Why does Stephen Hawking say black holes don't exist?

This is really a footnote to the accepted answer.

Light cannot escape from an event horizon. But how can you check that light can never escape? You can watch the surface for some time $T$, but all you have proved is that light can't escape in the time $T$. This is what we mean by an apparent horizon, i.e. it is a surface from which light can't escape within a time $T$.

To prove the surface really was an event horizon you would have to watch it for an infinite time. The problem is that Hawking radiation means that no event horizon can exist for an infinite time. The conclusion is that only apparent horizons can exist, though the time $T$ associated with them can be exceedingly long, e.g. many times longer than the current age of the universe.

A point worth mentioning because it's easy to overlook: when you start learning about black holes you'll start with a solution to Einstein's equations called the Schwarzschild metric, and this has a true horizon. However the Schwarzschild metric is time independent so it would only describe a real black hole if that black hole had existed for an infinite time and would continue to exist for an infinite time. Both of these are not possible in the real universe. So the Schwarzschild metric is only an approximate description of a real black hole, though we expect it to be a very good approximation. 

The paper by Dr. Stephen Hawking doesn't say that black holes don't exist. What he says is that black holes can exist without "event horizons". To understand what an event horizon is, we first have to understand what is meant by escape velocity. This last one is the speed you need to escape a body. Now, here is where the event horizon and the escape velocity comes in play: the event horizon is the boundary between where the speed needed to escape a black hole is less than that of light, and where the speed needed to escape a black hole is greater than the speed of light.

So Hawking says that instead of event horizon, there may be "apparent horizons" that would hold light and information only temporarily before releasing them back into space in a "garbled form".

Craig Feinstein asked: Does Stephen Hawking believe that General Relativity is wrong?

Here is my answer (I will shift my answer there if some one reopen that question):

Stephen Hawking did NOT say that black holes do not exist. Hawking used to think black holes are oblivious. Now he admits (like some other people do) black holes have perfect memory, just like any other quantum systems. So what Hawking said is that black holes are not forgetful.

In order for black holes to have perfect memory (ie satisfies unitary quantum evolution), the classical GR must be wrong. The black hole horizon in the classical GR must be modified. I believe that, in well defined quantum gravity (yet to be developed), the horizon carries degenerate states that give rise to black hole entropy. There is no such thing as inside horizon. In other words, black hole horizon behaves like a hard-wall (with nearly degenerate states). This picture contradicts with classical GR.

Hawking's picture for the horizon, I believe, is similar. This actually is a very old idea. See:

http://relativity.livingreviews.org/open?pubNo=lrr-2011-8&page=articlesu33.html

http://arxiv.org/abs/gr-qc/0303006

http://iopscience.iop.org/1742-6596/410/1/012137

So I believe that "Stephen Hawking believe that the classical General Relativity is wrong near the blackhole horizon". I think most people (not including me) believe that classical General Relativity is correct near the black hole horizon.

I believe that the classical General Relativity is wrong, since I believe that a well defined quantum theory (including quantum gravity) has a finite UV cutoff. A finite UV cutoff is incompatible with General Relativity principle. A finite UV cutoff is even incompatible Lorentz symmetry. A finite UV cutoff may modify our picture about the black hole horizon, but I do not know how.