Aren't black holes required to exist forever in our frame of reference?

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 ?

• Why is there a contradiction? The falling observer may well cross the event horizon in finite proper time along his trajectory (in the Schwarzschild metric), but throw in black hole evaporation and the black hole doesn't have to always be there. If the falling observer sees the black hole evaporate before reaching the event horizon, then no contradiction exists. – PhillS Feb 29 '16 at 14:14
• The contradiction I see is from our (= far-away) frame of reference: if the black hole has a finite lifetime, there is no trajectory in our frame for a falling object. All trajectories have to stop at the time of evaporation/explosion, before the horizon. – Stéphane Rollandin Feb 29 '16 at 14:47
• But why are you assuming that the falling observer has to see himself cross the event horizon? The falling observer will (presumably) see the black hole evaporate before he crosses the event horizon. Both observers will agree that the falling person never reached the event horizon before the black hole evaporated. (This is all supposition: I don't know of any metric describing an evaporating black hole, so don't know what the actual trajectories look like in that spacetime). – PhillS Feb 29 '16 at 15:15
• Why am I assuming that the falling observer sees himself cross the event horizon ? Because it is in that case that I have a problem. All matter that ever went into a black hole and made its horizon grow did cross the horizon from the free-falling point of view. And it seems to me that we need the black hole to have an infinite lifetime in order to account for this from our perspective. What am I missing ? – Stéphane Rollandin Feb 29 '16 at 15:57
• Why does topology come into question here? – user106422 Mar 4 '16 at 0:41