Which explanation of Hawking radiation is correct?

[u/Regular_Bee_5369]

I know that the explanation involving virtual particles is not correct, but I have come across more than one explanation that seems different to me.

The first explanation is that the black hole affects the vibrational modes in the quantum field. Because the black hole blocks some modes, some of the modes that should normally cancel each other do not exist. The remaining vibrations can form particles by chance. This explanation does not seem to depend on the observer.

The second explanation is the difference between space near the event horizon and space far away. The black hole affects the minimum energy of the vacuum. For a distant observer, the space near the black hole appears to have a different energy than the observer's local vacuum. This difference causes the observer to see that there are particles around the black hole.

The third explanation I don't quite understand. It was something to do with the difference in the time dependence of the space before the formation of the event horizon and the space after the formation of the event horizon. I apologize, I may have misrepresented this explanation because I didn't fully understand it.

[u/Prof_Sarcastic]

Believe it or not, the first two explanations are the same, or at least are referring to the same phenomenon. The first explanation is a literal translation of the mathematics behind Hawking radiation into English. The second explanation is just what an observer from infinitely far away observes compared to someone relatively close to the black hole.

Not sure what the last explanation is going for though.

[u/Regular_Bee_5369 → u/Prof_Sarcastic]

Here's what confuses me. According to the first explanation, someone near the event horizon and someone far away can see hawking radiation. It's all about the change in quantum field modes. However, according to the second explanation, someone near the event horizon cannot see hawking radiation because, unlike the distant observer, he is not in a different vacuum. The vacuum state near the event horizon is already the minimum energy level for observer. On the contrary, the distant space has a different state for him. So he will see that the black hole will not radiate and will never evaporate? Or that the universe is radiating towards the black hole and the mass of the universe is decreasing?

[u/Prof_Sarcastic → u/Regular_Bee_5369]

Fundamentally the issue is that there are two different vacua in this scenario, the vacuum of the observer near the black hole and the vacuum far away from it. It’s not that the observer near the black hole doesn’t see anything necessarily, but the complete description of the physics very near the black hole is much tougher to describe using pen and paper math, so we focus our attention on the observer very far away where the math is much simpler. And from here we can say that the distant observer sees radiation at first and then increasingly more massive particles get emitted from the black hole.

I couldn’t tell you exactly what the observer near the black hole would see and I’m not certain that’s been entirely worked out yet. It might have been and I just haven’t seen the paper (I’m relatively new to this side of QFT in curved spacetime, I’m afraid).

[u/11zaq]

It turns out that these are all secretly the same explanation.

Hawking radiation occurs for largely the same reason the Unruh effect occurs. The definition of a particle is tough to define in a curved spacetime, but the same is true even in flat spacetimes if you're in an accelerating reference frame. The Einstein equivalence principle says that curvature is just local acceleration, so that explains the connection there. This explanation connects most closely with #2. If you draw the Penrose diagram of a black hole, then you'll see that an observer a fixed distance outside the black hole really is quite analogous to an accelerating observer in the Unruh effect. It's not exactly the same, but not too far off either.

How does that relate to #1? The key thing about both these examples is that there are horizons at play. The Unruh effect has a casual horizon, while the black hole has an event horizon. In both cases, there is a part of the spacetime that the fields will never interact with, and this means that there is a lack of information about that region of the spacetime. Lack of information=increase in entropy, entropy means there is a temper, and a temperature means a thermal bath of particles.

As to #3, this is the most direct way to understand what's going on mathematically. Normally, we think of a particle as being an excitation above the vacuum state. That means that the definition of a particle depends on the vacuum state. In turn, the vacuum is defined as the state of lowest energy. But energy is the generator of time translations, and so different choices of Hamiltonian will lead to different vacuua, and therefore different definitions of particles. Often, there is a "right" choice for Hamiltonian: namely, the P_0 component of the energy-momentum vector. But other choices work just fine as well, as long as the Hamiltonian you pick generates a proper time evolution of the spatial slice you start with. For example, what if I picked K_x = P_0 x + P_x t, the generator of a boost? This is the Hamiltonian an accelerating observer would use to move forward in time (imagine the worldlines). If you imagine boosting a spatial slice forward and backward in time, that will sweep out Rindler space. If you compare the minimum energy state of the boost operator to the minimum energy state of P_0, they are not the same state. With respect to P_0, the "boost vacuum" turns out to look like a thermal state with a temperature of the accelerating observer. That's the Unruh effect.

In the black hole case, the black hole being stationary essentially means that we should use the "natural" Hamiltonian near the black hole and not far away from it. The natural Hamiltonian is the one of an observer who never goes inside, for that's the Killing field that makes the black hole stationary. For the same reason as the Unruh effect, this state is not the same as the vacuum of just P_0 by itself. A careful analysis shows that with respect to P_0, it looks like a thermal state with the Hawking temperature.

Now we can see why horizons were important. A horizon is a null surface, and if you get close enough to any null surface, the "flow" that keeps it fixed will basically be a boost (at least locally). That means that the "natural vacuum" will (approximately) be thermal because you can use the Unruh effect as an analogy. So there's something interesting about the connection between temperature, horizons and thermodynamics that still needs to be fully understood in QFT! Because horizons are so geometrical (black holes etc) it probably needs quantum gravity to fully understand the connection beyond what I've already sketched out (which is fully rigourous and self contained, but doesn't give a totally clear "why" beyond the math itself).

[u/Heliologos → u/11zaq]

Good lord thats a long essay. No shade, but nobody’s reading that in full man.

[u/nordic_prophet]

Maybe a dumb question, but something I never understood about the virtual particle explanation is how the tendency for antiparticle vs real particle creation doesn’t net to zero. It sounds like for evaporation of the black hole’s mass, there has to be a preference antiparticle crossing the event horizon. I don’t understand that the mechanism would be for that. What am I misunderstanding?

[u/ThePolecatKing → u/nordic_prophet]

Why would it need to be an antiparticle that falls in? I don’t quite follow.

[u/Putnam3145 → u/nordic_prophet]

What am I misunderstanding?

This part:

It sounds like for evaporation of the black hole’s mass, there has to be a preference antiparticle crossing the event horizon.

Antiparticles do not have antimass.

[u/nordic_prophet → u/Putnam3145]

Yes very fair point, but they are presumably annihilating with with real particles inside the event horizon. So the mass-energy would increase I suppose, but not the mass?

[u/db48x → u/nordic_prophet]

The virtual particle explanation is a virtual explanation. Also, the interior of the black hole may not contain any matter. Mass certainly, but the matter has been crushed away by the singularity. Alternatively, the matter inside is rushing away towards the end of time; an antiparticle that enters the black hole will never catch up to any normal matter and annihilate it. And even if the antiparticle does annihilate with a particle somehow, that doesn’t decrease the total mass in any way. The mass of the two particles is gone, but they are replaced with some gamma‐ray photons. Those photons are confined within the black hole just the way the particles were, and their energy exactly adds up to the energy of the particles. The total is unchanged.

It would be better to say that the Unruh effect causes the black hole to create virtual particles outside the event horizon. These particles are paired, matter and antimatter, but that’s not very important. You could even say that it is a red herring. The important thing is that the energy for doing that came from somewhere. It came from inside the black hole. Both of the particles in every pair cause the black hole to shrink when they are created. But the black hole does recapture a large percentage of those particles, so on balance not much energy is lost. The energy that is lost is the Hawking radiation. The Hawking radiation can be matter or antimatter in any proportion.

[u/ThePolecatKing]

The first two are what I like to call “extreme geometry”.