Words

Welcome to this week’s installment of the From Eternity to Here book club. Part Three of the book concludes with Chapter Eleven, “Quantum Time.”

Excerpt:

This distinction between “incomplete knowledge” and “intrinsic quantum indeterminacy” is worth dwelling on. If the wave function tells us there is a 75 percent chance of observing the cat under the table and a 25 percent chance of observing her on the sofa, that does not mean there is a 75 percent chance that the cat is under the table and a 25 percent chance that she is on the sofa. There is no such thing as “where the cat is.” Her quantum state is described by a superposition of the two distinct possibilities we would have in classical mechanics. It’s not even that “they are both true at once”; it’s that there is no “true” place where the cat is. The wave function is the best description we have of the reality of the cat.

It’s clear why this is hard to accept at first blush. To put it bluntly, the world doesn’t look anything like that. We see cats and planets and even electrons in particular positions when we look at them, not in superpositions of different possibilities described by wave functions. But that’s the true magic of quantum mechanics: What we see is not what there is. The wave function really exists, but we don’t see it when we look; we see things as if they were in particular ordinary classical configurations.

Title notwithstanding, the point of the chapter is not that there’s some “quantum” version of time that we have to understand. Some people labor under the impression that the transition from classical mechanics to quantum mechanics ends up “quantizing” everything, and turning continuous parameters into discrete ones, perhaps even including time. It doesn’t work that way; the conventional formalism of quantum mechanics (such as the Schrödinger equation) implies that time should be a continuous parameter. Things could conceivably change when we eventually understand quantum gravity, but they just as conceivably might not. In fact, I’d argue that the smart money is on time remaining continuous once all is said and done. (As a small piece of evidence, the context in which we understand quantum gravity the best is probably the AdS/CFT correspondence, where the Schrödinger equation is completely conventional and time is perfectly continuous.)

However, we still need to talk about quantum mechanics for the purposes of this book, for one very good reason: we’ve been making a big deal about how the fundamental laws of physics are reversible, but wave function collapse (under the textbook Copenhagen interpretation) is an apparent counterexample. Whether it’s a real counterexample, or simply an artifact of an inadequate interpretation of quantum mechanics, is a matter of much debate. I personally come down on the side that believes that there’s no fundamental irreversibility, only apparent irreversibility, in quantum mechanics. That’s basically the many-worlds interpretation, so I felt the book needed a chapter on what that was all about.

Along the way, I get to give my own perspective on what quantum mechanics really means. Unlike certain parts of the book, I’m pretty happy with how this one came out — feel free to correct me if you don’t completely agree. Quantum mechanics can certainly be tricky to understand, for the basic reason that what we see isn’t the same as what there is. I’m firmly convinced that most expositions of the subject make it seem even more difficult than it should be, by speaking as if “what we see” really does reflect “what there is,” even if we should know better.

So I present a number of colorful examples of two-state systems involving cats and dogs. Experts will recognize very standard treatments of the two-slit experiment and the EPR experiment, but in very different words. Things that seem very forbidding when phrased in terms of interference fringes and electron spins hopefully become a bit more accessible when we’re asking whether the cat is on the sofa or under the table. I did have to treat complicated macroscopic objects with many moving parts as if they could be described as very simple systems, but I judged that to be a worthwhile compromise in the interests of pedagogy. And no animals were harmed in the writing of this chapter! Let me know how you think the strategy worked.

Welcome to this week’s installment of the From Eternity to Here book club. This is a fun but crucial part of the book: Chapter Ten, “Recurrent Nightmares.”

Excerpt:

Fortunately, we (and Boltzmann) only need a judicious medium-strength version of the anthropic principle. Namely, imagine that the real universe is much bigger (in space, or in time, or both) than the part we directly observe. And imagine further that different parts of this bigger universe exist in very different conditions. Perhaps the density of matter is different, or even something as dramatic as different local laws of physics. We can label each distinct region a “universe,” and the whole collection is the “multiverse.” The different universes within the multiverse may or may not be physically connected; for our present purposes it doesn’t matter. Finally, imagine that some of these different regions are hospitable to the existence of life, and some are not. (That part is inevitably a bit fuzzy, given how little we know about “life” in a wider context.) Then—and this part is pretty much unimpeachable—we will always find ourselves existing in one of the parts of the universe where life is allowed to exist, and not in the other parts. That sounds completely empty, but it’s not. It represents a selection effect that distorts our view of the universe as a whole—we don’t see the entire thing, we only see one of the parts, and that part might not be representative. Boltzmann appeals to exactly this logic.

