Wave Propagation in Gravitational Systems: Completeness of Quasinormal Modes

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.

References in corpus (6)

• Wave Propagation in Gravitational Systems: Late Time Behavior
• The Collision of Two Black Holes
• Late Time Tail of Wave Propagation on Curved Spacetime
• Quasi-Normal Mode Expansion for Linearized Waves in Gravitational Systems
• The Escape of Gravitational Radiation from the Field of Massive Bodies
• The Emergence of Radiation from Gravitational Potential Wells: The Absence of $\omega M$ Effects

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