  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.