After the amusing diversions of the last chapter, here we resume again the main thread of argument. In Chapter Eight we talked a bit about the “reversibility objection” of Lohschmidt to Boltzmann’s attempts to derive the Second Law from kinetic theory in the 1870’s; now we pick up the historical thread in the 1890’s, when a similar controversy broke out over Zermelo’s “recurrence objection.” The underlying ideas are similar, but people have become a bit more sophisticated over the ensuing 20 years, and the arguments have become a bit more pointed. More importantly, they are still haunting us today.

One of the fun things about this chapter is the extent to which it is driven by direct quotations from great thinkers — Boltzmann, of course, but also Poincare, Nietzsche, Lucretius, Eddington, Feynman. That’s because the arguments they were making seem perfectly relevant to our present concerns, which isn’t always the case. Boltzmann tried very hard to defend his derivation of the Second Law, but by now it had sunk in that some additional ingredient was going to be needed — here we’re calling it the Past Hypothesis, but certainly you need something. He was driven to float the idea that the universe we see around us (which, to him, would have been our galaxy) was not representative of the wider whole, but was simply a local fluctuation away from equilibrium. It’s very educational to learn that ideas like “the multiverse” and “the anthropic principle” aren’t recent inventions of a new generation of postmodern physicists, but in fact have been part of respectable scientific discourse for over a century.

It’s in this chapter that we get to bring up the haunting idea of Boltzmann Brains — observers that fluctuate randomly out of thermal equilibrium, rather than arising naturally in the course of a gradual increase of entropy over billions of years. I tried my best to explain how such monstrosities would be the correct prediction of a model of an eternal universe with thermal fluctuations, but certainly are not observers like ourselves, which lets us conclude that that’s not the kind of world we live in. Hopefully the arguments made sense. One question people often ask is “how do we know we’re not Boltzmann Brains?” The realistic answer is that we can never prove that we’re not; but there is no reliable chain of argument that could ever convince us that we are, so the only sensible way to act is as if we are not. That’s the kind of radical foundational uncertainty that has been with us since Descartes, but most of us manage to get through the day without being overwhelmed by existential anxiety.

Welcome to this week’s installment of the From Eternity to Here book club. Now for something of a palate-cleanser, in the form of Chapter Nine, “Information and Life.”

Excerpt:

Schrödinger’s idea captures something important about what distinguishes life from non-life. In the back of his mind, he was certainly thinking of Clausius’s version of the Second Law: objects in thermal contact evolve toward a common temperature (thermal equilibrium). If we put an ice cube in a glass of warm water, the ice cube melts fairly quickly. Even if the two objects are made of very different substances—say, if we put a plastic “ice cube” in a glass of water—they will still come to the same temperature. More generally, nonliving physical objects tend to wind down and come to rest. A rock may roll down a hill during an avalanche, but before too long it will reach the bottom, dissipate energy through the creation of noise and heat, and come to a complete halt before very long.

Schrödinger’s point is simply that, for living organisms, this process of coming to rest can take much longer, or even be put off indefinitely. Imagine that, instead of an ice cube, we put a goldfish into our glass of water. Unlike the ice cube (whether water or plastic), the goldfish will not simply equilibrate with the water—at least, not within a few minutes or even hours. It will stay alive, doing something, swimming, exchanging material with its environment. If it’s put into a lake or a fish tank where food is available, it will keep going for much longer.

This chapter starts with something very important: the relationship between entropy and memory. Namely, the reason why we can “remember” the past and not the future is that the past features a low-entropy boundary condition, while the future does not. I don’t go into great detail about this, and we certainly don’t talk very specifically about how real memories are formed in the brain, or even in a computer. But when we get to the next chapter, about recurrences and Boltzmann brains, it will be crucial to understand how the assumption of a low-entropy boundary condition enables us to reconstruct the past. It’s hard for people to wrap their brains around the fact that, without such an assumption, our “memories” or records of the past will generally be unreliable — knowledge of the current macrostate wouldn’t allow us to reconstruct the past any better than it allows us to predict the future. (Which is only logical, since it’s only this hypothesis that breaks time-reversal symmetry.)

The rest of the chapter, meanwhile, is more about having fun and mentioning some ideas that are not directly related to our story, but certainly play a part in understanding the arrow of time. Information theory, life, complexity. I’m not an expert in any of these fields, but it was a lot of fun reading about them to pick out some things that fit into the broader narrative. The Maxwell’s Demon story, in particular, is one that every physicist should know (up through it’s relatively modern resolution), but relatively few do. And I think Jason Torchinsky did a great job with the illustrations of the Demon.

