The Fabric of the Cosmos: Space, Time, and the Texture of Reality
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A second and related insight is that the increasingly intense quantum jitters that arise on decreasing scales suggest that the notion of being able to divide distances or durations into ever smaller units likely comes to an end at around the Planck length (10 -33 centimeters) and Planck time (10 -43 seconds). We encountered this idea in Chapter 12, where we emphasized that, although the notion is thoroughly at odds with our usual experiences of space and time, it is not particularly surprising that a property relevant to the everyday fails to survive when pushed into the micro-realm. And since the arbitrary divisibility of space and time is one of their most familiar everyday properties, the inapplicability of this concept on ultrasmall scales gives another hint that there is something else lurking in the microdepths—something that might be called the bare-bones substrate of spacetime—the entity to which the familiar notion of spacetime alludes. We expect that this ur -ingredient, this most elemental spacetime stuff, does not allow dissection into ever smaller pieces because of the violent fluctuations that would ultimately be encountered, and hence is quite unlike the large-scale spacetime we directly experience. It seems likely, therefore, that the appearance of the fundamental spacetime constituents—whatever they may be—is altered significantly through the averaging process by which they yield the spacetime of common experience.
Thus, looking for familiar spacetime in the deepest laws of nature may be like trying to take in Beethoven's Ninth Symphony solely note by single note or one of Monet's haystack paintings solely brushstroke by single brushstroke. Like these masterworks of human expression, nature's spacetime whole may be so different from its parts that nothing resembling it exists at the most fundamental level.
Geometry in Translation
Another consideration, one physicists call geometrical duality, also suggests that spacetime may not be fundamental, but suggests it from a very different viewpoint. Its description is a little more technical than quantum averaging, so feel free to go into skim mode if at any point this section gets too heavy. But because many researchers consider this material to be among string theory's most emblematic features, it's worth trying to get the gist of the ideas.
In Chapter 13 we saw how the five supposedly distinct string theories are actually different translations of one and the same theory. Among other things, we emphasized that this is a powerful realization because, when translated, supremely difficult questions sometimes become far simpler to answer. But there is a feature of the translation dictionary unifying the five theories that I've so far neglected to mention. Just as a question's degree of difficulty can be changed radically by the translation from one string formulation to another, so, too, can the description of the geometrical form of spacetime. Here's what I mean.
Because string theory requires more than the three space dimensions and one time dimension of common experience, we were motivated in Chapters 12 and 13 to take up the question of where the extra dimensions might be hiding. The answer we found is that they may be curled up into a size that, so far, has eluded detection because it's smaller than we are able to probe experimentally. We also found that physics in our familiar big dimensions is dependent on the precise size and shape of the extra dimensions because their geometrical properties affect the vibrational patterns strings can execute. Good. Now for the part I left out.
The dictionary that translates questions posed in one string theory into different questions posed in another string theory also translates the geometry of the extra dimensions in the first theory into a different extra-dimensionalgeometry in the second theory. If, for example, you are studying the physical implications of, say, the Type IIA string theory with extra dimensions curled up into a particular size and shape, then every conclusion you reach can, at least in principle, be deduced by considering appropriately translated questions in, say, the Type IIB string theory. But the dictionary for carrying out the translation demands that the extra dimensions in the Type IIB string theory be curled up into a precise geometrical form that depends on —but generally differs from— the form given by the Type IIA theory. In short, a given string theory with curled-up dimensions in one geometrical form is equivalent to—is a translation of— another string theory with curled-up dimensions in a different geometrical form.
And the differences in spacetime geometry need not be minor. For example, if one of the extra dimensions of, say, the Type IIA string theory should be curled up into a circle, as in Figure 12.7, the translation dictionary shows that this is absolutely equivalent to the Type IIB string theory with one of its extra dimensions also curled up into a circle, but one whose radius is inversely proportional to the original. If one circle is tiny, the other is big, and vice versa—and yet there is absolutely no way to distinguish between the two geometries. (Expressing lengths as multiples of the Planck length, if one circle has radius R, the mathematical dictionary shows that the other circle has radius 1/ R ). You might think that you could easily and immediately distinguish between a big and a small dimension, but in string theory this is not always the case. All observations derive from the interactions of strings, and these two theories, the Type IIA with a big circular dimension and the Type IIB with a small circular dimension, are merely different translations of—different ways of expressing—the same physics. Every observation you describe within one string theory has an alternative and equally viable description within the other string theory, even though the language of each theory and the interpretation it gives may differ. (This is possible because there are two qualitatively different configurations for strings moving on a circular dimension: those in which the string is wrapped around the circle like a rubber band around a tin can, and those in which the string resides on a portion of the circle but does not wrap around it. The former have energies that are proportional to the radius of the circle [the larger the radius, the longer the wrapped strings are stretched, so the more energy they embody], while the latter have energies that are inversely proportional to the radius [the smaller the radius, the more hemmed in the strings are, so the more energetically they move because of quantum uncertainty]. Notice that if we were to replace the original circle by one of inverted radius, while also exchanging "wrapped" and "not wrapped" strings, physical energies—and, it turns out, physics more generally—would remain unaffected. This is exactly what the dictionary translating from the Type IIA theory to the Type IIB theory requires, and why two seemingly different geometries—a big and a small circular dimension—can be equivalent.)
