1.2 Outline, the asymptotics in the direction towards the singularity

Asymptotic behavior in the direction of the singularity in the polarized case. The two Gowdy cases in which results concerning strong cosmic censorship exist are the polarized case and the general T3-case. In Section 7, we focus on the polarized case. In particular, on finding asymptotic expansions of the metric components in the direction towards the singularity. One function appearing in the expansions has a special importance. From the wave-map point of view, it has a natural interpretation as the rate at which the solution tends to the boundary of hyperbolic space; see Section 7.4. As a consequence, it is referred to as the asymptotic velocity. Beyond having a natural interpretation, the asymptotic velocity has an additional important property. In fact, it can be used as a criterion for curvature blow up along causal curves going into the singularity; see Section 7.5.

Existence of solutions with specified asymptotics, Fuchsian methods. Due to the central importance of the asymptotic expansions in the polarized case, it is of interest to obtain expansions in the general T3-case. One way to proceed is to try to construct solutions with prescribed asymptotics. This is the subject of Section 8. Again, it is possible to define the concept of an asymptotic velocity. It has the same geometric interpretation and importance as in the polarized case. The results on existence of expansions depend on a restriction of the asymptotic velocity. We describe the results and motivate the restriction.

Spikes. The numerical studies indicate that for most spatial points, the asymptotic expansions presented in Section 8 constitute a good description of the asymptotics. However, they also indicate that there are spatial points where the behavior is very different. Due to the visual impression of plots of the solutions in the neighborhood of the exceptional points, the corresponding features have been referred to as “spikes”. In Section 9, we describe analytic constructions of solutions with spikes.

Existence of an asymptotic velocity in the general T3-Gowdy case. The analysis in the polarized case and the construction of solutions with prescribed asymptotics indicate the importance of the asymptotic velocity. Consequently, it is of interest to prove that the asymptotic velocity exists in general. This is the subject of Section 10. We also demonstrate that the asymptotic velocity can be viewed as a two-dimensional object in the disc model. Finally, we illustrate that it can be used as a criterion for the existence of expansions.

Definition of the generic set in the general T3-Gowdy case. As a preparation for the formulation of the theorem verifying that strong cosmic censorship holds in T3-Gowdy, we define the generic set of initial data in Section 11.


  Go to previous page Go up Go to next page