Complete the Kerr radiative-kernel decomposition

Complete the decomposition of Kerr plunge waveforms by explicitly treating isolated-\((\ell,m)\) angular-cut contributions, radial branch-cut tails, additional spheroidal and radiative modes, and other plunge trajectories.

Background

The numerical implementation obtains the final plunge waveform from a full real-axis inverse transform and assigns portions of it operationally to direct and post-front sectors. It does not separately evaluate the radial branch-cut tail or the angular-cut jump integrals, and the displayed calculations focus on the separated (s,ℓ,m)=(−2,2,2)(s,\ell,m)=(-2,2,2) mode.

A broader treatment is needed to obtain a systematic decomposition beyond the selected mode and trajectory, especially because isolated separated modes can possess angular cuts that obstruct the standard mode-by-mode causality argument.

References

However, a more complete treatment of isolated-$(\ell,m)$ angular cuts, explicit radial branch-cut tails, and extensions to additional modes and other plunge trajectories remains for future work.

— Gravitational Waves from Green's Function Decomposition for a Kerr black hole: I. Equatorial ISCO Plunge  (2608.17943 - Su et al., 18 Aug 2026) in Section 7, Conclusion

A full theoretical understanding of the convergence of the pole--zero representation, the role of the omitted nonrational response, and its relation to the full Green's function remains to be developed.

— From driven oscillations to free ringdown: a particle plunging into Kerr  (2609.09531 - Weller et al., 8 Sep 2026) in Conclusion