Question

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Black hole formation as seen by a distant observer

Given that matter can never cross the event horizon of a black hole (from an external observer point of view), if a black hole is "fed" with a large amount of matter then the new matter will eventually become extremely compressed, and presumably would be compressed below its Schwarzchild radius.

Would secondary black holes eventually form near the original black hole?

As an alternative one could also imagine that the combined mass of the original black hole and the new mass around the event horizon becomes contained within the Schwarzchild radius of both masses, and so a new event horizon forms, "swallowing" the new mass around the edge of the original black hole.

This mechanism would allow black holes to swallow mass in a finite time.
Would this contradict GR predictions?

Answer 1

Of course that black holes can swallow and do swallow additional matter and increase their mass in finite time, whatever reasonable coordinates we choose.

Yes, the Schwarzschild radius for the combined system smaller black hole + extra matter ("food") is larger than it is for the original smaller black hole which means that it's enough for the new matter to cross a critical surface that is outside the original event horizon of the smaller black hole. This occurs in finite time, even from the external observer's viewpoint, and the black hole simply grows in size.

It's a routine process we observe at many places. For example, the black hole at the galactic center devours new matter all the time.

Incidentally, there can't exist any "concentric pairs of black holes" solutions to the 3+1-dimensional general relativity. Once we know that it's the empty Schwarzschild metric up to the event horizon (from outside), then it is the black hole and there's no way for the metric to "unbecome" a black hole again. Everything that is inside the event horizon is, by definition, causally disconnected from infinity