There’s a major event in the life of every young book that marks its progression from mere draft on someone’s computer to a public figure in its own right. No, I’m not thinking about when the book gets published, or even when the final manuscript is sent to the publisher. I’m thinking of when a book gets its own page on amazon.com. (The right analogy is probably to “getting your drivers license” or something along those lines. Feel free to concoct your own details.)
So it’s with a certain parental joy that I can announce From Eternity to Here now has its own amazon page. My baby is all grown up! And, as a gesture of independence, has already chosen a different subtitle: “The Quest for the Ultimate Theory of Time.” The previous version, “The Origin of the Universe and the Arrow of Time,” was judged a bit too dry, and was apparently making the marketing people at Dutton scrunch up their faces in disapproval. I am told that “quests” are very hot right now.
All of which means, of course: you can buy it! For quite a handsome discount, I may add.
It also means: I really should finish writing it. Pretty darn close; the last chapters are finished, and I’m just touching up a couple of the previous ones that were abandoned in my rush to tell the end of the story. The manuscript is coming in at noticeably more words than I had anticipated — I suspect the “320 pages” listed on amazon is an underestimate.
And, yes, there is another book with almost the same title and an eerily similar cover, which just appeared. But very different content inside! Frank Viola’s subtitle is “Rediscovering the Ageless Purpose of God,” which should be a clue to the sharp-eyed shopper that the two works are not the same.
Writing a book is a big undertaking, in case no one before me had never noticed that before. I’m very grateful to my scientific collaborators for putting up with my extended disappearances along the way. It’s also very nerve-wracking to imagine sending it out there into the world all by itself. With blog posts there is immediate feedback in terms of comments and trackbacks; you can get a feel for what the reactions are, and revise and respond accordingly. But the book really has a life of its own. People will read and review it for goodness knows how long, and I won’t always be there to protect it.
Frankly, I’m not sure this “book” technology will ever catch on.
So I guess I also have trouble with your statement “If you have a state that is unstable to evolving into something else, by definition it is not maximum entropy.”
A state could have maximum entropy given a **fixed macrostate**. Indeed, that’s how we find the correct ensemble for a system: We write down known macrostate for the system as a set of constraint equations, and then we maximize the entropy function.
The system might still be unstable with respect to changes in the macrostate. As long as we hold the macrostate fixed, however, the system has maximal entropy with respect to that macrostate and is stable.
But if the macrostate can then change—in our case, the cells of initial size L_I get spatially larger due to inflation—then the maximally-allowed entropy changes as well, since now there’s a different set of accessible states consistent with the new macrostate.
Matt, I think there are two ways out of the paradox. One, again, is that there is no such thing as a state of maximum entropy.
The other is that your small cell simply cannot evolve into our universe. That’s certainly possible. You might think that we know it can, because of the usual story of inflation. But inflation requires that this tiny cell be dominated by potential energy with a very high energy density. That’s not at all what we expect in a high-entropy state. In a theory with gravity, high-entropy states are ones that are as empty as possible (at least as long as the vacuum energy is non-negative). Note that our present universe is evolving toward a very empty de Sitter phase — that’s high entropy.
So: either a tiny cell in a high-entropy state is never going to support inflation, or there is no such thing as a state of maximum entropy. Either possibility is certainly worth contemplating at this point.
Mike– yeah, I’m working on that right now. Hopefully it will make sense when all is done.
Follower– it’s clear that the total entropy of the early universe was low, so that’s the major challenge. What you want is a way of producing a low-entropy region, for example as part of a larger system. We are suggesting that the particular way the universe might do that is through the nucleation of baby universes, in which case there could be good reasons for the geometry to be smooth. (Lowest Euclidean action for the instanton, in technical terms.)
But all of that is a very tiny, speculative part of the book, which is mostly about much more down-to-earth things.
Sean–
Okay. You’ve changed your argument here, now that you’ve agreed to look just at spatially small states for the cell (consistent with the assumed size L_I in our definition of the cell’s macrostate) and saying that among those small states, ones with a high energy density needed to support inflation are rare and therefore unlikely.
