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 quasi normal 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 this type of perturbations. The explicit expression for the mode expansion are then used to obtain numerical values for some of the quasi normal mode complex frequencies. Some examples involving the numerical evaluation of the integral mode expansions are described and analyzed, and the quasi normal ringing displayed by the solutions is found to be in agreement with quasi normal modes found previously. Going back to the full relativistic equations of motion we find their general linear form by expanding 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 values 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 quasi normal ringing that results in the evolution of the system. We also discuss the relation of the present work to some recent results on the same model that have appeared in the literature.

On the evolution of the momentarily static radiation free data in the Apostolatos–Thorne cylindrical shell model

“…Once this is done we find no conflict between an arbitrary MSRF initial data and a corresponding final static configuration, confirming the conjecture in [1]. Furthermore, we consider MSRF data that is initially close to the static configuration, and, using some results obtained in [7], prove the stability of the evolution. We further apply a numerical integration procedure and obtain the detailed evolution of the system up to large times, finding somewhat unexpected features in the approach of the shell to its final static radius, that are described and discussed in the text below.…”

“…In this case we may consider a linearized expansion about the static configuration and apply some recently obtained results on the dynamics of the Apostolatos -Thorne model [7].…”

“…We may therefore expand (35) to first order in ǫ, and consider the resulting set as initial data for the linearized equations of motion for the shell. These were obtained in [7]. Briefly stated, one first writes the dynamical variables R, ψ − and ψ + in the form,…”

“…A different approach, where the evolution of the system is analyzed in the linear approximation was considered by Gleiser and Ramirez in [3]. The stability of the static configurations of the system under arbitrary perturbations was proved in an extended analysis given in [7], where the existence of quasi normal modes was also considered. A related analysis, but with a different interpretation was also given in [4].…”

“…We also show that, contrary to a simple expectation, the final approach to the static configuration, for fix radial coordinate, is very slow, with the appropriate quantities approaching zero only as 1/ ln(τ ), where τ is the proper time on the shell. Since the previous derivations give no information on the detailed evolution of the shell and the fields, in Section V we use a numerical integration procedure, developed in [7], to visualize, for some particular examples, the evolution from the initial state, up to large times. We find for the shell, as expected, an initial exponentially damped oscillatory stage, but, in agreement it our analysis in Section IV, not about the final static radius, but rather about a position that approaches the static radius as 1/ ln(τ ).…”