The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory - Brian Greene (2010)

Part III. The Cosmic Symphony

Chapter 8. More Dimensions Than Meet the Eye

Einstein resolved two of the major scientific conflicts of the past hundred years through special and then general relativity. Although the initial problems that motivated his work did not portend the outcome, each of these resolutions completely transformed our understanding of space and time. String theory resolves the third major scientific conflict of the past century and, in a manner that even Einstein would likely have found remarkable, it requires that we subject our conceptions of space and time to yet another radical revision. String theory so thoroughly shakes the foundations of modern physics that even the generally accepted number of dimensions in our universe—something so basic that you might think it beyond questioning—is dramatically and convincingly overthrown.

The Illusion of the Familiar

Experience informs intuition. But it does more than that: Experience sets the frame within which we analyze and interpret what we perceive. You would no doubt expect, for instance, that the "wild child" raised by a pack of wolves would interpret the world from a perspective that differs substantially from your own. Even less extreme comparisons, such as those between people raised in very different cultural traditions, serve to underscore the degree to which our experiences determine our interpretive mindset.

Yet there are certain things that we all experience. And it is often the beliefs and expectations that follow from these universal experiences that can be the hardest to identify and the most difficult to challenge. A simple but profound example is the following. If you were to get up from reading this book, you could move in three independent directions—that is, through three independent spatial dimensions. Absolutely any path you follow—regardless of how complicated—results from some combination of motion through what we might call the "left-right dimension," the "back-forth dimension," and the "up-down dimension." Every time you take a step you implicitly make three separate choices that determine how you move through these three dimensions.

An equivalent statement, as encountered in our discussion of special relativity, is that any location in the universe can be fully specified by giving three pieces of data: where it is relative to these three spatial dimensions. In familiar language, you can specify a city address, say, by giving a street (location in the "left-right dimension"), a cross street or an avenue (location in the "back-forth dimension"), and a floor number (location in the "up-down dimension"). And from a more modern perspective, we have seen that Einstein's work encourages us to think about time as another dimension (the "future-past dimension"), giving us a total of four dimensions (three space dimensions and one time dimension). You specify events in the universe by telling where and when they occur.

This feature of the universe is so basic, so consistent, and so thoroughly pervasive that it really does seem beyond questioning. In 1919, however, a little-known Polish mathematician named Theodor Kaluza from the University of Königsberg had the temerity to challenge the obvious—he suggested that the universe might not actually have three spatial dimensions; it might have more. Sometimes silly-sounding suggestions are plain silly. Sometimes they rock the foundations of physics. Although it took quite some time to percolate, Kaluza's suggestion has revolutionized our formulation of physical law. We are still feeling the aftershocks of his astonishingly prescient insight.

Kaluza's Idea and Klein's Refinement

The suggestion that our universe might have more than three spatial dimensions may well sound fatuous, bizarre, or mystical. In reality, though, it is concrete and thoroughly plausible. To see this, it's easiest to shift our sights temporarily from the whole universe and think about a more familiar object, such as a long, thin garden hose.

Imagine that a few hundred feet of garden hose is stretched across a canyon, and you view it from, say, a quarter of a mile away, as in Figure 8.1(a). From this distance, you will easily perceive the long, unfurled, horizontal extent of the hose, but unless you have uncanny eyesight, the thickness of the hose will be difficult to discern. From your distant vantage point, you would think that if an ant were constrained to live on the hose, it would have only one dimension in which to walk: the left-right dimension along the hose's length. If someone asked you to specify where the ant was at a given moment, you would need to give only one piece of data: the distance of the ant from the left (or the right) end of the hose. The upshot is that from a quarter of a mile away, a long piece of garden hose appears to be a one-dimensional object.

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Figure 8.1 (a) A garden hose viewed from a substantial distance looks like a one-dimensional object. (b) When magnified, a second dimension—one that is in the shape of a circle and is curled around the hose—becomes visible.

