Evaporating Black holes and entropy

1. I'm told that for Black Holes, when they radiate (Hawking radiation) particles and anti-particles) split at the event horizon, one going to infinity, the other into the BH. They then lose mass. How is that possible? Wouldn't their masses increase, since they are absorbing particle or anti-particles?

2. Also, their entropy varies as their masses, and since entropy increases, wouldn't their masses also be increasing, along with their area? How does that reconcile with case 1?

1) In the tunneling picture, there are two scenarios:

i)a virtual particle/antiparticle pair is created just outside of the horizon. The negative energy particle then tunnels into the horizon and the positive energy particle is radiated away.

ii) a virtual particle/antiparticle pair is created just inside of the horizon. The positive energy particle then tunnels out of the horizon and the negative energy particle remains inside.

In (i) although a particle is "falling in" to the horizon, it's a negative energy particle and hence results in a mass loss.

There are equivalent ways to derive the results without using these virtual particles though. These other ways involve working out the vacuum state of the asymptotic observer, who sees a flux of particles coming out of the hole. Either way the mass decreases.

2) The entropy of a closed system must not decrease. The BH on its own doesn't constitute a closed system. You need also to consider the states of the radiation. See the discussion of the "Generalized Second Law" here.

• Hi twistor, good to hear from you. What troubles me is this negative energy. As there would be as many particles and anti-particles falling into the BH. Similarly, an equal of them radiating out. Somehow, this is just ignored in all readings I've made on this subject. Jan 25, 2013 at 20:32
• Hi zaybu, I think the virtual particle pair picture is a bit of a heuristic device, and I'm not happy I understand how the creation process fits into the normal QFT pictures. See here. Another way to think of it is as having populated negative energy states inside the horizon which then tunnel out. The tunneling is modelled just as you would model barrier tunneling in elementary QM, using the WKB approximation. Jan 25, 2013 at 22:26
• If one invokes that only negative states are populated inside the horizon, one would need to invoke a principle that would select only those negative states. No such principle is known AFAIC. It seems to me the QFT arguments for Hawking radiation are tenuous at best. Hefler raises some serious questions on this. See arxiv.org/pdf/gr-qc/0304042v1.pdf Jan 26, 2013 at 14:57
• Whether one chooses the metric sig as (-+++) or (+---) is highly arbitrary. So that explanation is unsatisfactory. Besides, the metric cannot justify why only negative energy will fall into the BH since this process is happening at random and so either particles or anti-particles can fall into it. Indeed Hefler is raising other questions. It would seem that Hawking radiation is highly speculative at this point. Jan 27, 2013 at 13:19
• @ twistor59Yes, but that doesn't explain "this process is happening at random and so either particles or anti-particles can fall into it." The result should be: over a long time, no net gain in mass for the BH. Jan 27, 2013 at 16:57