Singular structures and causality of the Schwarzschild Green's function in the frequency domain

TL;DR

This paper analyzes the singular spectral components of the Schwarzschild black hole Green's function in the frequency domain, clarifying their physical interpretation and implications for gravitational wave signals.

Contribution

It provides the first analytical justification for phenomenological ringdown models based on greybody factors and clarifies the role of tail and quasinormal mode contributions in black hole perturbations.

Findings

Identifies two tail contributions activated with a time delay.

Shows corrections to Price's law are relevant at intermediate times.

Demonstrates redshifted components persist up to late times.

Abstract

We study two singular spectral components of the Green's function of a Schwarzschild black hole and their interpretation in the frequency domain: (i) the low-frequency branch cut, which yields corrections to Price's law tails in the form of inverse power laws weighted by logarithmic terms; and (ii) the quasinormal-mode spectrum, which generates a redshifted response for sources extended toward the horizon. We show that the frequency-domain Green's function can be naturally interpreted in terms of greybody factors, providing the first analytical justification for recent phenomenological ringdown models based on these quantities. For sources localized outside the peak of the potential barrier, we identify two tail contributions activated with a time delay, arising from backscattering of the prompt response and of the ringdown signal. We show that corrections to Price's law can be relevant at intermediate times, when the ringdown still dominates the waveform. For sources localized inside the potential barrier peak, the tail is suppressed and the signal is instead dominated by quasinormal frequencies. In this regime, these spectral components produce both the ordinary quasinormal-mode ringdown and an infinite tower of exponentially decaying terms governed by the horizon surface gravity, the so-called redshift terms. We demonstrate that this component is not screened by geometric features of the background spacetime and persists up to late times, as supported by numerical investigations of perturbative waveforms. Our results provide a mathematical foundation for phenomenological modeling of the branch-cut contribution at intermediate times, which is relevant for prospective observations of tails, and strong evidence for the presence of redshifted components from intermediate to late times.