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 some constraints. We find that the mass of the black hole in the resulting spacetime takes the form $Mpropto (p-p^*)^gamma$, where the critical exponent $gamma$ is equal to $0.5$. For the case $c eq 0$, 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 constructed from a different scalar field with continuous self-similarity, we obtain the same value of $gamma$. However, we show that the solution with $c eq 0$ is not self-similar. Therefore, we provide a rare example that a scalar field without self-similarity also possesses the features of critical collapse.