The Fabric of the Cosmos: Space, Time, and the Texture of Reality

by Brian Greene

  Thus, whether an event is simple, like a tennis ball arcing, or something more complex, like an egg splattering, the laws of physics show that what happens in one temporal direction can, at least in principle, also happen in reverse.

  Principle and Practice

  The stories of the tennis ball and the egg do more than illustrate the time-reversal symmetry of nature's laws. They also suggest why, in the real world of experience, we see many things happen one way but never in reverse. To get the tennis ball to retrace its path was not that hard. We grabbed it and sent it off with the same speed but in the opposite direction. That's it. But to get all the chaotic detritus of the egg to retrace its path would be monumentally more difficult. We'd need to grab every bit of splatter, and simultaneously send each off at the same speed but in the opposite direction. Clearly, that's beyond what we (or even all the King's horses and all the King's men) can really do.

  Have we found the answer we've been looking for? Is the reason why eggs splatter but don't unsplatter, even though both actions are allowed by the laws of physics, a matter of what is and isn't practical? Is the answer simply that it's easy to make an egg splatter—roll it off a counter—but extraordinarily difficult to make it unsplatter?

  Well, if it were the answer, trust me, I wouldn't have made it into such a big deal. The issue of ease versus difficulty is an essential part of the answer, but the full story within which it fits is far more subtle and surprising. We'll get there in due course, but we must first make the discussion of this section a touch more precise. And that takes us to the concept of entropy.

  Entropy

  Etched into a tombstone in the Zentralfriedhof in Vienna, near the graves of Beethoven, Brahms, Schubert, and Strauss, is a single equation, S = k log W, which expresses the mathematical formulation of a powerful concept known as entropy. The tombstone bears the name of Ludwig Boltzmann, one of the most insightful physicists working at the turn of the last century. In 1906, in failing health and suffering from depression, Boltzmann committed suicide while vacationing with his wife and daughter in Italy. Ironically, just a few months later, experiments began to confirm that ideas Boltzmann had spent his life passionately defending were correct.

  The notion of entropy was first developed during the industrial revolution by scientists concerned with the operation of furnaces and steam engines, who helped develop the field of thermodynamics. Through many years of research, the underlying ideas were sharply refined, culminating in Boltzmann's approach. His version of entropy, expressed concisely by the equation on his tombstone, uses statistical reasoning to provide a link between the huge number of individual ingredients that make up a physical system and the overall properties the system has. 4

  To get a feel for the ideas, imagine unbinding a copy of War and Peace, throwing its 693 double-sided pages high into the air, and then gathering the loose sheets into a neat pile. 5 When you examine the resulting stack, it is enormously more likely that the pages will be out of order than in order. The reason is obvious. There are many ways in which the order of the pages can be jumbled, but only one way for the order to be correct. To be in order, of course, the pages must be arranged precisely as 1, 2; 3, 4; 5, 6; and so on, up to 1,385, 1,386. Any other arrangement is out of order. A simple but essential observation is that, all else being equal, the more ways something can happen, the more likely it is that it will happen. And if something can happen in enormously more ways, like the pages landing in the wrong numerical order, it is enormously more likely that it will happen. We all know this intuitively. If you buy one lottery ticket, there is only one way you can win. If you buy a million tickets, each with different numbers, there are a million ways you can win, so your chances of striking it rich are a million times higher.

  Entropy is a concept that makes this idea precise by counting the number of ways, consistent with the laws of physics, in which any given physical situation can be realized. High entropy means that there are many ways; low entropy means there are few ways. If the pages of War and Peace are stacked in proper numerical order, that is a low-entropy configuration, because there is one and only one ordering that meets the criterion. If the pages are out of numerical order, that is a high-entropy situation, because a little calculation shows that there are 1245521984537783433660029353704988291633611012463890451368 8769126468689559185298450437739406929474395079418933875187 6527656714059286627151367074739129571382353800016108126465 3018234205620571473206172029382902912502131702278211913473 5826558815410713601431193221575341597338554284672986913981 5159925119085867260993481056143034134383056377136715110570 4786941333912934192440961051428879847790853609508954014012 5932850632906034109513149466389839052676761042780416673015 4945522818861025024633866260360150888664701014297085458481 5141598392546876231295293347829518681237077459652243214888 7351679284483403000787170636684623843536242451673622861091 9853939181503076046890466491297894062503326518685837322713 6370247390401891094064988139838026545111487686489581649140 3426444110871911844164280902757137738090672587084302157950 1589916232045813012950834386537908191823777738521437536312 2531641598589268105976528144801387748697026525462643937189 3927305921796747169166978155198569769269249467383642278227 3345776718073316240433636952771183674104284493472234779223 4027225630721193853912472880929072034271692377936207650190 4571097887744535443586803319160959249877443194986997700333 2494630732437553532290674481765795395621840329516814427104 2227608124289048716428664872403070364864934832509996672897 3446425310349300626622014604312051101093282396249251196897 8283306192150828270814393659987326849047994166839657747890 2124562796195600187060805768778947870098610692265944872693 4100008726998763399003025591685820639734851035629676461160 0225159200113722741273318074829547248192807653266407023083 2754286312646671501355905966429773337131834654748547607012 4233012872135321237328732721874825264039911049700172147564 7004992922645864352265011199999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999 99999999999999999999999—about 10 1878 —different out-of-order page arrangements. 6 If you throw the pages in the air and then gather them in a neat stack, it is almost certain that they will wind up out of numerical order, because such configurations have enormously higher entropy— there are many more ways to achieve an out-of-order outcome—than the sole arrangement in which they are in correct numerical order.

