From Eternity to Here: The Quest for the Ultimate Theory of Time

by Sean M. Carroll

  Science is about answering hard questions, but it’s also about pinpointing the right questions to ask. When it comes to understanding life, we’re not even sure what the right questions are. We have a bunch of intriguing concepts that we’re pretty sure will play some sort of role in an ultimate understanding—entropy, free energy, complexity, information. But we’re not yet able to put them together into a unified picture. That’s okay; science is a journey in which getting there is, without question, much of the fun.

  10

  RECURRENT NIGHTMARES

  Nature is a big series of unimaginable catastrophes.

  —Slavoj Žižek

  In Book Four of The Gay Science, written in 1882, Friedrich Nietzsche proposes a thought experiment. He asks us to imagine a scenario in which everything that happens in the universe, including our lives down to the slightest detail, will eventually repeat itself, in a cycle that plays out for all eternity.

  What if some day or night a demon were to steal into your loneliest loneliness and say to you: “This life as you now live it and have lived it you will have to live once again and innumerable times again; and there will be nothing new in it, but every pain and every joy and every thought and sigh and everything unspeakably small or great in your life must return to you, all in the same succession and sequence—even this spider and this moonlight between the trees, and even this moment and I myself. The eternal hourglass of existence is turned over again and again, and you with it, speck of dust!”167

  Nietzsche’s interest in an eternally repeating universe was primarily an ethical one. He wanted to ask: How would you feel about knowing that your life would be repeated an infinite number of times? Would you feel dismayed—gnashing your teeth is mentioned—at the horrible prospect, or would you rejoice? Nietzsche felt that a successful life was one that you would be proud to have repeated in an endless cycle.168

  The idea of a cyclic universe, or “eternal return,” was by no means original with Nietzsche. It appears now and again in ancient religions—in Greek myths, Hinduism, Buddhism, and some indigenous American cultures. The Wheel of Life spins, and history repeats itself.

  But soon after Nietzsche imagined his demon, the idea of eternal recurrence popped up in physics. In 1890 Henri Poincaré proved an intriguing mathematical theorem, showing that certain physical systems would necessarily return to any particular configuration infinitely often, if you just waited long enough. This result was seized upon by a young mathematician named Ernst Zermelo, who claimed that it was incompatible with Boltzmann’s purported derivation of the Second Law of Thermodynamics from underlying reversible rules of atomic motion.

  In the 1870s, Boltzmann had grappled with Loschmidt’s “reversibility paradox.” By comparison, the 1880s were a relatively peaceful time for the development of statistical mechanics—Maxwell had died in 1879, and Boltzmann concentrated on technical applications of the formalism he had developed, as well as on climbing the academic ladder. But in the 1890s controversy flared again, this time in the form of Zermelo’s “recurrence paradox.” To this day, the ramifications of these arguments have not been completely absorbed by physicists; many of the issues that Boltzmann and his contemporaries argued about are still being hashed out right now. In the context of modern cosmology, the problems suggested by the recurrence paradox are still very much with us.

  POINCARÉ’S CHAOS

  Oscar II, king of Sweden and Norway, was born on January 21, 1829. In 1887, the Swedish mathematician Gösta Mittag-Leffler proposed that the king should mark his upcoming sixtieth birthday in a somewhat unconventional way: by sponsoring a mathematical competition. Four different questions were proposed, and a prize would be given to whoever submitted the most original and creative solution to any of them.

  One of these questions was the “three-body problem”—how three massive objects would move under the influence of their mutual gravitational pull. (For two bodies it’s easy, and Newton had solved it: Planets move in ellipses.) This problem was tackled by Henri Poincaré, who in his early thirties was already recognized as one of the world’s leading mathematicians. He did not solve it, but submitted an essay that seemed to demonstrate a crucial feature: that the orbits of the planets would be stable. Even without knowing the exact solutions, we could be confident that the planets would at least behave predictably. Poincaré’s method was so ingenious that he was awarded the prize, and his paper was prepared for publication in Mittag-Leffler’s new journal, Acta Mathematica.169

  Figure 52: Henri Poincaré, pioneer of topology, relativity, and chaos theory, and later president of the Bureau of Longitude.

  But there was a slight problem: Poincaré had made a mistake. Edvard Phragmén, one of the journal editors, had some questions about the paper, and in the course of answering them Poincaré realized that he had left out an important case in constructing his proof. Such tiny mistakes occur frequently in complicated mathematical writing, and Poincaré set about correcting his presentation. But as he tugged at the loose thread, the entire argument became completely unraveled. What Poincaré ended up proving was the opposite of his original claim—three-body orbits were not stable at all. Not only are orbits not periodic; they don’t even approach any sort of regular behavior. Now that we have computers to run simulations, this kind of behavior is less surprising, but at the time it came as an utter shock. In his attempt to prove the stability of planetary orbits, Poincaré ended up doing something quite different—he invented chaos theory.

