BKL bounces outside homogeneity: Gowdy symmetric spacetimes

Abstract: We study the phenomenon of bounces, as predicted by Belinski, Khalatnikov and Lifshitz (BKL), as an instability mechanism within the setting of the Einstein vacuum equations in Gowdy symmetry. In particular, for a wide class of inhomogeneous initial data we prove that the dynamics near the $t = 0$ singularity are well-described by ODEs reminiscent of Kasner bounces. Unlike previous works regarding bounces, our spacetimes are not necessarily spatially homogeneous, and a crucial step is proving so-called asymptotically velocity term dominated (AVTD) behaviour, even in the presence of nonlinear BKL bounces and other phenomena such as spikes. (A similar phenomenon involving bounces and AVTD behaviour, though not spikes, can also be seen in our companion paper "BKL bounces outside homogeneity: Einstein-Maxwell-scalar field in surface symmetry", albeit in the context of the Einstein-Maxwell-scalar field model in surface symmetry.) One particular application is the study of (past) instability of certain polarized Gowdy spacetimes, including some Kasner spacetimes. Perturbations of such spacetimes are such that the singularity persists, but the intermediate dynamics -- between initial data and the singularity -- feature BKL bounces.