# Naked singularities for the Einstein vacuum equations: The exterior solution

### Abstract

In this work we initiate the mathematical study of naked singularities for the Einstein vacuum equations in $3+1$ dimensions by constructing solutions which correspond to the exterior region of a naked singularity. A key element is our introduction of a new type of self-similarity for the Einstein vacuum equations. Connected to this is a new geometric twisting phenomenon which plays the leading role in singularity formation.

Prior to this work, the only known examples of naked singularities were the solutions constructed by Christodoulou for the spherically symmetric Einstein-scalar-field system, as well as other solutions explored numerically for either the spherically symmetric Einstein equations coupled to suitable matter models or for the Einstein equations in higher dimensions.

## Authors

Igor Rodnianski

Department of Mathematics, Princeton University, Princeton, NJ, USA

Yakov Shlapentokh-Rothman

Department of Mathematics, University of Toronto, Toronto, ON, Canada and Department of Mathematical and Computational Sciences, University of Toronto Mississauga, Mississauga, ON, Canada