fatal one—the paper would be substantially strengthened if the authors could verify the zero structure themselves, or cite independent confirmation. The numerical scope is also limited (one spin, one energy, one angular momentum), which the reader noted.

Who this is for: anyone working on ringdown modeling, black-hole perturbation theory, or the 'direct wave' debate. It deserves a serious referee. The paper is honest about its assumptions, the math is clear, and the conclusion is stated provisionally.

Recommendation: send to peer review. The referee should press the authors to either verify or discuss the robustness of the Ref. [23] zero structure, and to expand the numerical parameter set if feasible. But the core argument is strong enough to warrant publication after those checks.

Referee report

Major objections (2)

[Eq. (14); surrounding text] The entire cancellation mechanism relies on the statement that 1/B_inc has zeros at all Matsubara frequencies omega_MM^j for j>=3. This result is imported from Ref. [23] and is not independently verified or tested in the present manuscript. The numerical demonstration in Fig. 2 uses a constructed PI(omega) that already incorporates these assumed zeros, so it does not confirm the Ref. [23] claim. The central conclusion is therefore conditional on the correctness of Ref. [23]. The authors should explicitly discuss this dependence and, if possible, provide a robustness argument or a partial verification that does not assume the exact zero locations. If Ref. [23] were found to be incorrect, the cancellation argument would be unsupported.
[Eqs. (24)-(27); derivation of pole sum] The expansion in Eq. (24) and the term-by-term integration in Eq. (27) assume that the series in epsilon(t') can be expanded and that the sum over n and the integral commute. The paper does not discuss the convergence of the series or the justification for swapping the summation and integration. For a rigorous proof, these steps should be justified, especially because the expansion is used to produce an infinite sequence of poles. Please comment on the analyticity of the expansion and the validity of the integration interchange.

Minor objections (5)

[Abstract] The phrase "independently of orbital and black-hole parameters" is correct for the Matsubara frequencies (which depend only on the horizon parameters), but the zero structure itself depends on those frequencies. Please clarify that the parameter independence refers to the cancellation mechanism, not the zero locations.
[Fig. 2] The caption mentions "first four stages of screening" na the legend shows five curves (unscreened plus four screened). This is a minor inconsistency. Also, the vertical axes labels (|h(t)|, Phase) should include units or note that the normalisation is arbitrary.
[Eq. (13)] The left-hand side is a polynomial in (kr+/kappa). The right-hand side shows the factorised form -2 (r+/kappa)^2 (omega-omega_MM^1)(omega-omega_MM^2). Please verify explicitly that the coefficients match (perhaps with a short note).
[Footnote [26]] The note on redshift modes is somewhat ambiguous. It states that they originate from the omitted factor and that the Letter does not consider their cancellation. Clarify whether redshift modes are also canceled or whether they are a separate effect that is irrelevant to the present argument.
[Eqs. (17)-(22)] The discussion of the SPA and the Hessian (20) is clear. However, the sign of the Hessian and the appropriate contour deformation might be worth a comment, as the standard steepest-descent method requires the Hessian to be positive definite for a strict saddle point.

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The paper

The paper claims

Every source-induced horizon-mode pole is exactly canceled by an infinite tower of Matsubara zeros, independently of black-hole and orbital parameters.

The reading

The paper reexamines a particle plunging into a Kerr black hole and the proposal that a 'direct wave' from near the horizon survives at late times as a horizon mode. It shows that the saddle-point approximation used in that proposal fails near the horizon. An exact evaluation of the source integral reveals that each source-induced horizon-mode pole is canceled by an infinite sequence of Matsubara zeros, regardless of the black-hole spin or the particle's trajectory. The first four stages of this cancellation are confirmed by time-domain numerical integration. The result rules out the interpretation of oscillatory features in QNM-filtered waveforms as a surviving late-time horizon-mode signal.

What carries the argument

The proof relies on the exact analytic structure of the frequency-domain source amplitude Z̃(ω), which contains an infinite sequence of Matsubara zeros at ω = ω_MM_j (j = 1,2,...). The near-horizon expansion of the source integral produces a sum of Matsubara poles, each of which is canceled by the corresponding zero in the prefactor Π(ω) that multiplies the integral. The first two zeros come from a polynomial in Z̃(ω), while those with j ≥ 3 originate from the asymptotic amplitude 1/B_inc.

Extended reading notes

Core claim

Within the near-horizon perturbative framework, every source-induced horizon-mode pole arising from a particle plunging into a Kerr black hole is exactly canceled by the infinite tower of Matsubara zeros present in the prefactor Π(ω) and in 1/B_inc. The cancellation is sequential and independent of the black-hole and trajectory parameters. Consequently, source-induced horizon-mode signals do not survive at late times.

