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From Eternity to Here
From Eternity to Here
Carroll, Sean
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1
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2010
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Penguin Group (USA)
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english
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9780525951339
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Table of Contents Title Page Copyright Page Dedication PART ONE  TIME, EXPERIENCE, AND THE UNIVERSE Chapter 1  THE PAST IS PRESENT MEMORY Chapter 2  THE HEAVY HAND OF ENTROPY Chapter 3  THE BEGINNING AND END OF TIME PART TWO  TIME IN EINSTEIN’S UNIVERSE Chapter 4  TIME IS PERSONAL Chapter 5  TIME IS FLEXIBLE Chapter 6  LOOPING THROUGH TIME PART THREE  ENTROPY AND TIME’S ARROW Chapter 7  RUNNING TIME BACKWARD Chapter 8  ENTROPY AND DISORDER Chapter 9  INFORMATION AND LIFE Chapter 10  RECURRENT NIGHTMARES Chapter 11  QUANTUM TIME PART FOUR  FROM THE KITCHEN TO THE MULTIVERSE Chapter 12  BLACK HOLES: THE ENDS OF TIME Chapter 13  THE LIFE OF THE UNIVERSE Chapter 14  INFLATION AND THE MULTIVERSE Chapter 15  THE PAST THROUGH TOMORROW Chapter 16  EPILOGUE APPENDIX: MATH NOTES BIBLIOGRAPHY Acknowledgements INDEX 5 TIME IS FLEXIBLE The reason why the universe is eternal is that it does not live for itself; it gives life to others as it transforms. —Lao Tzu, Tao Te Ching The original impetus behind special relativity was not a puzzling experimental result (although the MichelsonMorley experiment certainly was that); it was an apparent conflict between two preexisting theoretical frameworks.70 On the one hand you had Newtonian mechanics, the gleaming edifice of physics on which all subsequent theories had been based. On the other hand you had James Clerk Maxwell’s unification of electricity and magnetism, which came about in the middle of the nineteenth century and had explained an impressive variety of experimental phenomena. The problem was that these two marvelously successful theories didn’t fit together. Newtonian mechanics implied that the relative velocity of two objects moving past each other was simply the sum of their two velocities; Maxwellian electromagnetism implied that the speed of light was an exception to this rule. Special relativity managed to bring the two theories together into a single whole, by providing a framework for mechanics in which the speed of light; did play a special role, but which reduced to Newton’s model when particles were moving slowly. Like many dramatic changes of worldview, the triumph of special relativity came at a cost. In this case, the greatest single success of Newtonian physics—his theory of gravity, which accounted for the motions of the planets with exquisite precision—was left out of the happy reconciliation. Along with electromagnetism, gravity is the most obvious force in the universe, and Einstein was determined to fit it in to the language of relativity. You might expect that this would involve modifying a few equations here and there to make Newton’s equations consistent with invariance under boosts, but attempts along those lines fell frustratingly short. Eventually Einstein hit on a brilliant insight, essentially by employing the spaceship thought experiment we’ve been considering. (He thought of it first.) In describing our travels in this hypothetical sealed spaceship, I was careful to note that we are far away from any gravitational fields, so we wouldn’t have to worry about falling into a star or having our robot probes deflected by the pull of a nearby planet. But what if we were near a prominent gravitational field? Imagine our ship was, for example, in orbit around the Earth. How would that affect the experiments we were doing inside the ship? Einstein’s answer was: They wouldn’t affect them at all, as long as we confined our attention to relatively small regions of space and brief intervals of time. We can do whatever kinds of experiments we like—measuring the rates of chemical reactions, dropping balls and watching how they fall, observing weights on springs—and we would get exactly the same answer zipping around in lowEarth orbit as we would in the far reaches of interstellar space. Of course if we wait long enough we can tell we are in orbit; if we let a fork and a spoon freely float in front of our noses, with the fork just slightly closer to the Earth, the fork will then feel just a slightly larger gravitational pull, and therefore move just ever so slightly away from the spoon. But effects like that take time to accumulate; if we confine our attention to sufficiently small regions of space and time, there isn’t any experiment we can imagine doing that could reveal the presence of the gravitational pull keeping us in orbit around the Earth. Contrast the difficulty of detecting a gravitational field with, for example, the ease of detecting an electric field, which is a piece of cake. Just take your same fork and spoon, but now give the fork some positive charge, and the spoon some negative charge. In the presence of an electric field, the opposing charges would be pushed in opposite directions, so it’s pretty easy to check whether there are any electric fields in the vicinity. The difference with gravity is that there is no such thing as a “negative gravitational charge.” Gravity is universal—everything responds to it in the same way. Consequently, it can’t be detected in a small region of spacetime, only in the difference between its effects on objects at different events in spacetime. Einstein elevated this observation to the status of a law of nature, the Principle of Equivalence: No local experiment can detect the existence of a gravitational field. Figure 16: The gravitational field on a planet is locally indistinguishable from the acceleration of a rocket. I know what you’re thinking: “I have no trouble detecting gravity at all. Here I am sitting in my chair, and it’s gravity that’s keeping me from floating up into the room.” But how do you know it’s gravity? Only by looking outside and checking that you’re on the surface of the Earth. If you were in a spaceship that was accelerating, you would also be pushed down into your chair. Just as you can’t tell the difference between freely falling in interstellar space and freely falling in lowEarth orbit, you also can’t tell the difference between constant acceleration in a spaceship and sitting comfortably in a gravitational field. That’s the “equivalence” in the Principle of Equivalence: The apparent effects of the force of gravity are equivalent to those of living in an accelerating reference frame. It’s not the force of gravity that you feel when you are sitting in a chair; it’s the force of the chair pushing up on your posterior. According to general relativity, free fall is the natural, unforced state of motion, and it’s only the push from the surface of the Earth that deflects us from our appointed path. CURVING STRAIGHT LINES You or I, having come up with the bright idea of the Principle of Equivalence while musing over the nature of gravity, would have nodded sagely and moved on with our lives. But Einstein was smarter than that, and he appreciated what this insight really meant. If gravity isn’t detectable by doing local experiments, then it’s not really a “force” at all, in the same way that electricity or magnetism are forces. Because gravity is universal, it makes more sense to think of it as a feature of spacetime itself, rather than some force field stretching through spacetime. In particular, realized Einstein, gravity can be thought of as a manifestation of the curvature of spacetime. We’ve talked quite a bit about spacetime as a generalization of space, and how the time elapsed along a trajectory is a measure of the distance traveled through spacetime. But space isn’t necessarily rigid, flat, and rectilinear; it can be warped, stretched, and deformed. Einstein says that spacetime is the same way. It’s easiest to visualize twodimensional space, modeled, for example, by a piece of paper. A flat piece of paper is not curved, and the reason we know that is that it obeys the principles of good oldfashioned Euclidean geometry. Two initially parallel lines, for example, never intersect, nor do they grow apart. In contrast, consider the twodimensional surface of a sphere. First we have to generalize the notion of a “straight line,” which on a sphere isn’t an obvious concept. In Euclidean geometry, as we were taught in high school, a straight line is the shortest distance between two points. So let’s declare an analogous definition: A “straight line” on a curved geometry is the shortest curve connecting two points, which on a sphere would be a portion of a great circle. If we take two paths on a sphere that are initially parallel, and extend them along great circles, they will eventually intersect. That proves that the principles of Euclidean geometry are no longer at work, which is one way of seeing that the geometry of a sphere is curved. Figure 17: Flat geometry, with parallel lines extending forever; curved geometry, where initially parallel lines eventually intersect. Einstein proposed that fourdimensional spacetime can be curved, just like the surface of a twodimensional sphere. The curvature need not be uniform like a sphere, the same from point to point; it can vary in magnitude and in shape from place to place. And here is the kicker: When we see a planet being “deflected by the force of gravity,” Einstein says it is really just traveling in a straight line. At least, as straight as a line can be in the curved spacetime through which the planet is moving. Following the insight that an unaccelerated trajectory yields the greatest possible time a clock could measure between two events, a straight line through spacetime is one that does its best to maximize the time on a clock, just like a straight line through space does its best to minimize the distance read by an odometer. Let’s bring this down to Earth, in a manner of speaking. Consider a satellite in orbit, carrying a clock. And consider another clock, this one on a tower that reaches to the same altitude as the orbiting satellite. The clocks are synchronized at a moment when the satellite passes by the tower; what will they read when the satellite completes one orbit? (We can ignore the rotation of the Earth for the purposes of this utterly impractical thought experiment.) According to the viewpoint of general relativity, the orbiting clock is not accelerating; it’s in free fall, doing its best to move in a straight line through spacetime. The tower clock, meanwhile, is accelerating—it’s being prevented from freely falling by the force of the tower keeping it up. Therefore, the orbiting clock will experience more elapsed time per orbit than the tower clock—compared to the accelerated clock on the tower, the freely falling one in orbit appears to run more quickly. Figure 18: Time as measured on a tower will be shorter than that measured in orbit, as the former clock is on an accelerated (nonfreefalling) trajectory. There are no towers that reach to the heights of lowEarth orbit. But there are clocks down here at the surface that regularly exchange signals with clocks on satellites. That, for example, is the basic mechanism behind the Global Positioning System (GPS) that helps modern cars give driving directions in real time. Your personal GPS receiver gets signals from a number of satellites orbiting the Earth, and determines its position by comparing the time between the different signals. That calculation would quickly go astray if the gravitational time dilation due to general relativity were not taken into account; the GPS satellites experience about 38 more microseconds per day in orbit than they would on the ground. Rather than teaching your receiver equations from general relativity, the solution actually adopted is to tune the satellite clocks so that they run a little bit more slowly than they should if they were to keep correct time down here on the surface. EINSTEIN’S MOST IMPORTANT EQUATION The saying goes that every equation cuts your book sales in half. I’m hoping that this page is buried sufficiently deeply in the book that nobody notices before purchasing it, because I cannot resist the temptation to display another equation: the Einstein field equation for general relativity. Rµν  (t/2)Rgµν = 8πGTµν. This is the equation that a physicist would think of if you said “Einstein’s