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

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.

• The paper extends contour-based Green’s-function decomposition to Kerr, separating gravitational radiation into direct, quasinormal-mode, and late-time tail contributions while addressing frequency-dependent spheroidal harmonics and angular branch cuts.
• The authors validate the decomposition against independent time-domain Teukolsky evolutions, finding close agreement for Kerr spin a=0.4M with the direct component plus 32 quasinormal-mode overtones in the dominant (2,2) channel.
• The paper applies the method to equatorial ISCO plunges and finds that the direct waveform ends at spin-dependent retarded times of approximately 418.78M, 297.33M, and 180.03M for a/M=0.1, 0.4, and 0.7, after which radiation is quasinormal-mode dominated within the studied windows.

The paper presents a time-domain decomposition of the Kerr black-hole Green's function into a direct part, a quasinormal-mode (QNM) contribution, and a late-time tail, extending to rotating spacetimes the contour-based program previously carried out for Schwarzschild (Su et al., 29 Jan 2026). The decomposed Green's function is validated against independent time-domain Teukolsky evolutions and then applied to compute the linear gravitational waveform of a test particle on an equatorial ISCO plunge, providing a first-order-in-mass-ratio model of ringdown for spinning black holes.

Motivation and context

The motivation is grounded in observational precision: the event GW250114, with signal-to-noise ratio near 80, shows statistical evidence for fundamental and first-overtone modes, quadratic modes, and the prompt or "direct wave." As SNR improves, the traditional black-hole spectroscopy program must be upgraded to a complete ringdown test in which modal and non-modal components are separately identified. Leaver's original frequency-domain contour decomposition of the Schwarzschild Green's function left the direct part uncomputed because of difficulties with the large-arc integral; recent reformulations enabled explicit time-domain decompositions for Schwarzschild. This work extends that program to Kerr, where two new obstructions arise: the Teukolsky equation replaces Regge–Wheeler–Zerilli, and the spin-weighted spheroidal harmonics depend on frequency, introducing angular branch cuts absent in Schwarzschild.

Analytic structure of the Kerr Green's function

The full $(3+1)$-dimensional retarded Green's function is written as a sum over spheroidal channels with a Wronskian-constructed radial Green's function $G_{\ell m\omega}=R^{\rm in}(r_<)R^{\rm up}(r_>/W$. A key technical point is that at complex frequency the two angular factors must be paired bilinearly—without complex conjugation—for the integrand to remain analytic and usable for contour deformation.

Using the scattering identity, the radial Green's function splits as $\widetilde G=\widetilde G^++\widetilde G^-$, where only $\widetilde G^+$ contains QNM poles (zeros of $A^{\rm in}_{\rm inc}$) while $\widetilde G^-$ is pole-free; their sum carries only the negative-imaginary-axis cut. For fixed $(\ell,m)$, however, the angular eigenvalue and the rank-one projector onto the spheroidal mode develop square-root branching at exceptional points, producing angular cuts along rays $\omega=\rho\,\omega_{\rm EP}$, $\rho\geq1$, symmetric about the real axis. Consequently, no horizontal Bromwich line lies above all upper-half-plane angular cuts of an isolated mode, and the mode-by-mode proof of strict pre-arrival vanishing fails. The authors emphasize this is a property of the separated representation: the complete $\ell$ sum cancels the angular cuts and restores causality of the full retarded Green's function. Numerically, neglecting the angular-cut jump integrals produces no appreciable difference from time-domain evolution over the windows studied—an empirical result, not a proof.

Contour strategy and split times

Three causal regions govern the contour choice, generalizing the Schwarzschild construction: Region I ($G_{\ell m\omega}=R^{\rm in}(r_<)R^{\rm up}(r_>/W$0) encloses QNM poles and the NIA cut; Region II ($G_{\ell m\omega}=R^{\rm in}(r_<)R^{\rm up}(r_>/W$1, only for $G_{\ell m\omega}=R^{\rm in}(r_<)R^{\rm up}(r_>/W$2) closes $G_{\ell m\omega}=R^{\rm in}(r_<)R^{\rm up}(r_>/W$3 upward and $G_{\ell m\omega}=R^{\rm in}(r_<)R^{\rm up}(r_>/W$4 downward, yielding the direct part from imaginary-axis jumps plus a zero-frequency detour; Region III ($G_{\ell m\omega}=R^{\rm in}(r_<)R^{\rm up}(r_>/W$5) gives strict causality in Schwarzschild but is subtle in Kerr because of the angular cuts. Operationally the split time is still taken as $G_{\ell m\omega}=R^{\rm in}(r_<)R^{\rm up}(r_>/W$6.

