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My understanding is that if you fell into a black hole, i.e. crossing the event horizon time would be speed up infinitely fast. This means I could see my whole galaxy die, new blackholes develop and collapse again and so on.

My further understanding is, that time is relative - it is based on the position of the observer.

So if I am beyond the event horizon and times flies by,how can (for example) a observer on earth look at a any black hole in the universe and see it die in his "normal" speed of time - since the time inside the black hole is infinite?

It seems like a paradox to me, and I would love some further elaboration :)

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My understanding is that if you fell into a black hole, i.e. crossing the event horizon time would be speed up infinitely fast. This means I could see my whole galaxy die, new blackholes develop and collapse again and so on.

This is incorrect. See for example the answers to does-someone-falling-into-a-black-hole-see-the-end-of-the-universe? on Physics Stack Exchange and How much time passes in the outside universe when falling into a black hole? There is no extreme effect for a falling observer. They reach the singularity in a finite time and do not see much further into the future than if General Relativity were ignored.

My further understanding is, that time is relative - it is based on the position of the observer.

Time dilation is something that can only be expressed as being relative to the time measured on somebody else's clock. So yes, if someone else is at a different position then each will claim that the others clock is running at a different rate.

So if I am beyond the event horizon and times flies by, how can (for example) a observer on earth look at a any blackhole in the universe and see it die in his "normal" speed of time - since the time inside the blackhole is infinite?

None of this is true so there is no paradox and it is hard to see what you are asking about. Time doesn't fly by. An observer beyond the event horizon experiences time in exactly the same way. There is a time dilation effect with respect to a distant external clock, but it is not extreme for a falling observer and observers cannot do anything but move rapidly towards the singularity when they are inside the event horizon.

The commonly discussed example of a black hole and objects falling into a black hole "freezing" as they approach the event horizon is based on a time-invariant, eternal black hole, that has existed forever and will exist forever in an unchanging state. It does not accurately describe a collapsing star for example. If you are asking how can a black hole form if nothing can cross the event horizon according to a distant observer, then basically the answer is that different observers do not agree on what happens and on what timescales in General Relativity. There are several duplicates of this listed in the "related questions" - particularly Does matter accumulate just outside the event horizon of a black hole?

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What we see is the surface of the collapsing star becoming progressively more and more red shifted. In our space-time frame we don't see matter crossing the event horizon (as an infinite amount of time would have to pass) but we do see that matter becoming red-shifted.

The amount of red-shifting increases without limit and in seconds even gamma radiation gets red-shifted to radio waves with a wavelength that is as large as the observable universe. And so the black hole looks black.

The spacetime around the black hole can be well modelled with the Kerr metric. And so the black hole behaves much like the theoretical Kerr black hole, that describes a rotating region of spacetime and may be considered to be a model of the final state that a real black hole would tend towards. In practice, real black holes become indistinguishable from eternal but theoretical Kerr black holes very rapidly.

However the singularity doesn't form in our present time, and so the singularity can never be in our past, This is also true for infalling matter. The singularity is always in the future. (So you can't "see" it, because you can only see things that are in the past). You do, however reach the singularity in finite time. What happens then is unclear. General Relativity doesn't give a sensible answer, and the answer probably depends on which model of quantum gravity best describes reality.

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