equation”; that E = mc2 business is a minor thing, a special case of a broader principle. This one, in contrast, is a deep law of physics: It reveals how stuff in the universe causes spacetime to curve, and therefore causes gravity. Both sides of the equation are not simply numbers, but tensors—geometric objects that capture multiple things going on at once. (If you thought of them as 4x4 arrays of numbers, you would be pretty close to right.) The lefthand side of the equation characterizes the curvature of spacetime. The righthand side characterizes all the various forms of stuff that make spacetime curve—energy, momentum, pressure, and so on. In one fell swoop, Einstein’s equation reveals how any particular collection of particles and fields in the universe creates a certain kind of curvature in spacetime. According to Isaac Newton, the source of gravity was mass; heavier objects gave rise to stronger gravitational fields. In Einstein’s universe, things are more complicated. Mass gets replaced by energy, but there are also other properties that go into curving spacetime. Vacuum energy, for example, has not only energy, but also tension—a kind of negative pressure. A stretched string or rubber band has tension, pulling back rather than pushing out. It’s the combined effect of the energy plus the tension that causes the universe to accelerate in the presence of vacuum energy.71 The interplay between energy and the curvature of spacetime has a dramatic consequence: In general relativity, energy is not conserved. Not every expert in the field would agree with that statement, not because there is any controversy over what the theory predicts, but because people disagree on how to define “energy” and “conserved.” In a Newtonian absolute spacetime, there is a welldefined notion of the energy of individual objects, which we can add up to get the total energy of the universe, and that energy never changes (it’s the same at every moment in time). But in general relativity, where spacetime is dynamical, energy can be injected into matter or absorbed from it by the motions of spacetime. For example, vacuum energy remains absolutely constant in density as the universe expands. So the energy per cubic centimeter is constant, while the number of cubic centimeters is increasing—the total energy goes up. In a universe dominated by radiation, in contrast, the total energy goes down, as each photon loses energy due to the cosmological redshift. You might think we could escape the conclusion that energy is not conserved by including “the energy of the gravitational field,” but that turns out to be much harder than you might expect—there simply is no welldefined local definition of the energy in the gravitational field. (That shouldn’t be completely surprising, since the gravitational field can’t even be detected locally.) It’s easier just to bite the bullet and admit that energy is not conserved in general relativity, except in certain special circumstances.72 But it’s not as if chaos has been loosed on the world; given the curvature of spacetime, we can predict precisely how any particular source of energy will evolve. HOLES IN SPACETIME Black holes are probably the single most interesting dramatic prediction of general relativity. They are often portrayed as something relatively mundane: “Objects where the gravitational field is so strong that light itself cannot escape.” The reality is more interesting. Even in Newtonian gravity, there’s nothing to stop us from contemplating an object so massive and dense that the escape velocity is greater than the speed of light, rendering the body “black.” Indeed, the idea was occasionally contemplated, including by British geologist John Michell in 1783 and by PierreSimon Laplace in 1796.73 At the time, it wasn’t clear whether the idea quite made sense, as nobody knew whether light was even affected by gravity, and the speed of light didn’t have the fundamental importance it attains in relativity. More important, though, there is a very big distinction hidden in the seemingly minor difference between “an escape velocity greater than light” and “light cannot escape.” Escape velocity is the speed at which we would have to start an object moving upward in order for it to escape the gravitational field of a body without any further acceleration . If I throw a baseball up in the air in the hopes that it escapes into outer space, I have to throw it faster than escape velocity. But there is absolutely no reason why I couldn’t put the same baseball on a rocket and gradually accelerate it into space without ever reaching escape velocity. In other words, it’s not necessary to reach escape velocity in order to actually escape; given enough fuel, you can go as slowly as you like. But a real black hole, as predicted by general relativity, is a lot more dramatic than that. It is a true region of no return—once you enter, there is no possibility of leaving, no matter what technological marvels you have at your disposal. That’s because general relativity, unlike Newtonian gravity or special relativity, allows spacetime to curve. At every event in spacetime we find light cones that divide space into the past, future, and places we can’t reach. But unlike in special relativity, the light cones are not fixed in a rigid alignment; they can tilt and stretch as spacetime curves under the influence of matter and energy. In the vicinity of a massive object, light cones tilt toward the object, in accordance with the tendency of things to be pulled in by the gravitational field. A black hole is a region of spacetime where the light cones have tilted so much that you would have to move faster than the speed of light to escape. Despite the similarity of language, that’s an enormously stronger statement than “the escape velocity is larger than the speed of light.” The boundary defining the black hole, separating places where you still have a chance to escape from places where you are doomed to plunge ever inward, is the event horizon. Figure 19: Light cones tilt in the vicinity of a black hole. The event horizon, demarcating the edge of the black hole, is the place where they tip over so far that nothing can escape without moving faster than light. There may be any number of ways that black holes could form in the real world, but the standard scenario is the collapse of a sufficiently massive star. In the late 1960s, Roger Penrose and Stephen Hawking proved a remarkable feature of general relativity: If the gravitational field becomes sufficiently strong, a singularity must be formed.74 You might think that’s only sensible, since gravity becomes stronger and stronger and pulls matter into a single point. But in Newtonian gravity, for example, it’s not true. You can get a singularity if you try hard enough, but the generic result of squeezing matter together is that it will reach some point of maximum density. But in general relativity, the density and spacetime curvature increase without limit, until they form a singularity of infinite curvature. Such a singularity lies inside every black hole. It would be wrong to think of the singularity as residing at the “center” of the black hole. If we look carefully at the representation of spacetime near a black hole shown in Figure 19, we see that the future light cones inside the event horizon keep tipping toward the singularity. But that light cone defines what the observer at that event would call “the future.” Like the Big Bang singularity in the past, the black hole singularity in the future is a moment of time, not a place in space. Once you are inside the event horizon, you have absolutely no choice but to continue on to the grim destiny of the singularity, because it lies ahead of you in time, not in some direction in space. You can no more avoid hitting the singularity than you can avoid hitting tomorrow. When you actually do fall through the event horizon, you might not even notice. There is no barrier there, no sheet of energy that you pass through to indicate that you’ve entered a black hole. There is simply a diminution of your possible future life choices; the option of “returning to the outside universe” is no longer available, and “crashing into the singularity” is your only remaining prospect. In fact, if you knew how massive the black hole was, you could calculate precisely how long it will take (according to a clock carried along with you) before you reach the singularity and cease to exist; for a black hole with the mass of the Sun, it would be about onemillionth of a second. You might try to delay this nasty fate, for example, by firing rockets to keep yourself away from the singularity, but it would only be counterproductive. According to relativity, unaccelerated motion maximizes the time between two events. By struggling, you only hasten your doom.75 There is a definite moment on your infalling path when you cross the event horizon. If we imagine that you had been sending a constant stream of radio updates to your friends outside, they will never be able to receive anything sent after that time. They do not, however, see you wink out of existence; instead, they receive your signals at longer and longer intervals, increasingly redshifted to longer and longer wavelengths. Your final moment before crossing the horizon is (in principle) frozen in time from the point of view of an external observer, although it becomes increasingly dimmer and redder as time passes. Figure 20: As an object approaches an event horizon, to a distant observer it appears to slow down and become increasingly redshifted. The moment on the object’s world line when it crosses the horizon is the last moment it can be seen from the outside. WHITE HOLES: BLACK HOLES RUN BACKWARD If you think a bit about this blackhole story, you’ll notice something intriguing: time asymmetry. We have been casually tossing around terminology that assumes a directionality to time; we say “once you pass the event horizon you can never leave,” but not “once you leave the event horizon you can never return.” That’s not because we have been carelessly slipping into temporally asymmetric language; it’s because the notion of a black hole is intrinsically timeasymmetric. The singularity is unambiguously in your future, not in your past. The time asymmetry here isn’t part of the underlying physical theory. General relativity is perfectly timesymmetric; for every specific spacetime that solves Einstein’s equation, there is another solution that is identical except that the direction of time is reversed. A black hole is a particular solution to Einstein’s equation, but there are equivalent solutions that run the other way: white holes. The description of a white hole is precisely the same as that of a black hole, if we simply reverse the tenses of all words that refer to time. There is a singularity in the past, from which light cones emerge. The event horizon lies to the future of the singularity, and the external world lies to the future of that. The horizon represents a place past which, once you exit, you can never return to the whitehole region. Figure 21: The spacetime of a white hole is a timereversed version of a black hole. So why do we hear about black holes in the universe all the time, and hardly ever hear about white holes? For one thing, notice that we can’t “make” a white hole. Since we are in the external world, the singularity and event horizon associated with a white hole are necessarily in our past. So it’s not a matter of wondering what we would do to create a white hole; if we’re going to find one, it will already have been out there in the universe from the beginning. But in fact, thinking slightly more carefully, we should be suspicious of that word make. Why, in a world governed by reversible laws of physics, do we think of ourselves as “making” things that persist into the future, but not things that extend into the past? It’s the same reason why we believe in free will: A lowentropy boundary condition in the past dramatically fixes what possibly could have happened, while the absence of any corresponding future boundary condition leaves what can yet happen relatively open. So when we ask, “Why does it seem relatively straightforward to make a black hole, while white holes are something that we would have to find already existing in the universe?” the answer should immediately suggest itself: because a black hole tends to have more entropy than the things from which you would make it. Actually calculating what the entropy is turns