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

Abstract

In this work, we identify stable perturbations of k-self-similar naked singularities, in the full parameter range \(k^2 \in (0, \frac{1}{3}).\) 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.