Collapse of a cylindrically symmetric, self-similar scalar field with non-minimal coupling: Solutions with a regular axis

TL;DR

This paper analyzes self-similar solutions to Einstein's equations with a scalar field in cylindrical symmetry, revealing a family of solutions with a regular axis and exploring their evolution near singularities.

Contribution

It demonstrates the existence and uniqueness of a two-parameter family of solutions with a regular axis in cylindrically symmetric scalar fields with non-minimal coupling.

Findings

Existence of a 2-parameter family of solutions with a regular axis.

Solutions exhibit a singularity at the scaling origin.

Characterization of the evolution of solutions from the axis toward the past null cone.

Abstract

We investigate self-similar scalar field solutions to the Einstein equations in whole cylinder symmetry. Imposing self-similarity on the spacetime gives rise to a set of single variable functions describing the metric. Furthermore, it is shown that the scalar field is dependent on a single unknown function of the same variable and that the scalar field potential has exponential form. The Einstein equations then take the form of a set of ODEs. Self-similarity also gives rise to a singularity at the scaling origin. We discuss the number of degrees of freedom at an arbitrary point and prove existence and uniqueness of a 2-parameter family of solutions with a regular axis. We discuss the evolution of these solutions away from the axis toward the past null cone of the singularity, determining the maximal interval of existence in each case.