Naked Singularities for the Einstein Vacuum Equations: The Interior Solution

TL;DR

This paper constructs interior solutions to the Einstein vacuum equations that, when combined with exterior solutions, form a naked singularity, utilizing a novel self-similarity approach and analyzing complex PDEs.

Contribution

It introduces a new method for constructing interior solutions to Einstein vacuum equations that complement previous exterior solutions, enabling the formation of naked singularities.

Findings

Successfully glued interior and exterior solutions to form a naked singularity.

Developed a new class of quasilinear PDEs of mixed degenerate elliptic-hyperbolic type.

Demonstrated the applicability of a novel self-similarity in Einstein vacuum equations.

Abstract

In a previous work [I. Rodnianski and Y. Shlapentokh-Rothman, Naked Singularities for the Einstein Vacuum Equations: The Exterior Solution, arXiv:1912.08478] we constructed solutions to the Einstein vacuum equations in 3+1 dimensions which corresponded to the exterior region of a naked singularity. In this work we construct solutions which correspond to the interior region and show that the two solutions may be glued together to produce a naked singularity. Fundamental to our construction is the novel type of self-similarity for the Einstein vacuum equations that we introduced in our previous work and also the study of a new class of quasilinear PDE's of mixed degenerate elliptic-hyperbolic type.