Critical Behavior in a Massless Scalar Field Collapse with Self-interaction Potential

Understand

We examine a one-parameter family of analytical solutions representing spherically symmetric collapse of a nonlinear massless scalar field with self-interaction in an asymptotically flat spacetime.

• The time evolution exhibits a type of critical behavior.
• Depending on the scalar charge parameter $q$ as compared to a critical value $q^*$, the incoming scalar wave collapses either to a globally naked central singularity if $q<q^*$ (weak field) or to a scalar-hairy black hole if $q>q^*$ (strong field), both having finite asymptotic masses.
• Near the critical evolution, the black hole mass follows a product-logarithmic scaling law: $-M^2\ln M \sim q-q^*$ with $0<M\ll 1$ and $q>q^*$.