Was Stephen Hawking's explanation of Hawking Radiation in "A Brief History of Time" not entirely accurate?

Question

I've been looking into black holes and Hawking radiation recently (just on the surface level) and was reading "A Brief History in Time" by Stephen Hawking to understand the basics of Hawking radiation. Then I came across a website online which stated that Hawking's explanation (in the book) was not entirely accurate. The website seems pretty reliable, but Hawking was the one who developed the theory and I'm pretty new to this topic so just thought I should ask.

There are 3 claims made about Hawking begin incorrect.

1. The radiation arises not from the event horizon exclusively, but from the entirety of the curved space around it. (up to around 10–20 Schwarzschild radii)

2. There are no virtual particles (or antiparticles) with negative energy falling into the black hole. The curved space around the black hole is constantly emitting radiation due to the curvature gradient around it

3. Black holes are not decaying because there’s an infalling virtual particle carrying negative energy... Instead, black holes are decaying, and losing mass over time, because the energy emitted by this Hawking radiation is slowly reducing the curvature of space in that region.

Is there any validity to these claims?

Answer 1

Hawking's explanation in A Brief History of Time is pictorial. It is meant to illustrate the effect, but doesn't really explain it. He actually introduced this illustration in the original article, but mentions

It should be emphasized that these pictures of the mechanism responsible for thermal emission and area decrease are heuristic only and should not be taken too literally.

In fact, in curved spacetime, we can't even define in a proper way what is meant by a particle. Particles are an observer-dependent concept that become very subtle when you start considering noninertial observer or curved spacetimes. Quantum fields, on the other hand, are more abstract, but objective.

Considering your points.

The radiation arises not from the event horizon exclusively, but from the entirety of the curved space around it. (up to around 10–20 Schwarzschild radii)

Indeed, the radiation can't exactly be traced back to the horizon. Particle creation by black holes is in fact a global effect. The particles appear because the very concept of particle is changing, not because they are being created at some point.

There are no virtual particles (or antiparticles) with negative energy falling into the black hole. The curved space around the black hole is constantly emitting radiation due to the curvature gradient around it

Sort of. While particles are "heuristic only and should not be taken too literally", there is negative energy falling into the black hole in a very well-defined manner. One can find a negative energy flux going into the black hole.

Black holes are not decaying because there’s an infalling virtual particle carrying negative energy... Instead, black holes are decaying, and losing mass over time, because the energy emitted by this Hawking radiation is slowly reducing the curvature of space in that region.

As with point 2, sort of. Indeed, the particle interpretation is merely heuristic, but there is a negative energy flux. Perhaps this is what the author meant by "the energy emitted is slowly reducing the curvature of space", but I think it is important to mention that you do have a negative energy flux.

I mention quite often in this site how particles in curved spacetime are merely pictorial, and perhaps you will be interested in checking some of my other posts. Another interesting website discussing how Hawking's interpretation is just an illustration is this blog post by Sabine Hossenfelder.

Answer 2

All of the positive claims that you listed:

1. The radiation arises not from the event horizon exclusively, but from the entirety of the curved space around it

2. The curved space around the black hole is constantly emitting radiation due to the curvature gradient around it

3. black holes are [...] losing mass over time because the energy emitted by this Hawking radiation is slowly reducing the curvature of space in that region

are true, but none of them imply that the virtual-particle picture of black hole evaporation is wrong. Perhaps they would, if virtual particles were like classical point particles, but it's a mistake to think of virtual particles that way in any context, not only in the context of black hole evaporation. Virtual particles are just another way of talking about fields; they are spread out in the same way as fields, and the virtual-particle picture isn't incompatible with a nonlocal origin for Hawking radiation.

There are parts of the article you linked that are clearly completely wrong:

• He says the virtual-particle picture implies that the emitted radiation should be an equal mix of particles and antiparticles, but "Hawking radiation is made almost exclusively of photons, not a mix of particles and antiparticles." This makes no sense. A bunch of photons is an equal mix of particles and antiparticles, because photons are their own antiparticle.

