Universality and Criticality in Mass-less Scalar Field Collapse
Koushiki, Rituparno Goswami, Pankaj S. Joshi
TL;DR
The paper addresses the problem of universal critical behavior in the collapse of a massless scalar field by developing a covariant, coordinate-free treatment using the 1+1+2 semi-tetrad formalism in spherically symmetric spacetimes. The authors decompose the scalar-field gradient into collapsing and dispersing modes, $\Psi_-$ and $\Psi_+$, and derive covariant propagation and evolution equations together with the Klein-Gordon dynamics, using the Misner-Sharp mass to locate and characterize the apparent horizon. A key result is that the AH slope near the central singularity is governed by a single dimensionless parameter $\eta=2\mathcal{M}\Psi_-$, yielding a universal, scale-invariant description of the end-state: dispersal, a zero-mass black hole, a black hole with AH before singularity, or a locally naked null singularity as $\eta\to0$; in all cases the geometric mass on the singularity vanishes. This framework establishes a covariant, ansatz-independent route to critical collapse and highlights the universality of the collapse dynamics across MSF families, with potential implications for CCC and primordial black-hole formation.
Abstract
In this paper, we observe the collapse of a mass-less scalar field covariantly. We show that the strengths of the collapsing and dispersing modes of this scalar field will decide whether the collapse will end up in a black-hole or disperse. We find a locally naked null singularity as a critical case between these two and confirm that there is a single dimensionless parameter that determine the end state. This work is ansatz-independent, hence, true for all mass-less scalar field families. We also show that the geometrical mass of these singularities go to zero.