On Schwarzschild black hole singularity formation

Abstract

We examine whether the Schwarzschild black hole can emerge as the continuous end state of gravitational collapse from a non-singular configuration. Employing a time dependent extension of the regular Schwarzschild metric, we track the evolution of the geometry during collapse and find that the process cannot remain continuous. The metric function develops a discontinuity at the origin, marking a breakdown of spacetime smoothness, an effect identified as ``Minkowski breaking.'' Before the Schwarzschild point source can form at $r=0$, curvature singularities appear and the Cauchy horizon disappears. These results strongly suggest that spacetime may not evolve smoothly toward the Schwarzschild geometry. Instead, the formation of a Schwarzschild black hole appears to entail a discrete change in the structure of spacetime, pointing to the need for a noncontinuous, possibly quantized, framework to describe the emergence or regularization of gravitational singularities.

Figures (3)

• Figure 1: Minkowski breaking for the lapse function $f(v,r)$. Since the event horizon must form before the singularity evolves, in agreement with the weak Cosmic Censorship Conjecture, the formation of a regular configuration with its corresponding inner horizon is unavoidable prior to the emergence of the singularity, as represented by the solid line. The dashed curve corresponds to the singular Schwarzschild solution. In order to transition continuously from the regular configuration to the Schwarzschild solution, the condition $f(v,0)=1$ must be abandoned. At present, however, we are not aware of any dynamical mechanisms that could explain this transition. The radial coordinate $r$ is expressed in units of ${\cal M}$.
• Figure 2: Evolution of the metric function \ref{['sol1']} from a regular BH configuration with $n(v)>2$ down to the critical threshold $n(v)=2$. The evolution then proceeds to a second threshold, $n(v)=0$, marking the onset of Minkowski breaking $[f(v,0)\neq1]$, and thereby signaling the loss of spacetime continuity. The horizon remains fixed at $h=2{\cal M}\neq h(v)$. The radial coordinate $r$ is expressed in units of ${\cal M}$.
• Figure 3: Evolution of the mass function \ref{['m1']} from a regular BH configuration with $n(v)>2$ down to the critical threshold $n(v)=2$. The evolution then continues until a second threshold is reached at $n(v)=-1$, where a sudden collapse of all matter ${\cal M}$ into $r=0$ occurs, producing the point-like solution (Schwarzschild). The horizon remains fixed at $h=2{\cal M}\neq h(v)$. The radial coordinate $r$ is expressed in units of ${\cal M}$.