From a reference frame outside the black hole, observing an object falling into the black hole, don't we observe time slowing down as the object approaches the event horizon? In this reference frame, does it ever cross the event horizon, or does it just asymptotically approach it? If it doesn't cross the event horizon, how can anything ever fall into the black hole?
(From the reference frame of the object falling into the black hole, my understanding was that the event horizon shrinks away from you as you approach it.)
Please correct/inform me!