Hawking radiation is not produced at the black hole horizon.

Stephen Hawking’s “Brief History of Time” was one of the first popular science books I read, and I hated it. I hated it because I didn’t understand it. My frustration with this book is a big part of the reason I’m a physicist today – at least I know who to blame.

I don’t hate the book any more – admittedly Hawking did a remarkable job of sparking public interest in the fundamental questions raised by black hole physics. But every once in a while I still want to punch the damned book. Not because I didn’t understand it, but because it convinced so many other people they did understand it.

In his book, Hawking painted a neat picture for black hole evaporation that is now widely used. According to this picture, black holes evaporate because pairs of virtual particles nearby the horizon are ripped apart by tidal forces. One of the particles gets caught behind the horizon and falls in, the other escapes. The result is a steady emission of particles from the black hole horizon. It’s simple, it’s intuitive, and it’s wrong.

Hawking’s is an illustrative picture, but nothing more than that. In reality – you will not be surprised to hear – the situation is more complicated.

The pairs of particles – to the extent that it makes sense to speak of particles at all – are not sharply localized. They are instead blurred out over a distance comparable to the black hole radius. The pairs do not start out as points, but as diffuse clouds smeared all around the black hole, and they only begin to separate when the escapee has retreated from the horizon a distance comparable to the black hole’s radius. This simple image that Hawking provided for the non-specialist is not backed up by the mathematics. It contains an element of the truth, but take it too seriously and it becomes highly misleading.

That this image isn’t accurate is not a new insight – it’s been known since the late 1970s that Hawking radiation is not produced in the immediate vicinity of the horizon. Already in Birrell and Davies’ textbook it is clearly spelled out that taking the particles from the far vicinity of the black hole and tracing them back to the horizon – thereby increasing (“blueshifting”) their frequency – does not deliver the accurate description in the horizon area. The two parts of the Hawking-pairs blur into each other in the horizon area, and to meaningfully speak of particles one should instead use a different, local, notion of particles. Better even, one should stick to calculating actually observable quantities like the stress-energy tensor.

That the particle pairs are not created in the immediate vicinity of the horizon was necessary to solve a conundrum that bothered physicists back then. The temperature of the black hole radiation is very small, but this is in the far distance to the black hole. For this radiation to have been able to escape, it must have started out with an enormous energy close by the black hole horizon. But if such an enormous energy was located there, then an infalling observer should notice and burn to ashes. This however violates the equivalence principle, according to which the infalling observer shouldn’t notice anything unusual upon crossing the horizon.

This problem is resolved by taking into account that tracing back the outgoing radiation to the horizon does not give a physically meaningful result. If one instead calculates the stress-energy in the vicinity of the horizon, one finds that it is small and remains small even upon horizon crossing. It is so small that an observer would only be able to tell the difference to flat space on distances comparable to the black hole radius (which is also the curvature scale). Everything fits nicely, and no disagreement with the equivalence principle comes about.

[I know this sounds very similar to the firewall problem that has been discussed more recently but it’s a different issue. The firewall problem comes about because if one requires the outgoing particles to carry information, then the correlation with the ingoing particles gets destroyed. This prevents a suitable cancellation in the near-horizon area. Again however one can criticize this conclusion by complaining that in the original “firewall paper” the stress-energy wasn’t calculated. I don’t think this is the origin of the problem, but other people do.]

The actual reason that black holes emit particles, the one that is backed up by mathematics, is that different observers have different notions of particles.

We are used to a particle either being there or not being there, but this is only the case so long as we move relative to each other at constant velocity. If an observer is accelerated, his definition of what a particle is changes. What looks like empty space for an observer at constant velocity suddenly seems to contain particles for an accelerated observer. This effect, named after Bill Unruh – who discovered it almost simultaneously with Hawking’s finding that black holes emit radiation – is exceedingly tiny for accelerations we experience in daily life, thus we never notice it.

The Unruh effect is very closely related to the Hawking effect by which black holes evaporate. Matter that collapses to a black hole creates a dynamical space-time that gives rise to an acceleration between observers in the past and in the future. The result is that the space-time around the collapsing matter, that did not contain particles before the black hole was formed, contains thermal radiation in the late stages of collapse. This Hawking-radiation that is emitted from the black hole is the same as the vacuum that initially surrounded the collapsing matter.

That, really, is the origin of particle emission from black holes: what is a “particle” depends on the observer. Not quite as simple, but dramatically more accurate.

The image provided by Hawking with the virtual particle pairs close by the horizon has been so stunningly successful that now even some physicists believe it is what really happens. The knowledge that blueshifting the radiation from infinity back to the horizon gives a grossly wrong stress-energy seems to have gotten buried in the literature. Unfortunately, misunderstanding the relation between the flux of Hawking-particles in the far distance and in the vicinity of the black hole leads one to erroneously conclude that the flux is much larger than it is. Getting this relation wrong is for example the reason why Mersini-Houghton came to falsely conclude that black holes don’t exist.

It seems about time someone reminds the community of this. And here comes Steve Giddings.

Steve Giddings is the nonlocal hero of George Musser’s new book “Spooky Action at a Distance.” For the past two decades or so he’s been on a mission to convince his colleagues that nonlocality is necessary to resolve the black hole information loss problem. I spent a year in Santa Barbara a few doors down the corridor from Steve, but I liked his papers better when we was still on the idea that black hole remnants keep the information. Be that as it may, Steve knows black holes inside and out, and he has a new note on the arxiv that discusses the question where Hawking radiation originates.

In his paper, Steve collects the existing arguments why we know the pairs of the Hawking radiation are not created in the vicinity of the horizon, and he adds some new arguments. He estimates the effective area from which Hawking-radiation is emitted and finds it to be a sphere with a radius considerably larger than the black hole. He also estimates the width of wave-packets of Hawking radiation and shows that it is much larger than the separation of the wave-packet’s center from the horizon. This nicely fits with some earlier work of his that demonstrated that the partner particles do not separate from each other until after they have left the vicinity of the black hole.

All this supports the conclusion that Hawking particles are not created in the near vicinity of the horizon, but instead come from a region surrounding the black hole with a few times the black hole’s radius.

Steve’s paper has an amusing acknowledgement in which he thanks Don Marolf for confirming that some of their colleagues indeed believe that Hawking radiation is created close by the horizon. I can understand this. When I first noticed this misunderstanding I also couldn’t quite believe it. I kept pointing towards Birrell-Davies but nobody was listening. In the end I almost thought I was the one who got it wrong. So, I for sure am very glad about Steve’s paper because now, rather than citing a 40 year old textbook, I can just cite his paper.

If Hawking’s book taught me one thing, it’s that sticky visual metaphors that can be a curse as much as they can be a blessing.