out to be a tricky business involving Hawking radiation, as we’ll see in Chapter Twelve. But the key point is that black holes have a lot of entropy. Black holes turn out to provide the strongest connection we have between gravitation and entropy—the two crucial ingredients in an ultimate explanation of the arrow of time. 4 TIME IS PERSONAL Time travels in divers paces with divers persons. —William Shakespeare, As You Like It When most people hear “scientist,” they think “Einstein.” Albert Einstein is an iconic figure; not many theoretical physicists attain a level of celebrity in which their likeness appears regularly on Tshirts. But it’s an intimidating, distant celebrity. Unlike, say, Tiger Woods, the precise achievements Einstein is actually famous for remain somewhat mysterious to many people who would easily recognize his name.53 His image as the rumpled, absentminded professor, with unruly hair and baggy sweaters, contributes to the impression of someone who embodied the life of the mind, disdainful of the mundane realities around him. And to the extent that the substance of his contributions is understood—equivalence of mass and energy, warping of space and time, a search for the ultimate theory—it seems to be the pinnacle of abstraction, far removed from everyday concerns. The real Einstein is more interesting than the icon. For one thing, the rumpled look with the Don King hair attained in his later years bore little resemblance to the sharply dressed, wellgroomed young man with the penetrating stare who was responsible for overturning physics more than once in the early decades of the twentieth century.54 For another, the origins of the theory of relativity go beyond armchair speculations about the nature of space and time; they can be traced to resolutely practical concerns of getting persons and cargo to the right place at the right time. Figure 10: Albert Einstein in 1912. His “miraculous year” was 1905, while his work on general relativity came to fruition in 1915. Special relativity, which explains how the speed of light can have the same value for all observers, was put together by a number of researchers over the early years of the twentieth century. (Its successor, general relativity, which interpreted gravity as an effect of the curvature of spacetime, was due almost exclusively to Einstein.) One of the major contributors to special relativity was the French mathematician and physicist Henri Poincaré. While Einstein was the one who took the final bold leap into asserting that the “time” as measured by any moving observer was as good as the “time” measured by any other, both he and Poincaré developed very similar formalisms in their research on relativity.55 Historian Peter Galison, in his book Einstein’s Clocks, Poincaré’s Maps: Empires of Time, makes the case that Einstein and Poincaré were as influenced by their earthbound day jobs as they were by esoteric considerations of the architecture of physics.56 Einstein was working at the time as a patent clerk in Bern, Switzer land, where a major concern was the construction of accurate clocks. Railroads had begun to connect cities across Europe, and the problem of synchronizing time across great distances was of pressing commercial interest. The more senior Poincaré, meanwhile, was serving as president of France’s Bureau of Longitude. The growth of sea traffic and trade led to a demand for more accurate methods of determining longitude while at sea, both for the navigation of individual ships and for the construction of more accurate maps. And there you have it: maps and clocks. Space and time. In particular, an appreciation that what matters is not questions of the form “Where are you really?” or “What time is it actually?” but “Where are you with respect to other things?” and “What time does your clock measure?” The rigid, absolute space and time of Newtonian mechanics accords pretty well with our intuitive understanding of the world; relativity, in contrast, requires a certain leap into abstraction. Physicists at the turn of the century were able to replace the former with the latter only by understanding that we should not impose structures on the world because they suit our intuition, but that we should take seriously what can be measured by real devices. Special relativity and general relativity form the basic framework for our modern understanding of space and time, and in this part of the book we’re going to see what the implications of “spacetime” are for the concept of “time.”57 We’ll be putting aside, to a large extent, worries about entropy and the Second Law and the arrow of time, and taking refuge in the clean, precise world of fundamentally reversible laws of physics. But the ramifications of relativity and spacetime will turn out to be crucial to our program of providing an explanation for the arrow of time. LOST IN SPACE Zen Buddhism teaches the concept of “beginner’s mind”: a state in which one is free of all preconceptions, ready to apprehend the world on its own terms. One could debate how realistic the ambition of attaining such a state might be, but the concept is certainly appropriate when it comes to thinking about relativity. So let’s put aside what we think we know about how time works in the universe, and turn to some thought experiments (for which we know the answers from real experiments) to figure out what relativity has to say about time. To that end, imagine we are isolated in a sealed spaceship, floating freely in space, far away from the influence of any stars or planets. We have all of the food and air and basic necessities we might wish, and some high schoollevel science equipment in the form of pulleys and scales and so forth. What we’re not able to do is to look outside at things far away. As we go, we’ll consider what we can learn from various sensors aboard or outside the ship. But first, let’s see what we can learn just inside the spaceship. We have access to the ship’s controls; we can rotate the vessel around any axis we choose, and we can fire our engines to move in whatever direction we like. So we idle away the hours by alternating between moving the ship around in various ways, not really knowing or caring where we are going, and playing a bit with our experiments. Figure 11: An isolated spaceship. From left to right: freely falling, accelerating, and spinning. What do we learn? Most obviously, we can tell when we’re accelerating the ship. When we’re not accelerating, a fork from our dinner table would float freely in front of us, weightless; when we fire the rockets, it falls down, where “down” is defined as “away from the direction in which the ship is accelerating.”58 If we play a bit more, we might figure out that we can also tell when the ship is spinning around some axis. In that case, a piece of cutlery perfectly positioned on the rotational axis could remain there, freely floating; but anything at the periphery would be “pulled” to the hull of the ship and stay there. So there are some things about the state of our ship we can determine observa tionally, just by doing simple experiments inside. But there are also things that we can’t determine. For example, we don’t know where we are. Say we do a bunch of experiments at one location in our unaccelerated, nonspinning ship. Then we fire the rockets for a bit, zip off somewhere else, kill the rockets so that we are once again unaccelerated and nonspinning, and do the same experiments again. If we have any skill at all as experimental physicists, we’re going to get the same results. Had we been very good record keepers about the amount and duration of our acceleration, we could possibly calculate the distance we had traveled; but just by doing local experiments, there doesn’t seem to be any way to distinguish one location from another. Likewise, we can’t seem to distinguish one velocity from another. Once we turn off the rockets, we are once again freely floating, no matter what velocity we have attained; there is no need to decelerate in the opposite direction. Nor can we distinguish any particular orientation of the ship from any other orientation, here in the lonely reaches of interstellar space. We can tell whether we are spinning or not spinning; but if we fire the appropriate guidance rockets (or manipulate some onboard gyroscopes) to stop whatever spin we gave the ship, there is no local experiment we can do that would reveal the angle by which the ship had rotated. These simple conclusions reflect deep features of how reality works. Whenever we can do something to our apparatus without changing any experimental outcomes—shift its position, rotate it, set it moving at a constant velocity—this reflects a symmetry of the laws of nature. Principles of symmetry are extraordinarily powerful in physics, as they place stringent restrictions on what form the laws of nature can take, and what kind of experimental results can be obtained. Naturally, there are names for the symmetries we have uncovered. Changing one’s location in space is known as a “translation”; changing one’s orientation in space is known as a “rotation”; and changing one’s velocity through space is known as a “boost.” In the context of special relativity, the collection of rotations and boosts are known as “Lorentz transformations,” while the entire set including translations are known as “Poincaré transformations.” The basic idea behind these symmetries far predates special relativity. Galileo himself was the first to argue that the laws of nature should be invariant under what we now call translations, rotations, and boosts. Even without relativity, if Galileo and Newton had turned out to be right about mechanics, we would not be able to determine our position, orientation, or velocity if we were floating freely in an isolated spaceship. The difference between relativity and the Galilean perspective resides in what actually happens when we switch to the reference frame of a moving observer. The miracle of relativity, in fact, is that changes in velocity are seen to be close relatives of changes in spatial orientation; a boost is simply the spacetime version of a rotation. Before getting there, let’s pause to ask whether things could have been different. For example, we claimed that one’s absolute position is unobservable, and one’s absolute velocity is unobservable, but one’s absolute acceleration can be measured. 59 Can we imagine a world, a set of laws of physics, in which absolute position is unobservable, but absolute velocity can be objectively measured?60 Sure we can. Just imagine moving through a stationary medium, such as air or water. If we lived in an infinitely big pool of water, our position would be irrelevant, but it would be straightforward to measure our velocity with respect to the water. And it wouldn’t be crazy to think that there is such a medium pervading space.61 After all, ever since the work of Maxwell on electromagnetism we have known that light is just a kind of wave. And if you have a wave, it’s natural to think that there must be something doing the waving. For example, sound needs air to propagate; in space, no one can hear you scream. But light can travel through empty space, so (according to this logic, which will turn out not to be right) there must be some medium through which it is traveling. So physicists in the late nineteenth century postulated that electromagnetic waves propagated through an invisible but allimportant medium, which they called the “aether.” And experimentalists set out to actually detect the stuff. But they didn’t succeed—and that failure set the stage for special relativity. THE KEY TO RELATIVITY Imagine we’re back out in space, but this time we’ve brought along some more sophisticated experimental apparatus. In particular, we have an impressivelooking contraption, complete with stateoftheart laser technology, that measures the speed of light. While we are freely falling (no acceleration), to calibrate the thing we check that we get the same answer for the speed of light no matter how we orient our experiment. And indeed we do. Rotational invariance is a property of the propagation of light, just as we suspected. But now we try to measure the speed of light while moving at different velocities. That is, first we do the experiment, and then we fire our rockets a bit and turn them off so that we’ve established some constant velocity with respect to our initial motion, and then we do the experiment again. Interestingly, no matter how much velocity we picked