A lot of big ideas here, of course, and much of this stuff is still very much in the working-out stage, not the settled-understanding stage. We’re still arguing about basic things like the definition of “complexity” and “life.” It’s relatively easy to state the Second Law and explain how the arrow of time is related to the growth of entropy, but there’s a tremendous amount of work still to be done before we completely understand the way in which the universe actually evolves from low entropy to high.

Welcome to this week’s installment of the From Eternity to Here book club. Finally we dig into the guts of the matter, as we embark on Chapter Eight, “Entropy and Disorder.”

Excerpt:

Why is mixing easy and unmixing hard? When we mix two liquids, we see them swirl together and gradually blend into a uniform texture. By itself, that process doesn’t offer much clue into what is really going on. So instead let’s visualize what happens when we mix together two different kinds of colored sand. The important thing about sand is that it’s clearly made of discrete units, the individual grains. When we mix together, for example, blue sand and red sand, the mixture as a whole begins to look purple. But it’s not that the individual grains turn purple; they maintain their identities, while the blue grains and the red grains become jumbled together. It’s only when we look from afar (“macroscopically”) that it makes sense to think of the mixture as being purple; when we peer closely at the sand (“microscopically”) we see individual blue and red grains.

Okay cats and kittens, now we’re really cooking. We haven’t exactly been reluctant throughout the book to talk about entropy and the arrow of time, but now we get to be precise. Not only do we explain Boltzmann’s definition of entropy, but we give an example with numbers, and even use an equation. Scary, I know. (In fact I’d love to hear opinions about how worthwhile it was to get just a bit quantitative in this chapter. Does the book gain more by being more precise, or lose by intimidating people away just when it was getting good?)

In case you’re interested, here is a great simulation of the box-of-gas example discussed in the book. See entropy increase before your very eyes!

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From Eternity to Book Club: Chapter EightRead More »

Welcome to this week’s installment of the From Eternity to Here book club. We next take a look at Chapter Seven, “Running Time Backward.” Now we’re getting serious! (Where “serious” means “fun.”)

Excerpt:

The important concept isn’t “time reversal” at all, but the similar-sounding notion of reversibility–our ability to reconstruct the past from the present, as Laplace’s Demon is purportedly able to do, even if it’s more complicated than simply reversing time. And the key concept that ensures reversibility is conservation of information–if the information needed to specify the state of the world is preserved as time passes, we will always be able to run the clock backward and recover any previous state. That’s where the real puzzle concerning the arrow of time will arise.

With this chapter we begin Part Three of the book, which is the most important (and my favorite) of the four parts. Over the course of the next five chapters we’ll be exploring the statistical definition of entropy and its various implications, as well as the puzzles it raises.

But before getting to entropy, and the arrow of time that depends on it, we first have to understand life without an arrow of time. The only reason the Second Law is puzzling is because the rules of fundamental physics don’t exhibit an arrow of time on their own — they’re perfectly reversible. In this chapter we discuss what “reversible” really means, and contrast it with “time reversal invariance,” which is related by not quite the same. If a theory is both reversible and time-translation invariant (same rules at all times), it’s always possible to define time reversal so that your theory is invariant under it. (E.g. in most quantum field theories, “CPT” does the trick.)

Reversibility is a very deep idea; it implies that the state of the universe at any one moment in time is sufficient (along with the laws of physics) to precisely determine the state at any other time, past or future. But not many popular physics books spend much time explaining this idea. So we reach all the way back to very simplified models of discrete systems on a lattice (“checkerboard world”). What we’re after is an understanding of what it really means to have “laws of physics” in the first place — rules that the universe obeys as it evolves through time. That lets us explore different kinds of rules, in particular ones that are and are not reversible.

Along the way we talk about time-reversal invariance in the weak interactions of particle physics, and emphasize how this is not related to the thermodynamic arrow of time that is our concern in this book. Which gives me a good excuse to quote a touching passage from C.S. Wu. This chapter has everything, I tell you.

Welcome to this week’s installment of the From Eternity to Here book club. Chapter Six is entitled “Looping Through Time.” It’s about both the logical paradoxes presented by time travel, and some of the obstacles to actually building a time machine (closed timeline curves) in general relativity.

Excerpt:

Everyone knows what a time machine looks like: something like a steampunk sled with a red velvet chair, flashing lights, and a giant spinning wheel on the back. For those of a younger generation, a souped-up stainless-steel sports car is an acceptable substitute; our British readers might think of a 1950s style London police box. Details of operation vary from model to model, but when one actually travels in time, the machine ostentatiously dematerializes, presumably to be re-formed many millennia in the past or future.

