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

Given a static Schwarzschild spacetime of ADM mass M , it is wellknown 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. ∗Support by the Institut Mittag-Leffler (Djursholm, Sweden) is gratefully acknowledged.