Theoretical Aspects of Direct Waves in Kerr Black Holes: Pole-Splitting Method for Ringdown Analysis

Hiroki Takeda; Nao Nakamoto; Naritaka Oshita

REVIEW

Theoretical Aspects of Direct Waves in Kerr Black Holes: Pole-Splitting Method for Ringdown Analysis

Nao Nakamoto, Naritaka Oshita, Hiroki Takeda

General Relativity and Quantum Cosmology · Submitted 10 Sep 2026

Pith's verdict: Pole-splitting is a genuine technical step forward, but the quantitative validation doesn't yet support the strong claim that the direct wave is 'established as a probe' — the residuals are the same size as unquantified neglected terms.

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arXiv ID2609.11908 (version 3)

Pith IDU2RCPYRQ, Pith's permanent record of this paper and its review

CitationHiroki Takeda, Nao Nakamoto, Naritaka Oshita. "Theoretical Aspects of Direct Waves in Kerr Black Holes: Pole-Splitting Method for Ringdown Analysis." arXiv:2609.11908 (2026). https://pith.science/paper/U2RCPYRQ

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Editor's note

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This paper does two useful things. First, it gives a clean theoretical derivation of the direct-wave frequency ω_G and its screening in Kerr, extending the horizon-mode picture. Second, it introduces a pole-splitting method that separates QNM and non-QNM parts of the waveform without introducing the group-delay distortion that plagues QNM filtering. That method is a genuine technical step forward, and the theory is well-organized.

The soft spot is in the validation. The authors claim to "establish the DW as a probe of the ergoregion and of the redshift effect" based on the agreement between the non-pole instantaneous frequency and ω_G (or ω_screen). But the actual agreement is partial: for χ=0.8 the imaginary part is only qualitative, and for χ=0.9,0.95 the preference for ω_screen over ω_G is shown only visually — no model-selection statistic is given. More importantly, the paper itself acknowledges (Eq. 76, Section IV) that the predicted instantaneous frequency omits the full S-matrix prefactor and the branch-cut/large-arc integrals are left unevaluated. The stress-test is right: the residuals in Figures 6-7 are of the same apparent order as those neglected corrections, so the validation is not yet a closed loop. The interpretation of the non-pole sector as tracking ω_G is plausible, and the pole-spliting method makes it testable in a clean way, but the claim of "establishing" outruns hte evidence currently on the table.

That said, the paper is not sloppy. The authors discuss the limitations honestly (the imaginary-part discrepancy, the necessary approximations, the unmodeled frequency-dependence of C(ω)). They just draw a stronger conclusion than those limitations warrant. For a reserch community that works on black-hole pertrubation theory and waveform analysis, this is a solid theoretical contibution with a promising numerical technique. I would bring it to reading group, and I would cite it for the pole-spliting method and the ω_G formalism.

Peer review: yes, but the authors should be asked to either (1) estimate the ngelected terms quantitatively (even an order-of-magnitude bound woud help) or (2) downgrade the claim from "establishing a probe" to "demonstrating consistency with the DW picture." The theory and method are worth pbulishing, and the claim can be fixed.

Referee report

Significance

If the DW identification were quantitatively established, it would offer a new observational probe of frame-dragging and redshift near a Kerr horizon, complementing standard QNM spectroscopy. The pole-splitting method itself is a clean, filter-free decomposition that has independent value for ringdown analysis. However, the paper's central claim is currently supported only by qualitative agreement; the observed residuals are comparable in size to unquantified neglected contributions, and no model-selection statistics are provided for the higher-spin cases. The theoretical derivation is self-consistent under the stated approximations, and the numerical implementation is well-documented, making this a useful contribution even if the stronger interpretive claim requires further validation.

Major objections (3)

[Section IIE, Eq. (56) and surrounding text] The non-pole waveform ψ_non is defined as the sum of an exponential term at ω_{G,i} and the unevaluated branch-cut and large-arc integrals. The authors state that they 'assume that their contributions to the waveform and its time derivative are negligible over the interval of interest,' but provide no estimate of their magnitude. Because the instantaneous frequency extraction involves a ratio of the waveform to its time derivative, even subdominant contributions can bias the result. A quantitative bound (e.g., ratio of integral magnitude to the leading term) is needed to justify this central approximation.
[Section IV, Eq. (76) and discussion of imaginary-part agreement] The analytic model for the instantaneous frequency ω_non omits the full frequency-dependent S-matrix factor C[ω_G(u)] and its derivative. The paper attributes the systematically poorer agreement in the imaginary part 'at least partly' to this omission, but does not quantify whether the omitted term is indeed comparable to the observed discrepancy (e.g., for χ=0.8 near u≈-30, the decay-rate difference is ~0.2κ while κ itself is ~0.1). For the claim that the DW picture is 'established,' it must be shown that the omitted term is controlled and subdominant, or the claim should be softened.
[Section IV, Figures 6 and 7] For χ=0.9 and 0.95, the preference for ω_screen over ω_G is asserted only by visual inspection of the frequency and decay-rate plots. No quantitative metric (e.g., mean-square residual, correlation coefficient, or Bayesian evidence ratio) is given. Since the two curves are close and the residuals are of the same order as the unquantified neglected contributions (Major Comment 1), a statistical measure is required to support the consistency claim.

