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I'm writing a piece on Hawking radiation, and find I have something of a problem. The "given" explanation which I find on Wikipedia and elsewhere is unsatisfactory:

"Physical insight into the process may be gained by imagining that particle–antiparticle radiation is emitted from just beyond the event horizon. This radiation does not come directly from the black hole itself, but rather is a result of virtual particles being "boosted" by the black hole's gravitation into becoming real particles[10]. As the particle–antiparticle pair was produced by the black hole's gravitational energy, the escape of one of the particles lowers the mass of the black hole[11]. An alternative view of the process is that vacuum fluctuations cause a particle–antiparticle pair to appear close to the event horizon of a black hole. One of the pair falls into the black hole while the other escapes. In order to preserve total energy, the particle that fell into the black hole must have had a negative energy..."

It relies on virtual particles and an negative-energy particles. However vacuum fluctuations are not the same thing as virtual particles, which only exist in the mathematics of the model, and we know of know negative-energy particles. So I'm looking for a better explanation. The Wikpedia article also says this:

"In another model, the process is a quantum tunnelling effect, whereby particle–antiparticle pairs will form from the vacuum, and one will tunnel outside the event horizon[10]."

However that suggests pair production is occurring inside the event horizon, which seems to disregard the infinite gravitational time dilation, and that one of them a) appears outside of the event horizon and b) escapes as Hawking radiation when pair production typically involves the creation of an electron and a positron. Again it's unsatisfactory. So:

Is there a better explanation of Hawking radiation?

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    $\begingroup$ The particle falling in doesn't require negative energy. All that matters is that some photons escape to infinity, which means that some of the energy that was "borrowed" from the gravitational field is lost (in the form of those photons). So the gravitational field weakens, which reduces the apparent mass/energy. But "apparent" is just what we see as distant observers. What happens inside the event horizon is...in the range of conjectural to nothing. That said, I don't think there's a majority opinion on how the radiation arises, or if it even exists... $\endgroup$
    – zibadawa timmy
    Commented Mar 22, 2017 at 10:03
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    $\begingroup$ You might find more on Physics SE given the fairly esoteric nature of this material. $\endgroup$
    – StephenG - Help Ukraine
    Commented Mar 22, 2017 at 10:54
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    $\begingroup$ Noted Stephen. @zibadawa timmy : but how does you "borrow" energy from a gravitational field? And if you do, how does energy then leak out of the event horizon for more of the same until you end up with no black hole at all? $\endgroup$
    – John Duffield
    Commented Mar 22, 2017 at 12:36
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    $\begingroup$ John, from your questions it sounds like you don't understand the concepts of potential energy or energy stored in fields (gravitational, electric, etc). I'd start by reading about those concepts. $\endgroup$
    – Carl Witthoft
    Commented Mar 22, 2017 at 12:57
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    $\begingroup$ 1. All these verbal explanations are just metaphors. The real deal is doing the Hawking calculations - that's the real explanation. 2. Here's another metaphor: The black hole is nothing but a tremendous spacetime curvature tied into itself - and the name we have for spacetime curvature is "gravity". The black hole is nothing but gravity, intense enough to persist itself. The p / anti-p pairs are brought into existence the same way that any extremely intense field can generate particles: when you have plenty of energy, particles can pop out of it. E.g. electromagnetic radiation could do it too. $\endgroup$
    – Florin Andrei
    Commented Mar 22, 2017 at 18:35

2 Answers 2

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Andy Gould proposed a classical derivation of Hawking radiation in a somewhat obscure paper from 1987. The essential argument is that a black hole must have a finite, non-zero entropy (otherwise you could violate the second law of thermodynamics with a black hole). Moreover, the entropy of the black hole must depend only on its area (otherwise you could change the area of a black hole via the Penrose process and lower its entropy and make a perpetual motion machine). If a black hole has an entropy and a mass, then it has a temperature. If it has a temperature, then it must radiate thermally (otherwise you could again violate the second law of thermodynamics).

Of course, if you look at the Hawking radiation temperature, there's a Planck's constant in there, so it has to know something about quantum mechanics, right? But it turns out that it's actually thermodynamics in general that knows about quantum mechanics, not general relativity --- Planck's constant is only needed to keep entropies finite (and therefore temperatures non-zero). This is true of black holes and blackbodies alike.

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    $\begingroup$ It was an interesting read, but I noted this on page 5: ”One may now consider doing an experiment first proposed by Geroch$^{[8]}$. One adiabatically lowers a perfectly reflecting box filled with electromagnetic radiation at a temperature T >> T$_{BH}$ to a Schwarzschild radius r, close to the event horizon. One then exchanges radiation with the hole…” Surely there’s no exchange because of the infinite gravitational time dilation? Geroch’s gedankenexperiment from the 1971 Princeton colloquium seems to be widely referenced but unpublished. An interesting lead, thanks again. $\endgroup$
    – John Duffield
    Commented Mar 23, 2017 at 13:12
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    $\begingroup$ You don't lower the box exactly down to the event horizon, only close to the event horizon. So there is time dilation, but it is not infinite and radiation can be exchanged. $\endgroup$
    – J. O'Brien Antognini
    Commented Mar 23, 2017 at 17:02
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    $\begingroup$ I'm missing something here. If you lower the gedanken box to some place near the event horizon then exchange radiation with the hole then when you pull the box up there's no radiation in it. Assuming the black hole swallowed the radiation (or at least some of it) the black hole mass increases. I'll see if I can find another explanation of Geroch's scenario. $\endgroup$
    – John Duffield
    Commented Mar 23, 2017 at 20:19
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    $\begingroup$ I found this, see page 2, but it's wrong. When you lower the box and do work, at the event horizon the box has half the energy it started with. And ouch, I found this too: arxiv.org/abs/physics/0501056. $\endgroup$
    – John Duffield
    Commented Mar 23, 2017 at 20:55
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    $\begingroup$ I wouldn't trust the Arxiv paper you linked --- it's about 12 years old but was never published in a peer reviewed journal and has no citations. It looks kooky to me. And in the first (more trustworthy) reference, I'm not sure where you're getting that the box has half the energy that it started with. $\endgroup$
    – J. O'Brien Antognini
    Commented Mar 23, 2017 at 23:06
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There is quite a nice explanation on this web page. A key passage is this:

in curved spacetime there aren't these "best" co-ordinate systems, the inertial ones. So even very reasonable different choices of co-ordinates can give disagreements about particles vs antiparticles, or what's the vacuum. These disagreements don't mean that "everything is relative", because there are nice formulas for how to translate between the descriptions in different co-ordinate systems. These are Bogoliubov transformations.

In particular, he goes on to say

on the one hand we can split solutions of Maxwell's equations into positive frequency in the most blitheringly obvious way that someone far from the black hole and far in the future would do it...

and on the other hand we can split solutions of Maxwell's equations into positive frequency in the most blitheringly obvious way that someone far in the past, before the collapse into a black hole has happened would do it.

Thus what the observer in the far past thought was genuinely empty space with no (non virtual) particles or antiparticles, an observer in the far future might see as space with perfectly good particles (and antiparticles) in it. Those particles are Hawking radiation.

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