But for a patch of such a tiny size, there are only 10^12 possible states in total, and with so vastly many such tiny patches in our bath, it is highly likely that one of them will hit upon an inflationary state. That’s the argument. And the odds that **some** such tiny patch will fluctuate into a proto-inflationary state is still vastly higher than finding a Boltzmann brain somewhere in the bath.
It’s like the following analogy. Suppose you’re a young TV reporter whose dream is to do a story on a lottery winner. But you decide that maybe you should just give up hope, because if you take a particular person, his odds of winning are one-in-a-million.
But then you remember that there are millions people in your city, and you realize that one of them is almost certainly going to win! The rest can then be ignored.
That’s the story I’m telling here in my cosmological scenario. Restricting ourselves to states of size L_I, states in a proto-inflationary state are rare. But there’s zillions of these cells in our bath!
So what’s the problem? Where am I making any unnatural assumptions?
Indeed, you wrote “But inflation requires that this tiny cell be dominated by potential energy with a very high energy density. That’s not at all what we expect in a high-entropy state.”
Sure. We may not expect that any particular cell will begin inflating. But my point is that there are zillions of these tiny cells in our bath, and we only need *one* of these cells to hit the correct state needed to begin inflation. So what’s the problem here?
And you write “In a theory with gravity, high-entropy states are ones that are as empty as possible (at least as long as the vacuum energy is non-negative).”
But, again, that’s only true if our macrostate doesn’t specify the spatial size of our system. But in our case, the spatial size is part of the definition of the cell’s macrostate!
Given that these cells are, by our very mental partitioning of the bath, of fixed tiny size L_I, I’m not so sure that an “empty” such cell is the maximally-entropy state with respect to the cell’s macrostate!
Matt, I’m not changing the argument, I’m trying to explain what is necessarily entailed by the belief that there exists an equilibrium state with maximum entropy (a belief I don’t share). If that’s true, inflation can’t start in a small cell, because if it did, the entropy would go up, and that contradicts the hypothesis. The only allowed processes are those that decrease the total entropy.
Right. I follow what you’re saying. I think maybe we’re speaking at cross-purposes.
I can lay this out in steps, and you can tell me which step is wrong, and why.
Step 1 is that we’ve got a large maximally entropic bath, say, much bigger than the present-day observable universe and with its own entropy very large, say, 10^300.
Step 2 is that this bath can always be mentally partitioned into cells of tiny size L_I.
Step 3 is that these cells each individually have entropy bounded by 10^12, which is just the number of microstates consistent with the macrostate (including the specification of the size) of the cell.
Step 4 is that there are lots of these cells.
Step 5 is that, because there are so many of these cells, one of these cells will inevitably fluctuate into an inflationary state, whether or not such an inflationary state is rare for a single cell, given its macrostate.
Step 6 is that the cell now grows via inflation.
Step 7 is that entropy in the cell is pushed to the edge (or outside of) the cell’s de Sitter horizon due to inflation, so that the interior of the cell looks less and less entropic (but there’s no true decrease in global entropy).
Step 8 is that the curvature plummets as well, and we get all the rest of the goodies from inflation.
Step 9 is that entropy comes back into the cell after inflation, due to Hawking radiation and due to the enlarging of the de Sitter horizon as inflation ends. That’s how we seed cosmic structure, and so forth.
If I’m not mistaken, the trouble seems to be Step 6. From what I understand, every other step is uncontroversial.
So the question is whether a cell can, in fact, inflate without violating the laws of thermodynamics and our assumption that the bath in which the cell lives is already maximally entropic.
Now, when discussing the creation of baby universes, we usually have in mind the following story. There’s an infinite empty space, and a tiny patch of space fluctuates into an inflationary state. From the outside, we see a black hole that quickly evaporates away to nothing. From the inside, we’re in an exponentially inflating de Sitter phase. The disappearance of the black hole seen from the outside really corresponds to a “pinching off” of the baby universe, like a “spore.”
In my cosmology, the ambient space is a very large bath, rather than empty space. If there’s thermal equilibrium between the ambient bath and the newly formed black hole, then the black hole just remains at fixed size and temperature and doesn’t wink out at all. Or since it’s so small, it eventually winks out due to another small fluctuation and returns the entropy it stole.
Either way, the whole time the ambient bath outside remains at essentially fixed entropy. From the outside, in the bath, we see an extremely tiny black hole form and remain, or then disappear and return its entropy back into the bath.