Minor objections (7)

[Section I, second paragraph] The phrase 'extreme-mass mer’ should be 'extreme-mass-ratio merger' for clarity.
[Section IIB, Eq. (12)] The bilinear normalization convention is unconventional and should be briefly justified, as it differs from the standard inner product ⟨S|S⟩=1 and affects the analytic continuation of the spheroidal harmonics.
[Section IIIA, Eq. (67)] The source factor T(ω) defined via R_in/Bout is used in the pole splitting, but earlier in Eq. (13) the source integral is expressed with R_in only. Please clarify the relation between these two definitions and ensure notation is consistent.
[Appendix A] The mismatch threshold M_tol = 10^{-4} is chosen to truncate the frequency integration. Please provide a brief justification or cite a precedent from the literature.
[Section IIF] The prediction that the late-time DW decays as e^{-3κu} is derived from the condition m+n-1 ≥ 3 for s=-2. This result is interesting but is not tested in the paper. Add a sentence noting that verification is left for future work (already present, but could be explicit).
[Table I] The term 'pre-plunge QNMs' may be misleading because QNMs are conventionally defined for ringdown; consider using 'QNM-like excitations during inspiral' or adding a footnote.
[General] Several references (e.g., [10], [13], [17]) are explicitly discussed in the text but their precise relation to this work could be summarized more concisely in the introduction to reduce redundancy.

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The paper

The paper claims

Pole-splitting method isolates direct wave from Kerr ringdown, revealing frame-dragging and redshift imprints.

The reading

This paper aims to establish that a source-driven gravitational-wave component called the direct wave (DW) exists in extreme-mass-ratio mergers and that its complex frequency evolution encodes the frame-dragging and gravitational redshift near a Kerr black hole. The authors derive the DW frequency analytically using a near-horizon expansion and introduce a pole-splitting method that separates the waveform into a quasinormal-mode (QNM) pole part and a non-pole part without the frequency-dependent time shifts of traditional QNM filtering. By applying this method to numerical simulations of quasi-circular plunges into Kerr black holes with spins 0.8, 0.9, and 0.95, they show that the instantaneous frequency and decay rate of the non-pole waveform follow the predicted DW frequency omega_G or its screened version omega_screen. If true, this provides a direct observational probe of the ergoregion and the horizon properties of black holes.

What carries the argument

The pole-splitting method, which decomposes the black-hole response function D^hat(omega) = B_out / (2 i omega B_in) into a sum over QNM pole residues (pole sector) and the remainder (non-pole sector), allowing the waveform to be separated into Psi_pole and Psi_non on the same retarded-time coordinate without the group-delay distortion inherent in rational filtering.

Extended reading notes

Core claim

The paper claims that the non-pole sector of the ringdown waveform, obtained by subtracting the QNM pole contributions from the transfer function, contains a component whose instantaneous complex frequency evolves in time according to the source-driven frequency omega_G(u) = (m $\Omega$ - $\beta$ Omega_H)/(1-$\beta$) + 2 i kappa $\beta$/(1-$\beta$), where $\Omega$ and $\beta$ are the angular velocity and radial infall parameter of the plunging particle. For the ell=m=2 mode and spins chi >= 0.8, the real part of this frequency approaches m Omega_H and the imaginary part approaches kappa as the particle nears the horizon, consistent with the frame-dragging and redshift effects. The screening by the potential barrier modifies the effective frequency to omega_screen when the source lies inside the QNM convergence radius, and both predictions are verified against numerical waveforms.

Load-bearing premise

The non-pole waveform is approximated as dominated by the omega_G exponental term, and its time derivative is computed neglecting the branch-cut and large-arc integrals, with the screening effect modeled only by the leading zero (omega_G - $omega_H^{{(1)}}$.

Editorial extensions

If this is right

The direct wave can be used as a direct probe of the horizon angular velocity Omega_H and surface gravity kappa, providing a new test of the Kerr metric.
Analysis of gravitational-wave data from extreme-mass-ratio inspirals can extract the DW component to infer black-hole spin and mass without relying on QNM spectroscopy alone.
The pole-splitting method offers a general tool to isolate non-QNM contributions in other contexts, such as prompt response or matter effects.
The late-time decay of the DW is predicted to follow a third-order horizon mode e^{-3kappa u}, which can be searched for in future numerical or observational data.

Reading between the lines

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

Because the pole-splitting method avoids the frequency-dependent time shift of QNM filtering, it may reveal systematic biases in earlier direct-wave searches that used rational filters.
The success of the screened frequency omega_screen for high-spin cases suggests that the effective potential barrier (light ring) plays a crucial role; this could be tested by varying the angular mode (ell,m) to change the barrier location.
The qualitative agreement in the imaginary part for chi=0.8 hints that the full frequency dependence of the S-matrix (beyond the screening zero) is needed for quantitative fits; future work could incorporate the C(omega) factor from Eq. (40) to improve the model.
If the pole-splitting method is applied to comparable-mass-ratio numerical relativity waveforms (once linearised), it may clarify whether the direct wave survives in that regime, which is still debated.

