A Simplified Mathematical Model for the Formation of Null Singularities Inside Black Holes I - Basic Formulation and a Conjecture

TL;DR

This paper introduces a simplified hyperbolic model inspired by Einstein's equations to study null singularity formation inside black holes, proposing that blowup on a characteristic line is a generic feature.

Contribution

It presents a new semi-linear hyperbolic system modeling black hole interiors and conjectures its solutions' behavior reflects generic null singularity formation in Einstein's equations.

Findings

Solutions exhibit black-hole-like structure with finite-time blowup.

Blowup occurs on a characteristic line analogous to the inner horizon.

Conjecture that this behavior is generic for the system.

Abstract

Einstein's equations are known to lead to the formation of black holes and spacetime singularities. This appears to be a manifestation of the mathematical phenomenon of finite-time blowup: a formation of singularities from regular initial data. We present a simple hyperbolic system of two semi-linear equations inspired by the Einstein equations. We explore a class of solutions to this system which are analogous to static black-hole models. These solutions exhibit a black-hole structure with a finite-time blowup on a characteristic line mimicking the null inner horizon of spinning or charged black holes. We conjecture that this behavior - namely black-hole formation with blow-up on a characteristic line - is a generic feature of our semi-linear system. Our simple system may provide insight into the formation of null singularities inside spinning or charged black holes in the full system of Einstein equations.