Wyman's solution, self-similarity and critical behaviour

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 $Σ$. If one fixes $M$ to a positive value, say $M_0$, and let $Σ^2$ take values along the real line we show that this solution exhibits critical behaviour. For $Σ^2 >0$ the space-times have eternal naked singularities, for $Σ^2 =0$ one has a Schwarzschild black hole of mass $M_0$ and finally for $-M_0^2 \leq Σ^2 < 0$ one has eternal bouncing solutions.

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