A similar idea also holds when circular dimensions are replaced with the more complicated Calabi-Yau shapes introduced in Chapter 12. A given string theory with extra dimensions curled up into a particular Calabi-Yau shape gets translated by the dictionary into a different string theory with extra dimensions curled up into a different Calabi-Yau shape (one that is called the mirror or dual of the original). In these cases, not only can the sizes of the Calabi-Yaus differ, but so can their shapes, including the number and variety of their holes. But the translation dictionary ensures that they differ in just the right way, so that even though the extra dimensions have different sizes and shapes, the physics following from each theory is absolutely identical. (There are two types of holes in a given Calabi-Yau shape, but it turns out that string vibrational patterns—and hence physical implications—are sensitive only to the difference between the number of holes of each type. So if one Calabi-Yau has, say, two holes of the first kind and five of the second, while another Calabi-Yau has five holes of the first kind and two of the second, then even though they differ as geometrical shapes, they can give rise to identical physics. 44 )
From another perspective, then, this bolsters the suspicion that space is not a foundational concept. Someone describing the universe using one of the five string theories would claim that space, including the extra dimensions, has a particular size and shape, while someone else using one of the other string theories would claim that space, including the extra dimensions, has a different size and shape. Because
the two observers would simply be using alternative mathematical descriptions of the same physical universe, it is not that one would be right and the other wrong. They would both be right, even though their conclusions about space— its size and shape—would differ. Note too, that it's not that they would be slicing up spacetime in different, equally valid ways, as in special relativity. These two observers would fail to agree on the overall structure of spacetime itself. And that's the point. If spacetime were really fundamental, most physicists expect that everyone, regardless of perspective—regardless of the language or theory used—would agree on its geometrical properties. But the fact that, at least within string theory, this need not be the case, suggests that spacetime may be a secondary phenomenon.
We are thus led to ask: if the clues described in the last two sections are pointing us in the right direction, and familiar spacetime is but a large-scale manifestation of some more fundamental entity, what is that entity and what are its essential properties? As of today, no one knows. But in the search for answers, researchers have found yet further clues, and the most important have come from thinking about black holes.
Wherefore the Entropy of Black Holes?
Black holes have the universe's most inscrutable poker faces. From the outside, they appear just about as simple as you can get. The three distinguishing features of a black hole are its mass (which determines how big it is—the distance from its center to its event horizon, the enshrouding surface of no return), its electric charge, and how fast it's spinning. That's it. There are no more details to be gleaned from scrutinizing the visage that a black hole presents to the cosmos. Physicists sum this up with the saying "Black holes have no hair," meaning that they lack the kinds of detailed features that allow for individuality. When you've seen one black hole with a given mass, charge, and spin (though you've learned these indirectly, through their effect on surrounding gas and stars, since black holes are black), you've definitely seen them all.
Nevertheless, behind their stony countenances, black holes harbor the greatest reservoirs of mayhem the universe has ever known. Among all physical systems of a given size with any possible composition, black holes contain the highest possible entropy. Recall from Chapter 6 that one rough way to think about this comes directly from entropy's definition as a measure of the number of rearrangements of an object's internal constituents that have no effect on its appearance. When it comes to black holes, even though we can't say what their constituents actually are— since we don't know what happens when matter is crushed at the black hole's center—we can say confidently that rearranging these constituents will no more affect a black hole's mass, charge, or spin than rearranging the pages in War and Peace will affect the weight of the book. And since mass, charge, and spin fully determine the face that a black hole shows the external world, all such manipulations go unnoticed and we can say a black hole has maximal entropy.
Even so, you might suggest one-upping the entropy of a black hole in the following simple way. Build a hollow sphere of the same size as a given black hole and fill it with gas (hydrogen, helium, carbon dioxide, whatever) that you allow to spread through its interior. The more gas you pump in, the greater the entropy, since more constituents means more possible rearrangements. You might guess, then, that if you keep on pumping and pumping, the entropy of the gas will steadily rise and so will eventually exceed that of the given black hole. It's a clever strategy, but general relativity shows that it fails. The more gas you pump in, the more massive the sphere's contents become. And before you reach the entropy of an equal-sized black hole, the increasingly large mass within the sphere will reach a critical value that causes the sphere and its contents to become a black hole. There's just no way around it. Black holes have a monopoly on maximal disorder.
