Unlike boxes full of atoms, we can’t make black holes with the same size but different masses. The size of a black hole is characterized by the “Schwarzschild radius,” which is precisely proportional to its mass. If you know the mass, you know the size; contrariwise, if you have a box of fixed size, there is a maximum mass black hole you can possibly fit into it. But if the entropy of the black hole is proportional to the area of its event horizon, that means there is a maximum amount of entropy you can possibly fit into a region of some fixed size, which is achieved by a black hole of that size.
That’s a remarkable fact. It represents a dramatic difference in the behavior of entropy once gravity becomes important. In a hypothetical world in which there was no such thing as gravity, we could squeeze as much entropy as we wanted into any given region; but gravity stops us from doing that.
It’s not surprising to find a chapter about black holes in a book that talks about relativity and cosmology and all that. But the point here is obviously a slightly different one than usual: we care about the entropy of the black hole, not the gruesome story of what happens if you fall into the singularity.
Black holes are important to our story for a couple of reasons. One is that gravity is certainly important to our story, because we care about the entropy of the universe and gravity plays a crucial role in how the universe evolves. But that raises a problem that people love to bring up: because we don’t understand quantum gravity (and in particular we don’t have a complete understanding of the space of microstates), we’re not really able to calculate the entropy of a system when gravity is important. The one shining counterexample to this is when the system is a black hole; Bekenstein and Hawking gave us a formula that allows us to calculate the entropy with confidence. It’s a slightly weird situation — we know how to calculate the entropy of a system when gravity is completely irrelevant, and we also know how to calculate the entropy when gravity is completely dominant and you have a black hole. It’s only the messy in-between situations that give us trouble.
The other reason black holes are important, of course, is that the answer that Bekenstein and Hawking derive is somewhat surprising, and ultimately game-changing. The entropy is not proportional to the volume inside the black hole (whatever that might have meant, anyway) — it’s proportional to the area of the event horizon. That’s the origin of the holographic principle, which is perhaps the most intriguing result yet to come out of the thought-experiment-driven world of quantum gravity.
The holographic principle is undoubtedly going to have important consequences for our ultimate understanding of spacetime and entropy, but how it will all play out is somewhat unclear right now. I felt it was important to cover this stuff in the book, although it doesn’t really lead to any neat resolutions of the problems we are tackling. Still, hopefully it was somewhat comprehensible.