So, a particle - anti particle pair appear out of nothingness, one just inside the black hole's event horizon, getting sucked in, and the other just outside, getting away. Wouldn't that increase the mass of the black hole by the mass of the particle that got sucked in? Wouldn't that also increase the mass of the universe by the mass of the two particles, therefore violating the conservation of mass and energy?

I'm probably just displaying my ignorance. Please straighten me out.

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"For the world is hollow and I have touched the sky"

This question has been dealt with elsewhere in the Forum, but my attempt to locate it just now failed, so I'll simply answer your question here and let it go at that.

The rigorous accurate answer to your question is complicated, involving quantum field theory and general relativity. A somewhat oversimplifued version of what actually happens is what you described: spontaneous appearance of a pair of particles just outside the event horizon of the black hole. The explanation that follows is based on that picture.

Everywhere in space, pairs of conjugate particles are continuously being created from energy "borrowed" from space itself under the requirement that the borrowed energy must be returned to the vacuum within a very short time. If one particle is an electron, the other must be a positron, and so on. The larger the combined masses of the particles, the briefer the time within which the energy must be returned to the vacuum. When such a particle pair appears spontaneously just outside the event horizon and one of the particles falls through the horizon while the other escapes from the vicinity of he event horizon, the one escaping carries mass away from the black hole. Both particles must find conjugate particles with which to annihilate. The annihilation energy of the escaping particle has a wavelength determined by the combined energies of the escaping particle and the one with which it annihilates. Over time, the spectra of escaping particles and those with which they annihilate adds up to that of a black body with a very low temperature. The same happens to the body that fell into the black hole, but that energy does not escape from the black hole. It's the mass of the escaping particles that is lost from the black hole.

OK, let me rephrase what you said to see if I understand it correctly. Both particles must annihilate something. The one that fall into the black hole annihilates a piece of the black hole equal to its mass, thus decreasing the mass of the black hole, and the other particle also annihilates an amount of mass equal to its own somewhere outside, so the overall mass of the universe remains the same.

Wait a minute, wouldn't that mean the overall mass of the universe is reduced by the mass of the 2 particles? Hmm, it seems the universe's mass + energy is slightly increased, though. Then again, if the particles had annihilated normally, that would have increased the total energy of the universe, too. I think it was mentioned in another episode that that energy might be an explanation of dark energy.

__________________
"For the world is hollow and I have touched the sky"

The mass of the particles annihilated outside the black hole is converted into blackbody radiation experienced by the outside world. That energy is equivalent to the combined mass of the two particles, and it represents mass lost from the black hole. The particles annihilated instead the black hole also produce a similar amount of radiation, but that radiation is trapped inside the black hole so does not result in loss of mass by the black hole because of the equivalence of mass and energy.