Regimes of (in-)stability for self-similar naked singularities

Abstract

Our main aim in this thesis is to study the stability properties of the family of k-self-similar solutions to the Einstein-scalar field equations. These solutions were identified by Christodoulou, and model the gravitational collapse of matter without a corresponding event horizon. As a result, they are of interest in the context of the weak cosmic censorship conjecture. In Chapter 4 we apply techniques of Rodnianski–Shlapentokh-Rothman to sharply characterize instabilities in the exterior region. Within a scale of Hölder spaces, we establish nonlinear instability for regularities slightly above BV. This extends Christodoulou's foundational result which identified a blue-shift instability in low-regularity. We then turn our attention to the interior region, where the role of the blue-shift instability is less well understood. In Chapter 5 we give a backwards construction of asymptotically k-self-similar interiors which achieve a given asymptotic profile on the light-cone incident to the singularity. In Chapter 6 we study the asymptotics of solutions to the linear wave equation in the interior, with characteristic initial data posed to the past of the singularity. The presence of finite regularity data poses considerable problems, requiring a mixture of physical-space energy methods and frequency-space scattering techniques, as well as the restriction to small-mass spacetimes. We show that sufficiently regular data leads to self-similar asymptotics, raising intriguing questions about the diminished role of the blue-shift instability in high regularity function spaces. In Chapter 7 we leave the study of k-self-similar spacetimes and consider the related topic of C1 extension principles for the spherically symmetric Einstein-scalar field system. Our result, joint with Xinliang An (NUS) and Haoyang Chen (NUS), is a modest generalization of the well-known small-μ extension principle to cover solutions which (a) globally satisfy μ < 3/8 , and (b) exhibit finite blue-shift along ingoing null-cones.