A construction of approximately self-similar naked singularities for the spherically symmetric Einstein-scalar field system

Abstract

In this work we investigate the stability and instability properties of a class of naked singularity spacetimes. The first rigorous study of naked singularity formation in the spherically symmetric Einstein-scalar field system was due to Christodoulou, who constructed a family $(\overline{g}_k, \overline{\phi}_k)$ of $k$-self-similar solutions, for any $k^2 \in (0,\frac{1}{3})$. We extend the construction to produce examples of interior and exterior regions of naked singularity spacetimes locally modeled on the $(\overline{g}_k, \overline{\phi}_k)$, without requiring exact self-similarity. The main result is a global stability statement under fine-tuned data perturbations, for a class of naked singularity spacetimes satisfying self-similar bounds. Given the well-known blueshift instability for suitably regular naked singularities in the Einstein-scalar field model, we require non-generic conditions on the data perturbations. In particular, the scalar field perturbation along the past lightcone of the singular point $\mathcal{O}$ vanishes to high order near $\mathcal{O}$. Technical difficulties arise from the singular behavior of the background solution, as well as regularity considerations at the axis and past lightcone of the singularity. The interior region is constructed via a backwards stability argument, thereby avoiding activating the blueshift instability. The extension to the exterior region is treated as a global existence problem to the future of $\mathcal{O}$, adapting techniques of Rodnianski and Shlapentokh-Rothman for vacuum spacetimes.