Prompt Response from Plunging Sources in Schwarzschild Spacetime

Abstract: Gravitational waves generated by moving sources in Schwarzschild spacetime can be decomposed into three principal components: quasinormal modes, tail, and prompt response. While the first two have been extensively studied, a systematic and exact treatment of the prompt response has received comparatively little attention. In this work, building on recent progress in elucidating the structure of the Green's function of the Regge-Wheeler equation, we place the prompt response on a firm theoretical footing and investigate its morphology for sources inspiraling and plunging into a Schwarzschild black hole. We find that during the inspiral phase, the prompt response is stronger than the dynamical excitation of quasinormal modes by a factor of ~1.2, with both contributions modulated by the instantaneous orbital motion. Near the waveform peak, the prompt response rapidly decays, while the quasinormal modes transition into the ringdown regime. By combining the prompt response, quasinormal modes, and tail contributions, we achieve an accurate reconstruction of the full time-domain inspiral-merger-ringdown waveform at the 99$\%$ level, thereby providing strong support for the accuracy of this decomposition. These results offer new insight into the transition from inspiral to merger and ringdown.

• The paper develops a first-principles Green’s-function method that converts the inaccessible high-frequency contribution into a positive-imaginary-axis branch-cut integral, enabling exact prompt-response calculations for plunging sources.
• The prompt, quasinormal-mode, and tail components reconstruct the full waveform with below 2% error for an ISCO plunge and below 10⁻³ error for radial infall, while the prompt signal dominates late inspiral before rapidly fading near the horizon.
• The results identify a sharp source-dependent transition near r≈2.5569, quantify destructive interference between prompt and QNM radiation, and show why high-overtone excitation coefficients can remain time-dependent.

Motivation and scope

The gravitational radiation from a perturbed Schwarzschild black hole is conventionally attributed to quasinormal modes (QNMs) and late-time tails, both of which are well characterized. A third component — the prompt response, which propagates directly from source to observer without backscattering — has been identified in Green's function analyses but lacks a systematic treatment for sources near the horizon. This paper places that component on a firm footing for plunging sources in Schwarzschild spacetime (2604.08680). Building on a recent contour-deformation proposal for the Regge–Wheeler Green's function [Su:2026fvj], the authors compute the prompt response exactly for two representative trajectories — a quasi-circular plunge from the innermost stable circular orbit (ISCO) and radial infall — and demonstrate that prompt response plus QNMs plus tail reconstructs the full time-domain waveform to approximately 99% accuracy.

The motivation is partly phenomenological: direct-wave components have been identified in numerical-relativity and extreme-mass-ratio waveforms via rational filtering [Oshita:2025qmn; Lu:2025vol], and modeled phenomenologically through steepest-descent methods and the Backwards One Body approach [McWilliams:2018ztb; Kankani:2026byb], both rooted in eikonal reasoning. The present work supplies the first-principles Green's function counterpart to those constructions.

Structure of the Green's function decomposition

For the frequency-domain Green's function of the Regge–Wheeler equation with ingoing boundary conditions at the horizon and outgoing at infinity, Leaver's contour deformation gives QNM pole residues plus a branch-cut tail integral along the negative imaginary axis once $u > |r_*'|$, where $r_*$ is the tortoise coordinate of the source. At intermediate retarded times $-r_*' < u < |r_*'|$ the large arc does not vanish, and its direct evaluation is numerically intractable.

The key technical step is to split the in-solution into down-going and up-going pieces, $$R^{\rm in} = A^{\rm inc}_{\rm in} R^{\rm down} + A^{\rm ref}_{\rm in} R^{\rm up}$$, decomposing the Green's function into $G^- = R^{\rm down}/(2i\omega)$ and $$G^+ = A^{\rm ref}_{\rm in} R^{\rm up}/(2i\omega A^{\rm inc}_{\rm in})$$. For $G^-$, the lower-half-plane large arc vanishes and there is no negative-imaginary-axis branch cut, leaving only a small arc around $\omega=0$ — the step-function contribution familiar from flat spacetime and prior leading-order work [Andersson:1996cm; Hui:2019aox; Lagos:2022otp; DeAmicis:2026tus]. For $G^+$, by contrast, the lower-half-plane arc is nonvanishing and grows as the source approaches the horizon; closing instead in the upper half-plane converts it entirely to an integral along the positive imaginary axis, computable with established branch-cut techniques.

Numerically, using the Mano–Suzuki–Takasugi (MST) formalism, the sum $G^+ + G^-$ reproduces the full time-domain Green's function throughout the intermediate window for sources at $r_*$0, $r_*$1, $r_*$2, and $r_*$3. Near the light ring, $r_*$4 individually reach magnitudes of order $r_*$5–$r_*$6 but cancel almost completely, leaving a net amplitude of order unity — a delicate cancellation that the method nonetheless captures accurately. The prompt window shrinks as the source approaches $r_*$7, beyond which the entire signal is carried by QNM and tail contributions, joined smoothly to the prompt piece at $r_*$8. This provides an unambiguous, source-location-independent temporal boundary between the prompt-response and QNM+tail regimes.

Waveform decomposition for plunging sources

Convoluting this Green's function with a point-particle source $r_*$9, the waveform at null infinity splits into a prompt integral over the emission window $-r_*' < u < |r_*'|$0 defined implicitly by $-r_*' < u < |r_*'|$1 and $-r_*' < u < |r_*'|$2, plus a QNM+tail integral over earlier times. The Penrose-diagram interpretation is transparent: each observer time receives prompt radiation from a finite interval of the trajectory and backscattered radiation from everything before it.