  In principle, we could use the laws of classical physics to figure out exactly where each page will land after the whole stack has been thrown in the air. So, again in principle, we could precisely predict the resulting arrangement of the pages 7 and hence (unlike in quantum mechanics, which we ignore until the next chapter) there would seem to be no need to rely on probabilistic notions such as which outcome is more or less likely than another. But statistical reasoning is both powerful and useful. If War and Peace were a pamphlet of only a couple of pages we just might be able to successfully complete the necessary calculations, but it would be impossible to do this for the real War and Peace. 8 Following the precise motion of 693 floppy pieces of paper as they get caught by gentle air currents and rub, slide, and flap against one another would be a monumental task, well beyond the capacity of even the most powerful supercomputer.

  Moreover—and this is critical—having the exact answer wouldn't even be that useful. When you examine the resulting stack of pages, you are far less interested in the exact details of which page happens to be where than you are in the general question of whether the pages are in the correct order. If they are, great. You could sit down and continue reading about Anna Pavlovna and Nikolai Ilych Rostov, as usual. But if you found that the pages were not in their correct order, the precise details of the page arrangement are something you'd probably care little about. If you've seen one disordered page arrangement, you've pretty much seen them all. Unless for some strange reason you get mired in the minutiae of which pages happen to appear here or there in the stack, you'd hardly notice if someone further jumbled an out-of-order page arrangement you'd initially been given. The initi
al stack would look disordered and the further jumbled stack would also look disordered. So not only is the statistical reasoning enormously easier to carry out, but the answer it yields— ordered versus disordered—is more relevant to our real concern, to the kind of thing of which we would typically take note.

  This sort of big-picture thinking is central to the statistical basis of entropic reasoning. Just as any lottery ticket has the same chance of winning as any other, after many tosses of War and Peace any particular ordering of the pages is just as likely to occur as any other. What makes the statistical reasoning fly is our declaration that there are two interesting classes of page configurations: ordered and disordered. The first class has one member (the correct page ordering 1, 2; 3, 4; and so on) while the second class has a huge number of members (every other possible page ordering). These two classes are a sensible set to use since, as above, they capture the overall, gross assessment you'd make on thumbing through any given page arrangement.

  Even so, you might suggest making finer distinctions between these two classes, such as arrangements with just a handful of pages out of order, arrangements with only pages in the first chapter out of order, and so on. In fact, it can sometimes be useful to consider these intermediate classes. However, the number of possible page arrangements in each of these new subclasses is still extremely small compared with the number in the fully disordered class. For example, the total number of out-of-order arrangements that involve only the pages in Part One of War and Peace is 10 -178 of 1 percent of the total number of out-of-order arrangements involving all pages. So, although on the initial tosses of the unbound book the resulting page arrangement will likely belong to one of the intermediate, not fully disordered classes, it is almost certain that if you repeat the tossing action many times over, the page order will ultimately exhibit no obvious pattern whatsoever. The page arrangement evolves toward the fully disordered class, since there are so many page arrangements that fit this bill.

  The example of War and Peace highlights two essential features of entropy. First, entropy is a measure of the amount of disorder in a physical system. High entropy means that many rearrangements of the ingredients making up the system would go unnoticed, and this in turn means the system is highly disordered (when the pages of War and Peace are all mixed up, any further jumbling will hardly be noticed since it simply leaves the pages in a mixed-up state). Low entropy means that very few rearrangements would go unnoticed, and this in turn means the system is highly ordered (when the pages of War and Peace start in their proper order, you can easily detect almost any rearrangement). Second, in physical systems with many constituents (for instance, books with many pages being tossed in the air) there is a natural evolution toward greater disorder, since disorder can be achieved in so many more ways than order. In the language of entropy, this is the statement that physical systems tend to evolve toward states of higher entropy.

