I know that for an observer far away, nothing ever crosses a black hole horizon (due to time dilation), while in the frame of reference of a falling observer the horizon is nothing special on its way in; and there is no contradiction because in the far away frame of reference the horizon is reached in the infinite future. Now considering that a black hole evaporates in a finite time, it cannot exist in the infinite future of a far away observer. So it seems there is a contradiction here, where the observers (free-falling and far-away) do not agree on the mere fact that the horizon is reached at all by anything during the whole black hole lifetime. Thus my question: by the very nature of their topology in spacetime, aren't black holes required to live forever in our frame of reference ?
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