Well, I know basic General Relativity but I still learning the basics of Quantum Field Theory (therefore without mentions on curved spacetimes). But, I'm trying once and a while to become more used to quantum fields and curved spaces.

Now, the notion of Hawking Radiation is paramount but my knowledge is semi-technical. Reading $[1]$, the author said:

At first sight, black hole radiance seems paradoxical, for nothing can apparently escape from within the event horizon. However, inspection of $(8.36)$ shows that the average wavelength of the emitted quanta is $\approx M$, i.e., comparable with the size of the hole. As it is not possible to localize a quantum to within one wavelength, it is therefore meaningless to trace the origin of the particles to any particular region near the horizon.

What worries me is the final consideration when he says "therefore meaningless to trace the origin of the particles to any particular region near the horizon". So, knowing that phrase from $[1]$, I would like to ask:

Hawking Radiation isn't a phenomena that occurs near the Horizon?

$$ * * * $$

$[1]$ Birrell.N.D. Quantum Fields in Curved Space. page $264$.


Hawking radiation is observed at future null infinity. One can try to trace back where the radiation observed originated. A common method for this is to use the “geometric optics” approximation, where radiation follows null geodesics. Doing so with the modes of the Hawking radiation suggests that the modes originate arbitrarily close to the event horizon.

However, the geometric optics approximation only works in the limit that the wavelength is small compared to the curvature length scale. As Birrrell points out this isn’t the case for the dominant modes in the Hawking radiation, which have wavelengths comparable to the size of the horizon. The geometric optics results should therefore be taken with a grain of salt. All we can really say is that the modes of the Hawking radiation originate in vicinity of the horizon.

That is the point Birrell is trying to make (I think).


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