Critical phenomena in gravitational collapse of Husain–Martinez–Nunez scalar field

We construct analytical models to study the critical phenomena in gravitational collapse of the Husain-Martinez-Nunez massless scalar field. We first use the cut-and-paste technique to match the conformally flat solution ($$c=0$$ ) onto an outgoing Vaidya solution. To guarantee the continuity of the metric and the extrinsic curvature, we prove that the two solutions must be joined at a null hypersurface and the metric function in Vaidya spacetime must satisfy certain constraints. We find that the mass of the black hole in the resulting spacetime takes the form $$M\propto (p-p^*)^\gamma $$, where the critical exponent $$\gamma $$ is equal to 0.5. For the case $$c\ne 0$$, we show that the scalar field must be joined onto two pieces of Vaidya spacetimes to avoid a naked singularity. We also derive the power-law mass formula with $$\gamma =0.5$$. Compared with previous analytical models which were constructed from a different scalar field with continuous self-similarity, we obtain the same value of $$\gamma $$. However, we show that the solution with $$c\ne 0$$ is not self-similar. Therefore, we provide a rare example that a scalar field without self-similarity also possesses the features of critical collapse.