The asymptotic behaviour in Schwarzschild time of Vlasov matter in spherically symmetric gravitational collapse

TL;DR

This paper investigates the asymptotic behavior of Vlasov matter during gravitational collapse in Schwarzschild spacetime, revealing conditions under which matter takes infinite Schwarzschild time to cross the event horizon, contrasting with classical expectations.

Contribution

It constructs initial data leading to black hole formation where matter crosses the horizon at infinite Schwarzschild time and introduces a new method for proving global existence with non-compact initial data.

Findings

Matter can cross the event horizon at infinite Schwarzschild time.

A necessary condition for geodesic completeness of the horizon is matter crossing at infinite time.

New proof technique for Einstein-Vlasov system with non-compact support.

Abstract

Given a static Schwarzschild spacetime of ADM mass M, it is well-known that no ingoing causal geodesic starting in the outer domain r>2M will cross the event horizon r=2M in finite Schwarzschild time. In the present paper we show that in gravitational collapse of Vlasov matter this behaviour can be very different. We construct initial data for which a black hole forms and all matter crosses the event horizon as Schwarzschild time goes to infinity, and we show that this is a necessary condition for geodesic completeness of the event horizon. In addition to a careful analysis of the asymptotic behaviour of the matter characteristics our proof requires a new argument for global existence of solutions to the spherically symmetric Einstein-Vlasov system in an outer domain, since our initial data have non-compact support in the radial momentum variable and previous methods break down.