A WKB analysis in the appendix quantifies why this is approximate rather than exact. At large complex $G_{\ell m\omega}=R^{\rm in}(r_<)R^{\rm up}(r_>/W$7, the spheroidal spectrum separates into oblate-type branches ($G_{\ell m\omega}=R^{\rm in}(r_<)R^{\rm up}(r_>/W$8) and prolate-type branches ($G_{\ell m\omega}=R^{\rm in}(r_<)R^{\rm up}(r_>/W$9), each contributing angle-dependent exponential factors to the large arc. Since different portions of a large semicircle sample different sectors, the Kerr correction to the Schwarzschild phase is bounded by $\widetilde G=\widetilde G^++\widetilde G^-$0 but sector-dependent, so there exists a window of width $\widetilde G=\widetilde G^++\widetilde G^-$1 around the Schwarzschild transition time within which neither half-plane closure is uniformly valid. The paper concedes plainly that inside this window the large-arc contribution is not understood, owing to the absence of a full description of spheroidal harmonics at large complex frequency.

Validation against time-domain evolution

Two fixed-source tests validate the construction. In Schwarzschild, a single $\widetilde G=\widetilde G^++\widetilde G^-$2, $\widetilde G=\widetilde G^++\widetilde G^-$3 Teukolsky mode driven by a Gaussian-approximated delta source agrees closely with a $\widetilde G=\widetilde G^++\widetilde G^-$4-dimensional evolution, though a narrow gap remains between the end of the direct interval and QNM-sum convergence—a gap attributed to the two extra time derivatives in $\widetilde G=\widetilde G^++\widetilde G^-$5, which weight each overtone by roughly $\widetilde G=\widetilde G^++\widetilde G^-$6 and slow convergence relative to metric perturbations. Adding overtones narrows the gap systematically, but computing sufficiently many Teukolsky overtones is impractical.

For Kerr ($\widetilde G=\widetilde G^++\widetilde G^-$7), the spherical-mode transfer Green's function—the object coupling source spherical mode $\widetilde G=\widetilde G^++\widetilde G^-$8 to observer mode $\widetilde G=\widetilde G^++\widetilde G^-$9 through intermediate spheroidal channel $\widetilde G^+$0—is compared against a compactified $\widetilde G^+$1-dimensional time-domain solver using horizon-penetrating hyperboloidally compactified coordinates with spectral discretization. Using the direct part plus 32 QNM overtones of the diagonal $\widetilde G^+$2 channel alone reproduces the full evolution closely, confirming that the strongly diagonal mixing matrix makes higher-$\widetilde G^+$3 channels negligible here. Radial solutions combine MST series (160-bit precision, 400 terms) with Jaffé series near the horizon, and a Riccati solver handles $\widetilde G^+$4.

Radiative kernel and real-axis prescription

For extended sources, the fixed-source radiative kernel $\widetilde G^+$5 is defined so that all normalization factors are fixed unambiguously, with the point-particle source operator $\widetilde G^+$6 acting on the IN solution following Sasaki–Tagoshi. Applying $\widetilde G^+$7 to the scattering identity yields $\widetilde G^+$8, with QNM poles confined to $\widetilde G^+$9, giving the three-way split $A^{\rm in}_{\rm inc}$0.

The central numerical finding concerns convergence at causal fronts. The truncated contour representation converges nonuniformly there: writing $A^{\rm in}_{\rm inc}$1 on the positive imaginary axis, the factor $A^{\rm in}_{\rm inc}$2 progressively cancels radial exponential suppression as $A^{\rm in}_{\rm inc}$3, leaving a front spike—in one example reaching about $A^{\rm in}_{\rm inc}$4 times the peak of the physical kernel. This spike is explicitly identified as a numerical artifact of the truncated integral, not physical amplification. Away from fronts, the 42-pole QNM residue sum matches the real-axis inverse transform in phase, amplitude, and decay envelope, including in the instructive case $A^{\rm in}_{\rm inc}$5 where no direct part exists and the post-front match tests the residues directly.

