Staticity, Self-Similarity and Critical Phenomena in a Self-Gravitating Nonlinear Sigma Model

TL;DR

This thesis investigates self-similar solutions and critical phenomena in self-gravitating nonlinear sigma models, revealing how these phenomena depend on the coupling constant and identifying transitions between different types of critical solutions.

Contribution

It provides the first numerical construction and analysis of CSS and DSS solutions in these models, and describes the bifurcation and transition between them based on the coupling strength.

Findings

CSS solutions are stable for small couplings.

DSS solutions with one unstable mode exist for large couplings.

Transition from CSS to DSS occurs at a critical coupling.

Abstract

The main part of the thesis deals with continuously and discretely self-similar solutions and type II critical phenomena in a family of self-gravitating non-linear sigma-models. The phenomena strongly depend on the dimensionless coupling constant. For small couplings we numerically construct continuously self-similar (CSS) solutions and analyze their stability properties. For large couplings we construct a discretely self-similar (DSS) solution with one unstable mode. We argue that at some critical coupling the DSS solution bifurcates from the first CSS excitation in a heteroclinic loop bifurcation. We study critical phenomena between dispersal and singularity formation (at very small couplings) respectively black hole formation (for larger couplings). We give numerical evidence that for very small couplings the generic end state of ``intermediately strong'' data is the stable CSS ground state. For small couplings the critical solution is the first CSS excitation whereas for strong couplings the threshold of black hole formation is governed by the DSS solution. We describe the phenomena occurring at intermediate couplings where the critical solution changes from CSS to DSS. Content and references are at the state of October 2001.