Spacelike singularities in General Relativity and the BKL proposal

Abstract

In this dissertation, we present several results related to spacelike singularities arising in Einstein’s theory of General Relativity, and their connection to the heuristics of Belinski, Khalatnikov and Lifshitz (BKL) in the physics literature. BKL proposed that spacelike singularities arising from the Einstein equations resemble an anisotropic Kasner metric, but where the rates and directions of compression and stretching depend on space and time. Further, they describe the time dependence as a cascade of Kasner-like regimes connected by BKL bounces, where the rates and directions transition rapidly from one regime to another – often termed BKL’s chaotic and oscillatory approach to singularity. Our thesis has three main parts, each of which provides support of BKL’s proposal within various contexts. In the first part, we show that a large class of spherically symmetric solutions to the Einstein–(Maxwell)–scalar field system possess a Kasner-like singularity in the above sense. Though the scalar field and symmetry assumptions suppress the above oscillations, remnants of their effects remain upon introducing the Maxwell field. This is explored further in joint work with M. Van de Moortel, where we study homogeneous black hole interiors with charged scalar hair. In the second part, we characterize a wide class of symmetric, but spatially inhomogeneous spacetimes, both in vacuum and within the Einstein–Maxwell–scalar field model, which feature BKL bounces near their singularity. These are the first inhomogeneous examples of such bounces, and although our current work allows for only one bounce rather than a chaotic cascade, our method – combining nonlinear ODE analysis with energy estimates – is robust and we outline several potential further applications. The final part analyzes the linearized Einstein–scalar field system near exact Kasner spacetimes and exhibits a scattering isomorphism between suitable Cauchy initial data and the asymptotic quantities (e.g. Kasner-like rates and directions) at the singularity. The scattering map reveals interesting gains and losses of derivatives, which depend sensitively on the anisotropy of the Kasner background.