Because of the front pathology, the production plunge calculation uses the full real-axis inverse Fourier transform—sampled on a 26,465-node nonuniform grid with degree-11 Hermite interpolation evaluated analytically per panel—and classifies the resulting kernel in time via causal indicators $A^{\rm in}_{\rm inc}$6 and $A^{\rm in}_{\rm inc}$7. The authors state clearly that this prescription is equivalent to the complete contour deformation only when all pieces (angular-cut jumps, NIA tail, zero-frequency terms) are included; the tail and angular cuts are never computed separately, and their effects enter only implicitly through the real-axis transform. Over the plotted windows the tail is numerically negligible.

ISCO plunge waveforms

The source is a point mass on a finite-start prograde equatorial plunge retaining ISCO energy and angular momentum, started at $A^{\rm in}_{\rm inc}$8 to avoid the multiple root of the marginal-orbit potential, and integrated to $A^{\rm in}_{\rm inc}$9. Trajectory-integrated $\widetilde G^-$0 metric waveforms are computed for $\widetilde G^-$1, $\widetilde G^-$2, and $\widetilde G^-$3 at observer angles $\widetilde G^-$4, showing slowly evolving near-ISCO oscillations, rapid plunge modulation, and damped ringdown, with comparable amplitudes across spins but strongly spin-dependent phase evolution.

A structurally significant result is the finite termination of the direct part. Because a source point contributes to the direct-part orbit integral only when $\widetilde G^-$5 and $\widetilde G^-$6, the union of direct intervals has finite supremum

$\widetilde G^-$7

approached as the particle crosses $\widetilde G^-$8. Numerically, $\widetilde G^-$9, $(\ell,m)$0, and $(\ell,m)$1 for $(\ell,m)$2, $(\ell,m)$3, $(\ell,m)$4 respectively. Beyond this endpoint the disappearance of the direct component is geometric, not due to damping or numerical cancellation; deeper source points simply have no direct interval. All later radiation belongs to the post-front QNM-dominated sector.

The paper also draws a sharp distinction between its direct part and the phenomenological "direct wave" (NODW) prescription of Oshita et al. (Oshita et al., 11 Sep 2025): the two describe completely different parts of the signal and should not be conflated despite sharing the word "direct."

Limitations and open questions

Several limitations are stated explicitly. First, the exact split time in Kerr remains undetermined: the $(\ell,m)$5 window around $(\ell,m)$6 is not controlled by the present asymptotics, and the full large-complex-frequency behavior of spheroidal harmonics—including whether more than the two known asymptotic sectors exist—is unresolved. Second, the isolated-$(\ell,m)$7 angular-cut jump integrals are never computed; any future complex-frequency contour reconstruction of an isolated mode must either include discontinuities across every crossed angular cut or be formulated with closed-cluster Riesz projectors, whose square-root discontinuities cancel. Third, the NIA branch-cut tail is not evaluated separately, so the identification of the post-front sector with "QNM" rests on the empirical negligibility of the tail over the plotted windows. Fourth, the results cover only the $(\ell,m)$8 mode, equatorial prograde plunges, and future null infinity; finite-radius waveforms, additional modes, and other trajectories are deferred. Finally, the whole construction is linear perturbation theory, i.e., leading order in mass ratio, and nonlinear couplings among modal and non-modal components remain outside its scope.

Conclusion

This work establishes the first numerical extraction of the branch-cut direct part in Kerr spacetime and demonstrates that the direct/QNM/tail organization of the Green's function survives rotation, with split times set by spin and emitter-receiver geometry up to an uncontrolled $(\ell,m)$9 window. The agreement between contour and real-axis constructions away from causal fronts, together with the trajectory-integrated plunge waveforms and their geometrically determined direct-part termination times, provides a first-principles template for decomposing ringdown signals at leading order in mass ratio. The open problems it leaves—exact Kerr split times, angular-cut contributions for isolated modes, explicit tails, and nonlinear extensions—are well posed and follow directly from the analysis presented.