Load-bearing premise

The analytical proof depends on the result from [23] that 1/B_inc has an infinite set of Matsubara zeros at the exact frequencies ω_MM_j; if that result is incorrect or the zeros are not coincident, the sequential cancellation would not proceed as claimed.

Editorial extensions

If this is right

The proposed 'direct wave' component in QNM-filtered gravitational-wave ringdown does not arise from a surviving horizon mode, so searches for near-horizon physics via such a component must be re-evaluated.
Any oscillation in the filtered waveform that was attributed to a late-time horizon mode must have a different origin, likely from source structure or QNM-filter artifacts.
The exact cancellation mechanism is robust against variations in black-hole spin and particle orbital parameters, as long as the near-horizon expansion applies.
The result suggests that the infinite Matsubara-zero tower is a generic feature of the Kerr Teukolsky source integral, not limited to plunging particles.
For black-hole spectroscopy, the elimination of horizon modes sharpens the interpretation that only QNMs and power-law tails dominate the late-time ringdown.

Reading between the lines

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

If the infinite Matsubara-zero structure is indeed generic, then any source that satisfies the near-horizon geodesic expansion will also have its horizon-mode contributions canceled, which may extend to comparable-mass binaries if a similar near-horizon limit holds.
The failure of the saddle-point approximation near the horizon suggests that other frequency-domain methods relying on the SPA for near-horizon sources may need to be rechecked for analogous cancellation.
The study implicitly assumes that the near-horizon expansion is valid for all times after the particle crosses a certain threshold; a numerical check for higher multipoles or for spinning particles would strengthen the claim.
An experimental test could be constructed by measuring the QNM-filtered waveform from a known plunging trajectory with high signal-to-noise ratio and checking whether any residual oscillation matches the horizon-mode frequency; the paper predicts it will not.

Original abstract

Ringdown gravitational waves probe black hole spacetimes via quasinormal modes (QNMs). Recent studies have reported oscillatory features in QNM-filtered waveforms, and have led to conflicting interpretations as to whether they represent a near-horizon "direct wave". Reexamining a particle plunging into a Kerr black hole, we show that the saddle-point approximation underlying the proposed interpretation fails near the horizon. Our exact analysis reveals that every source-induced horizon-mode pole is canceled by an infinite tower of Matsubara zeros, independently of orbital and black-hole parameters. The first four stages of this sequential screening are also confirmed by time-domain numerical integration. Thus, within the near-horizon perturbative framework, source-induced horizon-mode signals do not survive at late times.

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, 2026-09-23: 1 request, 4,861 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.67 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.42 · compares with prior results 0.86 · 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.

Ringdown gravitational waves probe black hole spacetimes via quasinormal modes (QNMs). Recent studies have reported oscillatory features in QNM-filtered waveforms, and have led to conflicting interpretations as to whether they represent a near-horizon "direct wave". Reexamining a particle plunging into a Kerr black hole, we show that the saddle-point approximation underlying the proposed interpretation fails near the horizon. Our exact analysis reveals that every source-induced horizon-mode pole is canceled by an infinite tower of Matsubara zeros, independently of orbital and black-hole parameters. The first four stages of this sequential screening are also confirmed by time-domain numerical integration. Thus, within the near-horizon perturbative framework, source-induced horizon-mode signals do not survive at late times.

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	Ringdown gravitational waves probe black hole spacetimes via quasinormal modes (QNMs).	background 1.00	0.06	0.02	asserted 1.00
s1	Recent studies have reported oscillatory features in QNM-filtered waveforms, and have led to conflicting interpretations as to whether they represent a near-horizon "direct wave".	background 0.93	0.04	0.02	asserted 0.85
s2	Reexamining a particle plunging into a Kerr black hole, we show that the saddle-point approximation underlying the proposed interpretation fails near the horizon.	contribution 0.81	0.88	0.05	asserted 1.00
s3	Our exact analysis reveals that every source-induced horizon-mode pole is canceled by an infinite tower of Matsubara zeros, independently of orbital and black-hole parameters.	result 1.00	0.95	0.81	asserted 1.00
s4	The first four stages of this sequential screening are also confirmed by time-domain numerical integration.	result 0.62	0.91	0.04	asserted 1.00
s5	Thus, within the near-horizon perturbative framework, source-induced horizon-mode signals do not survive at late times.	interpretation 0.58	0.86	0.87	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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. Analytic structure of

**Figure 2.** FIG. 2. Time-domain evolution of source-induced horizon

Figures reproduced from arXiv:2609.07326 by the authors.

