A Construction of Approximately Self-Similar Naked Singularities for the Spherically Symmetric Einstein-Scalar Field System

In this work, we identify stable perturbations of k-self-similar naked singularities, in the full parameter range k2 is an element of(0,13).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k<^>2 \in (0, \frac{1}{3}).$$\end{document} As a consequence, we provide a dynamical construction of naked singularity interior and exterior regions outside of exact self-similarity, extending the work of Chrisotodoulou (Ann Math 140:607-653, 1994). More generally, we consider a wide range of singular spacetimes satisfying self-similar bounds, and parameterize perturbations in terms of fine-tuned data for the scalar field along the past lightcone of the singularity. This data is required to satisfy conditions consistent with the absence of a blue-shift instability and is therefore non-generic. The argument combines a backwards stability argument in the interior region with a global existence problem in the exterior region, adapting techniques developed by Rodnianski and Shlapentokh-Rothman (Naked singularities for the Einstein vacuum equations: the exterior solution. arXiv:1912.08478. 2019) to the spherically symmetric setting.