general relativity - Black hole formation as seen by a distant observer

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How can anything ever fall into a black hole as seen from an outside observer?

Is black hole formation observable for a distant observer in finite amount of time? Specifically, let's take uniform non rotating distribution of 10 solar masses within solar radius, assume no outward pressure, and calculate its gravitational collapse. I assume distant observer would never witness complete formation of black hole with event horizon corresponding to entire 10 solar masses. What would be observable in finite amount of time, then? For example, would there be a black hole with smaller radius, while the rest of mass is still falling into its event horizon?

Answer

There is no unique way to answer this question. The problem is the following. You are asking if, at a specific proper time, a black hole of some radius exists. However, to do this you have to say something like "the black hole has a 10m radius at the same time that the observer's proper time is 5sec". Such simultaneous events are not well defined in GR unless you make an arbitrary choice of a coordinate system with a time coordinate. Nonetheless, if you make such a choice, it is true that the black hole will start with a radius of 0 inside of the infalling matter and then get larger. Eventually it will reach a maximum radius and then get smaller through Hawking radiation.

So what will an observer literally see if she observes the star collapsing? The answer is that the observer will see infalling matter slow down and thermalize. The matter will never appear to "fall in" as far as this observer is concerned. However, the particles/light emitted will be redshifted more and more (and before long the only outgoing particles of reasonably not-low energy will be from Hawking radiation).