Wyman's Solution, Self-similarity and Critical Behaviour

We show that the Wyman’s solution may be obtained from the fourdimensional 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 M0, and let Σ 2 take values along the real line we show that this solution exhibits critical behaviour. For Σ2 > 0 the space-times have eternal naked singularities, for Σ2 = 0 one has a Schwarzschild black hole of mass M0 and finally for −M2 0 ≤ Σ2 < 0 one has eternal bouncing solutions. 04.20.Dw,04.40.Nr,04.70.Bw Typeset using REVTEX Email: [email protected] Email:[email protected] 1