Naked singularities in the Einstein-Euler system

TL;DR

This paper rigorously proves the existence of naked singularities in the Einstein-Euler system by analyzing self-similar solutions and overcoming the complexities of the sonic hypersurface, advancing understanding of gravitational collapse.

Contribution

It provides the first rigorous proof of self-similar spacetimes forming naked singularities in the Einstein-Euler system, using novel dynamical system and geometric methods.

Findings

Existence of self-similar naked singularities proven rigorously.

Construction of solutions connecting sonic hypersurface to Friedmann solution.

Development of a nonlinear extension method for asymptotically flat spacetimes.

Abstract

In 1990, based on numerical and formal asymptotic analysis, Ori and Piran predicted the existence of self-similar spacetimes, called relativistic Larson-Penston solutions, that can be suitably flattened to obtain examples of spacetimes that dynamically form naked singularities from smooth initial data, and solve the radially symmetric Einstein-Euler system. Despite its importance, a rigorous proof of the existence of such spacetimes has remained elusive, in part due to the complications associated with the analysis across the so-called sonic hypersurface. We provide a rigorous mathematical proof. Our strategy is based on a delicate study of nonlinear invariances associated with the underlying non-autonomous dynamical system to which the problem reduces after a self-similar reduction. Key technical ingredients are a monotonicity lemma tailored to the problem, an ad hoc shooting method developed to construct a solution connecting the sonic hypersurface to the so-called Friedmann solution, and a nonlinear argument to construct the maximal analytic extension of the solution. Finally, we reformulate the problem in double-null gauge to truncate the self-similar profile and thus obtain an asymptotically flat spacetime with an isolated naked singularity.