  But the story doesn’t quite end there. Mittag-Leffler, convinced that Poincaré would be able to fix things up in his prize essay, had gone ahead and printed it. By the time he heard from Poincaré that no such fixing-up would be forthcoming, the journal had already been mailed to leading mathematicians throughout Europe. Mittag-Leffler swung into action, telegraphing Berlin and Paris in an attempt to have all copies of the journal destroyed. He basically succeeded, but not without creating a minor scandal in elite mathematical circles across the continent.

  In the course of revising his argument, Poincaré established a deceptively simple and powerful result, now known as the Poincaré recurrence theorem. Imagine you have some system in which all of the pieces are confined to some finite region of space, like planets orbiting the Sun. The recurrence theorem says that if we start with the system in a particular configuration and simply let it evolve according to Newton’s laws, we are guaranteed that the system will return to its original configuration—again and again, infinitely often into the future.

  That seems pretty straightforward, and perhaps unsurprising. If we have assumed from the start that all the components of our system (planets orbiting the Sun, or molecules bouncing around inside a box) are confined to a finite region, but we allow time to stretch on forever, it makes sense that the system is going to keep returning to the same state over and over. Where else can it go?

  Things are a bit more subtle than that. The most basic subtlety is that there can be an infinite number of possible states, even if the objects themselves don’t actually run away to infinity.170 A circular orbit is confined to a finite region, but there are an infinite number of points along it; likewise, there are an infinite number of points inside a finite-sized box of gas. In that case, a system will typically not return to precisely the original state. What Poincaré realized is that this is a case where “almost” is good enough. If you decide ahead of time how close two states need to be so that you can’t tell the difference between them, Poincaré proved that the system would return at least that close to the original state an infinite number of times.

  Consider the three inner planets of the Solar System: Mercury, Venus, and Earth. Venus orbits the Sun once every 0.61520 years (about 225 days), while Mercury orbits once every 0.24085 years (about 88 days). As shown in Figure 53, imagine that we started in an arrangement where all three planets were arranged in a straight line. After 88 days have passed, Mercury will have returned to its starting point, but Venus and Earth will be a
t some other points in their orbits. But if we wait long enough, they will all line up again, or very close to it. After 40 years, for example, these three planets will be in almost the same arrangement as when they started.

  Poincaré showed that all confined mechanical systems are like that, even ones with large numbers of moving parts. But notice that the amount of time we have to wait before the system returns close to its starting point keeps getting larger as we add more components. If we waited for all nine of the planets to line up,171 we would have to wait much longer than 40 years; that’s partly because the outer planets orbit more slowly, but in large part it simply takes longer for more objects to conspire in the right way to re-create any particular starting configuration.

  This is worth emphasizing: As we consider more and more particles, the time it takes for a system to return close to its starting point—known, reasonably enough, as the recurrence time—quickly becomes unimaginably huge.172 Consider the divided box of gas we played with in Chapter Eight, where individual particles had a small chance of hopping from one side of the box to the other every second. Clearly if there are only two or three particles, it won’t take long for the system to return to where it started. But once we consider a box with 60 total particles, we find that the recurrence time has become as large as the current age of the observable universe.

  Figure 53: The inner Solar System in a configuration with Mercury, Venus, and Earth all aligned (bottom), and 88 days later (top). Mercury has returned to its original position, but Venus and Earth are somewhere else along their orbits.

  Real objects usually have a lot more than 60 particles in them. For a typical macroscopic-sized object, the recurrence time would be at least

  101,000,000,000,000,000,000,000,000 seconds.

  That’s a long time. For the total number of particles in the observable universe, the recurrence time would be even much longer—but who cares? The recurrence time for any interestingly large object is much longer than any time relevant to our experience. The observable universe is about 1018 seconds old. An experimental physicist who put in a grant proposal suggesting that they would pour a teaspoon of milk into a cup of coffee and then wait one recurrence time for the milk to unmix itself would have a very hard time getting funding.

  But if we waited long enough, it would happen. Nietzsche’s Demon isn’t wrong; it’s just thinking long-term.

  ZERMELO VERSUS BOLTZMANN

  Poincaré’s original paper in which he proved the recurrence theorem was mainly concerned with the crisp, predictable world of Newtonian mechanics. But he was familiar with statistical mechanics, and within a short while realized that the idea of eternal recurrence might, at first blush, be incompatible with attempts to derive the Second Law of Thermodynamics. After all, the Second Law says that entropy only ever goes one way: It increases. But the recurrence theorem seems to imply that if a low-entropy state evolves to a higher-entropy state, all we have to do is wait long enough and the system will return to its low-entropy beginnings. That means it must decrease somewhere along the way.