up, the speed of light that we measure is always the same. If there really were an aether medium through which light traveled just as sound travels through air, we should get different answers depending on our speed relative to the aether. But we don’t. You might guess that the light had been given some sort of push by dint of the fact that it was created within your moving ship. To check that, we’ll allow you to remove the curtains from the windows and let some light come in from the outside world. When you measure the velocity of the light that was emitted by some outside source, once again you find that it doesn’t depend on the velocity of your own spaceship. A realworld version of this experiment was performed in 1887 by Albert Michelson and Edward Morley. They didn’t have a spaceship with a powerful rocket, so they used the next best thing: the motion of the Earth around the Sun. The Earth’s orbital velocity is about 30 kilometers per second, so in the winter it has a net velocity of about 60 kilometers per second different from its velocity in the summer, when it’s moving in the other direction. That’s not much compared to the speed of light, which is about 300,000 kilometers per second, but Michelson designed an ingenious device known as an “interferometer” that was extremely sensitive to small changes in velocities along different directions. And the answer was: The speed of light seems to be the same, no matter how fast we are moving. Advances in science are rarely straightforward, and the correct way to interpret the MichelsonMorley results was not obvious. Perhaps the aether is dragged along with the Earth, so that our relative velocity remains small. After some furious backandforth theorizing, physicists hit upon what we now regard to be the right answer: The speed of light is simply a universal invariant. Everyone measures light to be moving at the same speed, independent of the motion of the experimenter.62 Indeed, the entire content of special relativity boils down to these two principles: • No local experiment can distinguish between observers moving at constant velocities. • The speed of light is the same to all observers. When we use the phrase the speed of light, we are implicitly assuming that it’s the speed of light through empty space that we’re talking about. It’s perfectly easy to make light move at some other speed, just by introducing a transparent medium—light moves more slowly through glass or water than it does through empty space, but that doesn’t tell us anything profound about the laws of physics. Indeed, “light” is not all that important in this game. What’s important is that there exists some unique preferred velocity through spacetime. It just so happens that light moves at that speed when it’s traveling through empty space—but the existence of a speed limit is what matters, not that light is able to go that fast. We should appreciate how astonishing all this is. Say you’re in your spaceship, and a friend in a faraway spaceship is beaming a flashlight at you. You measure the velocity of the light from the flashlight, and the answer is 300,000 kilometers per second. Now you fire your rockets and accelerate toward your friend, until your relative velocity is 200,000 kilometers per second. You again measure the speed of the light coming from the flashlight, and the answer is: 300,000 kilometers per second. That seems crazy; anyone in their right mind should have expected it to be 500,000 kilometers per second. What’s going on? The answer, according to special relativity, is that it’s not the speed of light that depends on your reference frame—it’s your notion of a “kilometer” and a “second.” If a meterstick passes by us at high velocity, it undergoes “length contraction”—it appears shorter than the meterstick that is sitting at rest in our reference frame. Likewise, if a clock moves by us at high velocity, it undergoes “time dilation”—it appears to be ticking more slowly than the clock that is sitting at rest. Together, these phenomena precisely compensate for any relative motion, so that everyone measures exactly the same speed of light.63 The invariance of the speed of light carries with it an important corollary: Nothing can move faster than light. The proof is simple enough; imagine being in a rocket that tries to race against the light being emitted by a flashlight. At first the rocket is stationary (say, in our reference frame), and the light is passing it at 300,000 kilometers per second. But then the rocket accelerates with all its might, attaining a tremendous velocity. When the crew in the rocket checks the light from the (now distant) flashlight, they see that it is passing them by at—300,000 kilometers per second. No matter what they do, how hard they accelerate or for how long, the light is always moving faster, and always moving faster by the same amount.64 (From their point of view, that is. From the perspective of an external observer, they appear to be moving closer and closer to the speed of light, but they never reach it.) However, while length contraction and time dilation are perfectly legitimate ways to think about special relativity, they can also get pretty confusing. When we think about the “length” of some physical object, we need to measure the distance between one end of it and the other, but implicitly we need to do so at the same time. (You can’t make yourself taller by putting a mark on the wall by your feet, climbing a ladder, putting another mark by your head, and proclaiming the distance between the marks to be your height.) But the entire spirit of special relativity tells us to avoid making statements about separated events happening at the same time. So let’s tackle the problem from a different angle: by taking “spacetime” seriously. SPACETIME Back to the spaceship with us. This time, however, instead of being limited to performing experiments inside the sealed ship, we have access to a small fleet of robot probes with their own rockets and navigation computers, which we can program to go on journeys and come back as we please. And each one of them is equipped with a very accurate atomic clock. We begin by carefully synchronizing these clocks with the one on our main shipboard computer, and verifying that they all agree and keep very precise time. Then we send out some of our probes to zip away from us for a while and eventually come back. When they return, we notice something right away: The clocks on the probe ships no longer agree with the shipboard computer. Because this is a thought experiment, we can rest assured that the difference is not due to cosmic rays or faulty programming or tampering by mischievous aliens—the probes really did experience a different amount of time than we did. Happily, there is an explanation for this unusual phenomenon. The time that clocks experience isn’t some absolute feature of the universe, out there to be measured once and for all, like the yard lines on a football field. Instead, the time measured by a clock depends on the particular trajectory that the clock takes, much like the total distance covered by a runner depends on their path. If, instead of sending out robot probes equipped with clocks from a spaceship, we had sent out robots on wheels equipped with odometers from a base located on the ground, nobody would be surprised that different robots returned with different odometer readings. The lesson is that clocks are kind of like odometers, keeping track of some measure of distance traveled (through time or through space) along a particular path. If clocks are kind of like odometers, then time is kind of like space. Remember that even before special relativity, if we believed in absolute space and time à la Isaac Newton, there was nothing stopping us from combining them into one entity called “spacetime.” It was still necessary to give four numbers (three to locate a position in space, and one time) to specify an event in the universe. But in a Newtonian world, space and time had completely separate identities. Given two distinct events, such as “leaving the house Monday morning” and “arriving at work later that same morning,” we could separately (and uniquely, without fear of ambiguity) talk about the distance between them and the time elapsed between them. Special relativity says that this is not right. There are not two different things, “distance in space” measured by odometers and “duration in time” measured by clocks. There is only one thing, the interval in spacetime between two events, which corresponds to an ordinary distance when it is mostly through space and to a duration measured by clocks when it is mostly through time. What decides “mostly”? The speed of light. Velocity is measured in kilometers per second, or in some other units of distance per time; hence, having some special speed as part of the laws of nature provides a way to translate between space and time. When you travel more slowly than the speed of light, you are moving mostly through time; if you were to travel faster than light (which you aren’t about to do), you would be moving mostly through space. Let’s try to flesh out some of the details. Examining the clocks on our probe ships closely, we realize that all of the traveling clocks are different in a similar way: They read shorter times than the one that was stationary. That is striking, as we were comforting ourselves with the idea that time is kind of like space, and the clocks were reflecting a distance traveled through spacetime. But in the case of good old ordinary space, moving around willynilly always makes a journey longer; a straight line is the shortest distance between two points in space. If our clocks are telling us the truth (and they are), it would appear that unaccelerated motion—a straight line through spacetime, if you like—is the path of longest time between two events. Figure 12: Time elapsed on trajectories that go out and come back is less than that elapsed according to clocks that stay behind. Well, what did you expect? Time is kind of like space, but it’s obviously not completely indistinguishable from space in every way. (No one is in any danger of getting confused by some driving directions and making a left turn into yesterday.) Putting aside for the moment issues of entropy and the arrow of time, we have just uncovered the fundamental feature that distinguishes time from space: Extraneous motion decreases the time elapsed between two events in spacetime, whereas it increases the distance traveled between two points in space. If we want to move between two points in space, we can make the distance we actually travel as long as we wish, by taking some crazy winding path (or just by walking in circles an arbitrary number of times before continuing on our way). But consider traveling between two events in spacetime—particular points in space, at particular moments in time. If we move on a “straight line”—an unaccelerated trajectory, moving at constant velocity all the while—we will experience the longest duration possible. So if we do the opposite, zipping all over the place as fast as we can, but taking care to reach our destination at the appointed time, we will experience a shorter duration. If we zipped around at precisely the speed of light, we would never experience any duration at all, no matter how we traveled. We can’t do exactly that, but we can come as close as we wish.65 That’s the precise sense in which “time is kind of like space”—spacetime is a generalization of the concept of space, with time playing the role of one of the dimensions of spacetime, albeit one with a slightly different flavor than the spatial dimensions. None of this is familiar to us from our everyday experience, because we tend to move much more slowly than the speed of light. Moving much more slowly than light is like being a running back who only marched precisely up the football field, never swerving left or right. To a player like that, “distance traveled” would be identical to “number of yards gained,” and there would be no ambiguity. That’s what time is like in our everyday experience; because we and all of our friends move much more slowly than the speed of light, we naturally assume that time is a universal feature of the universe, rather than a measure of the spacetime interval along our particular trajectories. STAYING INSIDE YOUR LIGHT CONE One way of coming to terms with the workings of spacetime according to special relativity is to make a map: draw a picture of space and time, indicating