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

Abstract

We study the evolution of the "Momentarily Static and Radiation Free" (MSRF) initial data for the Apostolatos - Thorne cylindrical shell model. We analyze the relation between the parameters characterizing the MSRF data those for the corresponding final static configuration, and show that there is a priori no conflict for any choice of initial MSRF data, in contrast with some recent results of Nakao, Ida and Kurita. We also consider the problem in the linear approximation, and show that the evolution is stable in all cases. We find that the approach to the final state is very slow, with an inverse logarithmic dependence on time at fixed radius. To complement these results we introduce a numerical computation procedure that allows us to visualize the explicit form of the evolution of the shell and of the gravitational field up to large times. The results are in agreement with the qualitative behaviour conjectured by Apostolatos and Thorne, with an initial damped oscillatory stage, but with oscillations about a position that approaches slowly that of the static final state, as indicated by our analysis. We also include an Appendix, where we prove the existence of solutions of the cylindrical wave equation with vanishing initial value for \(r > R_0\), (\(R_0 > 0\) some finite constant), that approach a constant value for large times. This result is crucial for the proof of compatibility of arbitrary MSRF initial data and a final static configuration for the system.