Gravitational Waves from Green's Function Decomposition for a Kerr black hole: I. Equatorial ISCO Plunge

Abhishek Chowdhuri; Huan Yang; Junquan Su; Marc Casals; Neev Khera

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Gravitational Waves from Green's Function Decomposition for a Kerr black hole: I. Equatorial ISCO Plunge

Junquan Su, Neev Khera, Abhishek Chowdhuri, Marc Casals, Huan Yang

General Relativity and Quantum Cosmology · Submitted 18 Aug 2026

Pith's verdict: A careful and genuinely new Kerr Green's function decomposition, but the headline spin-dependent termination time is inherited from an assumed split time, not derived.

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arXiv ID2608.17943 (version 2)

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CitationAbhishek Chowdhuri, Huan Yang, Junquan Su, Marc Casals, Neev Khera. "Gravitational Waves from Green's Function Decomposition for a Kerr black hole: I. Equatorial ISCO Plunge." arXiv:2608.17943 (2026). https://pith.science/paper/NHB4NWQD

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Dear [colleague],

The one thing to know: this paper extends the Schwarzschild Green's function split to Kerr and gives the first numerical look at the branch-cut direct part in Kerr, but the headline spin-dependent termination time in Eq. (90) is not derived. It follows from setting the Kerr split time equal to the Schwarzschild value |r*|+|r'*|, and the authors admit as much. The abstract, however, presents it more confidently.

What is genuinely new: the WKB analysis of large-frequency spheroidal harmonics indicating an O(a) window of uncertainty around the causal front; the treatment of angular branch cuts in the separated (l,m) modes; and fixed-source comparisons in which direct part plus truncated QNM sum reproduces the real-axis inverse transform away from the fronts. The numerical work looks careful: MST and Jaffé series, a Riccati handoff, Hermite interpolation in frequency, and no FFT periodicity artifacts. The application to an ISCO plunge and the explicit contrast with Oshita's 'direct wave' are useful.

Where the paper is soft: the split time. The authors set it to the Schwarzschild value everywhere, while their own Appendix E shows sector-dependent O(a) phases in the large-arc integrand. Within a window of width roughly a around |r*|+|r'*|, contour closure is not established. The final plunge waveform is not obtained by contour integration; it is a real-axis inverse transform partitioned by causal step functions. So u_D,end inherits the assumed boundary, and for a/M=0.7 the six-decimal precision in Eq. (90) is not justified — the uncertainty is of order 0.7M. The fixed-source tests validate the split away from the fronts; they do not constrain the front neighborhood, which is exactly where the endpoint lives. The angular-cut jump integrals and the radial tail are not evaluated separately; the authors say they are negligible over the displayed windows, but that is not quantified.

Does this invalidate the paper? No. The fixed-source checks are not circular — they use independent time-domain evolutions and external QNM frequencies — and they give real support to the direct/QNM/tail organization. The main gap is that the spin dependence of the endpoint is not established, and the abstract's 'excellent agreement' oversells the body's 'qualitative agreement.'

This is for people working on black-hole spectroscopy, prompt versus modal ringdown content, and Kerr Green's functions. A serious referee should press on the split-time assumption and on the overstatement in the abstract. My recommendation: send it out. It is a technically solid paper with an honest limitation that needs to be surfaced before Eq. (90) is used as a prediction.

Referee report

Significance

If the construction were fully justified, this would be the first numerical extraction of the branch-cut direct part in Kerr and a useful first-order-in-mass-ratio model of ringdown excitation. The numerical implementation is substantial: MST and Jaffé radial solutions, a high-order Hermite interpolation of a 26,465-node frequency grid, externally supplied Kerr QNM frequencies, and an independent HPHC time-domain solver. The fixed-source comparisons are credible sanity checks, and the paper usefully distinguishes its direct part from the 'direct wave' of Oshita et al. However, because the final plunge split is a time-windowing of one real-axis transform, and because the causal boundary is assumed rather than derived, the new quantitative claims, especially Eq. (90), are not yet established. The significance is therefore conditional: high value if the split-time ambiguity and the omitted contributions are resolved, but not sufficient as it stands.

