Green function of the P\"{o}schl-Teller potential

TL;DR

This paper computes the exact time-domain Green function for black hole perturbations using the Pöschl-Teller potential approximation, revealing new exponentially growing modes and analyzing waveform components.

Contribution

It provides an exact Green function calculation for a simplified black hole model, uncovering additional early-time modes and waveform structure.

Findings

Discovered an early-time exponentially growing mode in the Green function.

Decomposed the waveform into instantaneous and historical components.

Showed the Green function is regular at redshift mode frequencies.

Abstract

We use an approximation of the Regge-Wheeler-Zerilli potential, known as P\"{o}schl-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.