With so many cells around in the bath, and such a tiny entropy of 10^12 involved for each cell, this doesn’t seem like an unlikely series of events to me!
My point is that the entropy of the bath itself hardly changes at all the whole time. As seen from the bath, there’s no real increase in entropy at any time. And as seen from the inflating patch, there’s no global decrease in entropy anywhere. Entropy is pushed away during inflation, and then comes back in again afterward.
So is there really a problem here?
Matt, I don’t think you are following what I am saying! You are getting caught up in potentially weird things that curved spacetime can do, and neglecting the underlying point.
Look at it this way: right now, in our observable universe, the entropy isn’t nearly as high as it could be. It would be very easy, without changing the overall size of the universe, to dramatically increase its entropy, just by putting more of the existing matter into black holes. Therefore, it is perfectly clear that our universe right now is *not* part of a larger universe that is in a maximum entropy state, because it’s easy to see how the entropy could be increased. Either there is no such thing as a maximum-entropy state, and we are not part of it, or there is, and we represent a *lower* entropy than that state has, presumably because of some downward fluctuation in entropy.
I’m purposefully avoiding all mention of inflation etc. in that description because it’s completely beside the point. The geometry of spacetime has nothing to do with it. It’s just a statement about entropy.
I think your specific confusion is in Step 7. In inflation, the interior of the inflating bubble grows, but not the external geometry of the “cell” from which it arose. It doesn’t push into the rest of space, it just grows from inside. So the entropy would go up if that were to happen, which contradicts the hypothesis that you started from maximum entropy.
The problem with Boltzmann’s first idea for where our low-entropy observable universe came from was that it required a large spatial region of a bath to fluctuate way down to a much, much smaller entropy than its natural, maximum entropy. Indeed, that led to the Boltzmann brain argument, that it was far more likely that a human-brain-sized patch would fluctuate into an isolated human brain.
The miracle of inflation is that we don’t need the bath to undergo a big fluctuation in entropy. We don’t need a large spatial patch of the bath to fluctuate way, way down in entropy.
We just need a tiny, tiny patch, one of zillions of such patches in the bath, to fluctuate a tiny bit, to get the tiny patch into an inflationary state. From the outside, in the bath, this blip is hardly noticeable; yes, it’s a somewhat rare event for any one patch, but likely to occur *somewhere* given how many tiny patches there are, and how small (10^12) is the total number of states available to each patch (or “cell”). And then as seen from the outside, in the bath, the patch either remains (looking like a black hole) for a while or quickly disappears again. The bath hardly notices any of it.
Please let me know where’s my mistake. Thanks!
Your mistake is believing that having inflation begin in one small patch is compatible with the hypothesis that you started in a state of maximum entropy.
If you give up on that hypothesis, you just described the scenario I’ve been pushing.
I see your point. You’re saying that if our observable universe today is part of a larger bath, then because our universe could have a higher entropy than it does today, so could the bath, and hence the bath does not have maximal entropy after all.
But I’m not so sure this works. If our whole observable universe is inside a black hole as seen from outside in the bath, then the bath has no clue what’s going on inside our universe. As far as the bath is concerned, the entropy of our little cell is maximal all the time, whatever is happening inside our cell.
The interior of our cell could get blown up exponentially like a balloon so that the entropy is fixed but its density goes way down. But the bath knows none of this, always assigns the same entropy to the tiny black hole (as long as the black hole exists), and stays at maximal entropy all the time.
If the black hole decides to evaporate away, then the entropy originally hidden in the black hole comes back out, and our baby universe pinches off and goes on its merry way, free to go up in entropy to its heart’s desire still without affecting the bath from which it was born. But no significant amount of entropy was ever created or destroyed as far as the bath was concerned.
I’m always very wary of any arguments that invoke entropy as their lynchpin. Entropy is a very slippery concept, as you well know! There are many different kinds of entropy, and it’s very easy to slide between the different kinds of entropy if we’re not careful. And it’s very easy to make a mistake when discussing entropy and information in the context of horizons, since information may be able to exist in multiple (but casually disconnected!) places without violating the no-cloning theorem.