• He says that the escaping virtual particles would have to have a huge kinetic energy to get out of the gravity well, but Hawking particles have low kinetic energy. This makes no sense either. They would only have a high kinetic energy before they escape. The Hawking radiation is by definition measured at infinity, after they escape. Theory predicts that a thermometer close to the hole will measure a temperature higher than the Hawking temperature, which diverges to infinity at the event horizon, consistent with the idea that the particles do originate from near there with huge kinetic energy (although that seems implausible for other reasons).

Those are two of his three reasons why the virtual-particle picture can't be right. His third is that it implies that the radiation originates close to the horizon, which, as I said, it doesn't.

I don't know if the virtual-particle picture is correct. As Níckolas Alves said, Hawking mentioned it in his original paper with the caveat "these pictures [...] are heuristic only and should not be taken too literally", and I don't know whether anyone has converted it into a rigorous argument since then. But none of the arguments against it in the article that you linked make sense. If it's wrong, it isn't for those reasons.

Answer 3

That Big Think article draws a number of incorrect conclusions from Hawkings particle-antiparticle picture of Hawking radiation, and then goes on to debunk these strawman conclusions and ascribing their blunders to Hawking.

More specifically they write:

If that explanation were true, then that would mean:

1. Hawking radiation was composed of a 50/50 mix of particles and antiparticles, since which member falls and which one escapes will be random,
2. that all of the Hawking radiation, which causes black holes to decay, will be emitted from the event horizon itself, and
3. that every quantum of Hawking radiation that’s emitted by the black hole must possess a tremendous amount of energy: enough to escape from the incredible gravitational pull of black hole from just outside the event horizon.

There first conclusion is actually true (for a Schwarzschild black hole), but they wrongfully conclude that this is at odds with most of the Hawking radiation consisting of photons. Photons are their own anti-particle, being an equal mix of particle/anti-particles simply translates to their being an equal mix of both polarisation states.

The second conclusion is simply false. The particle/anti-particle picture actually suggests that the pair can be created anywhere outside of the horizon as long as one of the pair crosses the horizon before reuniting. There is no-reason to assume the particle creation happens exactly at the horizon.

The third conclusion only follows if you assume the particle creation happens close to the horizon (see previous point). There is some truth to it. In order to escape to infinity Hawking radiation needs to climb out of the potential well it was created, arriving at infinity with a much lower energy at infinity. This is the intuitive understanding behind the appearance of so-called "greybody factors" in the Hawking radiation spectrum, which also show up in more rigorous field theoretic derivations.

So, in all the article is a rather sad attempt by Ethan Siegel to gain clout by pissing over the reputation of a highly regarded scientist. In the process they achieve nothing but soiling their own. Shame.

Answer 4

1. Quantum field theory does not really answer this question, because particle states are defined for far away from the black hole. It basically says that there are particles far away from the black hole, but does not say where they came from.

2. Virtual particles don’t exist, so yes

3. This is the same claim as 2.

Hawking radiation, calculated in the original paper, works by the black hole disturbing the quantum field around it. This disturbance causes a flux of particles far away from it.

Answer 5

To paraphrase a well known aphorism “All explanations are inaccurate, but some are useful”. And the picture of Hawking radiation as tunneling of one particle from the pair through the horizon while possibly “not entirely accurate” is certainly useful.

Note, that when Hawking suggested this tunneling picture it was indeed a “heuristic only” (and it remained such when his book was published), but since then a formalism was developed that put this explanation on a much firmer ground and linked it with a well known technique of quantum theory, the WKB approximation. So nowadays, this explanation not only helps with the intuition, but also produces quantitative predictions including Planck's spectrum of radiation (with the temperature coinciding with one obtained by QFT methods).

