Causal Green function decomposition for quantum black hole seismology

The growing sensitivity of gravitational-wave detectors enables increasingly precise tests of black hole (BH) ringdown spectroscopy. BH quasinormal modes (QNMs) are, however, spectrally unstable: small near-horizon modifications can produce a drastically different QNM spectrum, while causality requires the prompt ringdown to remain BH-like until the reflected signal returns. Quantum BHs with substantial interior reflection provide a natural setting for this tension, yet the relation between their time-domain waveform and different QNM spectra still lacks a consistent picture. In this work, we systematically examine the time-domain Green function for quantum BHs, considering sources located outside and inside the light-ring potential barrier. By decomposing the Green function into causally distinct components and choosing the corresponding inverse-Laplace contours consistently, we clarify how the response is built from different sets of QNMs. We find that the quantum BH QNM reconstruction always faithfully describes the signal once the curved spacetime is probed, but its practical efficiency depends strongly on the evolutionary stage. Before interior reflection becomes relevant, we prove that this basis is formally equivalent to the BH QNM and tail expansions, with convergence properties sensitive to source location. At late times, the long-lived modes provide an efficient basis. Time-domain simulations confirm these results, providing a unified causal framework for BH spectroscopy and quantum BH seismology.