Quasinormal modes, echoes and the causal structure of the Greens function

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 Greens 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 Greens function, demonstrating under what conditions two very different quasinormal spectra give rise to very similar ringdown waveforms.