Hawking Radiation

In 1975 Hawking published a shocking result: if one takes quantum theory into account, it seems that black holes are not quite black!  Instead, they should glow slightly with "Hawking radiation", consisting of photons, neutrinos, and to a lesser extent all sorts of massive particles.  This has never been observed, since the only black holes we have evidence for are those with lots of hot gas falling into them, whose radiation would completely swamp this tiny effect.  Indeed, if the mass of a black hole is M solar masses, Hawking predicted it should glow like a blackbody of temperature

                     6 × 10-8/M kelvins,

so only for very small black holes would this radiation be significant.  Still, the effect is theoretically very interesting, and folks working on understanding how quantum theory and gravity fit together have spent a lot of energy trying to understand it and its consequences.  The most drastic consequence is that a black hole, left alone and unfed, should radiate away its mass, slowly at first but then faster and faster as it shrinks, finally dying in a blaze of glory like a hydrogen bomb.  However, the total lifetime of a black hole of M solar masses works out to be

                     1071 M3 seconds

so don't wait around for a big one to give up the ghost.  (People have looked for the death of small ones that could have formed in the big bang, but they haven't seen any.)
How does this work?  Well, you'll find Hawking radiation explained this way in a lot of "pop-science" treatments:

Virtual particle pairs are constantly being created near the horizon of the black hole, as they are everywhere.  Normally, they are created as a particle-antiparticle pair and they quickly annihilate each other.  But near the horizon of a black hole, it's possible for one to fall in before the annihilation can happen, in which case the other one escapes as Hawking radiation.

In fact this argument also does not correspond in any clear way to the actual computation.  Or at least I've never seen how the standard computation can be transmuted into one involving virtual particles sneaking over the horizon, and in the last talk I was at on this it was emphasized that nobody has ever worked out a "local" description of Hawking radiation in terms of stuff like this happening at the horizon.  I'd gladly be corrected by any experts out there...  Note: I wouldn't be surprised if this heuristic picture turned out to be accurate, but I don't see how you get that picture from the usual computation.
The usual computation involves Bogoliubov transformations.  The idea is that when you quantize (say) the electromagnetic field you take solutions of the classical equations (Maxwell's equations) and write them as a linear combination of positive-frequency and negative-frequency parts.  Roughly speaking, one gives you particles and the other gives you antiparticles.  More subtly, this splitting is implicit in the very definition of the vacuum of the quantum version of the theory!  In other words, if you do the splitting one way, and I do the splitting another way, our notion of which state is the vacuum may disagree!
This should not be utterly shocking, just pretty darn shocking.  The vacuum, after all, can be thought of as the state of least energy.  If we are using really different co-ordinate systems, we'll have really different notions of time, hence really different notions of energy—since energy is defined in quantum theory to be the operator H such that time evolution is given by exp(-itH).  So on the one hand, it's quite conceivable that we'll have different notions of positive and negative frequency solutions in classical field theory—a solution that's a linear combination of those with time dependence exp(-iωt) is called positive or negative frequency depending on the sign of ω—but of course this depends on a choice of time co-ordinate t.  And on the other hand, it's quite conceivable that we'll have different notions of the lowest-energy state.
Now when we are in good old flat Minkowski spacetime, a la special relativity, there are a bunch of "inertial frames" differing by Lorentz transformations.  These give different time co-ordinates, but one can check that the difference is never so bad that different co-ordinates give different notions of positive or negative frequency solutions of Maxwell's equations.  Nor will different people using these co-ordinate systems ever disagree about what's the lowest-energy state.  So all inertial observers agree about what's a particle, what's an antiparticle, and what's the vacuum.
But in curved spacetime there aren't these "best" co-ordinate systems, the inertial ones.  So even very reasonable different choices of co-ordinates can give disagreements about particles vs antiparticles, or what's the vacuum.  These disagreements don't mean that "everything is relative", because there are nice formulas for how to translate between the descriptions in different co-ordinate systems.  These are Bogoliubov transformations.
So if there is a black hole around...
on the one hand we can split solutions of Maxwell's equations into positive frequency in the most blitheringly obvious way that someone far from the black hole and far in the future would do it...
and on the other hand we can split solutions of Maxwell's equations into positive frequency in the most blitheringly obvious way that someone far in the past, before the collapse into a black hole has happened would do it.
That'd be the heuristic explanation I'd give that most closely corresponds to the usual computation.  There are additional things to say about the fact that the guy far in the future and far away from the black hole can't see what's in the hole, so he has incomplete information about the state, so he sees a state with entropy, in fact a thermal state.  (Here I'm assuming the black hole was NOT eternal, so the guy way back in the past didn't have the black hole to contend with.  Apparently Hawking's original computation dealt with this case, but people subsequently watered down his explanation by assuming the black hole was there eternally, to simplify the math.  This is what the guy at the talk said...  I'd only seen the watered-down version!)
Now in fact when you do a Bogoliubov transformation to the vacuum you get a state in which there are pairs of particles and antiparticles, so this is possibly the link between the math and the heuristic explanation.  Hopefully whoever made up the usual heuristic explanation understood the link better than I do!