Assuming they exist (which seems likely according to our current best theories) black holes are very strange things indeed. In particular, the fate of something heading over the black hole horizon depends seems to be different depending on whether you cross the horizon or not.
In the frame of reference of somebody passing through the horizon, the horizon doesn't seem very special at all - you could cross the horizon of a sufficiently large black hole without even noticing. After passing the horizon, however, escape is impossible and you would inevitably end up spaghettified by the massive tidal forces near the central singularity.
In a frame of reference of somebody outside the black hole, the picture is very different. From that perspective, the gravitational time dilation at the horizon means that the progress of the falling object becomes slower and slower, grinding to a complete stop at the horizon itself. The red-shifting of any photons emitted from close to the horizon also means that the object looks redder and redder, going through infra-red, terahertz radiation, microwaves, and radio waves of increasing wavelength until they are too low energy to detect but they never quite stop altogether.
In the meantime, the black hole evaporates by the process of Hawking Radiation. This takes an unimaginably long time - as long as 10100 years for the largest black holes but if you wait there long enough it will happen. Supposing that you could somehow detect the infalling observer during the entire period of the evaporation, you'd see the infalling observer crossing the horizon at the exact moment that the black hole disappeared altogether in a bright flash of energy (the smaller the black hole, the more it radiates). But of course, at that moment the infalling observer has zero mass-energy (as does the black hole as a whole) so how can it be the same observer in its own frame of reference (when its mass is the same as it was before the experiment began)?
Clearly our current theories of physics are insufficient here - we don't yet have a consistent theory of quantum gravity so we just don't know what happens near a tiny black hole when the spacetime curvature is high enough to be in the quantum regime.
One way out of the apparent paradox is simply that the two observers can never compare notes because whatever information the infalling observer collects is inevitably destroyed at the singularity, and no information from inside the black hole can ever be transmitted to the outside universe. The interior of the black hole is causally disconnected from the outside - it can be considered to exist infinitely far in the future (long after the black hole evaporates) or can even be considered to be an entirely different universe in its own right. If the two observers can never get back together to compare notes (and find that they disagree) there isn't really any disagreement. It's a bit like the Many Worlds Interpration of quantum mechanics - observers in different parallel universes can't subsequently interact so they can't disagree about the observable outcome of some experiment.
But in both of these cases there is a philosophical problem - if the universe is qualitatively different for different observers then it seems to make a mockery of the very idea of an objective reality. Just as the IO Monad theory of subjective experience has no problems with observers disagreeing on matters that make no difference in an objective sense, it seems like general relativity may require that we have no problems with disagreements between causally disconnected objective realities.
Interesting read (as always). But as I understand it (I admit I haven't given it much thought yet) this sounds just like a different version of the Zeno paradox to me.
The outside observer would, at some point, simply see the falling object vanish (as soon as the photons have lost enough energy due to gravitational pull). The fact that an outside observer doesn't actually 'see' the object 'fall into the black hole' doesn't mean it's staying on the event horizon for all time. Light doesn't get slower, it only gets shifted towards red.
-Darkstar
The analogy with Zeno's paradox doesn't really hold up here because it can be resolved with calculus, and applying the same calculus to this situation still gives the behaviour I've described here. The light from the infalling observer really does continue to get redshifted indefinitely. It may get redshifted to a wavelength many times the size of the observable universe (and thus become impossible to practically detect) but that doesn't mean that the infalling observer has crossed the horizon by that point - the outside observer can in theory continue to receive messages from them right up until the black hole collapses. There is a never a time before then (no matter how you compute or measure it) when the infalling observer can be said to have crossed the horizon, in any sense.
Well, you can't shift light to arbitrary low wavelengths since a photon can only occupy discrete energy levels. After that it just vanishes.
I still understand it that the fact that the photons get redshifted doesn't mean that light takes an endless time to cross the event horizon. That's the part I think is an analogy to Zenos paradox
It seems I really need to think about this a bit longer... The last relativity classes I joined are quite a few years past now ;-)
It's only bound states where photon energies are discrete - free photons can go to arbitrarily low wavelengths (at least according to our current best theories). Of course, once we have a theory of quantum gravity it may limit the process in various ways.
Another confusing part about this is the time dilation - the closer to the singularity someone is the slower time seems to move for them (relative to a distant observer). So if your identical twin went to hang out near a black hole for a while when you met up again you'd be much older. At the horizon itself, the time dilation is infinite.
Are you sure that the observer actually crosses the event horizon when the black hole evaporates?
It would be nice to know if someone has done the maths to figure out the amount of time dilation near the event horizon and singularity and see if anything can actually reach them before Hawking radiation undoes the black hole.
What if time dilation actually prevents anything from crossing the event horizon? Hawking radiation will eventually shrink both the black hole and the event horizon. As a result the black hole would evaporate in front of anything that ever fell into the black hole - before anything actually crosses the event horizon.
Hi Timo,
Nobody's sure of anything where black holes are concerned, but our best guess based on what's currently known is that no, a black hole wouldn't evaporate out from underneath a falling observer - the Hawking radiation is negligible to the in-falling observer. No quantum corrections are required here - this result can be obtained using classical GR and assuming only that the black hole can form in the first place.
A more complete discussion of this can be found at http://johanw.home.xs4all.nl/PhysFAQ/Relativity/BlackHoles/fall_in.html under "What about Hawking radiation? Won't the black hole evaporate before you get there?" about 2/3 the way down the page.
Andrew