In reality, we know that the hose does have thickness. You might have trouble resolving this from a quarter mile, but by using a pair of binoculars you can zoom in on the hose and observe its girth directly, as shown in Figure 8.1(b). From this magnified perspective, you see that a little ant living on the hose actually has two independent directions in which it can walk: along the left-right dimension spanning the length of the hose as already identified, and along the "clockwise-counterclockwise dimension" around the circular part of the hose. You now realize that to specify where the tiny ant is at any given instant, you must actually give two pieces of data: where the ant is along the length of the hose, and where the ant is along its circular girth. This reflects the fact the surface of the garden hose is two-dimensional.1

Nonetheless, there is a clear difference between these two dimensions. The direction along the length of the hose is long, extended, and easily visible. The direction circling around the thickness of the hose is short, "curled up," and harder to see. To become aware of the circular dimension, you have to examine the hose with significantly greater precision.

This example underscores a subtle and important feature of spatial dimensions: they come in two varieties. They can be large, extended, and therefore directly manifest, or they can be small, curled up, and much more difficult to detect. Of course, in this example you did not have to exert a great deal of effort to reveal the "curled-up" dimension encircling the thickness of the hose. You merely had to use a pair of binoculars. However, if you had a very thin garden hose—as thin as a hair or a capillary—detecting its curled-up dimension would be more difficult.

In a paper he sent to Einstein in 1919, Kaluza made an astounding suggestion. He proposed that the spatial fabric of the universe might possess more than the three dimensions of common experience. The motivation for this radical thesis, as we will discuss shortly, was Kaluza's realization that it provided an elegant and compelling framework for weaving together Einstein's general relativity and Maxwell's electromagnetic theory into a single, unified conceptual framework. But, more immediately, how can this proposal be squared with the apparent fact that we see precisely three spatial dimensions?

The answer, implicit in Kaluza's work and subsequently made explicit and refined by the Swedish mathematician Oskar Klein in 1926, is that the spatial fabric of our universe may have both extended and curled-up dimensions. That is, just like the horizontal extent of the garden hose, our universe has dimensions that are large, extended, and easily visible—the three spatial dimensions of common experience. But like the circular girth of a garden hose, the universe may also have additional spatial dimensions that are tightly curled up into a tiny space—a space so tiny that it has so far eluded detection by even our most refined experimental equipment.

To gain a clearer image of this remarkable proposal, let's reconsider the garden hose for a moment. Imagine that the hose is painted with closely spaced black circles along its girth. From far away, as before, the garden hose looks like a thin, one-dimensional line. But if you zoom in with binoculars, you can detect the curled-up dimension, even more easily after our paint job, and you see the image illustrated in Figure 8.2. This figure emphasizes that the surface of the garden hose is two-dimensional, with one large, extended dimension and one small, circular dimension. Kaluza and Klein proposed that our spatial universe is similar, but that it has three large, extended spatial dimensions and one small, circular dimension—for a total of four spatial dimensions. It is difficult to draw something with that many dimensions, so for visualization purposes we must settle for an illustration incorporating two large dimensions and one small, circular dimension. We illustrate this in Figure 8.3, in which we magnify the fabric of space in much the same way that we zoomed in on the surface of the garden hose.

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Figure 8.2 The surface of the garden hose is two-dimensional: one dimension (its horizontal extent), emphasized by the straight arrow, is long and extended; the other dimension (its circular girth), emphasized by the circular arrow, is short and curled up.

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Figure 8.3 As in Figure 5.1, each subsequent level represents a huge magnification of the spatial fabric displayed in the previous level. Our universe may have extra dimensions—as we see by the fourth level of magnification—so long as they are curled up into a space small enough to have as yet evaded direct detection.

The lowest image in the figure shows the apparent structure of space—the ordinary world around us—on familiar distance scales such as meters. These distances are represented by the largest set of grid lines. In the subsequent images, we zoom in on the fabric of space by focusing our attention on ever smaller regions, which we sequentially magnify in order to make them easily visible. At first as we examine the fabric of space on shorter distance scales, not much happens; it appears to retain the same basic form as it has on larger scales, as we see in the first three levels of magnification. However, as we continue on our journey toward the most microscopic examination of space—the fourth level of magnification in Figure 8.3—a new, curled-up, circular dimension becomes apparent, much like the circular loops of thread making up the pile of a tightly woven piece of carpet. Kaluza and Klein suggested that the extra circular dimension exists at every point in the extended dimensions, just as the circular girth of the garden hose exists at every point along its unfurled, horizontal extent. (For visual clarity, we have drawn only an illustrative sample of the circular dimension at regularly spaced points in the extended dimensions.) We show a close-up of the Kaluza-Klein vision of the microscopic structure of the spatial fabric in Figure 8.4.

