Quasi-Normal Mode Expansion for Linearized Waves in Gravitational Systems

TL;DR

This paper investigates the conditions under which quasinormal modes form a complete basis for gravitational systems modeled by the Klein-Gordon equation, providing criteria and regularization methods for their expansion.

Contribution

It establishes the conditions for QNM completeness in gravitational systems with effective potentials, confirming a conjecture and addressing divergence issues.

Findings

QNM sum is complete when spatial discontinuities exist.

Procedures for regularizing divergent QNM sums are provided.

Completeness depends on the presence of discontinuities like stellar surfaces.

Abstract

The quasinormal modes (QNM's) of gravitational systems modeled by the Klein-Gordon equation with effective potentials are studied in analogy to the QNM's of optical cavities. Conditions are given for the QNM's to form a complete set, i.e., for the Green's function to be expressible as a sum over QNM's, answering a conjecture by Price and Husain [Phys. Rev. Lett. {\bf 68}, 1973 (1992)]. In the cases where the QNM sum is divergent, procedures for regularization are given. The crucial condition for completeness is the existence of spatial discontinuities in the system, e.g., the discontinuity at the stellar surface in the model of Price and Husain.