Eternal black holes and Hawking radiation I have a fairly simple question which is confusing me a lot.
As Hawking showed, a black hole originated by collapse will emit Hawking radiation. This process will reduce the mass of the black hole until it will evaporate. But what about black holes which are not originated by collapse, i.e. eternal black holes?
In case we have a black hole in AdS, the black hole will be indeed eternal because all the emitted Hawking radiation will eventually come back to the black hole, since AdS acts like a confining box.
But if we consider a eternal black hole in flat space there is no process that ensures the stability of the black hole. So it would mean that at a certain point it will evaporate. How is this black hole eternal? To put it in a different way, how can a geometry which is invariant under time translations identify a precise time, the one at which the black hole evaporates?
 A: Hawking's original calculation used a semi-classical approximation. The quantum calculation was done in a fixed curved spacetime background and the back reaction of the Hawking radiation on the black hole was ignored. This should be a good, actually exceedingly good, approximation for any macroscopic black hole because the Hawking temperature is so low.
So the calculation does not calculate the evolution of the black hole. However given that we know what energy is being lost due to Hawking radiation we know how fast the black hole will shrink. Again, this is an approximation, but as long as the temperature is low and the rate of shrinkage very small we expect that once again this will be an excellent approximation.
As we approach the final stages of evaporation we expect that the semi-classical approximation will no longer be useful, and some theory of quantum gravity will have to take over. I have seen lectures where string theorists attempted to model the very final stages, but to be honest it was way over my head and in any case I got the impression it was all highly speculative and poorly understood at best.
A: The black hole would only be "eternal" from the standpoint of General Relativity alone.  Hawking adds the effects of the quantum field in empty space.  So a black hole starting with mass $M$ would have that mass decrease over time, and the spacetime geometry would change accordingly.
It'd be like setting an ice cube with mass $m$ on a surface maintained at 33°F.  The problem is inherently time dependent.