where we are allowed to go. Let’s warm up by drawing a picture of Newtonian spacetime. Because Newtonian space and time are absolute, we can uniquely define “moments of constant time” on our map. We can take the four dimensions of space and time and slice them into a set of threedimensional copies of space at constant time, as shown in Figure 13. (We’re actually only able to show twodimensional slices on the figure; use your imagination to interpret each slice as representing threedimensional space.) Crucially, everyone agrees on the difference between space and time; we’re not making any arbitrary choices. Figure 13: Newtonian space and time. The universe is sliced into moments of constant time, which unambiguously separate time into past and future. World lines of real objects can never double back across a moment of time more than once. Every Newtonian object (a person, an atom, a rocket ship) defines a world line—the path the object takes through spacetime. (Even if you sit perfectly still, you still move through spacetime; you’re aging, aren’t you?66) And those world lines obey a very strict rule: Once they pass through one moment of time, they can never double backward in time to pass through the same moment again. You can move as fast as you like—you can be here one instant, and a billion lightyears away 1 second later—but you have to keep moving forward in time, your world line intersecting each moment precisely once. Relativity is different. The Newtonian rule “you must move forward in time” is replaced by a new rule: You must move more slowly than the speed of light. (Unless you are a photon or another massless particle, in which case you always move exactly at the speed of light if you are in empty space.) And the structure we were able to impose on Newtonian spacetime, in the form of a unique slicing into moments of constant time, is replaced by another kind of structure: light cones. Figure 14: Spacetime in the vicinity of a certain event x. According to relativity, every event comes with a light cone, defined by considering all possible paths light could take to or from that point. Events outside the light cone cannot unambiguously be labeled “past” or “future.” Light cones are conceptually pretty simple. Take an event, a single point in spacetime, and imagine all of the different paths that light could take to or from that event; these define the light cone associated with that event. Hypothetical light rays emerging from the event define a future light cone, while those converging on the event define a past light cone, and when we mean both we just say “the light cone.” The rule that you can’t move faster than the speed of light is equivalent to saying that your world line must remain inside the light cone of every event through which it passes. World lines that do this, describing slowerthanlight objects, are called “timelike”; if somehow you could move faster than light, your world line would be “spacelike,” since it was covering more ground in space than in time. If you move exactly at the speed of light, your world line is imaginatively labeled “lightlike.” 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 in the future. Figure 15: Light cones replace the moments of constant time from Newtonian spacetime. World lines of massive particles must come to an event from inside the past light cone, and leave inside the future light cone—a timelike path. Spacelike paths move faster than light and are therefore not allowed. It’s that last bit that really gets people. In our reflexively Newtonian way of thinking about the world, we insist that some faraway event happened either in the past, or in 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.67 There is a very strong temptation, when drawing maps of spacetime such as shown in Figure 15, to draw a vertical axis labeled “time,” and a horizontal axis (or two) labeled “space.” The absence of those axes on our version of the spacetime diagram is completely intentional. The whole point of spacetime according to relativity is that it is not fundamentally divided up into “time” and “space.” The light cones, demarcating the accessible past and future of each event, are not added on top of the straightforward Newtonian decomposition of spacetime into time and space; they replace that structure entirely. Time can be measured along each individual world line, but it’s not a builtin feature of the entire spacetime. It would be irresponsible to move on without highlighting one other difference between time and space: There is only one dimension of time, whereas there are three dimensions of space.68 We don’t have a good understanding of why this should be so. That is, our understanding of fundamental physics isn’t sufficiently developed to state with confidence whether there is some reason why there couldn’t be more than one dimension of time, or for that matter zero dimensions of time. What we do know is that life would be very different with more than one time dimension. With only one such dimension, physical objects (which move along timelike paths) can’t help but move along that particular direction. If there were more than one, nothing would force us to move forward in time; we could move in circles, for example. Whether or not one can build a consistent theory of physics under such conditions is an open question, but at the least, things would be different. EINSTEIN’S MOST FAMOUS EQUATION Einstein’s major 1905 paper in which he laid out the principles of special relativity, “On the Electrodynamics of Moving Bodies,” took up thirty pages in Annalen der Physik, the leading German scientific journal of the time. Soon thereafter, he published a twopage paper entitled “Does the Inertia of a Body Depend upon Its Energy Content?”69 The purpose of this paper was to point out a straightforward but interesting consequence of his longer work: The energy of an object at rest is proportional to its mass. (Mass and inertia are here being used interchangeably.) That’s the idea behind what is surely the most famous equation in history, E = mc2. Let’s think about this equation carefully, as it is often misunderstood. The factor c2 is of course the speed of light squared. Physicists learn to think, Aha, relativity must be involved, whenever they see the speed of light in an equation. The factor m is the mass of the object under consideration. In some places you might read about the “relativistic mass,” which increases when an object is in motion. That’s not really the most useful way of thinking about things; it’s better to consider m as the onceandforall mass that an object has when it is at rest. Finally, E is not exactly “the energy”; in this equation, it specifically plays the role of the energy of an object at rest. If an object is moving, its energy will certainly be higher. So Einstein’s famous equation tells us that the energy of an object when it is at rest is equal to its mass times the speed of light squared. Note the importance of the innocuous phrase an object. Not everything in the world is an object! For example, we’ve already spoken of dark energy, which is responsible for the acceleration of the universe. Dark energy doesn’t seem to be a collection of particles or other objects; it pervades spacetime smoothly. So as far as dark energy is concerned, E = mc2 simply doesn’t apply. Likewise, some objects (such as a photon) can never be at rest, since they are always moving at the speed of light. In those cases, again, the equation isn’t applicable. Everyone knows the practical implication of this equation: Even a small amount of mass is equivalent to a huge amount of energy. (The speed of light, in everyday units, is a really big number.) There are many forms of energy, and what special relativity is telling us is that mass is one form that energy can take. But the various forms can be converted back and forth into one another, which happens all the time. The domain of validity of E = mc2 isn’t limited to esoteric realms of nuclear physics or cosmology; it’s applicable to every kind of object at rest, on Mars or in your living room. If we take a piece of paper and burn it, letting the photons produced escape along with their energy, the resulting ashes will have a slightly lower mass (no matter how careful we are to capture all of them) than the combination of the original paper plus the oxygen it used to burn. E = mc2 isn’t just about atomic bombs; it’s a profound feature of the dynamics of energy all around us. 8 ENTROPY AND DISORDER Nobody can imagine in physical terms the act of reversing the order of time. Time is not reversible. —Vladimir Nabokov, Look at the Harlequins! Why is it that discussions of entropy and the Second Law of Thermodynamics so often end up being about food? Here are some popular (and tasty) examples of the increase of entropy in irreversible processes: • Breaking eggs and scrambling them. • Stirring milk into coffee. • Spilling wine on a new carpet. • The diffusion of the aroma of a freshly baked pie into a room. • Ice cubes melting in a glass of water. To be fair, not all of these are equally appetizing; the icecube example is kind of bland, unless you replace the water with gin. Furthermore, I should come clean about the scrambledeggs story. The truth is that the act of cooking the eggs in your skillet isn’t a straightforward demonstration of the Second Law; the cooking is a chemical reaction that is caused by the introduction of heat, which wouldn’t happen if the eggs weren’t an open system. Entropy comes into play when we break the eggs and whisk the yolks together with the whites; the point of cooking the resulting mixture is to avoid salmonella poisoning, not to illustrate thermodynamics. The relationship between entropy and food arises largely from the ubiquity of mixing. In the kitchen, we are often interested in combining together two things that had been kept separate—either two different forms of the same substance (ice and liquid water) or two altogether different ingredients (milk and coffee, egg whites and yolks). The original nineteenthcentury thermodynamicists were extremely interested in the dynamics of heat, and the melting ice cube would have been of foremost concern to them; they would have been less fascinated by processes where all the ingredients were at the same temperature, such as spilling wine onto a carpet. But clearly there is some underlying similarity in what is going on; an initial state in which substances are kept separate evolves into a final state in which they are mixed together. It’s easy to mix things and hard to unmix them—the arrow of time looms over everything we do in the kitchen. 