On the existence of self-similar spherically symmetric wave maps coupled to gravity

TL;DR

This paper analytically investigates self-similar solutions in the SU(2) sigma model coupled to gravity, revealing a family of solutions that include naked singularities and relate to black hole formation thresholds.

Contribution

It provides a rigorous proof of the existence of a countable family of self-similar solutions with analytic properties and explores their implications for naked singularities and critical phenomena.

Findings

Existence of a countable family of solutions with analytic properties.

Solutions can have regular future self-similarity horizons.

One solution is identified as a critical solution at black hole formation threshold.

Abstract

We present a detailed analytical study of spherically symmetric self-similar solutions in the SU(2) sigma model coupled to gravity. Using a shooting argument we prove that there is a countable family of solutions which are analytic inside the past self-similarity horizon. In addition, we show that for sufficiently small values of the coupling constant these solutions possess a regular future self-similarity horizon and thus are examples of naked singularities. One of the solutions constructed here has been recently found as the critical solution at the threshold of black hole formation.