What if you try to further increase the entropy in the space inside the black hole itself by continuing to pump in yet more gas? Entropy will indeed continue to rise, but you'll have changed the rules of the game. As matter takes the plunge across a black hole's ravenous event horizon, not only does the black hole's entropy increase, but its size increases as well. The size of a black hole is proportional to its mass, so as you dump more matter into the hole, it gets heavier and bigger. Thus, once you max out the entropy in a region of space by creating a black hole, any attempt to further increase the entropy in that region will fail. The region just can't support more disorder. It's entropy-sated. Whatever you do, whether you pump in gas or toss in a Hummer, you will necessarily cause the black hole to grow and hence surround a larger spatial region. Thus, the amount of entropy contained within a black hole not only tells us a fundamental feature of the black hole, it also tells us something fundamental about space itself: the maximum entropy that can be crammed into a region of space—any region of space, anywhere, anytime—is equal to the entropy contained within a black hole whose size equals that of the region in question.
So, how much entropy does a black hole of a given size contain? Here is where things get interesting. Reasoning intuitively, start with something more easily visualized, like air in a Tupperware container. If you were to join together two such containers, doubling the total volume and number of air molecules, you might guess that you'd double the entropy. Detailed calculations confirm 1 this conclusion and show that, all else being equal (unchanging temperature, density, and so on), the entropies of familiar physical systems are proportional to their volumes. A natural next guess is that the same conclusion would also apply to less familiar things, like black holes, leading us to expect that a black hole's entropy is also proportional to its volume.
But in the 1970s, Jacob Bekenstein and Stephen Hawking discovered that this isn't right. Their mathematical analyses showed that the entropy of a black hole is not proportional to its volume, but instead is proportional to the area of its event horizon—roughly speaking, to its surface area. This is a very different answer. Were you to double the radius of a black hole, its volume would increase by a factor of 8 (2 3 ) while its surface area would increase by only a factor of 4 (2 2 ); were you to increase its radius by a factor of a hundred, its volume would increase by a factor of a million (100 3 ), while its surface area would increase only by a factor of 10,000 (100 2 ). Big black holes have much more volume than they do surface area. 2 Thus, even though black holes contain the greatest entropy among all things of a given size, Bekenstein and Hawking showed that the amount of entropy they contain is less than what we'd naïvely guess.
That entropy is proportional to surface area is not merely a curious distinction between black holes and Tupperware, about which we can take note and swiftly move on. We've seen that black holes set a limit to the amount of entropy that, even in principle, can be crammed into a region of space: take a black hole whose size precisely equals that of the region in question, figure out how much entropy the black hole has, and that is the absolute limit on the amount of entropy the region of space can contain. Since this entropy, as the works of Bekenstein and Hawking showed, is proportional to the black hole's surface area—which equals the surface area of the region, since we chose them to have the same size—we conclude that the maximal entropy any given region of space can contain is proportional to the region's surface area. 3
The discrepancy between this conclusion and that found from thinking about air trapped in Tupperware (where we found the amount of entropy to be proportional to the Tupperware's volume, not its surface area) is easy to pinpoint: Since we assumed the air was uniformly spread, the Tupperware reasoning ignored gravity; remember, when gravity matters, things clump. To ignore gravity is fine when densities are low, but when you are considering large entropy, densities are high, gravity matters, and the Tupperware reasoning is no longer valid. Instead, such extreme conditions require the gravity-based calculations of Bekenstein and Hawking, with the conclusion that the maximum entropy potential for a region of space is proportional to its surface area, not its volume.
All right, but why should we care? There are two rea
sons.
First, the entropy bound gives yet another clue that ultramicroscopic space has an atomized structure. In detail, Bekenstein and Hawking found that if you imagine drawing a checkerboard pattern on the event horizon of a black hole, with each square being one Planck length by one Planck length (so each such "Planck square" has an area of about 10 -66 square centimeters), then the black hole's entropy equals the number of such squares that can fit on its surface. 4 It's hard to miss the conclusion to which this result strongly hints: each Planck square is a minimal, fundamental unit of space, and each carries a minimal, single unit of entropy. This suggests that there is nothing, even in principle, that can take place within a Planck square, because any such activity could support disorder and hence the Planck square could contain more than the single unit of entropy found by Bekenstein and Hawking. Once again, then, from a completely different perspective we are led to the notion of an elemental spatial entity. 5
Second, for a physicist, the upper limit to the entropy that can exist in a region of space is a critical, almost sacred quantity. To understand why, imagine you're working for a behavioral psychiatrist, and your job is to keep a detailed, moment-to-moment record of the interactions between groups of intensely hyperactive young children. Every morning you pray that the day's group will be well behaved, because the more bedlam the children create, the more difficult your job. The reason is intuitively obvious, but it's worth saying explicitly: the more disorderly the children are, the more things you have to keep track of. The universe presents a physicist with much the same challenge. A fundamental physical theory is meant to describe everything that goes on—or could go on, even in principle—in a given region of space. And, as with the children, the more disorder the region can contain—even in principle—the more things the theory must be capable of keeping track of. Thus, the maximum entropy a region can contain provides a simple but incisive litmus test: physicists expect that a truly fundamental theory is one that is perfectly matched to the maximum entropy in any given spatial region. The theory should be so tightly in tune with nature that its maximum capacity to keep track of disorder exactly equals the maximum disorder a region can possibly contain, not more and not less.