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

TL;DR

This paper analyzes a family of solutions describing scalar field collapse, revealing a critical transition between naked singularities and black holes, with unique mass scaling laws near the critical point.

Contribution

It introduces a new analytical solution for scalar field collapse with self-interaction, demonstrating critical behavior without self-similarity and establishing a novel mass scaling law.

Findings

Collapse outcomes depend on scalar charge relative to critical value

Black hole mass scales as -M^2 ln M near criticality

Solutions satisfy null and strong energy conditions

Abstract

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 $q>q^*$. The solution admits no self-similarity and satisfies the null and the strong energy conditions.