The heuristic argument for Hawking Radiation is, that a virtual pair-production happens just at the event horizon. One particle goes into the black hole, while the other can be observed as radiation.
I never quite understood this explanation. Say, a virtual pair is created. From the point of view of the "radiation particle" the other particle will need an infinite amount of time in order to reach the event horizon. But the virtual pair production violates energy conservation, so the particles can only exist for a finite amount of time (Heisenberg uncertainty principle). Then, they must annihilate each other again, without emitting any radiation.
I often heard the "explanation" that one particle "tunnels" through the event horizon. But isn't this again a flawed argument? Tunnelling is an effect known from flat spacetime and it makes particles cross potentials that they could not cross classically. How could this help to "jump over" an infinite time interval? For me it seems like superimposing ideas from flat spacetime on curved spacetime without any justification.
While I understand, that the original derivation from Hawking follows a different argument, one should, in principle, be able to attach his results to a physical process. I worked through his derivation and while I must admit that I didn't understand every step, it seems to me that a lot of assumptions have to be made along the way in order to produce the results. Also the whole derivation is based on Quantum Field Theory in curved spacetime, which stands on shaky grounds, since nobody can say, where it is valid and where not - we don't have a complete theory of Quantum Gravity and we simply can't say in which situations QFT in curved spacetime is a good approximation.
As you can see, I am really confused about this issue. Every sort of help is appreciated!