Do the null geodesics of photons emitted by Hawking radiation arise from the event horizon? It is a well-known explanation of Hawking radiation that it originates from the quantum fluctuations near the horizon. Does it mean that one can look at the photons (part of the radiation) and follow their null geodesics back to the horizon?
Reference would be great, thanks.
 A: If in an eternal black hole spacetime, you trace back the origin of the "photons" in the Hawking radiation using the geometric optics approximation you will find that some of them pass arbitrarily close to the future event horizon, but never cross it any point. Instead the lightrays eventually cross the past event horizon and seemingly emanate from the white hole region in the past of any eternal black hole.
Of course, an astrophysical black hole will not have a past event horizon or white hole region, which reside in the part of spacetime replaced by whatever collapsing mass distribution (e.g. a dying star) that formed the black hole. So in astrophysically realistic situations the lightrays never cross any event horizon.
An important caveat to this discussion is that the geometric optics approximation only applies to lightrays whose wavelength is much smaller than the curvature length scale of the black hole. The typical wavelength of light in the Hawking radiation is of the same length scale as the event horizon of the black hole. Consequently, any conclusions drawn from the geometric optics approximation about the origin of Hawking photons need to be taken with copious amounts of salt. The more conservative conclusion is that you cannot really say anything about the location of the origin of the Hawking radiation other than that it comes from somewhere around the vicinity of the event horizon.
A: No. There are two possible situations in which we could consider your question: either in an eternal black hole, or in a gravitational collapse black hole. Both cases yield a negative answer, albeit for different reasons.
Eternal Black Hole
For an eternal Schwarzschild black hole, the physically reasonable state of the quantum field that will be thermal is the so-called Hartle—Hawking vacuum. This state is thermal at the Hawking temperature and has the property that a stationary observer will see particles coming from all around them: the particles don't come only from the black hole, but from infinity as well. This is in complete analogy with the Unruh effect and, in fact, is known as the Unruh effect in curved spacetime, not the Hawking effect.
Hence, in this case, the particles don't come only from near the horizon. The Unruh effect in curved spacetime is probably discussed in most texts in Quantum Field Theory in Curved Spacetime. Wald's book is my favorite reference on the topic.
Gravitational Collapse
In a gravitational collapse scenario, the field can be put in the so-called Unruh state, which corresponds to having no radiation incoming from infinity. This state is not available in the eternal black hole scenario because it is not well-behaved near the past event horizon (i.e., if one tried to compute physical observables close to the white hole in the eternal solution, they would obtain unphysical infinities). However, in a gravitational collapse scenario, there is no past event horizon, and hence the Unruh state ends up being well behaved everywhere.
Nevertheless, one has a different issue. To define the meaning of the world "particle" in Quantum Field Theory in Curved Spacetime, one needs to have a timelike Killing vector field to provide a notion of a time-translation symmetry. In the absence of such a symmetry, there is no way to define what one even means by the word "particle". While the gravitational collapse scenario allows for asymptotic symmetries, i.e., while it allows for one to define particles at infinity, it does not possess time-translation symmetry close to the black hole, since it is, well, collapsing. As a consequence, it doesn't really makes sense to speak of a photon close to the event horizon, and the question becomes ill-posed.
A less extreme view would be to define particles in a neighborhood of the observer, as Hawking's original paper does if I recall correctly, but this point of view has already been discussed in the previous answer to this question, so I'll prefer to leave the reminder that the notion of particle is not at all fundamental in the quantum theory of fields.
For this case, I particularly recommend Hawking's original paper (the beginning of sec. 4 discusses the issue of it being a global process) and also Sabine Hossenfelder's blog post on the matter.