The similarity with the garden hose is manifest, although there are some important differences. The universe has three large, extended space dimensions (only two of which we have actually drawn), compared with the garden hose's one, and, more important, we are now describing the spatial fabric of the universe itself, not just an object, like the garden hose, that exists within the universe. But the basic idea is the same: Like the circular girth of the garden hose, if the additional curled-up, circular dimension of the universe is extremely small, it is much harder to detect than the manifest, large, extended dimensions. In fact, if its size is small enough, it will be beyond detection by even our most powerful magnifying instruments. And, of utmost importance, the circular dimension is not merely a circular bump within the familiar extended dimensions as the illustration might lead you to believe. Rather, the circular dimension is a new dimension, one that exists at every point in the familiar extended dimensions just as each of the up-down, left-right, and back-forth dimensions exists at every point as well. It is a new and independent direction in which an ant, if it were small enough, could move. To specify the spatial location of such a microscopic ant, we would need to say where it is in the three familiar extended dimensions (represented by the grid) and also where it is in the circular dimension. We would need four pieces of spatial information; if we add in time, we get a total of five pieces of spacetime information—one more than we normally would expect.

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Figure 8.4 The grid lines represent the extended dimensions of common experience, whereas the circles are a new, tiny, curled-up dimension. Like the circular loops of thread making up the pile of a carpet, the circles exist at every point in the familiar extended dimensions—but for visual clarity we draw them as spread out on intersecting grid lines.

And so, rather surprisingly, we see that although we are aware of only three extended spatial dimensions, Kaluza's and Klein's reasoning shows that this does not preclude the existence of additional curled-up dimensions, at least if they are very small. The universe may very well have more dimensions than meet the eye.

How small is "small?" Cutting-edge equipment can detect structures as small as a billionth of a billionth of a meter. So long as an extra dimension is curled up to a size less than this tiny distance, it is too small for us to detect. In 1926 Klein combined Kaluza's initial suggestion with some ideas from the emerging field of quantum mechanics. His calculations indicated that the additional circular dimension might be as small as the Planck length, far shorter than experimental accessibility. Since then, physicists have called the possibility of extra tiny space dimensions Kaluza-Klein theory.2

Comings and Goings on a Garden Hose

The tangible example of the garden hose and the illustration in Figure 8.3 are meant to give you some sense of how it is possible that our universe has extra spatial dimensions. But even for researchers in the field, it is quite difficult to visualize a universe with more than three spatial dimensions. For this reason, physicists often hone their intuition about these extra dimensions by contemplating what life would be like if we lived in an imaginary lower-dimensional universe—following the lead of Edwin Abbott's enchanting 1884 classic popularization Flatland3—in which we slowly realize that the universe has more dimensions than those of which we are directly aware. Let's try this by imagining a two-dimensional universe shaped like our garden hose. Doing so requires that you relinquish an "outsider's" perspective that views the garden hose as an object in our universe. Rather, you must leave the world as we know it and enter a new Garden-hose universe in which the surface of a very long garden hose (you can think of it as being infinitely long) is all there is as far as spatial extent. Imagine that you are a tiny ant living your life on its surface.

Let's start by making things even a little more extreme. Imagine that the length of the circular dimension of the Garden-hose universe is very short—so short that neither you nor any of your fellow Hose-dwellers are even aware of its existence. Instead, you and everyone else living in the Hose universe take one basic fact of life to be so evident as to be beyond questioning: the universe has one spatial dimension. (If the Garden-hose universe had produced its own ant-Einstein, Hose-dwellers would say that the universe has one spatial and one time dimension.) In fact, this feature is so self-evident that Hose-dwellers have named their home Lineland, directly emphasizing its having one spatial dimension.

