Quasinormal modes, echoes and the causal structure of the Green's function

TL;DR

This paper investigates how quasinormal modes relate to the global structure of spacetime, revealing their sensitivity to distant perturbations, the role of echoes, and the causal limits on their influence in black hole perturbations.

Contribution

It introduces a composition law for quasinormal frequencies with disjoint potentials and analyzes the causal structure and echo expansion of the Green's function.

Findings

Quasinormal modes are sensitive to global spacetime structure.

Echoes influence the Green's function and are causally limited.

Different quasinormal spectra can produce similar waveforms.

Abstract

Quasinormal modes describe the return to equilibrium of a perturbed system, in particular the ringdown phase of a black hole merger. But as globally-defined quantities, the quasinormal spectrum can be highly sensitive to global structure, including distant small perturbations to the potential. In what sense are quasinormal modes a property of the resulting black hole? We explore this question for the linearized perturbation equation with two potentials having disjoint bounded support. We give a composition law for the Wronskian that determines the quasinormal frequencies of the combined system. We show that over short time scales the evolution is governed by the quasinormal frequencies of the individual potentials, while the sensitivity to global structure can be understood in terms of echoes. We introduce an echo expansion of the Green's function and show that, as expected on general grounds, at any finite time causality limits the number of echoes that can contribute. We illustrate our results with the soluble example of a pair of $\delta$-function potentials. We explicate the causal structure of the Green's function, demonstrating under what conditions two very different quasinormal spectra give rise to very similar ringdown waveforms.