Simplified mathematical model for the formation of null singularities inside black holes. I. Basic formulation and a conjecture

Abstract

Einstein’s equations are known to lead to the formation of black holes and space time 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 semilinear 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 blowup on a characteristic line—is a generic feature of our semilinear 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.