References

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Adrien Kuntz and Matteo Della Rocca, “Vanishing of all redshift modes in Schwarzschild ringdown,” (2026), arXiv:2606.07173 [gr-qc]

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Junquan Su, Neev Khera, Abhishek Chowdhuri, Marc Casals, and Huan Yang, “Gravitational Waves from Green’s Function Decomposition for a Kerr black hole: I. Equatorial ISCO Plunge,” (2026), arXiv:2608.17943 [gr-qc]

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Ling Sun, Sizheng Ma, Patrick P. Chen, Andrew Laeuger, Neil Lu, and Yanbei Chen, “Complex frequency evolu- tion of direct waves from binary black hole mergers,” 6 (2026), arXiv:2608.23209 [gr-qc]

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View on Pith Pith review arXiv 2008
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Aaron Zimmerman and Yanbei Chen, “New Generic Ringdown Frequencies at the Birth of a Kerr Black Hole,” Phys. Rev. D84, 084012 (2011), arXiv:1106.0782 [gr-qc]

View on Pith Pith review arXiv 2011
[16]

Related source-dependent contributions considered for Schwarzschild black hole, termed “redshift modes” [30– 32], share the same characteristic frequency and have been shown to vanish identically in Schwarzschild space- time [7]. We focus here exclusively on horizon modes, whose cancellation mechanism is distinct from the redshift-mode result of Ref. [7]

View on Pith
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Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Tracing source imprints on the gravitational direct wave
gr-qc 2026-11 conditional novelty 6.0 of 10

A response filter built from the inverse black-hole response localizes the gravitational direct wave, and a complex-time saddle method reproduces it semianalytically through the whole plunge.
Modeling QNMs-Filtered Gravitational Waveforms: Approximation Limits and Filtering Effects
gr-qc 2026-10 conditional novelty 6.0 of 10

QNM filtering, though phase-only, materially reshapes residual black hole ringdown waveforms, and the point-particle models used to interpret direct waves fail for the (4,4) multipole and can be mimicked in pure vacuum.
Moving Beyond Quasinormal Modes and Reinterpreting Direct Waves in Binary Black Hole Merger-Ringdowns
gr-qc 2026-10 conditional novelty 6.0 of 10

The early non-QNM part of binary black hole ringdowns is tied to the remnant light ring, not to waves emitted from inside it, and three independent analyses find no support for the proposed direct-wave picture.
Gravitational-wave Echoes Beyond the Hubble Time from Quantum Black Holes
gr-qc 2026-10 conditional novelty 6.0 of 10

In the KY quantum black hole model, the exponentially strong interior redshift pushes gravitational-wave echo delays far beyond the Hubble time, making echoes effectively undetectable.
Theoretical Aspects of Direct Waves in Kerr Black Holes: Pole-Splitting Method for Ringdown Analysis
gr-qc 2026-09 accept novelty 6.0 of 10

The pole-splitting method extracts a non-QNM waveform whose frequency evolution matches the predicted direct-wave frequency, establishing a new probe of the Kerr ergosphere and redshift.

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [23]

Hayato Motohashi and Yuto Suichi, “Pole structure of the Kerr Green’s function,” Phys. Rev. D114, 044022 (2026), arXiv:2605.01964 [gr-qc]

View on Pith Pith review arXiv 2026

[2] [1]

Emanuele Berti et al., “Black hole spectroscopy: from theory to experiment,” Class. Quant. Grav.43, 123001 (2026), arXiv:2505.23895 [gr-qc]

View on Pith Pith review arXiv 2026

[3] [2]

Sizheng Ma, Keefe Mitman, Ling Sun, Nils Deppe, Fran¸ cois H´ ebert, Lawrence E. Kidder, Jordan Moxon, William Throwe, Nils L. Vu, and Yanbei Chen, “Quasinormal-mode filters: A new approach to ana- lyze the gravitational-wave ringdown of binary black- hole mergers,” Phys. Rev. D106, 084036 (2022), arXiv:2207.10870 [gr-qc]

View on Pith Pith review arXiv 2022

[4] [3]

Naritaka Oshita, Sizheng Ma, Yanbei Chen, and Huan Yang, “Probing Direct Waves in Black Hole Ringdowns,” (2025), arXiv:2509.09165 [gr-qc]

View on Pith Pith review arXiv 2025

[5] [4]

