Quantifying excitations of quasinormal mode systems

TL;DR

This paper develops a mathematical framework to quantify quasinormal mode excitations in black hole gravitational waveforms, introduces the excitation coefficient, and evaluates its effectiveness through a model problem.

Contribution

It introduces the excitation coefficient as a tool for quantifying QN content and proves completeness for a modified model, advancing understanding of QN systems.

Findings

The excitation coefficient has limited utility in the model problem.

Complete QN systems can be constructed from incomplete ones.

The paper discusses fundamental differences between normal modes and QN systems.

Abstract

Computations of the strong field generation of gravitational waves by black hole processes produce waveforms that are dominated by quasinormal (QN) ringing, a damped oscillation characteristic of the black hole. We describe here the mathematical problem of quantifying the QN content of the waveforms generated. This is done in several steps: (i) We develop the mathematics of QN systems that are complete (in a sense to be defined) and show that there is a quantity, the ``excitation coefficient,'' that appears to have the properties needed to quantify QN content. (ii) We show that incomplete systems can (at least sometimes) be converted to physically equivalent complete systems. Most notably, we give a rigorous proof of completeness for a specific modified model problem. (iii) We evaluate the excitation coefficient for the model problem, and demonstrate that the excitation coefficient is of limited utility. We finish by discussing the general question of quantification of QN excitations, and offer a few speculations about unavoidable differences between normal mode and QN systems.