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.