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But I'd like to understand properly what happens to the length contraction experienced by an observer falling together with that clock.

Would he experience length contraction ?

As length contraction reaches extreme levels, what would he be able to see? The whole Universe shrank to a miniscule size ?

I remember reading in another post that time dilation and length contraction always happen together. But I've only seen this explained for the case of relativistic speeds, and never for the case of a black hole.

Thanks