Major objections (4)

[Sec. III C and Eqs. (21)-(22), (83)-(84), (89)-(90); Appendix E] The Kerr split time is assumed rather than derived, and this assumption propagates into the endpoint u_D,end. In Sec. III C the authors state that they 'operationally set the split time to be |r*|+|r'_*| as the Schwarzschild limit' and that in an O(a) window around the front the large-arc contributions are not understood; Appendix E confirms that the new phase T(r',θ',r,θ,arg ω)=O(a) is sector-dependent. Because the final plunge waveform is not obtained by contour deformation but by windowing the full real-axis inverse transform with I_D and I_L, every direct-part interval and Eq. (89) inherit the assumed boundary. The supremum in Eq. (90) is controlled by source points with r'_* near zero, where the width 2r'_* of the direct interval is smallest and the O(a) uncertainty is relatively largest. For a/M=0.7, an O(a) shift exceeds both the six-decimal precision of Eq. (90) and the Δu=0.05M sampling of the waveforms. The abstract's statement that split times are determined by the spin and positions is therefore not supported; this is the central quantitative claim of the paper.
[Secs. III B, V C, and VII] The decomposition is incomplete for the isolated (ℓ,m)=(2,2) mode. The paper states that the angular-cut jump integrals and the NIA branch-cut tail are not evaluated, and that the post-front sector is treated as QNM+Tail without a separate tail computation. For a single separated mode, Sec. III B shows that angular cuts obstruct the upper-half-plane analyticity argument that would enforce strict pre-front vanishing; consequently, the assignment of the real-axis response to 'direct' versus 'QNM+Tail' by Eqs. (83)-(84) may misclassify contributions in the front neighborhood. Since the omitted pieces are exactly the ones that determine the nature of the split, their neglect should be quantified, or the claims in the abstract and conclusion should be weakened to an approximate time-windowing of the full response.
[Sec. VI and Fig. 8] The trajectory-integrated split is not validated independently. The total waveform in Eq. (87) is equal to direct plus QNM+Tail by construction, since both components are obtained by multiplying the same real-axis inverse transform by complementary causal indicators. A comparison of the decomposed components against a time-domain evolution of the finite-start plunge, analogous to the fixed-source checks of Figs. 3-6, would be needed to test whether the split has physical content. The current figure demonstrates internal bookkeeping consistency only.
[Sec. V B and Figs. 5-6] The fixed-source validation is qualitative. The lower panel of Fig. 5 is on a linear scale that hides the front spike of the truncated direct-part contour, which is about 2.97×10^5 times the real-axis peak in the upper panel, and the paper reports no quantitative error norms or convergence tests in the number of QNM overtones or contour parameters. The abstract's claim of 'excellent agreement' should be restricted to the away-from-front regime that is actually shown.

Minor objections (4)

[Appendix E] There are typos in Appendix E: 'retarted Green’s funtion' should read 'retarded Green's function'.
[Eq. (90)] The six-decimal entries are inconsistent with the Δu=0.05M sampling of the waveform; report u_D,end to the resolution of the observer grid or state that the extra digits come from the source-time interpolation.
[Fig. 8] The green curve is labeled H^{NODW}_{22} in the caption but the main text and legend refer to NODW; please use one notation consistently.
[Sec. III B and Appendix C] The text says that the complete ℓ sum restores upper-half-plane analyticity for each fixed m, but the numerical calculation retains only the ℓ=2 channel; the fixed-source agreement therefore also depends on the smallness of the ℓ>2 mixing, which should be stated and quantified at least once.

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The paper

The paper claims

The Kerr Green's function admits a clean split into prompt, quasinormal-mode, and tail parts, applied here to a test particle plunging into a rotating black hole.

The reading

This paper extends a time-domain decomposition of the point-source Green's function to the Kerr spacetime: the gravitational response splits into a direct (prompt) part, a quasinormal-mode (QNM) ringing part, and a late-time tail. The authors validate the split against time-domain Teukolsky simulations and then apply it to a test particle plunging from the innermost stable circular orbit (ISCO) into a Kerr black hole. For the (2,2) radiative metric mode they find that the direct part of the emitted waveform ends exactly at a spin-dependent retarded time, after which only QNM ringdown remains. They also show that this direct part is not the same as the 'direct wave' identified in earlier phenomenological ringdown models. The result matters because it offers a first-principles, leading-order-in-mass-ratio decomposition of ringdown into modal and non-modal pieces.

What carries the argument

The central object is the frequency-domain Teukolsky radial Green's function split through the scattering identity R_in = A_ref R_up + A_inc R_down, which separates the response into G+ (carrying the QNM poles) and G- (direct-part and tail). After a contour deformation in the complex-frequency plane, the causal sectors are identified by step-function 'causal indicators' whose fronts sit at U = ±r'_* and U = |r'_*|; the direct part is the support between -r'_* and r'_* for sources with r'_*>0. Because Kerr's spheroidal harmonics are frequency dependent, the paper also confronts angular branch cuts (exceptional-point cuts in the angular eigenvalue spectrum), which block the usual mode-by-mode causality argument, and its WKB analysis in Appendix E shows an O(a)-wide window around the Schwarzschild split time in which the large-arc contour is not under control.

