Does the positive energy particle emitted from Hawking radiation directly equate to the amount of mass contained within the black hole?

Question

With virtual particle-antiparticle pairs in Hawking radiation, one member of the pair falls into the black hole, typically with negative energy, while the other escapes, typically with positive energy. Does the positive energy of the escaping particle correspond directly to the decrease in mass caused by the negative energy particle falling into the black hole?

Answer 1

I should mention that the particle-antiparticle pair description is just pictorial, and shouldn't be taken seriously. If you look up my answers on Hawking radiation you'll see a number of posts on this. I won't dive into this theme in this answer because it would be missing the point.

Anyway, yes, the energy that is carried away by Hawking radiation precisely corresponds to the decrease in mass of the black hole. This is actually how we calculate that the black hole evaporates: we compute how much energy is being carried out to infinity due to Hawking radiation and use energy conservation to conclude that this energy is coming from the black hole: it is actually due to an influx of negative energy into the black hole (which you could interpret pictorially as the negative-energy particles going in the black hole), so everything is causal and nothing really escapes the black hole.

Answer 2

Even if there was such a thing as particles with negative energy, using them to explain the loss of mass by a black hole by Hawking radiation is flawed, because it is equally likely that a particle with positive energy falls into the black hole while the negative energy particle radiates away from the black hole. The net change in mass or energy of the black hole due to the creation of these hypothetical positive/negative particle pairs is zero. That is unless someone can come up with a explanation why it should be the hypothetical particle with negative energy that preferentially falls into the black hole, instead of the particle with positive energy.

The notion of negative energy particles falling into the black hole was introduced by Hawking himself in his popular book "A Brief History of Time" even though he gave a different explanation in his more formal paper black hole radiation. Even the respected Schutz. (see page 323 of "A First Course in General Relativity") talks about particles with negative energy, but nowadays this not considered a serious explanation. A better, but not too formal description of what is really going on is given here: (Article by Forbes).

In normal flat space, QM and the uncertainty principle predict that virtual pairs of particles are randomly created and they annihilate each other after a brief period. These virtual particles are not real and they differ from real particle-antiparticle pairs in that when when a real pair annihilate, they produce real photons with measurable energy equivalent to the total energy of the real pair. When virtual pairs annihilate, they do not produce any energy. Now we know from the Unruh effect, that when an observer is experiencing strong proper acceleration these virtual particles can appear to be very real while an inertial observer in the same vicinity will not see any radiation at all. By the equivalence principle, in the vicinity of the black hole the gravitational acceleration can be strong enough to create real pairs of particles (both with positive energy) and the energy of the black hole that creates the gravitational field outside and it is the energy of the blackhole that effectively creates these real pairs. The black hole effectively has to expend $2mc^2$ of energy, to create the real pair with mass 2m. This loss of energy by the black hole results in the event horizon shrinking ever so slightly.

The energy loss of the black hole is not dependent on any real particles falling in. The energy was lost when the real particle pair was created outside the black hole as mentioned above. Neither of the particles have to fall into the black hole. Quite often the real pair annihilate and create a pair of photons and both photons escape to infinity. Very rarely and under very extreme curvature and acceleration, both a real particle and real antiparticle, both with real mass can both escape. (BTW. recent experiments have demonstrated that antiparticles fall downwards just the same as normal particles.) Conversely, sometimes both created particles fall into the black hole and cancel out the lost energy of the black hole. So, two, one or none of the particles from the pair might fall into the black hole. In the former case we only break even and in the latter two cases there is a net loss of energy from the black hole.