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?
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.