Perturbative evolution of the static configurations, quasinormal modes and quasinormal ringing in the Apostolatos-Thorne cylindrical shell model

Abstract

We study the perturbative evolution of the static configurations, quasinormal modes and quasinormal ringing in the Apostolatos–Thorne cylindrical shell model. We consider first an expansion in harmonic modes and show that it provides a complete solution for the characteristic value problem for the finite perturbations of a static configuration. As a consequence of this completeness, we obtain a proof of the stability of static solutions under these types of perturbations. The explicit expressions for the mode expansion are then used to obtain numerical values for some of the quasinormal mode complex frequencies. Some examples involving the numerical evaluation of the integral mode expansions are described and analyzed, and the quasinormal ringing displayed by the solutions is found to be in agreement with quasinormal modes found previously. Going back to the full relativistic equations of motion, we find their general linear form by expanding them to first order about a static solution. We then show that the resulting set of coupled ordinary and partial differential equations for the dynamical variables of the system can be used to set an initial plus boundary value problem, and prove that there is an associated positive definite constant of the motion that puts absolute bounds on the dynamic variables of the system, establishing the stability of the motion of the shell under arbitrary, finite perturbations. We also show that the problem can be solved numerically, and provide some explicit examples that display the complete agreement between the purely numerical evolution and that obtained using the mode expansion, in particular regarding the quasinormal ringing that results in the evolution of the system. We also discuss the relation of this work to some recent results on the same model that have appeared in the literature.