The pop-sci explanation is that virtual particle-antiparticle pairs are created and one part of the pair falls into the black hole and the other part escapes as radiation. This is how Hawking explained it in his book A Brief History of Time, but unfortunately, this explanation is too simplified to the point that it’s incorrect.
The correct explanation, as best as I can put it in ELI5 terms, is that the event horizon creates “boundary conditions” for the quantum fields that exist outside of the black hole. When I press my finger on a guitar string, it makes sure that the string isn’t moving there. That’s a kind of boundary condition, and it affects how the string vibrates. The black hole does the same kind of thing for the quantum fields that exist outside of it, and just like how pressing my finger changes the note my guitar plays, the black hole constrains the quantum fields so that radiation appears on the outside.
The mathematically-inclined reader can find Hawking’s original argument [here](https://www.brainmaster.com/software/pubs/physics/Hawking%20Particle%20Creation.pdf). To be honest I don’t understand it myself, but I take it on faith in my physicist friends that the explanation I just gave is more accurate than the usual pop-sci one. But it’s still at best an approximation of the actual math.
A couple things are happening.
First, there’s a lot of radiation coming off the matter surrounding the black hole as it falls inward. That radiation isn’t coming *from* the black hole, but it helps us to see where black holes are.
Second, there’s radiation coming off of the outer surface, or ‘event horizon’, of the black hole itself. The mechanism which produces this radiation is a lot weirder. It arises from the microscopic quantum fluctuations which are constantly happening all over the place in empty space, which create a phenomenon called “virtual particles.” These particles are always created in equal-and-opposite pairs, and usually cancel themselves out before they have a chance to go anywhere, or affect anything, or be observed. But when the fluctuations happen *right* at the edge of the region where radiation can’t escape, sometimes they don’t cancel out. Instead, one half falls into the black hole, and the other half escapes. This is called “Hawking radiation.”
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