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We know stars take infinite time in our coordinates to actually collapse into a black hole due to time dilation. See, e.g., How can anything ever fall into a black hole as seen from an outside observer?

My question is: why is the black hole information paradox a problem given this fact? In our coordinates, all information is stuck at the event horizon for infinite time, and all information is on that surface. There is no paradox at all.

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    $\begingroup$ "We know stars take infinite time in our coordinates to actually collapse into a black hole due to time dilation." There are perfectly fine coordinates for us in which the collapse of a black hole happens in a finite time. $\endgroup$
    – TimRias
    Commented Dec 9, 2021 at 8:06
  • $\begingroup$ @safesphere No, I mean, if there is no real blackhole in the universe, why does its evaporation brings a problem? $\endgroup$
    – Gerry Che
    Commented Dec 9, 2021 at 14:58
  • $\begingroup$ @safesphere Any of the coordinate systems used in well-posed 3+1 decompositions of GR as used in numerical relativity would be an example. $\endgroup$
    – TimRias
    Commented Dec 10, 2021 at 10:55
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    $\begingroup$ More than 90 astrophysical black holes have been detected indirectly, by the elliptical orbits still followed by their former partners is binary pairs. (Most stars are in binary pairs.) One vastly larger black hole, formed by dust collapse, has been detected, in Sagitarrius A. Maybe you're thinking of Schwarzschild black holes, which are usually considered to be an entirely hypothetical conceptual device, because they don't rotate: A star lacking any rotation would be extraordinarily rate, and I don't believe any have been observed. $\endgroup$
    – Edouard
    Commented Dec 12, 2021 at 19:54

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The paradox is basically independent of this issue. Suppose you collapse a star into a black hole and let it evaporate. Hawking's calculation (the source of the information paradox) says that the radiation emitted during evaporation is completely unrelated to the state of the star. Once the black hole has evaporated completely, you're only left with Hawking radiation, which contains no information about the star. Information has been lost, hence the paradox.

(Note that you don't have to wait for the black hole to evaporate completely to get a paradox. There is a contradiction as soon as the black hole has lost half its mass to evaporation).

Edit: To the question of whether a black hole will form in the first place, take a look at this related question (in particular the answer by FrankH). Infalling matter will fade out exponentially fast, so that after a very short time it will be virtually indistinguishable from a black hole with nothing at the surface. This object emits Hawking radiation, whether you call it a black hole or not.

But more to the point of the original question, one way to derive Hawking radiation (though this is not how his original calculation worked) is to consider the frame of reference of an infalling observer. In this frame (I think it's just Kruskal-Szekeres Coordinates), nothing is frozen at the event horizon and you see nothing but vacuum. The vacuum has quantum mechanical fluctuations (often described as virtual particles appearing and annihilating). The strong gravity of the black hole 'stretches' these fluctuations in a way that produces Hawking radiation (often described as pulling apart the virtual particles so that they don't get a chance to annihilate).

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    $\begingroup$ The problem is that the star takes infinite time to collapse into a black hole, so no information will be stored behind the event horizon. I am not sure if there is still evaporation in this case. $\endgroup$
    – Gerry Che
    Commented Dec 9, 2021 at 14:53

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