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

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.

Figures (5)

• Figure 1: Schematic representation of the singular structure of the retarded Green's function in the complex $\omega$ plane. The branch cut along the negative imaginary axis, the zero-frequency singularity, and the QNM poles are indicated.
• Figure 2: Schematic picture of the two causally distinct contributions $G^{(1)}_{\ell m}$ and $G^{(2)}_{\ell m}$ for a source $S$ supported outside the light ring ($r'>3M$) and an observer $O$ at future null infinity, $O\in\mathscr{I}^+$. The effective potential barrier peaked near the light ring acts as a scattering region. The direct channel, associated with $G^{(1)}_{\ell m}$, propagates outward from $S$ to $O$ without probing the barrier and therefore activates the prompt response at $t_0^{1}$. The scattered channel, associated with $G^{(2)}_{\ell m}$, initially propagates inward, interacts with the barrier, and is backscattered to infinity. It reaches $O$ only after a time delay $\Delta t=t_0^{2}-t_0^{1}=2r_*'-4M\log\!(f(r'))$ (see Eq. \ref{['eq:delay']} and Sec. \ref{['sec:greybody']}). This causal delay underlies the fact that the ringdown contribution (quasinormal ringing plus late-time tails) is governed by $G^{(2)}_{\ell m}$ and is modulated by the reflection coefficient, directly related to the greybody factor.
• Figure 3: Late-time response of the full Green's function at $\mathscr{I}^+$ for several source locations $r'$ with $r'>3M$ for the $(\ell=2,m=1)$ axial mode. Black: raw numerical signal. Purple: rationally filtered signal after removing two QNM poles (fundamental mode and first overtone). Pink: Leaver's leading order result for the power-law tail. Magenta (dashed): analytic late-time expansion of the tail based on Eq. \ref{['eq:greenintegral']}, truncated to leading order plus the first two subleading orders.
• Figure 4: Late-time response along selected source trajectories for the (22) mode. We consider one eccentric trajectory ($e_0=0.5$) and a radial infall. Black: numerical waveform. Purple: rationally filtered signal, after the removal of 2 QNMs (fundamental mode and first overtone). Magenta (dashed): analytic tail from Eq. \ref{['eq:greenintegral']} (leading order plus the first two subleading orders). The analytic prediction accurately tracks the filtered tail down to times that overlap with the ringdown-dominated regime.
• Figure 5: Representative examples suggesting the presence of a redshift contribution in the ringdown signal. Left: real part of the waveform $h_{22}$ produced by a test particle in radial infall starting from $r_0=2.75M$, together with the fundamental QNM contribution $h_{220}^{\rm (QNMs)}$. The amplitude and phase of the fundamental mode are obtained from a fit performed for several starting times $t_{\rm start}$; the value shown corresponds to a region where the fitted parameters become approximately stable (around $t_{\rm start}\gtrsim 40M$). Middle: residual signal $h_{22}-h_{220}^{\rm (QNMs)}$ compared with two reference decays, $e^{-t/(4M)}$ (the first Schwarzschild redshift term) and $e^{-|\Im(\omega_{221})|t/M}$ (the first overtone). The residual appears qualitatively more consistent with the slower decay associated with the redshift term. Right: rescaled residuals $( {\rm Re}\,h_{22}-h_{220}^{\rm (QNMs)} )e^{t/(4M)}$ and $( {\rm Re}\,h_{22}-h_{220}^{\rm (QNMs)} )e^{|\Im(\omega_{221})|t/M}$. The latter remains approximately constant only during the first two oscillations, whereas at later times the decay becomes slower, suggesting a behavior closer to that expected from the redshift contribution.