Green function of the Pöschl-Teller potential

TL;DR

This work uses a Pöschl-Teller surrogate to obtain an exact time-domain Green function for black-hole perturbations, enabling a causally consistent decomposition of the waveform. The analysis reveals an unexpected early-time exponential-growth piece prior to ringdown and splits the signal into an instantaneous light-cone term and a history-dependent component, including redshift/horizon-mode effects. The study also contrasts the PT results with full RWZ behavior, noting the absence of tails in the PT model and highlighting the challenges posed by branch cuts in the true RWZ problem, which motivates future exploration of the analytic structure. Overall, the approach yields valuable physical intuition about prompt and early-time responses in BH perturbations and clarifies the role of causality and redshift modes in the waveform.

Abstract

We use an approximation of the Regge-Wheeler-Zerilli potential, known as Pöschl-Teller, to exactly compute the time-domain Green function of black hole perturbations in this simplified model, taking into account all causality conditions. We find the existence of an additional early times piece in the Green function, contributing to new exponentially growing modes just before the signal interacts with the maximum of the potential. The waveform itself is decomposed as an instantaneous piece traveling exactly on the light-cones of the Green function and a historical piece depending on the past trajectory of the system inside the light-cone. We also study redshift modes and show that the Regge-Wheeler-Zerilli Green function is regular at their frequency, with no zero nor pole.

Figures (4)

• Figure 1: Comparison of the Zerilli potential $V_\mathrm{Z}$ for $\ell=2$ and the Pöschl-Teller potential $V_\mathrm{PT}$ as a function of the tortoise coordinate $x$.
• Figure 2: Closing the integration contour according to causality conditions. For $t<x-\bar{x}$, the contour can be closed in the upper half-plane and the Green function is zero. For $t>x+|\bar{x}|$ the contour can be closed in the lower half-plane and picks up the residues at QNM frequencies. Finally, for $x-\bar{x}<t<x+\bar{x}$ (this only exists provided $\bar{x}>0$), the Green function is decomposed as a sum of two contours in the upper and lower half-planes picking up residues on the imaginary axis.
• Figure 3: Comparison of the analytical Green function \ref{['eq:totGreen']} (shifted to ensure that the maximum of the potential is at $x = x_m$) and the numerical results for an observer situated at $x=100$. The upper panel shows the results for an initial perturbation at $\bar{x} = \bar{x}_\mathrm{ISCO} \simeq 3.69$ at the ISCO, while the lower panel has $\bar{x} = 40$. The continuous lines show the analytical results with an increasing number of terms $N$ in the sums (\ref{['eq:GQNM']},\ref{['eq:GI']}). The black dashed line shows the result of the numerical integration with the Pöschl-Teller potential, while the gray dot-dashed line shows the numerical result for the Zerilli potential. As expected, the Zerilli Green function is quite different from the Pöschl-Teller one for $\bar{x} = 40$, because at large distances the two potentials are different (Zerilli has power-law tails, while Pöschl-Teller does not).
• Figure 4: Real part of the wavefunction $\psi$ for a radial infall and a plunge trajectory. We plot each component of the waveform in Eqs. (\ref{['eq:psiInst']},\ref{['eq:psiHist']}) separately, where $\psi_\mathrm{QNM}$ refers to the first line of Eq. \ref{['eq:psiHist']} and $\psi_\mathrm{E}$ to the second line of Eq. \ref{['eq:psiHist']}. We also plot the sum of these three waveforms and the numerical solution to the Zerilli equation using the same trajectory and the code described in Section \ref{['sec:summary_numerical']}. $N=20$ means the wavefunction obtained by summing 20 terms in Eqs. (\ref{['eq:GQNM']},\ref{['eq:GI']}), while "Direct Source" refers to the simplified expression \ref{['eq:psiInstSimplified']}. The divergence of $\psi_\mathrm{Inst.}$ close to $u=0$ is the same than the one discussed below Figure \ref{['fig:GF_comparison']}.