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I was reading up on Hawking Radiation and here's what I read in exact words:

Physical vacuum constantly produces particle-antiparticle pairs and normally annihilate within a time given by Heisenberg's Uncertainty Principle. When this occurs near a black hole, one of them may fall into it whereas the other would not. The other particle would get annihilated, but outside the black hole, and emit radiation. This comes from the rest mass energy of the black hole.

My questions are as follows:

  1. How can a particle-antiparticle pair arise out of nothingness? Does this not violate energy conservation?
  2. If the source of the pair is vacuum, how is it that the black hole's mass decreases? On the contrary, it should increase because one member of the pair gets sucked into the black hole.
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  • $\begingroup$ Where did you read this? It is always best to give the source of any quoted material since its context can sometimes inform the answer. $\endgroup$ Commented Feb 10, 2023 at 15:20

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Your questions are specifically related to the formulation of Hawking radiations you cited (without source), but unfortunately, this explanation is completely off (and the two other answers too). I will explain what is wrong with this pop-science picture of Hawking radiation, and then attempt to answer your questions.

Firstly, why would people say something wrong when an accurate explanation exists? Because of Hawking himself. This explanation of his so-called "Hawking radiations" comes from his article. He used this virtual particle metaphor to explain in simple words the phenomenon he stumbled upon. But shortly after saying so, he also said:

"It should be emphasized that these pictures of the mechanism responsible for the thermal emission and area decrease are heuristic only and should not be taken too literally." : Hawking himself (https://link.springer.com/article/10.1007/bf02345020) page 202

In short, this is wrong. And in fact, it is more than wrong: it doesn't make sense at all because virtual particles are internal lines of Feynman diagrams. They do not have any sort of reality as they are purely mathematical entities. Virtual particles do not exist.

What is happening, then?

What happens is a perfect example of what we call "relativity of the concept of particle number", an effect of quantum field theory in an accelerated reference frames or in curved space-time. And this is simply a consequence of the "relativity of the measure of energy". In short, you have two reference frames, $A$ and $B$. $A$ is near the black hole and $B$ at infinity. The question one should ask to uncover Hawking radiation is:

What does the vacuum in $A$ look like for the observer at $B$?

The answer is: "a thermal bath" as exposed in the Scholarpedia article http://www.scholarpedia.org/article/Hawking_radiation. Note that now, the problem is stated exactly as an Unruh effect. And though mathematics is exactly the same as the Unruh effect, physics is different. Indeed, according to the Scholarpedia article on the Unruh effect http://www.scholarpedia.org/article/Unruh_effect:

The mathematical relationship between Rindler and Minkowski coordinates in flat space is practically identical to that between Schwarzschild and Kruskal coordinates in a nonrotating black hole, and hence many of the elements of the Unruh theory have counterparts in the black-hole theory, with the important difference that Schwarzschild coordinates become inertial in the limit of large distance and hence the analogs of Unruh-Rindler particles are "real" (at infinity) rather than effects of an observer's acceleration.

This is because of this very last part that black holes are shrinking. Indeed since the particles created by the black hole are observable, their energy must come from somewhere. Note that it is meaningless to trace back the particles until the horizon of the black hole, as Hawking radiation may not come from the actual horizon (see Birrell & Davies). So, since black holes are just particular configurations of space-time, and since "at infinity" it is observed that energy is coming from it (or its surrounding), then we can safely say that black holes lose energy.

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