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. The great insight of the pioneers of kinetic theory—Daniel Bernoulli in Swit zerland, Rudolf Clausius in Germany, James Clerk Maxwell and William Thomson in Great Britain, Ludwig Boltzmann in Austria, and Josiah Willard Gibbs in the United States—was to understand all liquids and gases in the same way we think of sand: as collections of very tiny pieces with persistent identities. Instead of grains, of course, we think of liquids and gases as composed of atoms and molecules. But the principle is the same. When milk and coffee mix, the individual milk molecules don’t combine with the individual coffee molecules to make some new kind of molecule; the two sets of molecules simply intermingle. Even heat is a property of atoms and molecules, rather than constituting some kind of fluid in its own right—the heat contained in an object is a measure of the energy of the rapidly moving molecules within it. When an ice cube melts into a glass of water, the molecules remain the same, but they gradually bump into one another and distribute their energy evenly throughout the molecules in the glass. Without (yet) being precise about the mathematical definition of “entropy,” the example of blending two kinds of colored sand illustrates why it is easier to mix things than to unmix them. Imagine a bowl of sand, with all of the blue grains on one side of the bowl and the red grains on the other. It’s pretty clear that this arrangement is somewhat delicate—if we disturb the bowl by shaking it or stirring with a spoon, the two colors will begin to mix together. If, on the other hand, we start with the two colors completely mixed, such an arrangement is robust—if we disturb the mixture, it will stay mixed. The reason is simple: To separate out two kinds of sand that are mixed together requires a much more precise operation than simply shaking or stirring. We would have to reach in carefully with tweezers and a magnifying glass to move all of the red grains to one side of the bowl and all of the blue grains to the other. It takes much more care to create the delicate unmixed state of sand than to create the robust mixed state. That’s a point of view that can be made fearsomely quantitative and scientific, which is exactly what Boltzmann and others managed to do in the 1870s. We’re going to dig into the guts of what they did, and explore what it explains and what it doesn’t, and how it can be reconciled with underlying laws of physics that are perfectly reversible. But it should already be clear that a crucial role is played by the large numbers of atoms that we find in macroscopic objects in the real world. If we had only one grain of red sand and one grain of blue sand, there would be no distinction between “mixed” and “unmixed.” In the last chapter we discussed how the underlying laws of physics work equally well forward or backward in time (suitably defined). That’s a microscopic description, in which we keep careful track of each and every constituent of a system. But very often in the real world, where large numbers of atoms are involved, we don’t keep track of nearly that much information. Instead, we make simplifications—thinking about the average color or temperature or pressure, rather than the specific position and momentum of each atom. When we think macroscopically, we forget (or ignore) detailed information about every particle—and that’s where entropy and irreversibility begin to come into play. SMOOTHING OUT The basic idea we want to understand is “how do macroscopic features of a system made of many atoms evolve as a consequence of the motion of the individual atoms?” (I’ll use “atoms” and “molecules” and “particles” more or less interchangeably, since all we care is that they are tiny things that obey reversible laws of physics, and that you need a lot of them to make something macroscopic.) In that spirit, consider a sealed box divided in two by a wall with a hole in it. Gas molecules can bounce around on one side of the box and will usually bounce right off the central wall, but every once in a while they will sneak through to the other side. We might imagine, for example, that the molecules bounce off the central wall 995 times out of 1,000, but onehalf of 1 percent of the time (each second, let’s say) they find the hole and move to the other side. Figure 41: A box of gas molecules, featuring a central partition with a hole. Every second, each molecule has a tiny chance to go through the hole to the other side. This example is pleasingly specific; we can examine a particular instance in detail and see what happens.123 Every second, each molecule on the left side of the box has a 99.5 percent chance of staying on that side, and a 0.5 percent chance of moving to the other side; likewise for the right side of the box. This rule is perfectly timereversal invariant; if you made a movie of the motion of just one particle obeying this rule, you couldn’t tell whether it was being run forward or backward in time. At the level of individual particles, we can’t distinguish the past from the future. In Figure 42 we have portrayed one possible evolution of such a box; time moves upward, as always. The box has 2,000 “air molecules” in it, and starts at time t = 1 with 1,600 molecules on the lefthand side and only 400 on the right. (You’re not supposed to ask why it starts that way—although later, when we replace “the box” with “the universe,” we will start asking such questions.) It’s not very surprising what happens as we sit there and let the molecules bounce around inside the box. Every second, there is a small chance that any particular molecule will switch sides; but, because we started with a much larger number of molecules on the one side, there is a general tendency for the numbers to even out. (Exactly like temperature, in Clausius’s formulation of the Second Law.) When there are more molecules on the left, the total number of molecules that shift from left to right will usually be larger than the number that shift from right to left. So after 50 seconds we see that the numbers are beginning to equal out, and after 200 seconds the distribution is essentially equal. Figure 42: Evolution of 2,000 molecules in a divided box of gas. We start with 1,600 molecules on the left, 400 on the right. After 50 seconds, there are about 1,400 on the left and 600 on the right; by the time 200 seconds have passed, the molecules are distributed equally between the two sides. This box clearly displays an arrow of time. Even if we hadn’t labeled the different distributions in the figure with the specific times to which they corresponded, most people wouldn’t have any trouble guessing that the bottom box came first and the top box came last. We’re not surprised when the air molecules even themselves out, but we’d be very surprised if they spontaneously congregated all (or even mostly) on one side of the box. The past is the direction of time in which things were more segregated, while the future is the direction in which they have smoothed themselves out. It’s exactly the same thing that happens when a teaspoon of milk spreads out into a cup of coffee. Of course, all of this is only statistical, not absolute. That is, it’s certainly possible that we could have started with an even distribution of molecules to the right and left, and just by chance a large number of them could jump to one side, leaving us with a very uneven distribution. As we’ll see, that’s unlikely, and it becomes more unlikely as we get more and more particles involved; but it’s something to keep in mind. For now, let’s ignore these very rare events and concentrate on the most likely evolution of the system. ENTROPY À LA BOLTZMANN We would like to do better than simply saying, “Yeah, it’s pretty obvious that the molecules will most likely move around until they are evenly distributed.” We want to be able to explain precisely why we have that expectation, and turn “evenly distributed” and “most likely” into rigorously quantitative statements. This is the subject matter of statistical mechanics. In the immortal words of Peter Venkman: “Back off, man, I’m a scientist.” Boltzmann’s first major insight was the simple appreciation that there are more ways for the molecules to be (more or less) evenly distributed through the box than there are ways for them to be all huddled on the same side. Imagine that we had numbered the individual molecules, 1 through 2,000. We want to know how many ways we can arrange things so that there are a certain number of molecules on the left and a certain number on the right. For example, how many ways are there to arrange things so that all 2,000 molecules are on the left, and zero on the right? There is only one way. We’re just keeping track of whether each molecule is on the left or on the right, not any details about its specific position or momentum, so we simply put every molecule on the left side of the box. But now let’s ask how many ways there are for there to be 1,999 molecules on the left and exactly 1 on the right. The answer is: 2,000 different ways—one for each of the specific molecules that could be the lucky one on the right side. If we ask how many ways there are to have 2 molecules on the right side, we find 1,999,000 possible arrangements. And when we get bold and consider 3 molecules on the right, with the other 1,997 on the left, we find 1,331,334,000 ways to make it happen.124 It should be clear that these numbers are growing rapidly: 2,000 is a lot bigger than 1, and 1,999,000 is a lot bigger than 2,000, and 1,331,334,000 is bigger still. Eventually, as we imagine moving more and more molecules to the right and emptying out the left, they would begin to go down again; after all, if we ask how many ways we can arrange things so that all 2,000 are on the right and zero are on the left, we’re back to only one unique way. The situation corresponding to the largest number of different possible arrangements is, unsurprisingly, when things are exactly balanced: 1,000 molecules on the left and 1,000 molecules on the right. In that case, there are—well, a really big number of ways to make that happen. We won’t write out the whole thing, but it’s approximately 2 × 10600 different ways; a 2 followed by 600 zeroes. And that’s with only 2,000 total particles. Imagine the number of possible arrangements of atoms we could find in a real roomful of air or even a glass of water. (Objects you can hold in your hand typically have about 6 × 1023 molecules in them—Avogadro’s Number.) The age of the universe is only about 4 × 1017 seconds, so you are welcome to contemplate how quickly you would have to move molecules back and forth before you explored every possible allowed combination. This is all very suggestive. There are relatively few ways for all of the molecules to be hanging out on the same side of the box, while there are very many ways for them to be distributed more or less equally—and we expect that a highly uneven distribution will evolve easily into a fairly even one, but not vice versa. But these statements are not quite the same. Boltzmann’s next step was to suggest that, if we didn’t know any better, we should expect systems to evolve from “special” configurations into “generic” ones—that is, from situations corresponding to a relatively small number of arrangements of the underlying particles, toward arrangements corresponding to a larger number of such arrangements. Boltzmann’s goal in thinking this way was to provide a basis in atomic theory for the Second Law of Thermodynamics, the statement that the entropy will always increase (or stay constant) in a closed system. The Second Law had already been formulated by Clausius and others, but Boltzmann wanted to derive it from some simple set of underlying principles. You can see how this statistical thinking leads us in the right direction—“systems tend to evolve from uncommon arrangements into common ones” bears a family resemblance to “systems tend to evolve from lo wentropy configurations into highentropy ones.” So we’re tempted to define “entropy” as “the number of ways we can rearrange the microscopic components of a system that will leave it macroscopically unchanged.” In our dividedbox example, that would correspond to the number of ways we could rearrange individual molecules that would leave the total number on each side unchanged. That’s almost right, but not quite. The pioneers of thermodynamics actually knew more about entropy than simply “it tends to go up.” For example, they knew that if you took two different systems and put them into contact next to each other, the total entropy would simply be the sum of the individual entropies of the two systems. Entropy is additive, just like the number of particles (but not, for example, like the temperature). But the number of rearrangements is certainly not additive; if you combine two boxes of gas, the number of ways you can rearrange the molecules between the two boxes is enormously larger than the number of ways you can rearrange them within each box. Boltzmann was able to crack the puzzle of how to define entropy in terms of microscopic rearrangements. We use the letter W—from the German Wahrscheinlichkeit , meaning “probability” or “likelihood”—to represent the number of ways we can rearrange the microscopic constituents of a system without changing its macroscopic appearance. Boltzmann’s final step was to take the logarithm of W and proclaim that the result is proportional to the entropy. The word logarithm sounds very highbrow, but it’s just a way to express how many digits it takes to express a number. If the number is a power of 10, its logarithm is just that power.125 So the logarithm of 10 is 1, the logarithm of 100 is 2, the logarithm of 1,000,000 is 6, and so on. In the Appendix, I discuss some of the mathematical niceties in more detail. But those niceties aren’t crucial to the bigger picture; if you just glide quickly past any appearance of the word logarithm, you won’t be missing much. You only really need to know two things: • As numbers get bigger, their logarithms get bigger. • But not very fast. The logarithm of a number grows slowly as the number itself gets bigger and bigger. One billion is much greater than 1,000, but 9 (the logarithm of 1 billion) is not much greater than 3 (the logarithm of 1,000). That last bit is a