  Of course, in making the concept of entropy precise and universal, the physics definition does not involve counting the number of page rearrangements of one book or another that leave it looking the same, either ordered or disordered. Instead, the physics definition counts the number of rearrangements of fundamental constituents—atoms, subatomic particles, and so on—that leave the gross, overall, "big-picture" properties of a given physical system unchanged. As in the example of War and Peace, low entropy means that very few rearrangements would go unnoticed, so the system is highly ordered, while high entropy means that many rearrangements would go unnoticed, and that means the system is very disordered. 12

  For a good physics example, and one that will shortly prove handy, let's think about the bottle of Coke referred to earlier. When gas, like the carbon dioxide that was initially confined in the bottle, spreads evenly throughout a room, there are many rearrangements of the individual molecules that will have no noticeable effect. For example, if you flail your arms, the carbon dioxide molecules will move to and fro, rapidly changing positions and velocities. But overall, there will be no qualitative effect on their arrangement. The molecules were spread uniformly before you flailed your arms, and they will be spread uniformly after you're done. The uniformly spread gas configuration is insensitive to an enormous number of rearrangements of its molecular constituents, and so is in a state of high entropy. By contrast, if the gas were spread in a smaller space, as when it was in the bottle, or confined by a barrier to a corner of the room, it has significantly lower entropy. The reason is simple. Just as thinner books have fewer page reorderings, smaller spaces provide fewer places for molecules to be located, and so allow for fewer rearrangements.

  But when you twist off the bottle's cap or remove the barrier, you open up a whole new universe to the gas molecules, and through their bumping and jostling they quickly disperse to explore it. Why? It's the same statistical reasoning as with the pages of War and Peace. No doubt, some of the jostling will move a few gas molecules purely within the initial blob of gas or nudge a few that have left the blob back toward the initial dense gas cloud. But since the volume of the room exceeds that of the initial cloud of gas, there are many more rearrangements available to the molecules if they disperse out of the cloud than there are if they remain within it. On average, then, the gas molecules will diffuse from the initial cloud and slowly approach the state of being spread uniformly throughout the room. Thus, the lower-entropy initial configuration, with the gas all bunched in a small region, naturally evolves toward the higher-entropy configuration, with the gas uniformly spread in the larger space. And once it has reached such uniformity, the gas will tend to maintain this state of high entropy: bumping and jostling still causes the molecules to move this way and that, giving rise to one rearrangement after another, but the overwhelming majority of these rearrangements do not affect the gross, overall appearance of the gas. That's what it means to have high entropy. 9

  In principle, as with the pages of War and Peace, we could use the laws of classical physics to determine precisely where each carbon dioxide molecule will be at a given moment of time. But because of the enormous number of CO 2 molecules—about 10 24 in a bottle of Coke—actually carrying out such calculations is practically impossible. And even if, somehow, we were able to do so, having a list of a million billion billion particle positions and velocities would hardly give us a sense of how the molecules were distributed. Focusing on big-picture statistical features— is the gas spread out or bunched up, that is, does it have high or low entropy?—is far more illuminating.

  Entropy, the Second Law, and the Arrow of Time

  The tendency of physical systems to evolve toward states of higher entropy is known as the second law of thermodynamics. (The first law is the familiar conservation of energy.) As above, the basis of the law is simple statistical reasoning: there are more ways for a system to have higher entropy, and "more ways" means it is more likely that a system will evolve into one of these high-entropy configurations. Notice, though, that this is not a law in the conventional sense since, although such events are rare and unlikely, something can go from a state of high entropy to one of lower entropy. When you toss a jumbled stack of pages into the air and then gather them into a neat pile, they can turn out to be in perfect numerical order. You wouldn't want to place a high wager on its happening, but it is possible. It is also possible that the bumping and jostling will be just right to cause all the dispersed carbon dioxide molecules to move in concert and swoosh back into your open bottle of Coke. Don't hold your breath waiting for this outcome either, but it can happen. 10

  The large number of pages in War and Peace and the large number of gas molecules in the room are what makes the entropy difference between the disordered and ordered arrangements so huge, and what causes low-entropy outcomes to be so terribly unlikely. If you tossed only two double-sided pages in the air over and over again, you'd find that they landed in the correct order about 12.5 percent of the time. With three pages this would drop to about 2 percent of the tosses, with four pages it's a
bout .3 percent, with five pages it's about .03 percent, with six pages it's about .002 percent, with ten pages it's .000000027 percent, and with 693 pages the percentage of tosses that would yield the correct order is so small—it involves so many zeros after the decimal point—that I've been convinced by the publisher not to use another page to write it out explicitly. Similarly, if you dropped only two gas molecules side by side into an empty Coke bottle, you'd find that at room temperature their random motion would bring them back together (within a millimeter of each other), on average, roughly every few seconds. But for a group of three molecules, you'd have to wait days, for four molecules you'd have to wait years, and for an initial dense blob of a million billion billion molecules it would take a length of time far greater than the current age of the universe for their random, dispersive motion to bring them back together into a small, ordered bunch. With more certainty than death and taxes, we can count on systems with many constituents evolving toward disorder.