Wyman's solution, self-similarity and critical behaviour

TL;DR

This paper demonstrates how Wyman's solution arises from Einstein's equations with a scalar field using self-similarity, revealing critical behavior and different spacetime structures depending on scalar charge values.

Contribution

It shows that Wyman's solution can be derived from Einstein's equations via self-similarity and analyzes its critical behavior based on scalar charge parameters.

Findings

Solutions exhibit naked singularities for positive scalar charge squared.

Black hole solutions occur at zero scalar charge.

Bouncing solutions appear for negative scalar charge squared within a specific range.

Abstract

We show that the Wyman's solution may be obtained from the four-dimensional Einstein's equations for a spherically symmetric, minimally coupled, massless scalar field by using the continuous self-similarity of those equations. The Wyman's solution depends on two parameters, the mass $M$ and the scalar charge $\Sigma$. If one fixes $M$ to a positive value, say $M_0$, and let $\Sigma^2$ take values along the real line we show that this solution exhibits critical behaviour. For $\Sigma^2 >0$ the space-times have eternal naked singularities, for $\Sigma^2 =0$ one has a Schwarzschild black hole of mass $M_0$ and finally for $-M_0^2 \leq \Sigma^2 < 0$ one has eternal bouncing solutions.