Some Remarks on the Horizon in the Dust Cloud Collapse

Abstract

We examine the existence of an apparent horizon in the collapse of an isolated dust cloud in spherically symmetric spacetime. The interior geometry is expressed in the Lemaître–Tolman–Bondi coordinates, while the outside geometry is the standard Schwarzschild. The novelty of the approach lies in using the Lemaître coordinates for the Schwarzschild part of the geometry when discussing junction conditions. We provide general formulas for expansions in the marginally bound cloud case and use them to determine the position of the apparent horizon. Our results indicate that in the region of spacetime far from the gravitational singularity, the system under consideration has a horizon, and thus the singularity is not naked. The results are expected to be correct away from the singularity, where quantum effects can be neglected. Near the singularity, general relativity breaks down, and quantum gravity would be of primary importance. The classical formulas for expansion scalars can serve as a starting point for more detailed quantum analysis. © 2026 by the authors.