huge help, of course, when it comes to the gigantic numbers we are dealing with in this game. The number of ways to distribute 2,000 particles equally between two halves of a box is 2 × 10600, which is an unimaginably enormous quantity. But the logarithm of that number is just 600.3, which is relatively manageable. Boltzmann’s formula for the entropy, which is traditionally denoted by S (you wouldn’t have wanted to call it E, which usually stands for energy), states that it is equal to some constant k, cleverly called “Boltzmann’s constant,” times the logarithm of W, the number of microscopic arrangements of a system that are macroscopically indistinguishable.126 That is: S = k log W. This is, without a doubt, one of the most important equations in all of science—a triumph of nineteenthcentury physics, on a par with Newton’s codification of dynamics in the seventeenth century or the revolutions of relativity and quantum mechanics in the twentieth. If you visit Boltzmann’s grave in Vienna, you will find this equation engraved on his tombstone (see Chapter Two).127 Taking the logarithm does the trick, and Boltzmann’s formula leads to just the properties we think something called “entropy” should have—in particular, when you combine two systems, the total entropy is just the sum of the two entropies you started with. This deceptively simple equation provides a quantitative connection between the microscopic world of atoms and the macroscopic world we observe.128 BOX OF GAS REDUX As an example, we can calculate the entropy of the box of gas with a small hole in a divider that we illustrated in Figure 42. Our macroscopic observable is simply the total number of molecules on the left side or the right side. (We don’t know which particular molecules they are, nor do we know their precise coordinates and momenta.) The quantity W in this example is just the number of ways we could distribute the 2,000 total particles without changing the numbers on the left and right. If there are 2,000 particles on the left, W equals 1, and log W equals 0. Some of the other possibilities are listed in Table 1. Table 1: The number of arrangements W, and the logarithm of that number, corresponding to a divided box of 2,000 particles with some on the left side and some on the right side. In Figure 43 we see how the entropy, as defined by Boltzmann, changes in our box of gas. I’ve scaled things so that the maximum possible entropy of the box is equal to 1. It starts out relatively low, corresponding to the first configuration in Figure 42, where 1,600 molecules were on the left and only 400 on the right. As molecules gradually slip through the hole in the central divider, the entropy tends to increase. This is one particular example of the evolution; because our “law of physics” (each particle has a 0.5 percent chance of switching sides every second) involved probabilities, the details of any particular example will be slightly different. But it is overwhelmingly likely that the entropy will increase, as the system tends to wander into macroscopic configurations that correspond to larger numbers of microscopic arrangements. The Second Law of Thermodynamics in action. So this is the origin of the arrow of time, according to Boltzmann and his friends. We start with a set of microscopic laws of physics that are timereversal invariant: They don’t distinguish between past and future. But we deal with systems featuring large numbers of particles, where we don’t keep track of every detail necessary to fully specify the state of the system; instead, we keep track of some observable macroscopic features. The entropy characterizes (by which we mean, “is proportional to the logarithm of”) the number of microscopic states that are macroscopically indistinguishable. Under the reasonable assumption that the system will tend to evolve toward the macroscopic configurations that correspond to a large number of possible states, it’s natural that entropy will increase with time. In particular, it would be very surprising if it spontaneously decreased. The arrow of time arises because the system (or the universe) naturally evolves from rare configurations into more common configurations as time goes by. Figure 43: The evolution of the entropy of a divided box of gas. The gas starts with most molecules on the left, and the distribution evens out in time, as we saw in Figure 42. The entropy correspondingly rises, as there are more ways for the molecules to be distributed evenly than to be mostly on one side or the other. For convenience we have plotted the entropy in terms of the maximum entropy, so the maximum value attainable on this plot is 1. All of this seems superficially plausible and will turn out to be basically true. But along the way we made some “reasonable” leaps of logic, which deserve more careful examination. For the rest of this chapter we will bring to light the various assumptions that go into Boltzmann’s way of thinking about entropy, and try to decide just how plausible they are. USEFUL AND USELESS ENERGY One interesting feature of this boxofgas example is that the arrow of time is only temporary. After the gas has had a chance to even itself out (at around time 150 in Figure 43), nothing much happens anymore. Individual molecules will continue to bounce between the right and left sides of the box, but these will tend to average out, and the system will spend almost all of its time with approximately equal numbers of molecules on each side. Those are the kinds of configurations that correspond to the largest number of rearrangements of the individual molecules, and correspondingly have the highest entropy the system can possibly have. A system that has the maximum entropy it can have is in equilibrium. Once there, the system basically has nowhere else to go; it’s in the kind of configuration that is most natural for it to be in. Such a system has no arrow of time, as the entropy is not increasing (or decreasing). To a macroscopic observer, a system in equilibrium appears static, not changing at all. Richard Feynman, in The Character of Physical Law, tells a story that illustrates the concept of equilibrium.129 Imagine you are sitting on a beach when you are suddenly hit with a tremendous downpour of rain. You’ve brought along a towel, but that also gets wet as you dash to cover. Once you’ve reached some cover, you start to dry yourself with your towel. It works for a little while because the towel is a bit drier than you are, but soon you find that the towel has gotten so wet that rubbing yourself with it is keeping you wet just as fast as it’s making you dry. You and the towel have reached “wetness equilibrium,” and it can’t make you any drier. Your situation maximizes the number of ways the water molecules can arrange themselves on you and the towel.130 Once you’ve reached equilibrium, the towel is no longer useful for its intended purpose (drying you off). Note that the total amount of water doesn’t change as you dry yourself off; it is simply transferred from you to the towel. Similarly, the total energy doesn’t change in a box of gas that is isolated from the rest of the world; energy is conserved, at least in circumstances where we can neglect the expansion of space. But energy can be arranged in more or less useful ways. When energy is arranged in a lowentropy configuration, it can be harnessed to perform useful work, like propelling a vehicle. But the same amount of energy, when it’s in an equilibrium configuration, is completely useless, just like a towel that is in wetness equilibrium with you. Entropy measures the uselessness of a configuration of energy.131 Consider our divided box once again. But instead of the divider being a fixed wall with a hole in it, passively allowing molecules to move back and forth, imagine that the divider is movable, and hooked up to a shaft that reaches outside the box. What we’ve constructed is simply a piston, which can be used to do work under the right circumstances. In Figure 44 we’ve depicted two different situations for our piston. The top row shows a piston in the presence of a lowentropy configuration of some gas—all the molecules on one side of the divider—while the bottom row shows a highentropy configuration—equal amounts of gas on both sides. The total number of molecules, and the total amount of energy, is assumed to be the same in both cases; the only difference is the entropy. But it’s clear that what happens in the two cases is very different. In the top row, the gas is all on the left side of the piston, and the force of the molecules bumping into it exerts pressure that pushes the piston to the right until the gas fills the container. The moving piston shaft can be used to do useful work—run a flywheel or some such thing, at least for a little while. That extracts energy from the gas; at the end of the process, the gas will have a lower temperature. (The pistons in your car engine operate in exactly this way, expanding and cooling the hot vapor created by igniting vaporized gasoline, performing the useful work of moving your car.) Figure 44: Gas in a divided box, used to drive a cylinder. On the top, gas in a lowentropy state pushes the cylinder to the right, doing useful work. On the bottom, gas in a highentropy state doesn’t push the cylinder in either direction. On the bottom row in the figure, meanwhile, we imagine starting with the same amount of energy in the gas but in an initial state with a much higher entropy—an equal number of particles on each side of the divider. High entropy implies equilibrium, which implies that the energy is useless, and indeed we see that our piston isn’t going anywhere. The pressure from gas on one side of the divider is exactly canceled by pressure coming from the other side. The gas in this box has the same total energy as the gas in the upper left box, but in this case we can’t harness that energy to make the piston move to do something useful. This helps us understand the relationship between Boltzmann’s viewpoint on entropy and that of Rudolf Clausius, who first formulated the Second Law. Remember that Clausius and his predecessors didn’t think of entropy in terms of atoms at all; they thought of it as an autonomous substance with its own dynamics. Clausius’s original version of the Second Law didn’t even mention entropy; it was the simple statement that “heat never flows spontaneously from a colder object to a hotter one.” If we put two objects with different temperatures into contact with each other, they will both evolve toward a common middle temperature; if we put two objects with the same temperature into contact with each other, they will simply stay that way. (They’re in thermal equilibrium.) From the point of atoms, this all makes sense. Consider the classic example of two objects at different temperatures in contact with each other: an ice cube in a glass of warm water, discussed at the end of the previous chapter. Both the ice cube and the liquid are made of precisely the same kind of molecules, namely H2O. The only difference is that the ice is at a much lower temperature. Temperature, as we have discussed, measures the average energy of motion in the molecules of a substance. So while the molecules of the liquid water are moving relatively quickly, the molecules in the ice are moving slowly. But that kind of condition—one set of molecules moving quickly, another moving slowly—isn’t all that different, conceptually, from two sets of molecules confined to different sides of a box. In either case, there is a broadbrush limitation on how we can rearrange things. If we had just a glass of nothing but water at a constant temperature, we could exchange the molecules in one part of the glass with molecules in some other part, and there would be no macroscopic way to tell the difference. But when we have an ice cube, we can’t simply exchange the molecules in the cube for some water molecules elsewhere in the glass—the ice cube would move, and we would certainly notice that even from our everyday macroscopic perspective. The division of the water molecules into “liquid” and “ice” puts a serious constraint on the number of rearrangements we can do, so that configuration has a low entropy. As the temperature between the water molecules that started out as ice equilibrates with that of the rest of the glass, the entropy goes up. Clausius’s rule that temperatures tend to even themselves out, rather than spontaneously flowing from cold to hot, is precisely equivalent to the statement