Strong cosmic censorship in $T^{3}$-Gowdy spacetimes


Einstein’s vacuum equations can be viewed as an initial value problem, and given initial data there is one part of spacetime, the so-called maximal globally hyperbolic development (MGHD), which is uniquely determined up to isometry. Unfortunately, it is sometimes possible to extend the spacetime beyond the MGHD in inequivalent ways. Consequently, the initial data do not uniquely determine the spacetime, and in this sense the theory is not deterministic. It is then natural to make the strong cosmic censorship conjecture, which states that for generic initial data, the MGHD is inextendible. Since it is unrealistic to hope to prove this conjecture in all generality, it is natural to make the same conjecture within a class of spacetimes satisfying some symmetry condition. Here, we prove strong cosmic censorship in the class of $T^{3}$-Gowdy spacetimes. In a previous paper, we introduced a set $\mathcal{G}_{i,c}$ of smooth initial data and proved that it is open in the $C^{1}\times C^{0}$-topology. The solutions corresponding to initial data in $\mathcal{G}_{i,c}$ have the following properties. First, the MGHD is $C^{2}$-inextendible. Second, following a causal geodesic in a given time direction, it is either complete, or a curvature invariant, the Kretschmann scalar, is unbounded along it (in fact the Kretschmann scalar is unbounded along any causal curve that ends on the singularity). The purpose of the present paper is to prove that $\mathcal{G}_{i,c}$ is dense in the $C^{\infty}$-topology.