Indeed, that’s why a tiny inflating patch that starts with only 10^12 units of entropy can unitarily evolve into a vast multiverse with many universes that each have much more entropy than that. Due to the phenomenon of entanglement entropy (which becomes very important when we have lots of horizons around), the entropy of a subsystem in quantum mechanics is by no means whatsoever limited by the entropy of the full system! A pure state, having zero entropy, can easily have subsystems with large entropy, all while not violating unitarity!
And because we’re trying to treat gravity classically here, I’m very, very worried that we’re missing out on the hugely important quantum effects like entanglement entropy that can radically alter our logic.
(These side comments on quantum-mechanical effects in understanding entropy are a separate point. I hope you can also still address my questions regarding our discussion of our two cosmological scenarios!)
Sean–
So your basic point seems to be the following: Suppose that the larger bath cannot grow in entropy, and suppose that our own universe is a subsystem of that bath; then it’s a paradox that our own universe can have a much higher entropy than it does, because that would imply that the bath should be able to have a higher entropy as well, a contradiction. Indeed, as the entropy of our universe gets larger and larger, the entropy of the bath to which it belongs should get larger and larger as well. This seems to be the crux of your argument that there cannot be such a bath and that inflation cannot solve the cosmic entropy problem on its own.
I guess what I’m saying is that I’m not convinced that it’s impossible under unitary time evolution for a subsystem of the larger bath to grow in entropy while leaving the entropy of the bath unchanged, especially if that subsystem, while perhaps open to the larger subsystem at certain times, is otherwise causally hidden from the larger bath—as our inflating baby universe would be.
And besides, in quantum-mechanical theories it’s easy to find simple examples in which a subsystem grows in entropy—under perfectly unitary time evolution—while the larger system to which it belongs stays at constant entropy. As a trivial instance, consider a pure state consisting of two particles that interact and grow mutually entangled—each particle’s state becomes increasing mixed and entropic, but time evolution is always globally unitary. Things get even more interesting when there are horizons that shield subsystems from each other after they’ve become entangled—and horizons abound in our cosmological scenarios. These issues are invisible in any scenario that treats gravity classically, because these kinds of quantum effects are then invisible.
So what am I missing here?
Thanks again!
Matt — sure, anything is possible. Basically you would have to imagine some dramatic non-locality (at the level of the Hamiltonian, not just kinematics of the wave function) that would ensure that any rise in entropy within our observable patch is compensated by a decrease in entropy elsewhere, and vice-versa. I can’t rule that out, but it’s no theory I’ve ever heard proposed before.
Sean–
My point was that the entropy of a subsystem can increase without changing the entropy of the larger system to which it belongs. That happens all the time in quantum mechanics. There’s no need for the entropy increase of the subsystem to be “compensated by a decrease in entropy elsewhere.” There are lots of examples of this.
Back to the specific case of cosmology, the bath sees a tiny cell whose entropy is bounded by 10^12 turn into a tiny 10^12 entropy black hole of roughly the same size, and then the black hole either stays around for a while, still at 10^12 entropy, in thermal equilibrium with its surroundings in the bath or evaporates again, with the 10^12 entropy coming back out. That’s all the bath ever sees, so where is there any marked increase in the entropy of the bath?
At worst, there’s a shift of entropy by 10^12 during this whole process (if that much), but with zillions of cells in the bath and with the bath’s own entropy of order, say, 10^300, that’s not an unlikely fluctuation at all.
So I’m still confused about where there’s a problem here. Where does the bath experience some large change in entropy anywhere? Who cares what happens in the causally disconnected baby universe? It’s fully consistent with dynamically local quantum mechanics for a subsystem to grow in entropy while leaving the full system unchanged in entropy—no compensation needed. And especially so when the subsystem remains causally disconnected from the larger system, as in our case!
So I’m still not certain I see the problem here. Please let me know where I’ve gone wrong.
Thanks!
Furthermore, there’s this point:
There’s a tiny 10^12 entropy black hole for a while in the bath, and then the black hole evaporates and is gone. The baby universe is not located anywhere inside the bath anymore. As far as the bath is concerned, there is no baby universe anymore. It’s gone—finito.
So why does an increase in the entropy of the baby universe now have anything to do with the entropy of the bath?
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Sounds like another cool book to add to my shelf! Thanks for the heads up!
very cool book, i’ll be waiting!