that the entropy as defined by Boltzmann never decreases in a closed system. None of this means that it’s impossible to cool things down, of course. But in everyday life, where most things around us are at similar temperatures, it takes a bit more ingenuity than heating them up. A refrigerator is a more complicated machine than a stove. (Refrigerators work on the same basic principle as the piston in Figure 44, expanding a gas to extract energy and cool it off.) When Grant Achatz, chef of Chicago’s Alinea restaurant, wanted a device that would rapidly freeze food in the same way a frying pan rapidly heats food up, he had to team with culinary technologist Philip Preston to create their own. The result is the “antigriddle,” a microwaveovensized machine with a metallic top that attains a temperature of 34 degrees Celsius. Hot purees and sauces, poured on the antigriddle, rapidly freeze on the bottom while remaining soft on the top. We have understood the basics of thermodynamics for a long time now, but we’re still inventing new ways to put them to good use. DON’T SWEAT THE DETAILS You’re out one Friday night playing pool with your friends. We’re talking about realworld pool now, not “physicist pool” where we can ignore friction and noise.132 One of your pals has just made an impressive break, and the balls have scattered thoroughly across the table. As they come to a stop and you’re contemplating your next shot, a stranger walks by and exclaims, “Wow! That’s incredible!” Somewhat confused, you ask what is so incredible about it. “Look at these balls at those exact positions on the table! What are the chances that you’d be able to put all the balls in precisely those spots? You’d never be able to repeat that in a million years!” The mysterious stranger is a bit crazy—probably driven slightly mad by reading too many philosophical tracts on the foundations of statistical mechanics. But she does have a point. With several balls on the table, any particular configuration of them is extremely unlikely. Think of it this way: If you hit the cue ball into a bunch of randomly placed balls, which rattled around before coming to rest in a perfect arrangement as if they had just been racked, you’d be astonished. But that particular arrangement (all balls perfectly arrayed in the starting position) is no more or less unusual than any other precise arrangement of the balls.133 What right do we have to single out certain configurations of the billiard balls as “astonishing” or “unlikely,” while others seem “unremarkable” or “random”? This example pinpoints a question at the heart of Boltzmann’s definition of entropy and the associated understanding of the Second Law of Thermodynamics: Who decides when two specific microscopic states of a system look the same from our macroscopic point of view? Boltzmann’s formula for entropy hinges on the idea of the quantity W, which we defined as “the number of ways we can rearrange the microscopic constituents of a system without changing its macroscopic appearance.” In the last chapter we defined the “state” of a physical system to be a complete specification of all the information required to uniquely evolve it in time; in classical mechanics, it would be the position and momentum of every single constituent particle. Now that we are considering statistical mechanics, it’s useful to use the term microstate to refer to the precise state of a system, in contrast with the macrostate, which specifies only those features that are macroscopically observable. Then the shorthand definition of W is “the number of microstates corresponding to a particular macrostate.” For the box of gas separated in two by a divider, the microstate at any one time is the position and momentum of every single molecule in the box. But all we were keeping track of was how many molecules were on the left, and how many were on the right. Implicitly, every division of the molecules into a certain number on the left and a certain number on the right defined a “macrostate” for the box. And our calculation of W simply counted the number of microstates per macrostate.134 The choice to just keep track of how many molecules were in each half of the box seemed so innocent at the time. But we could imagine keeping track of much more. Indeed, when we deal with the atmosphere in an actual room, we keep track of a lot more than simply how many molecules are on each side of the room. We might, for example, keep track of the temperature, and density, and pressure of the atmosphere at every point, or at least at some finite number of places. If there were more than one kind of gas in the atmosphere, we might separately keep track of the density and so on for each different kind of gas. That’s still enormously less information than the position and momentum of every molecule in the room, but the choice of which information to “keep” as a macroscopically measurable quantity and which information to “forget” as an irrelevant part of the microstate doesn’t seem to be particularly well defined. The process of dividing up the space of microstates of some particular physical system (gas in a box, a glass of water, the universe) into sets that we label “macroscopically indistinguishable” is known as coarsegraining. It’s a little bit of black magic that plays a crucial role in the way we think about entropy. In Figure 45 we’ve portrayed how coarsegraining works; it simply divides up the space of all states of a system into regions (macrostates) that are indistinguishable by macroscopic observations. Every point within one of those regions corresponds to a different microstate, and the entropy associated with a given microstate is proportional to the logarithm of the area (or really volume, as it’s a very highdimensional space) of the macrostate to which it belongs. This kind of figure makes it especially clear why entropy tends to go up: Starting from a state with low entropy, corresponding to a very tiny part of the space of states, it’s only to be expected that an ordinary system will tend to evolve to states that are located in one of the largevolume, highentropy regions. Figure 45 is not to scale; in a real example, the lowentropy macrostates would be much smaller compared to the highentropy macrostates. As we saw with the dividedbox example, the number of microstates corresponding to highentropy macrostates is enormously larger than the number associated with lowentropy macrostates. Starting with low entropy, it’s certainly no surprise that a system should wander into the roomier highentropy parts of the space of states; but starting with high entropy, a typical system can wander for a very long time without ever bumping into a lowentropy condition. That’s what equilibrium is like; it’s not that the microstate is truly static, but that it never leaves the highentropy macrostate it’s in. Figure 45: The process of coarsegraining consists of dividing up the space of all possible microstates into regions considered to be macroscopically indistinguishable, which are called macrostates. Each macrostate has an associated entropy, proportional to the logarithm of the volume it takes up in the space of states. The size of the lowentropy regions is exaggerated for clarity; in reality, they are fantastically smaller than the highentropy regions. This whole business should strike you as just a little bit funny. Two microstates belong to the same macrostate when they are macroscopically indistinguishable. But that’s just a fancy way of saying, “when we can’t tell the difference between them on the basis of macroscopic observations.” It’s the appearance of “we” in that statement that should make you nervous. Why should our powers of observation be involved in any way at all? We like to think of entropy as a feature of the world, not as a feature of our ability to perceive the world. Two glasses of water are in the same macrostate if they have the same temperature throughout the glass, even if the exact distribution of positions and momenta of the water molecules are different, because we can’t directly measure all of that information. But what if we ran across a race of superobservant aliens who could peer into a glass of liquid and observe the position and momentum of every molecule? Would such a race think that there was no such thing as entropy? There are several different answers to these questions, none of which is found satisfactory by everyone working in the field of statistical mechanics. (If any of them were, you would need only that one answer.) Let’s look at two of them. The first answer is, it really doesn’t matter. That is, it might matter a lot to you how you bundle up microstates into macrostates for the purposes of the particular physical situation in front of you, but it ultimately doesn’t matter if all we want to do is argue for the validity of something like the Second Law. From Figure 45, it’s clear why the Second Law should hold: There is a lot more room corresponding to highentropy states than to lowentropy ones, so if we start in the latter it is natural to wander into the former. But that will hold true no matter how we actually do the coarsegraining. The Second Law is robust; it depends on the definition of entropy as the logarithm of a volume within the space of states, but not on the precise way in which we choose that volume. Nevertheless, in practice we do make certain choices and not others, so this transparent attempt to avoid the issue is not completely satisfying. The second answer is that the choice of how to coarsegrain is not completely arbitrary and socially constructed, even if some amount of human choice does come into the matter. The fact is, we coarsegrain in ways that seem physically natural, not just chosen at whim. For example, when we keep track of the temperature and pressure in a glass of water, what we’re really doing is throwing away all information that we could measure only by looking through a microscope. We’re looking at average properties within relatively small regions of space because that’s what our senses are actually able to do. Once we choose to do that, we are left with a fairly welldefined set of macroscopically observable quantities. Averaging within small regions of space isn’t a procedure that we hit upon randomly, nor is it a peculiarity of our human senses as opposed to the senses of a hypothetical alien; it’s a very natural thing, given how the laws of physics work.135 When I look at cups of coffee and distinguish between cases where a teaspoon of milk has just been added and ones where the milk has become thoroughly mixed, I’m not pulling a random coarsegraining of the states of the coffee out of my hat; that’s how the coffee looks to me, immediately and phenomenologically. So even though in principle our choice of how to coarsegrain microstates into macrostates seems absolutely arbitrary, in practice Nature hands us a very sensible way to do it. RUNNING ENTROPY BACKWARD A remarkable consequence of Boltzmann’s statistical definition of entropy is that the Second Law is not absolute—it just describes behavior that is overwhelmingly likely. If we start with a mediumentropy macrostate, almost all microstates within it will evolve toward higher entropy in the future, but a small number will actually evolve toward lower entropy. It’s easy to construct an explicit example. Consider a box of gas, in which the gas molecules all happened to be bunched together in the middle of the box in a lo wentropy configuration. If we just let it evolve, the molecules will move around, colliding with one another and with the walls of the box, and ending up (with overwhelming probability) in a much higherentropy configuration. Now consider a particular microstate of the above box of gas at some moment after it has become highentropy. From there, construct a new state by keeping all of the molecules at exactly the same positions, but precisely reversing all of the velocities. The resulting state still has a high entropy—it’s contained within the same macrostate as we started with. (If someone suddenly reversed the direction of motion of every single molecule of air around you, you’d never notice; on average there are equal numbers moving in every direction.) Starting in this state, the motion of the molecules will exactly retrace the path that they took from the previous lowentropy state. To an external observer, it will look as if the entropy is spontaneously decreasing. The fraction of highentropy states that have this peculiar property is