Original abstract

We develop a theoretical description of direct waves (DWs) from a particle plunging into a Kerr black hole in the extreme-mass-ratio regime. A DW is characterized by a source-driven complex frequency $\omega_{\rm G}$ that reflects the particle's orbital motion, including motion inside the ergoregion. Hence, it is expected to be a probe of rich structures of Kerr black holes. We show that the contribution of the quasinormal-mode (QNM) pole sector to the DW is part of dynamical QNM excitation. We also show that the apparent horizon-mode poles cancel at every order of a consistent near-horizon expansion of the trajectory, homogeneous radial solution, and complete point-particle source. This result assumes that the point particle follows a geodesic near the horizon and leaves open the possibility of a transient DW while the particle passes near the ergosurface or the vicinity of the potential barrier. To investigate this possibility in simulated plunge waveforms, we develop a pole-splitting method that decomposes the transfer function into QNM-pole and non-pole parts. The resulting waveform decomposition avoids the frequency-dependent time shift (group delay) introduced by the QNM filtering. For quasi-circular plunges into moderately and rapidly rotating Kerr black holes, the frequency and decay rate of the non-pole waveform show qualitative agreement with our theoretical prediction over finite time intervals. These results support the potential use of DWs as probes of the ergoregion and gravitational redshift effect around black holes.

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, 5,282 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.13 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.87 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We formulate the theoretical aspects of direct waves (DWs) in the case of extreme-mass merger. A DW is a source-driven waveform characterized by a complex frequency $\omega_{\rm G}$, which reflects the orbital motion of the particle in the vicinity of the black hole, including a part of the orbit inside the ergoregion: its real part is governed by frame dragging and its imaginary part by the redshift of the source. Using the Green's function technique, we derive the source-driven frequency $\omega_{\rm G}$, describe its screening by the potential barrier, and discuss its relation to dynamically excited quasinormal modes (QNMs). We then introduce a pole-splitting method that divides the full waveform into a QNM-pole sector and a non-QNM sector. Unlike QNM filtering, which multiplies the waveform spectrum by a filter function and thereby deforms it through a frequency-dependent time shift (i.e., group delay), our pole-splitting method merely divides the transfer function into pole and non-pole parts, separating the full waveform. Simulating a quasi-circular plunge into a Kerr black hole with medium and rapid spins, we find that the frequency and decay rate of the non-pole sector in the dominant mode, $\ell = m = 2$, evolve consistently with $\omega_{\rm G}$-or with its screened counterpart $\omega_{\rm screen}$-establishing the DW as a probe of the ergoregion and of the redshift effect around a black hole.

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 formulate the theoretical aspects of direct waves (DWs) in the case of extreme-mass merger.	contribution 0.98	0.91	0.04	asserted 0.98
s1	A DW is a source-driven waveform characterized by a complex frequency $\omega_{\rm G}$, which reflects the orbital motion of the particle in the vicinity of the black hole, including a part of the orbit inside the ergoregion: its real part is governed by frame dragging and its imaginary part by the redshift of the source.	background 0.86	0.16	0.06	asserted 1.00
s2	Using the Green's function technique, we derive the source-driven frequency $\omega_{\rm G}$, describe its screening by the potential barrier, and discuss its relation to dynamically excited quasinormal modes (QNMs).	method 0.77	0.95	0.06	asserted 1.00
s3	We then introduce a pole-splitting method that divides the full waveform into a QNM-pole sector and a non-QNM sector.	method 0.76	0.94	0.07	asserted 0.99
s4	Unlike QNM filtering, which multiplies the waveform spectrum by a filter function and thereby deforms it through a frequency-dependent time shift (i.e., group delay), our pole-splitting method merely divides the transfer function into pole and non-pole parts, separating the full waveform.	method 0.70	0.48	0.09	asserted 1.00
s5	Simulating a quasi-circular plunge into a Kerr black hole with medium and rapid spins, we find that the frequency and decay rate of the non-pole sector in the dominant mode, $\ell = m = 2$, evolve consistently with $\omega_{\rm G}$-or with its screened counterpart $\omega_{\rm screen}$-establishing the DW as a probe of the ergoregion and of the redshift effect around a black hole.	result 0.94	0.96	0.76	asserted 0.96

What kind of paper the classifier reports this is: method or tool 0.85 · evidence from simulation or computation 0.54.

What was examined, and what was not

sources read

arXiv listing abstract v2

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. Schematic picture of prompt response, DW, and

**Figure 2.** FIG. 2. Spin dependence of the Kerr geometry. The radial

**Figure 4.** FIG. 4. Contours used to evaluate the segment integral of Eq.

**Figure 8.** FIG. 8. Convergence of the pole (QNM) contribution in

Showing 7 of the paper's 8 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2609.11908 by the authors.

References

Forward citations

Cited by 3 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.
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.

Discussion

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