Universality and criticality in mass-less scalar field collapse

Abstract

In this paper, working in a geometrical, frame-independent and ansatz-independent formalism, we analytically investigate the gravitational collapse of a mass-less scalar field in a spherically symmetric space-time. This is one of the fundamental matter fields determined by the Lagrangian formulation. We identify a single dimensionless parameter, that determines the end-state to be a black-hole or complete dispersal and a locally naked null singularity, which exists as a critical case between the previous two end-states. The governing parameter, being dimensionless, makes the result scale independent, thus confirming the universality and criticality that was observed in extensive numerical studies by various authors. © 2026 IOP Publishing Ltd. All rights, including for text and data mining, AI training, and similar technologies, are reserved. This article is available under the terms of the https://publishingsupport.iopscience.iop.org/iop-standard/v1.