# If Black holes are maximal entropy how can they evaporate?

According to Hawking/Bekenstein a black hole represents the highest amount of entropy for a given volume, (actually the entropy is related to the surface area of the black hole but the fact that they represent maximal entropy is what counts to what I have to say)

It is supposed that there is literally no way to squeeze more information (entropy) into a given volume than that in a black hole occupying that volume as in doing so you would create a black hole, and any attempt to pour more information into the black hole (and thus increase its entropy) would just increase the size of the black hole.

But black holes evaporate...So how can a maximal entropy object evaporate without reducing entropy!

So how does this tie in with the second law of thermodynamics and the heat death of the universe?

Black holes radiate, and while now a large black hole takes in much more in the form of CMB than it radiates as Hawking radiation, eventually once the temperature of the CMB becomes lower than the temperature of the lonely black holes that constitute the only mass left in a trillion trillion years or whatever, these large cold black holes will actually start losing mass by hawking radiation faster than they are gaining it from CMB, but then given they were the maximal entropy state then doesn't that suggest that entropy of the universe is decreasing as these black holes radiate themselves away?

The only thing I can think of that would make this not entail a breaking of the second law of thermodynamics is that CMB temperature drop is result of universe expanding, and perhaps the expansion of the universe itself is creating more entropy than that which is being lost by the evaporation of black holes? Is this the resolution to my question?

Although if thats the case its easy to imagine a closed non-expanding universe that contains only a black hole which is radiating away. How can this be if the black hole represents the maximal entropy object of given size?

I guess this comes down to what happens to the information (entropy) of a black hole as it radiates away, some have posited that it leaves behind a nugget containing all that information but this flioes in the face of the idea that the black hole itself was maximal entropy as if this entropy is left inside a smaller object hopw could this be the case?

But you must keep in mind that the volume occupied by the radiation+BH system is larger than the volume of the black hole by itself. When the black hole initially forms the horizon has a radius $r$ which describes a volume of maximal entropy; after one second, the same amount of energy is represented as a sphere of radiation of radius $r+c$ with a black hole at the center. The entropy of this volume is certainly not maximal.