ISCO plunge. With geodesic parameters $-r_*' < u < |r_*'|$3, $-r_*' < u < |r_*'|$4, and a redshifted source profile $-r_*' < u < |r_*'|$5 for the $-r_*' < u < |r_*'|$6 mode, the reconstruction achieves relative error below 2% across the full inspiral–merger–ringdown evolution, using QNM overtones up to $-r_*' < u < |r_*'|$7 (adding more overtones does not improve accuracy). Two quantitative findings stand out:

• During late inspiral, the prompt response exceeds the dynamically excited QNM contribution by a factor of $-r_*' < u < |r_*'|$8, with both modulated at twice the orbital frequency.
• The two components differ in phase by $-r_*' < u < |r_*'|$9, producing partial destructive interference in the total signal.

Near the waveform peak the prompt response decays sharply, for two reasons: the prompt emission window $$R^{\rm in} = A^{\rm inc}_{\rm in} R^{\rm down} + A^{\rm ref}_{\rm in} R^{\rm up}$$0 collapses as the particle nears the horizon, and the source term vanishes under redshift. The QNM contribution then dominates and transitions smoothly into ringdown. The dynamical excitation coefficients $$R^{\rm in} = A^{\rm inc}_{\rm in} R^{\rm down} + A^{\rm ref}_{\rm in} R^{\rm up}$$1 grow exponentially during inspiral at rates set by the QNM decay rates — compensated by $$R^{\rm in} = A^{\rm inc}_{\rm in} R^{\rm down} + A^{\rm ref}_{\rm in} R^{\rm up}$$2 so the physical amplitudes stay roughly constant — and all coefficients intersect near $$R^{\rm in} = A^{\rm inc}_{\rm in} R^{\rm down} + A^{\rm ref}_{\rm in} R^{\rm up}$$3. Notably, modes with $$R^{\rm in} = A^{\rm inc}_{\rm in} R^{\rm down} + A^{\rm ref}_{\rm in} R^{\rm up}$$4 continue growing exponentially even at late times, because the QNM eigenfunction growth $$R^{\rm in} = A^{\rm inc}_{\rm in} R^{\rm down} + A^{\rm ref}_{\rm in} R^{\rm up}$$5 outpaces the near-horizon source decay $$R^{\rm in} = A^{\rm inc}_{\rm in} R^{\rm down} + A^{\rm ref}_{\rm in} R^{\rm up}$$6 ($$R^{\rm in} = A^{\rm inc}_{\rm in} R^{\rm down} + A^{\rm ref}_{\rm in} R^{\rm up}$$7); the resulting growth rate matches $$R^{\rm in} = A^{\rm inc}_{\rm in} R^{\rm down} + A^{\rm ref}_{\rm in} R^{\rm up}$$8. This behavior implies that high overtone amplitudes extracted from such decompositions are intrinsically time-dependent rather than settling to constants.

Radial infall. For a particle released at $$R^{\rm in} = A^{\rm inc}_{\rm in} R^{\rm down} + A^{\rm ref}_{\rm in} R^{\rm up}$$9 with $G^- = R^{\rm down}/(2i\omega)$0 and $G^- = R^{\rm down}/(2i\omega)$1, the reconstruction error is below $G^- = R^{\rm down}/(2i\omega)$2. Because backscattering requires finite propagation time, an early window (retarded times before $G^- = R^{\rm down}/(2i\omega)$3) contains only the prompt response — a clean regime in which the direct wave can be studied in isolation. The prompt contribution then decays to zero at $G^- = R^{\rm down}/(2i\omega)$4, when the particle crosses $G^- = R^{\rm down}/(2i\omega)$5; the tail is excited with opposite sign to the QNM component and is about an order of magnitude smaller before transitioning to power-law decay. The same $G^- = R^{\rm down}/(2i\omega)$6 exponential growth of $G^- = R^{\rm down}/(2i\omega)$7 appears, peaking when the prompt response vanishes.

Limitations and open questions

Several caveats qualify these results. The analysis is restricted to Schwarzschild spacetime and linear perturbation theory; extension to Kerr, self-force corrections, and more realistic source terms remains open, though the authors note these are needed for interpreting extreme-mass-ratio-inspiral data. The source profile $G^- = R^{\rm down}/(2i\omega)$8 is a specific proof-of-principle choice, and while qualitative features are expected to persist, quantitative factors such as the 1.2 amplitude ratio and the $G^- = R^{\rm down}/(2i\omega)$9 phase offset may depend on this prescription. The connection between the exact prompt response computed here and the eikonal-based steepest-descent and Backwards One Body models of direct waves is asserted but not demonstrated. Finally, whether the linear-order decomposition survives at nonlinear order — where quadratic QNMs and mode coupling enter — and how it compares against Cauchy–characteristic-matching extractions from binary black hole simulations are questions the paper identifies but does not resolve.

Conclusion

This paper establishes a controlled, first-principles computation of the prompt response for plunging sources in Schwarzschild spacetime, resolving the previously inaccessible high-frequency-arc contribution via conversion to a positive-imaginary-axis branch cut. The decomposition into prompt response, QNMs, and tail reproduces full waveforms at the 99% level for ISCO plunge and better than $$G^+ = A^{\rm ref}_{\rm in} R^{\rm up}/(2i\omega A^{\rm inc}_{\rm in})$$0 for radial infall, and it defines a sharp temporal boundary between prompt and ringdown regimes. By quantifying the prompt response's dominance over dynamically excited QNMs during late inspiral and its rapid decay near merger, the work offers a theoretically grounded alternative to QNM fitting and rational filters for analyzing the merger–ringdown transition, and a foundation for incorporating direct waves into tests of General Relativity.