Negative energy in Hawking radiation

[Demystifier]

At a descriptive level, negative energy quanta enter a black hole during Hawking radiation. But when one tries to understand it mathematically, it seems that negative "energies" appear in two very different senses, which seem to be totally unrelated to each other. At one level one has Bogoliubov modes with negative frequencies, i.e. modes proportional to $e^{-i\omega t}$ where $\omega$ is negative and $t$ is a certain coordinate time. At another level one has negative flux of energy through the horizon, computed from the renormalized expectation value of the energy-momentum tensor. But it seems to me that those two notions of "negative energy" are totally unrelated to each other. Or at least I don't see how they are related. Is there a relation between them that I miss?

 

[PeterDonis]

Is there a relation between them that I miss?

Not that I'm aware of. I believe the disconnect between the various heuristic descriptions of Hawking radiation has been commented on in the literature. Another disconnect is between both of your descriptions and the "pair of virtual particles with one falling into the hole and the other escaping to infinity" description.