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This is a kind of follow-up of my previous Phys.SE question Can something (again) ever fall through the event horizon? that seems to have established that, for an outside observer, things never fall into the black hole. Note that this is not to say that an outside observer never sees anything falling into a black hole (as in receiving the optical signal), but it is instead an stronger statement: no matter how long an outside observer wait for anything to fall into a black hole, there will always exist intersection between his own future light cone and the in-falling object's light cone outside the event horizon. That is what I mean by "things never fall into the black hole".
Assume an initial universe with no black holes (stellar or primordials), and thus, has no event horizons. Is there a solution to GR equations showing that the light cone of any amount of mass will inevitably reach a singularity, never intersecting with the light cone of an observer outside the would-be-black-hole? For my understanding, the Schwazschild metric describes an eternal, static, black hole, so I guess I am looking for something else here.
Please notice that I tried to be much more specific than the other questions about black hole formation, because the answers saying that in falling observers would reach the singularity in a finite amount of proper time isn't really helping me to understand the phenomenon from an outside perspective.