This is a question about Hawking radiation in a fixed Schwarzschild background. The radiation does not back react on the metric and the black hole can not grow or evaporate.
Now I've seen Hawking radiation derived using either a Bogoliubov transformation or by using path integral arguments, so I understand that the vacuum state for a quantum field in the Schwarzchild background will be in a thermal equilibrium state in terms of the Hamiltonian that generates time translations in the Schwarzschild coordinate time.
My question is how is this interpreted as radiation coming from the black hole? This is an equilibrium state that does not change with time and unless I'm missing something there are as many particles moving radially in to the black hole as out from it.
Where are the particles from infinity coming from? Does an observer far from the black hole observe as much radiation as an observer near it?
Basically my confusion is coming from the fact that the state is expressed as a thermal distribution of modes without obvious (to me) spatial dependence. So I'm wondering how people jumped to the interpretation that all of the radiation in this vacuum state is coming from the black hole.
There is a very similar question here: Eternal black holes and Hawking radiation. But I am aware of the fact that this is a fixed background metric, and this still doesn't answer my questions of how the radiation is interpreted.