The original work establishing the mathematical formulation of tunneling for Hawking radiation is:

• Parikh, M. K., & Wilczek, F. (2000). Hawking radiation as tunneling. Physical review letters, 85(24), 5042, doi:10.1103/PhysRevLett.85.5042, arXiv:hep-th/9907001.

An essay explaining the main ideas, while omitting technical details:

• Parikh, M. (2004). A secret tunnel through the horizon. International Journal of Modern Physics D, 13(10), 2351-2354, doi:10.1142/S0218271804006498, arXiv:hep-th/0405160.

And a review covering the first dozen years of further development of the tunneling methods:

• Vanzo, L., Acquaviva, G., & Di Criscienzo, R. (2011). Tunnelling methods and Hawking's radiation: achievements and prospects. Classical and Quantum Gravity, 28(18), 183001, doi:10.1088/0264-9381/28/18/183001, arXiv:1106.4153.

Specific criticisms:

1. The radiation arises not from the event horizon exclusively, but from the entirety of the curved space around it. (up to around 10–20 Schwarzschild radii)

It is possible to construct a static spacetime that fully coincides with the Schwarzschild metric outside event horizon for all $r>r_s+\epsilon$ for some small length $\epsilon$, yet does not have an event horizon. Such spacetime would not be emitting Hawking radiation.

On the other hand, we can imagine a black hole completely surrounded by a static screen impenetrable to radiation placed a small distance above the horizon. Inside this cavity a “thermal atmosphere” would emerge, consisting mostly of photons being emitted by horizon and falling back into it. If the walls of this cavity absorb some of the radiation and are thermally conductive then the black hole would be losing energy through vibrational degrees of freedom of the wall material. Theoretically it would be possible co construct a “heat sink” around the black hole that would remove energy from black hole much more efficiently than Hawking radiation in an empty space.

So the role of “entirety of the curved space” around the black hole is similar to the role of a lampshade around a lamp: it does affect radiation but as a passive element. The “active element” in the Hawking radiation is the horizon.

(Situation is more complicated for rotating black holes, where there are negative energy states strictly outside horizon. In this case tunneling overlaps with superradiance).

2. There are no virtual particles (or antiparticles) with negative energy falling into the black hole. …

The flux of negative energy inside the black hole horizon is a real thing and could be obtained by a variety of methods. How this flux can be partitioned into particles is observer dependent, but once we chose consistent procedure there would be a particle stream falling toward singularity. Also note, that for a hypothetical observer inside the black hole horizon those particles would be perfectly ordinary photons, their “negative energy” would be apparent only in reference to how time flows outside the horizon.

… The curved space around the black hole is constantly emitting radiation due to the curvature gradient around it

The logic is backward. Curvature gradient is responsible for the black hole having an event horizon with nonzero surface gravity. Event horizon produces radiation.

3. Black holes are not decaying because there’s an infalling virtual particle carrying negative energy... Instead, black holes are decaying, and losing mass over time, because the energy emitted by this Hawking radiation is slowly reducing the curvature of space in that region.

There is no “instead” here, energy conservation means that the flux of negative energy into the black hole equals up to the sign the flux of energy carried away by radiation and so we cannot have one without the other.

Overall remarks

It is good to have multiple ways to derive the same thing. This offers a way to check for consistency of other methods, some methods may be easier to use in some situations than the others and different methods might provide more insights for unsolved problems.

Here are some features of tunneling framework that might make it preferable to other (such as Bogoliubov transform) methods:

• it incorporates gravitational backreaction from the start, so it might work better in situations where black hole quickly radiates away considerable fraction of its mass;

• it is local in character, so just from the existence of this description we can conclude that the Hawking temperature of e.g. $\mathbb{RP}^3$ geon (a solution differing from Schwarzschild black hole by nontrivial topology of the interior) would be the same as Schwarzschild black hole;

• calculating a tunneling rate might be possible even in spacetimes with pathologies posing difficulties for constructing QFT (such as spacetimes with closed timelike curves)…

So the tunneling framework for Hawking radiation has its place.