Extended reading notes

Core claim

The paper claims that the Kerr Teukolsky Green's function admits a time-domain split into a direct part, QNM residues, and a branch-cut tail, with causal-front boundaries at split times inherited from the Schwarzschild analysis, namely |r*|+|r'*| for the scattered part and r*-r'* for the direct part. Applying this to a finite-start equatorial ISCO plunge, the trajectory-integrated (2,2) metric waveform separates into a direct part and a QNM+tail part, and the direct part has compact support: it terminates at the spin-dependent time u_D,end (Eq. 90) when the last source point with r'_*>0 crosses r'_*=0. The paper presents this as the first numerical extraction of the branch-cut direct part in Kerr spacetime and as a step toward a systematic decomposition of ringdown into modal and non-modal components.

Load-bearing premise

The paper's entire split, including the direct-part endpoint u_D,end, assumes the direct/QNM boundary sits at the Schwarzschild time |r*|+|r'*|, even though the authors' own WKB analysis shows a window of width O(a) around that time in which the large-arc contour contributions are not understood.

Editorial extensions

If this is right

The direct part of the (2,2) plunge waveform has compact support and vanishes identically after u_D,end = 418.78, 297.33, and 180.03 M for spin a/M = 0.1, 0.4, and 0.7 respectively (Eq. 90).
Outside a thin window near the causal fronts, the sum of the direct part and the truncated QNM overtone sum reproduces the full time-domain Teukolsky evolution; the gap near the front narrows as more overtones are included.
For source points with r'_* < 0 no direct part exists; the response starts directly in the QNM+tail sector, so the causal-indicator prescription determines where the split is applied.
The paper's direct part differs from the phenomenological 'direct wave' proposed in an earlier paper; the two describe different parts of the signal and should not be conflated.

Reading between the lines

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

If the split boundary is later corrected from the Schwarzschild value |r*|+|r'*| to the true Kerr value (shifted by O(a) according to the WKB sector analysis), then both the direct-part waveform and the QNM excitation coefficients would shift, and u_D,end would no longer coincide exactly with the r'_*=0 crossing on the worldline.
A natural testable extension is to apply the same causal-indicator split to eccentric or inclined plunge trajectories and check whether the direct part still has compact support with an endpoint given by the last r'_*>0 source crossing r'_*=0.
Since the decomposition is performed mode-by-mode for the (2,2) channel only, summing over all angular modes may shift the effective split time of the full 3+1 waveform; the paper notes the full sum restores causality, so a multi-mode sum would be a direct consistency check of the isolated-mode split.

Original abstract

We present a decomposition of the Kerr Green's function in the time domain, motivated by the frequency-domain split previously studied in the Schwarzschild limit. We show that the identification of a quasinormal-mode contribution, a direct part, and a late-time tail is still available, where the split times are determined by the black hole spin and positions of the emitter and receiver. We have checked this Green's function with time-domain Teukolsky numerical simulations and find excellent agreement. We also apply this decomposed Green's function in the time domain to a model problem with a test particle plunging into a Kerr black hole. The dynamically excited direct wave and quasinormal modes are obtained by convoluting the Green's function with the particle's source term, which may be viewed as the first order in mass ratio of a spinning black hole ringdown.

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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, 4,348 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.90 · mentions code or data availability 0.03.

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We present a decomposition of the Kerr Green's function in the time domain, motivated by the frequency-domain split previously studied in the Schwarzschild limit. We show that the identification of a quasinormal-mode contribution, a direct part, and a late-time tail is still available, where the split times are determined by the black hole spin and positions of the emitter and receiver. We have checked this Green's function with time-domain Teukolsky numerical simulations and find excellent agreement. We also apply this decomposed Green's function in the time domain to a model problem with a test particle plunging into a Kerr black hole. The dynamically excited direct wave and quasinormal modes are obtained by convoluting the Green's function with the particle's source term, which may be viewed as the first order in mass ratio of a spinning black hole ringdown.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a decomposition of the Kerr Green's function in the time domain, motivated by the frequency-domain split previously studied in the Schwarzschild limit.	contribution 0.97	0.92	0.12	asserted 0.99
s1	We show that the identification of a quasinormal-mode contribution, a direct part, and a late-time tail is still available, where the split times are determined by the black hole spin and positions of the emitter and receiver.	result 0.90	0.89	0.44	asserted 0.99
s2	We have checked this Green's function with time-domain Teukolsky numerical simulations and find excellent agreement.	result 0.71	0.97	0.10	asserted 1.00
s3	We also apply this decomposed Green's function in the time domain to a model problem with a test particle plunging into a Kerr black hole.	contribution 0.74	0.95	0.08	asserted 1.00
s4	The dynamically excited direct wave and quasinormal modes are obtained by convoluting the Green's function with the particle's source term, which may be viewed as the first order in mass ratio of a spinning black hole ringdown.	interpretation 0.40	0.89	0.20	hedged possibility 0.61

What kind of paper the classifier reports this is: method or tool 0.79 · evidence from simulation or computation 0.86.

