5 BKL, Fuchsian Methods and Asymptotic Expansions

The results on strong cosmic censorship in Gowdy vacuum spacetimes cover the polarized sub-case as well as the general T3-Gowdy case. However, to the best of our knowledge, there are no results concerning strong cosmic censorship in the general S3 and S2 × S1 cases. The method of proof, in all the situations in which results exist, consists of a detailed analysis of the asymptotics of solutions. As a consequence, we shall devote most of this review to a description of the analysis of the asymptotic behavior. Note that it might be possible to prove strong cosmic censorship without analyzing the asymptotics in detail. In fact, there are proofs under related symmetry assumptions, which are not based on a detailed analysis of the asymptotics [25262784].

The existence of asymptotic expansions in the direction towards the singularity has played a central role in proving strong cosmic censorship for T3 and polarized vacuum Gowdy spacetimes. In the latter case, to take one example, there is a computation of asymptotic expansions due to Isenberg and Moncrief [50Jump To The Next Citation Point]. This computation was then used in [21Jump To The Next Citation Point] in the proof of strong cosmic censorship in the polarized case. In the general T3-case, there is a large literature on asymptotic expansions, which we shall return to in Section 8; the starting point being the work of Grubišić and Moncrief [40Jump To The Next Citation Point]. It is worth noting that both in the case of [40Jump To The Next Citation Point] and [50Jump To The Next Citation Point], the ideas of Belinskii, Khalatnikov, and Lifshitz [5567] (henceforth BKL) played an important role. As a consequence, we wish to give a brief description of the BKL perspective as well as of some related proposals.

 5.1 The BKL picture
 5.2 Asymptotic expansions, Fuchsian methods
  5.2.1 From solutions to asymptotics
  5.2.2 From asymptotics to solutions
  5.2.3 Overview of results

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