Beyond quasinormal modes: a complete mode decomposition of black hole perturbations

TL;DR

This paper demonstrates that black hole perturbation Green's functions can be fully decomposed into convergent mode sums across spacetime, unifying early and late time behaviors with new analytical and numerical methods.

Contribution

It introduces a complete mode decomposition of black hole perturbations, showing convergence of mode sums everywhere and connecting early and late time regimes through novel analytical techniques.

Findings

Mode sums converge everywhere in spacetime

Late-time quasinormal mode sum converges

Early-time Matsubara sum converges

Abstract

We show that retarded Green's functions of black hole spacetimes can be expressed as a convergent mode sum everywhere in spacetime. At late times a quasinormal mode sum converges, while at early times a Matsubara (or, Euclidean) mode sum converges. The two regions are separated by a lightcone which scatters from the black hole potential. The Matsubara sum is a Fourier series on the Euclidean thermal circle associated to the early time region. We illustrate our results for P\"oschl-Teller, BTZ, and Schwarzschild. In the case of Schwarzschild, we express the branch cut contribution as a convergent sum of de Sitter quasinormal modes as $\Lambda\to 0^+$, and exploit recent exact solutions to the Heun connection problem. In each case we analytically show convergence by studying the asymptotics of residue sums and also provide numerical demonstrations.