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Figures

**Figure 1.** FIG. 1. Schematic transfer between spherical modes. For a fixed spheroidal/radial channel

**Figure 3.** FIG. 3. Real part of the radiative response

**Figure 5.** plots the fixed-source metric kernel 𝐾H with 𝑟 ′ ∗ = 2𝑀. The shaded interval is the direct-part support −2𝑀 < 𝑈 < 2𝑀. On the logarithmic magnitude scale, the finite-cutoff direct-part contour develops a narrow spike at 𝑈 ≃ 𝑟 ′ ∗ , about 2.97 × 105 times the peak of the real-axis result. This is not a physical amplification: it is the nonuniform light-front limit of the truncated imaginary-axis integral. The lower signed, linear-scale panel shows that the ordinary direct part remains of the same order as the real-axis kernel and that the QNM pole sum follows both the phase and…

**Figure 6.** FIG. 6. Real part of the fixed-source metric kernel at

**Figure 8.** FIG. 8. Trajectory-integrated

**Figure 9.** FIG. 9. Angular cuts obtained after analytic overlap continuation for

Showing 7 of the paper's 9 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2608.17943 by the authors.

References

Forward citations

Cited by 7 Pith papers

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

Cubic Quasinormal Modes in the Black-Hole Ringdown
gr-qc 2026-10 conditional novelty 8.0 of 10

Cubic (third-order) quasinormal-mode amplitudes for Kerr black holes are computed from nonlinear perturbation theory and claimed to contribute about 38 percent of the dominant strain at 3M after merger.
Exact Cancellation of Horizon Modes in Kerr Ringdown
gr-qc 2026-09 accept novelty 8.0 of 10

每个源诱导的地平线模极点都被无穷多个Matsubara零点依次抵消，彻底排除了直接波模型。
Dissection of a merger-ringdown waveform in the small-mass-ratio limit
gr-qc 2026-09 accept novelty 8.0 of 10

First-principles calculations of quadratic quasinormal modes and nonlinear memory in the small-mass-ratio merger-ringdown, showing that quadratic modes dominate over linear overtones for many mass ratios.
Bouncing and QNM convergence in Kerr
gr-qc 2026-11 accept novelty 7.0 of 10

The convergence region of QNM sums in Kerr is set by complex-time bouncing singularities, with prolate and oblate families giving different bounce radii and convergence boundaries.
Tidal Excitation of Relativistic Stars
gr-qc 2026-10 conditional novelty 7.0 of 10

A relativistic star's tidal response is decomposed into direct, quasinormal-mode, and tail sectors, each obeying a first-order driven oscillator equation, and the split is verified against a full numerical evolution.

Show 2 more citing papers

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.

How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 39 sentences without a confident label

Builds on 1 paper

Finally, we generalize the contour construction strategy from our previous work [ref] to the Teukolsky equation.

2601.22015 · Green's function decomposition and validation · 0.96

Uses the method or tool of 4 papers

In addition, we retain the continuation strategy developed in our previous work [ref] , whereby Eq. [ref] is solved along a continuous trajectory in the complex $ω$ plane.

2601.22015 · Renormalized Angular Momentum · 0.95
Therefore, when $Mω>0.4$, we switch to using a Riccati solver from the modified open-source package [ref] .

Recipes for computing radiation from a Kerr black hole using a generalized Sasaki-Nakamura formalism: Homogeneous solutions · Numerical implementation · 1.00
With these coefficients, the separated point-particle source for the radial Teukolsky equation can be written, following Ref. [ref] , as [equation] where $μ$ is the mass of the plunging particle.

Analytic black hole perturbation approach to gravitational radiation · Fixed-source kernel and its construction · 0.93

Compares with 9 papers

In particular, we compare the direct wave obtained from the Green's function calculation with the phenomenological direct-wave model proposed in Ref. [ref] (also see [ref] ).

Probing Direct Waves in Black Hole Ringdowns · Introduction · 1.00
In particular, we compare the direct wave obtained from the Green's function calculation with the phenomenological direct-wave model proposed in Ref. [ref] (also see [ref] ).

GW250114 reveals black hole horizon signatures · Introduction · 1.00
In particular, we compare the direct wave obtained from the Green's function calculation with the phenomenological direct-wave model proposed in Ref. [ref] (also see [ref] ).

2601.11706 · Introduction · 1.00

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