Wave Propagation and Quasinormal Mode Excitation on Schwarzschild Spacetime

TL;DR

This paper explores wave behavior and quasinormal modes on Schwarzschild spacetime, revealing how the Green function's singularities relate to lightcone caustics and providing asymptotic expressions for QN modes.

Contribution

It extends a recent expansion method to derive asymptotics for QN wavefunctions and links the Green function's singular structure to lightcone caustics in Schwarzschild spacetime.

Findings

QN mode sum is singular on the lightcone

Green function exhibits a four-fold singularity pattern at caustics

Asymptotic expressions for QN wavefunctions are obtained

Abstract

To seek a deeper understanding of wave propagation on the Schwarzschild spacetime, we investigate the relationship between (i) the lightcone of an event and its caustics (self-intersections), (ii) the large-$l$ asymptotics of quasinormal (QN) modes, and (iii) the singular structure of the retarded Green function (GF) for the scalar field. First, we recall that the GF has a (partial) representation as a sum over QN modes. Next, we extend a recently-developed expansion method to obtain asymptotic expressions for QN wavefunctions and their residues. We employ these asymptotics to show (approximately) that the QN mode sum is singular on the lightcone, and to obtain approximations for the GF which are valid close to the lightcone. These approximations confirm a little-known prediction: the singular part of the GF undergoes a transition each time the lightcone passes through a caustic, following a repeating four-fold sequence. We conclude with a discussion of implications and extensions of this work.