A while ago (in 2005, I think), I took part in a discussion in a (not astronomy-related) mailing list on the subject of the existence of black holes. The argument of the guy pushing this position was interesting and, for me, very relevant: black holes can't exist because there was no time for them to form.
See, the story is like this: when a super-massive star dies, its core collapses due to gravity and gets progressively smaller very quickly; as it gets smaller, gravity at its surface increases, and so does its escape velocity. When surface gravity increases past a certain point, the escape velocity becomes higher than the speed of light, which means that nothing can escape that body anymore, not even light. Presto, a black hole. After that, the core continues to shrink until it becomes a single point, with infinite density and gravity: a singularity.
However, that's the story as told from the point of view of the shrinking core. As Einstein told us in the theory of general relativity, clocks run slower when in a gravitational field, and the stronger the gravity (or acceleration) the clock is subjected to, the slower it runs. The effect is that, as seen from the outside, the shrinking core actually shrinks progressively more slowly as its surface gravity increases (and, from the point of view of the core, the universe outside moves progressively faster as it - the core - shrinks), and the speed of "shrinkage" tends to zero as the escape velocity approaches the speed of light. Therefore, from the point of view of anyone outside a black hole, the black hole never finishes forming: the escape velocity never actually reaches the speed of light.
It's a good argument, and I couldn't think of a good response to it. It turns out that this is a scientifically interesting problem, and a recent post by Phil Plait, the Bad Astronomer, points to a paper by astronomers from the Case Western Reserve University that argues that not only black holes never finish forming, they can't finish forming because Hawking radiation makes them evaporate before the event horizon forms (from the point of view of an external observer).
Read Phil's article on the subject, as he explains the whole situation much better than I can, and the comments make for excellent reading as well. The original paper, I have to say, was a bit too technical for me...
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