I was thinking about the following thought experiment, but wasn't sure about its outcome.
Suppose there is a black-hole and I enter it with a partitioned box containing two different gases on either side. After passing, the event horizon the radius of a black-hole increases as I added mass to the system (?). Now after I pass the event horizon, I open the partitioned box thereby increasing the entropy of the system. Will the area of the black hole also increase? If so, will the radius also increase? (Isn't the radius proportional to the mass of the black-hole?)
Best Answer
First I will describe a few facts about entropy and how it applies to black holes, and then I will apply these facts to your question.
Entropy is observer-dependent. Entropy is (roughly speaking) the logarithm of the number of microstates compatible with the system's macroscopic state. What the macroscopic state is depends on what you can measure. This is true even for mundane things like mixtures of gases - in this case the entropy simply depends on what instruments you have available for making measurements. (See Jaynes ,1996.)
In the case of an object falling in to a black hole, the entropy depends on whether you're falling in with it or watching from the outside. If you're watching from the outside, time dilation means that any radiation emitted by the object will rapidly be redshifted to unobservably low frequencies. This means that any information about the object besides its mass, charge and angular momentum is, for all practical purposes, lost. The object is still technically there, since nothing can pass the event horizon as seen from the perspective of an outside observer, but since you can't see it any more, the black hole's macrostate is said to consist only of its mass, charge and angular momentum. The number of microstates associated with a black hole's macrostate rapidly increases with its mass, which means that the entropy increases by quite a lot when something falls into it, when seen from outside.
However, for someone falling in with the object it's a different story. They can still see the object, so they don't lose any information about it, and its entropy doesn't do anything special as it passes the event horizon. The entropy increases for the outside observer but stays the same for the falling observer. There is no paradox, because entropy is observer-dependent.
We now have enough information to answer your question. From your point of view, falling into the black hole with your partitioned box of gases, nothing much happens when you pass the event horizon. Then you mix the gases, and watch their entropy increase just as it normally would. Nothing particularly special seems to happen as a result. Some time later, you and the box of gas hit the singularity and suddenly cease to exist.
From my point of view as an observer staying safely outside the black hole, it looks a bit different. I watch you falling towards the event horizon with your box of gas, but as you approach it you seem to move in slow motion. Your light gets red-shifted until it gets harder and harder to see you, and at some point I just give up and say you've become part of the black hole, which now has a greater mass and hence a higher entropy and a larger radius. (The radius and the entropy are always proportional to one another.)
In principle, if I could see you, I would see you frozen in time just before you passed the horizon; the gases would still be unmixed. Nothing you do after passing the horizon can ever have an effect on what I observe. But if you'd mixed the gases before passing the horizon it wouldn't have made any difference. The black hole's macrostate depends only on its mass, charge and angular momentum, and those are all the same whether the gases are mixed or separate. In general, an increase in entropy is a loss of information, but by the time you've fallen into the black hole I've already lost all the information about the gases' microstate, so mixing them together can't cause me to lose any more. So from the outside viewer's perspective, the radius of the black hole increases when you fall into it, and then that's it. The entropy is already at its maximum at this point, and there's nothing you can do that can increase it further.