BKL bounces outside homogeneity: Einstein-Maxwell-scalar field in surface symmetry

We study the phenomenon of bounces, as predicted by Belinski, Khalatnikov and Lifshitz (BKL) in the study of singularities arising from Einstein's equations, as an instability mechanism within the setting of the (inhomogeneous) Einstein-Maxwell-scalar field system in surface symmetry. This article can be viewed as a companion to our other article "BKL bounces outside homogeneity: Gowdy symmetric spacetimes" [27], where we study bounces for the Einstein vacuum equations in Gowdy symmetry. That is, we show many features of such bounces generalize to the matter model described, albeit in a different symmetry class. The articles may be read independently. In analogy to [27], we describe a wide class of inhomogeneous initial data which permit formation of a spacelike singularity, but such that the dynamics towards different spatial points at the singularity are well-described by independent nonlinear ODEs reminiscent of BKL bounces. A major ingredient is the proof of so-called Asymptotically Velocity Term Dominated behaviour even in the presence of such bounces, though one difference from [27] is that our model does not permit the existence of so-called "spikes". One particular application is the study of (past) instability of certain generalized Kasner spacetimes with no electromagnetic field present. Perturbations of such spacetimes are such that the singularity persists, but for perturbations with electromagnetism turned on the intermediate dynamics - between data and the singularity - features up to one BKL-like bounce. This is in analogy with the instability of polarized Gowdy spacetimes due to non-polarized perturbations in [27].