Black hole quantum atmosphere for freely falling observers

We analyze Hawking radiation as perceived by a freely-falling observer and try to draw an inference about the region of origin of the Hawking quanta. To do so, first we calculate the energy density from the stress energy tensor, as perceived by a freely-falling observer. Then we compare this with the energy density computed from an effective temperature functional which depends on the state of the observer. The two ways of computing these quantities show a mismatch at the light ring outside the black hole horizon. To better understand this ambiguity, we show that even taking into account the (minor) breakdown of the adiabatic evolution of the temperature functional which has a peak in the same region of the mismatch, is not enough to remove it. We argue that the appearance of this discrepancy can be traced back to the process of particle creation by showing how the Wentzel--Kramers--Brillouin approximation for the field modes breaks down between the light ring at $3M$ and $4M$, with a peak at $r=3.3M$ exactly where the energy density mismatch is maximized. We hence conclude that these facts strongly support a scenario where the Hawking flux does originate from a "quantum atmosphere" located well outside the black hole horizon.