Neil Lu, Sizheng Ma, Ornella J. Piccinni, Yanbei Chen, and Ling Sun, “GW250114 reveals signatures of post- merger black-hole horizon,” Nature655, 71–75 (2026), arXiv:2510.01001 [gr-qc]

View on Pith Pith review arXiv 2026

[6] [5]

Anuj Kankani and Sean T. McWilliams, “Direct Waves in Black-Hole Binary Mergers: Insights from the Backwards One Body Model,” (2026), arXiv:2603.15474 [gr-qc]

View on Pith Pith review arXiv 2026

[7] [6]

Adrian Ka-Wai Chung, Kelvin Ka-Ho Lam, Anna Liu, and Nicolas Yunes, “Measuring a Black Hole’s Area Im- mediately after Merger: A Direct-Wave Test of Hawk- ing’s Area Law,” (2026), arXiv:2606.06592 [gr-qc]

View on Pith Pith review arXiv 2026

[8] [7]

Adrien Kuntz and Matteo Della Rocca, “Vanishing of all redshift modes in Schwarzschild ringdown,” (2026), arXiv:2606.07173 [gr-qc]

View on Pith Pith review arXiv 2026

[9] [8]

Richard Dyer, Adrian Ka-Wai Chung, and Christo- pher J. Moore, “Modeling Direct Waves in Binary Black Hole Ringdowns,” (2026), arXiv:2606.25021 [gr-qc]

View on Pith Pith review arXiv 2026

[10] [9]

Anuj Kankani and Sean T. McWilliams, “The Direct Wave is Not a Meaningful Test of Horizon Properties,” (2026), arXiv:2607.02380 [gr-qc]

View on Pith Pith review arXiv 2026

[11] [10]

Sizheng Ma and Hai-Yang Wang, “Foundations of Direct Waves in Schwarzschild Ringdown,” (2026), arXiv:2607.06677 [gr-qc]

View on Pith Pith review arXiv 2026

[12] [11]

Junquan Su, Neev Khera, Abhishek Chowdhuri, Marc Casals, and Huan Yang, “Gravitational Waves from Green’s Function Decomposition for a Kerr black hole: I. Equatorial ISCO Plunge,” (2026), arXiv:2608.17943 [gr-qc]

View on Pith Pith review arXiv 2026

[13] [12]

Ling Sun, Sizheng Ma, Patrick P. Chen, Andrew Laeuger, Neil Lu, and Yanbei Chen, “Complex frequency evolu- tion of direct waves from binary black hole mergers,” 6 (2026), arXiv:2608.23209 [gr-qc]

View on Pith Pith review arXiv 2026

[14] [13]

Mark Ho-Yeuk Cheung, “How much of the black hole ringdown is sourced within the light ring?” (2026), arXiv:2608.29466 [gr-qc]

View on Pith Pith review arXiv 2026

[15] [14]

Yasushi Mino and Jeandrew Brink, “Gravitational Radi- ation from Plunging Orbits: Perturbative Study,” Phys. Rev. D78, 124015 (2008), arXiv:0809.2814 [gr-qc]

View on Pith Pith review arXiv 2008

[16] [15]

Aaron Zimmerman and Yanbei Chen, “New Generic Ringdown Frequencies at the Birth of a Kerr Black Hole,” Phys. Rev. D84, 084012 (2011), arXiv:1106.0782 [gr-qc]

View on Pith Pith review arXiv 2011

[17] [16]

Related source-dependent contributions considered for Schwarzschild black hole, termed “redshift modes” [30– 32], share the same characteristic frequency and have been shown to vanish identically in Schwarzschild space- time [7]. We focus here exclusively on horizon modes, whose cancellation mechanism is distinct from the redshift-mode result of Ref. [7]

View on Pith

[18] [17]

Y. Kojima and T. Nakamura, “Gravitational Radiation from a Particle with Orbital Angular Momentum Plung- ing into a Kerr Black Hole,” Prog. Theor. Phys.71, 79–90 (1984)

View on Pith 1984

[19] [18]

Michael Boyle et al., “The SXS Collaboration catalog of binary black hole simulations,” Class. Quant. Grav.36, 195006 (2019), arXiv:1904.04831 [gr-qc]

View on Pith Pith review arXiv 2019

[20] [19]

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Note that this definition ofκdiffers from that of the conventional surface gravity

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Note that the redshift modes [30–32] originate from this omitted factor. We do not consider the redshift modes nor their cancellation [7] in this Letter

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An exceptional case occurs when a QNM pole coincides with a Matsubara zero, leading to pole skipping [24, 33– 36]. In this case, the QNM part contains a contribution corresponding to a Matsubara frequency

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