The formation of black holes in spherically symmetric gravitational collapse

TL;DR

This paper analyzes conditions on initial data in the Einstein-Vlasov system that lead to black hole formation, demonstrating explicit criteria for event horizon development and asymptotic Schwarzschild behavior.

Contribution

It provides explicit initial data conditions for black hole formation in the Einstein-Vlasov system, including cases with global existence without smallness assumptions.

Findings

Black hole formation occurs under specific initial data conditions.

Solutions can exist globally for all r ≥ 0 in certain scenarios.

The spacetime asymptotically approaches Schwarzschild metric with mass M.

Abstract

We consider the spherically symmetric, asymptotically flat Einstein-Vlasov system. We find explicit conditions on the initial data, with ADM mass M, such that the resulting spacetime has the following properties: there is a family of radially outgoing null geodesics where the area radius r along each geodesic is bounded by 2M, the timelike lines $r=c\in [0,2M]$ are incomplete, and for r>2M the metric converges asymptotically to the Schwarzschild metric with mass M. The initial data that we construct guarantee the formation of a black hole in the evolution. We also give examples of such initial data with the additional property that the solutions exist for all $r\geq 0$ and all Schwarzschild time, i.e., we obtain global existence in Schwarzschild coordinates in situations where the initial data are not small. Some of our results are also established for the Einstein equations coupled to a general matter model characterized by conditions on the matter quantities.