A semi-linear wave model for critical collapse

TL;DR

This paper develops semi-linear wave models that exhibit self-similar critical solutions at the threshold of blow-up, extending understanding of critical phenomena in gravitational collapse beyond spherical symmetry.

Contribution

It introduces models based on deformations of the wave equation that demonstrate discretely self-similar threshold solutions, including in non-spherical contexts.

Findings

Threshold solutions are self-similar and unique in spherical symmetry.

Numerical evolutions show similar critical behavior in more general models.

Multiple blow-up topologies are possible beyond spherical symmetry.

Abstract

In spherical symmetry compelling numerical evidence suggests that in general relativity solutions near the threshold of black hole formation exhibit critical behavior. One aspect of this is that threshold solutions themselves are self-similar and are, in a certain sense, unique. To an extent yet to be fully understood, the same phenomena persist beyond spherical symmetry. It is therefore desirable to construct models that exhibit such symmetry at the threshold of blow-up. Starting with deformations of the wave equation, we discuss models which have discretely self-similar threshold solutions. We study threshold solutions in the past light cone of the blow-up point. In spherical symmetry there is a sense in which a unique critical solution exists. Spherical numerical evolutions are also presented for more general models, and exhibit similar behavior. Away from spherical symmetry threshold solutions attain more freedom. Different topologies of blow-up are possible, and even locally the critical solution needs reinterpretation as a parameterized family.