Blow-up for solutions of hyperbolic PDE and spacetime singularities

TL;DR

This paper explores the relationship between blow-up phenomena in nonlinear hyperbolic PDEs and the formation of spacetime singularities in general relativity, highlighting recent progress and future challenges.

Contribution

It reviews recent advances in applying hyperbolic PDE theory to construct solutions with singularities and discusses the complexity of generic singularity formation in Einstein equations.

Findings

Construction of large families of solutions with simple singularities

Application of hyperbolic PDE techniques to spacetime singularities

Indication of increased complexity in generic singularity formation

Abstract

An important question in mathematical relativity theory is that of the nature of spacetime singularities. The equations of general relativity, the Einstein equations, are essentially hyperbolic in nature and the study of spacetime singularities is naturally related to blow-up phenomena for nonlinear hyperbolic systems. These connections are explained and recent progress in applying the theory of hyperbolic equations in this field is presented. A direction which has turned out to be fruitful is that of constructing large families of solutions of the Einstein equations with singularities of a simple type by solving singular hyperbolic systems. Heuristic considerations indicate, however, that the generic case will be much more complicated and require different techniques.