That’s not how it would really work. And not because time travel is impossible and the whole thing is just silly; whether or not time travel is possible is more of an open question than you might suspect. I’ve emphasized that time is kind of like space. It follows that, if you did stumble across a working time machine in the laboratory of some mad inventor, it would simply look like a “space machine”—an ordinary vehicle of some sort, designed to move you from one place to another. If you want to visualize a time machine, think of launching a rocket ship, not disappearing in a puff of smoke.

There might not be too much new to say about this chapter, as part of it appeared as an excerpt in Discover and we’ve already talked about that. But maybe you weren’t reading that post, in which case, it’s new to you!

There were three main goals in this chapter. The first was to explain what time travel would and would not be, in the context of general relativity — in particular, it would be just another form of travel through spacetime, not involving any disappearing and rematerializing at some other point in the past. The second was to go through some of the possible ways to make closed timelike curves (with wormholes or cosmic strings) and see how difficult it really was.

But the third and most interesting goal was to connect time machines to the arrow of time and entropy. At this point in the book we’ve only introduced these concepts somewhat casually — the careful exploration of entropy is in Part Three, which begins next week — so one could argue that a more logical presentation would have delayed this discussion for later. But sometimes there are considerations beyond logic; in particular, once we built up momentum with the entropy discussion, a digression on time travel would have seemed like wandering too far afield. That was my feeling at the time, anyway.

This is a really interesting aspect of time travel, which I think is dramatically under-emphasized in discussions about it: the real reason why traveling backwards in time makes us nervous is that it becomes impossible to define a consistent arrow of time. The arrow is very ingrained in how we think about the world, including the sense that the past is set in stone while we can still make choices that affect the future. In the presence of a time machine part of our personal “future” is already in the “past,” which seems to compromise our free will.

So be it! Our free will was always an approximation, if we are good materialists who believe in the laws of physics. But it’s a highly useful approximation. It’s always worth emphasizing, when you start talking about the paradoxes of time travel: the simplest and most plausible way out is to imagine that the universe doesn’t (and won’t ever) actually have any time machines.

Welcome to this week’s installment of the From Eternity to Here book club. This week we’re tackling two chapters at once: Chapter Four, “Time is Personal,” and Chapter Five, “Time is Flexible.” That’s just because these chapters are relatively short; next time we’ll return to one chapter per week.

Excerpt:

Starting from a single event in Newtonian spacetime, we were able to define a surface of constant time that spread uniquely throughout the universe, splitting the set of all events into the past and the future (plus “simultaneous” events precisely on the surface). In relativity we can’t do that. Instead, the light cone associated with an event divides spacetime into the past of that event (events inside the past light cone), the future of that event (inside the future light cone), the light cone itself, and a bunch of points outside the light cone that are neither in the past nor the future.

It’s that last bit that really gets people. In our reflexively Newtonian way of thinking about the world, we insist that some far away event either happened in the past, the future, or at the same time as some event on our own world line. In relativity, for spacelike separated events (outside one another’s light cones), the answer is “none of the above.” We could choose to draw some surfaces that sliced through spacetime, and label them “surfaces of constant time,” if we really wanted to. That would be taking advantage of the definition of time as a coordinate on spacetime, as discussed in Chapter One. But the result reflects our personal choice, not a real feature of the universe. In relativity, the concept of “simultaneous faraway events” does not make sense.

These two chapters take on a task that is part of the responsibility of any good book on modern cosmology or gravity: explaining Einstein’s theory of relativity. Both special relativity and general relativity, hence two chapters. In retrospect they are pretty short, so an argument could be made that I should have just combined them into a single chapter.

The special challenge of these chapters is precisely that many readers — but not all — will already have read numerous other popular-level expositions of relativity. But you have to do it. Fortunately, my favorite way of talking about relativity is a little bit different from the standard one, and lines up well with the overarching goal of understanding the meaning of “time.” In particular, I try to make the point that the secret to relativity is to think locally — to compare things happening right next to each other in spacetime, not events that are widely separated. You’re allowed to compare separated events, of course, but the answers are necessarily dependent on arbitrary choices of coordinates, and that leads to endless confusion. So you won’t read a lot about “length contraction” or “time dilation,” but you will read a lot about the actual amount of time measured along a trajectory.

