Author: Sean Carroll

Welcome to this week’s installment of the From Eternity to Here book club. Today we have a look at Chapter Thirteen, “The Life of the Universe.”

Excerpt:

If our comoving patch defines an approximately closed system, the next step is to think about its space of states. General relativity tells us that space itself, the stage on which particles and matter move and interact, evolves over time. Because of this, the definition of the space of states becomes more subtle than it would have been in if spacetime were absolute. Most physicists would agree that information is conserved as the universe evolves, but the way that works is quite unclear in a cosmological context. The essential problem is that more and more things can fit into the universe as it expands, so—naively, anyway—it looks as if the space of states is getting bigger. That would be in flagrant contradiction to the usual rules of reversible, information-conserving physics, where the space of states is fixed once and for all.

Of course we’ve already looked a bit at the life of the universe, way back in Chapter Three. The difference is that we’re now focusing on how entropy evolves, given our hard-acquired understanding of what entropy is and how it works for black holes. This is where we review Roger Penrose’s well-known-yet-still-widely-ignored argument that the low entropy of the early universe is something that needs to be explained.

In a sense, this is pretty straightforward stuff, following directly from what we’ve already done in the book. But it’s also somewhat controversial among professional cosmologists. The reason why can be found in the slightly technical digression that begins on page 292, “Conservation of information in an expanding universe.”

The point is that physicists often think of “the space of states in a region of spacetime” as being equal to “the space of states we can describe by quantum field theory.” They know that’s not right, because gravity doesn’t fit into that description, but these are the states they know how to deal with. This collection of states isn’t fixed; it grows with time as the universe expands. You will therefore sometimes hear cosmologists talk about the high entropy of the early universe, under the misguided assumption that there were fewer states that could “fit” into the universe at that time. (Equivalently, that gravity can be ignored.) This approach has, in my opinion anyway, done great damage to how cosmologists think about fine-tuning problems. One of the major motivations for writing the book was to explain these issues, not only to the general reader but also to my scientist friends.

At the end of the chapter I deviate from Penrose’s argument a bit. He believes that a high-entropy state of the universe would be one that was highly inhomogeneous, full of black holes and white holes and what have you. I think that’s right if you are thinking about a very dense configuration of matter. But matter doesn’t have to be dense — the expansion of the universe can dilute it away. So I argue that the truly highest-entropy configuration is one where space is essentially empty, with nothing but vacuum energy. This is also very far from being widely accepted, and certainly relies on a bit of hand-waving. But again, I think the failure to appreciate this point has distorted how cosmologists think about the problems presented by the early universe. So hopefully they read this far in the book!

Welcome to this week’s installment of the From Eternity to Here book club. Part Four opens with Chapter Twelve, “Black Holes: The Ends of Time.”

Excerpt:

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.