Simplified mathematical model for the formation of null singularities inside black holes. II. Proof for a particular case

We study a simple system of two hyperbolic semilinear equations inspired by the Einstein equations. The system, which was introduced by Ori and Gorbonos [J. Math. Phys., 48, 092502 (2007)], is a model for singularity formation inside black holes. We show for a particular case of the equations that the system demonstrates a finite time blowup. The singularity that is formed is a null singularity. Then we show that in this particular case the singularity has features that are analogous to known features of models of black-hole interiors—which describe the inner-horizon instability. Our simple system may provide insight into the formation of null singularities inside spinning or charged black holes.