Wave Propagation in Gravitational Systems: Completeness of Quasinormal Modes

TL;DR

This paper systematically investigates the conditions under which quasinormal modes of the Klein-Gordon equation form complete sets in relativistic systems, with implications for wave analysis in black hole and star dynamics.

Contribution

It identifies the criteria on effective potentials for QNM completeness and proposes a method to achieve completeness via infinitesimal potential modifications.

Findings

QNM completeness depends on the structure of the effective potential

A complete set of QNM's can be obtained by small potential adjustments

The study enables wave analysis in systems with initially incomplete QNM spectra

Abstract

The dynamics of relativistic stars and black holes are often studied in terms of the quasinormal modes (QNM's) of the Klein-Gordon (KG) equation with different effective potentials $V(x)$. In this paper we present a systematic study of the relation between the structure of the QNM's of the KG equation and the form of $V(x)$. In particular, we determine the requirements on $V(x)$ in order for the QNM's to form complete sets, and discuss in what sense they form complete sets. Among other implications, this study opens up the possibility of using QNM expansions to analyse the behavior of waves in relativistic systems, even for systems whose QNM's do {\it not} form a complete set. For such systems, we show that a complete set of QNM's can often be obtained by introducing an infinitesimal change in the effective potential.