  In 1893, Poincaré wrote a short paper that examined this apparent contradiction more closely. He pointed out that the recurrence theorem implied that the entropy of the universe would eventually start decreasing:

  I do not know if it has been remarked that the English kinetic theories can extract themselves from this contradiction. The world, according to them, tends at first toward a state where it remains for a long time without apparent change; and this is consistent with experience; but it does not remain that way forever, if the theorem cited above is not violated; it merely stays that way for an enormously long time, a time which is longer the more numerous are the molecules. This state will not be the final death of the universe, but a sort of slumber, from which it will awake after millions of millions of centuries. According to this theory, to see heat pass from a cold body to a warm one, it will not be necessary to have the acute vision, the intelligence, and the dexterity of Maxwell’s demon; it will suffice to have a little patience.173

  By “the English kinetic theories,” Poincaré was presumably thinking of the work of Maxwell and Thomson and others—no mention of Boltzmann (or for that matter Gibbs). Whether it was for that reason or just because he didn’t come across the paper, Boltzmann made no direct reply to Poincaré.

  But the idea would not be so easily ignored. In 1896, Zermelo made a similar argument to Poincaré’s (referencing Poincaré’s long 1890 paper that stated the recurrence theorem, but not his shorter 1893 paper), which is now known as “Zermelo’s recurrence objection.”174 Despite Boltzmann’s prominence, atomic theory and statistical mechanics were not nearly as widely accepted in the late-nineteenth-century German-speaking world as they were in the English-speaking world. Like many German scientists, Zermelo thought that the Second Law was an absolute rule of nature; the entropy of a closed system would always increase or stay constant, not merely most of the time. But the recurrence theorem clearly implied that if entropy initially went up, it would someday come down as the system returned to its starting configuration. The lesson drawn by Zermelo was that the edifice of statistical mechanics was simply wrong; the behavior of heat and entropy could not be reduced to the motions of molecules obeying Newton’s laws.

  Zermelo would later go on to great fame within mathematics as one of the founders of set theory, but at the time he was a student studying under Max Planck, and Boltzmann didn’t take the young interloper’s objections very seriously. He did bother to respond, although not with great patience.

  Zermelo’s paper shows that my writings have been misunderstood; nevertheless it pleases me for it seems to be the first indication that these writings have been paid any attention in Germany. Poincaré’s theorem, which Zermelo explains at the beginning of his paper, is clearly correct, but his application of it to the theory of heat is not.175

  Oh, snap. Zermelo wrote another paper in response to Boltzmann, who replied again in turn.176 But the two were talking past each other, and never seemed to reach a satisfactory conclusion.

  Boltzmann, by this point, was completely comfortable with the idea that the Second Law was only statistical in nature, rather than absolute. The main thrust of his response to Zermelo was to distinguish between theory and practice. In theory, the whole universe could start in a low entropy state, evolve toward thermal equilibrium, and eventually evolve back to low entropy again; that’s an implication of Poincaré’s theorem, and Boltzmann didn’t deny it. But the actual time you would have to wait is enormous, much longer than what we currently think of as “the age of the universe,” and certainly much longer than any timescales that were contemplated by scientists in the nineteenth century. Boltzmann argued that we should accept the implications of the recurrence theorem as an interesting mathematical curiosity, but not one that was in any way relevant to the real world.

  TROUBLES OF AN ETERNAL UNIVERSE

  In Chapter Eight we discussed Loschmidt’s reversibility objection to Boltzmann’s H-Theorem: It is impossible to use reversible laws of physics to derive an irreversible result. In other words, there are just as many high-entropy states whose entropy will decrease as there are low-entropy states whose entropy will increase, because the former trajectories are simply the time-reverses of the latter. (And neither is anywhere near as numerous as high-entropy states that remain high-entropy.) The proper response to this objection, at least within our observable universe, is to accept the need for a Past Hypothesis—an additional postulate, over and above the dynamical laws of nature, to the effect that the early universe had an extremely low entropy.

  In fact, by the time of his clash with Zermelo, Boltzmann himself had cottoned on to this realization. He called his version of the Past Hypothesis “assumption A,” and had this to say about it:

  The second law will be explained mechanically by means of assumption A (which is of course unprovable) that the universe, considered as a mechanical system—or at least a very large part of it which
surrounds us—started from a very improbable state, and is still in an improbable state.177

  This short excerpt makes Boltzmann sound more definitive than he really is; in the context of this paper, he offers several different ways to explain why we see entropy increasing around us, and this is just one of them. But notice how careful he is—not only admitting up front that the assumption is unprovable, but even limiting consideration to “a very large part of [the universe] which surrounds us,” not the whole thing.

  Unfortunately, this strategy isn’t quite sufficient. Zermelo’s recurrence objection is closely related to the reversibility objection, but there is an important difference. The reversibility objection merely notes that there are an equal number of entropy-decreasing evolutions as entropy-increasing ones; the recurrence objection points out that the entropy-decreasing processes will eventually happen some time in the future. It’s not just that a system could decrease in entropy—if we wait long enough, it is eventually guaranteed to do so. That’s a stronger statement and requires a better comeback.