14.1 The polarized case

As has already been mentioned, the proof of strong cosmic censorship in the polarized Gowdy case proceeds via Conjecture 2. In other words, via a proof of the fact that for generic initial data, the Kretschmann scalar is unbounded in the incomplete directions of causal geodesics. Future causal geodesic completeness in the polarized T3-Gowdy case was announced in [21Jump To The Next Citation Point] and proven in [52, Corollary 21, p. 190]. Thus the main problem is that of proving that the curvature blows up at the singularities. This is achieved in two steps in [21Jump To The Next Citation Point]. First, the existence of a diffeomorphism between asymptotic data (i.e., v and ϕ in Equation (35View Equation)) and ordinary initial data is demonstrated; see [21Jump To The Next Citation Point, p. 1675]. Second, using the observations made concerning curvature blow up in Section 7.4, it can be shown that there is an open and dense subset of the set of asymptotic data such that the curvature of the corresponding solutions blows up everywhere on the singularity. Let us state the result as given in [21, p. 1673]:

Theorem 7 (Strong cosmic censorship for polarized Gowdy spacetimes) Let Σ3 = T3, S3, or S2 × S1, and let 𝒫 (Σ3 ) be the space of initial data for the polarized Gowdy spacetimes (with C ∞ topology). There exists an open dense subset 𝒫ˆ(Σ3) ⊂ 𝒫 (Σ3) such that the maximal development of any set of data in ˆ𝒫 (Σ3 ) is inextendible.

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