Unfortunately, a search for vivid examples of the maxim “freely-falling paths through spacetime experience the longest amount of proper time” led me directly to the most embarrassing mistake in the book. (At least, “most embarrassing mistake so far uncovered.”) Sordid details below the fold!

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From Eternity to Book Club: Chapters Four and FiveRead More »

From Eternity to Here addresses the problem of the arrow of time — why is the past different from the future? But Chapter Six is all about time travel, and in particular the interesting version in which you travel backwards in time. Whether it’s possible, what rules it would have to obey, and so on. And now — even though I’m sure there aren’t more than two or three of you out there who haven’t purchased the book already — you can get a sneak peek of part of that chapter. It’s going to be the cover story in the March issue of Discover, and the story is already available online.

And here’s a bit of multimedia bonus: to get the cool exploding-clock image, the intrepid editors worked with Biwa Studios to film high-speed video of exploding clocks, and you can see the whole videos online. They run the events forwards and backwards, just in case your personal arrow of time needs to be calibrated.

One may ask, why is there a chapter about time travel in a book about time’s arrow? Just couldn’t resist the temptation to talk about everything related to “time”? In fact there is a deeper reason. In the real world, the laws of physics may or may not allow for closed timelike curves — physicist-speak for time machines. (Probably not, but we’re not as sure as we could be.) But apart from the difficulty in constructing them, time machines boggle our minds by offering up logical paradoxes — what’s to prevent you from traveling into the past and killing your parents before they met? There is a consistent way to handle these paradoxes, simply by insisting that they never happen. (And we’re still hopeful that the folks at Lost adhere to this principle, regardless of the surface interpretation of last night’s Season Six premiere.)

The reason why that’s hard to swallow is because we can’t imagine anything that stops us from killing our parents, once we grant the existence of time machines. We conceptualize the past and future very differently — the past is settled once and for all, while we can still make choices about what happens in the future. That, of course, is the arrow of time. At the heart of what bothers us about time-travel paradoxes is the difficulty of establishing a uniform arrow of time in a universe where time loops back on itself.

Of course the easy, and probably correct, way out is to simply believe that time machines don’t and can’t exist. But disentangling the demands of logic from the demands of common sense is always a rewarding exercise in its own right.

Welcome to this week’s installment of the From Eternity to Here book club. Next up is Chapter Three: “The Beginning and End of Time.” Remember that next week we’re doing two chapters at once, Four and Five.

For those who missed them, here’s the Science Friday discussion, and here’s the Firedoglake book salon with Chad. I should also point to some substantive review/discussions: Wall Street Journal, New Scientist, USA Today, and Overcoming Bias.

Excerpt:

For the most part, people interested in statistical mechanics care about experimental situations in laboratories or kitchens here on Earth. In an experiment, we can control the conditions before us; in particular, we can arrange systems so that the entropy is much lower than it could be, and watch what happens. You don’t need to know anything about cosmology and the wider universe to understand how that works.

But our aims are more grandiose. The arrow of time is much more than a feature of some particular laboratory experiments; it’s a feature of the entire world around us. Conventional statistical mechanics can account for why it’s easy to turn an egg into an omelet, but hard to turn an omelet into an egg. What it can’t account for is why, when we open our refrigerator, we are able to find an egg in the first place. Why are we surrounded by exquisitely ordered objects such as eggs and pianos and science books, rather than by featureless chaos?

This chapter is a fairly straightforward review of the modern understanding of cosmology, with a particular eye on those issues that will become important later in the book. We zip through the expansion, structure formation, and dark energy. There I got to tell a fun personal story of my wager with Brian Schmidt. At least I think it’s fun — including personal stories is not my natural tendency, but at the right moments it can help to humanize all the forbidding science. Hopefully this was one such moment.

A few topics go beyond the standard cosmology summary. I discussed the Steady State theory a bit, because it’s a relevant historical example when we will much later turn to the question of what the universe should look like. I also dwell a bit on vacuum fluctuations and dark energy, because those will pay a crucial role in my personal favorite explanation for the arrow of time. And we close the chapter with a very brief overview of the evolution of entropy. It has to be brief, because we haven’t laid nearly enough groundwork to do the job right. This is a conscious choice, which may or may not work: rather than simply progressing on an absolutely logical path from foundations to conclusions, I felt free to mention points that would be important later, on the theory that they would come as less of a shock if we had established some familiarity. Again, hope that worked.

Tom Levenson, who is an actual writer, advised me to omit “smoking a pipe” from the caption to Figure 7, on the theory that what is shown should not also be told. I left it in anyway. It’s my book!