Life in Lineland is very different from life as we know it. For example, the body with which you are familiar cannot fit in Lineland. No matter how much effort you may put into body reshaping, one thing you can't get around is that you definitely have length, width, and breadth—spatial extent in three dimensions. In Lineland there is no room for such an extravagant design. Remember, although your mental image of Lineland may still be tied to a long, threadlike object existing in our space, you really need to think of Lineland as a universe—all there is. As an inhabitant of Lineland you must fit within its spatial extent. Try to imagine it. Even if you take on an ant's body, you still will not fit. You must squeeze your ant body to look more like a worm, and then further squeeze it until you have no thickness at all. To fit in Lineland you must be a being that has only length.

Imagine further that you have an eye on each end of your body. Unlike your human eyes, which can swivel around to look in all three dimensions, your eyes as a Linebeing are forever locked into position, each staring off into the one-dimensional distance. This is not an anatomical limitation of your new body. Instead, you and all other Linebeings recognize that since Lineland has but one dimension, there simply isn't another direction in which your eyes can look. Forward and backward exhaust the extent of Lineland.

We can try to go further in imagining life in Lineland, but we quickly realize that there's not much more to it. For instance, if another Linebeing is on one or the other side of you, picture how it will appear: you will see one of her eyes—the one facing you—but unlike human eyes, hers will be a single dot. Eyes in Lineland have no features and display no emotion—there is just no room for these familiar characteristics. Moreover, you will be forever stuck with this dotlike image of your neighbor's eye. If you wanted to pass her and explore the realm of Lineland on the other side of her body, you would be in for a great disappointment. You can't pass by her. She is fully "blocking the road," and there is no space in Lineland to go around her. The order of Linebeings as they are sprinkled along the extent of Lineland is fixed and unchanging. What drudgery.

A few thousand years after a religous epiphany in Lineland, a Linebeing named Kaluza K. Line offers some hope for the downtrodden Line-dwellers. Either from divine inspiration or from the sheer exasperation of years of staring at his neighbor's dot-eye, he suggests that Lineland may not be one-dimensional after all. What if, he theorizes, Lineland is actually two-dimensional, with the second space dimension being a very small circular direction that has, as yet, evaded direct detection because of its tiny spatial extent. He goes on to paint a picture of a vastly new life, if only this curled-up space direction would expand in size—something that is at least possible according to the recent work of his colleague, Linestein. Kaluza K. Line describes a universe that amazes you and your comrades and fills everyone with hope—a universe in which Linebeings can move freely past one another by making use of the second dimension: the end of spatial enslavement. We realize that Kaluza K. Line is describing life in a "thickened" Garden-hose universe.

In fact, if the circular dimension were to grow, "inflating" Lineland into the Garden-hose universe, your life would change in profound ways. Take your body, for example. As a Linebeing, anything between your two eyes constitutes the interior of your body. Your eyes, therefore, play the same role for your linebody as skin plays for an ordinary human body: They constitute the barrier between the inside of your body and the outside world. A doctor in Lineland can access the interior of your linebody only by puncturing its surface—in other words, "surgery" in Lineland takes place through the eyes.

But now imagine what happens if Lineland does, a la Kaluza K. Line, have a secret, curled-up dimension, and if this dimension expands to an observably large size. Now one Linebeing can view your body at an angle and thereby directly see into its interior, as we illustrate in Figure 8.5. Using this second dimension, a doctor can operate on your body by reaching directly inside your exposed interior. Weird! In time, Linebeings, no doubt, would develop a skinlike cover to shield the newly exposed interior of their bodies from contact with the outside world. And moreover, they would undoubtedly evolve into beings with length as well as breadth: Flatbeings sliding along the two-dimensional Garden-hose universe as illustrated in Figure 8.6. If the circular dimension were to grow very large this two-dimensional universe would be closely akin to Abbott's Flatland—an imaginary two-dimensional world Abbott suffused with a rich cultural heritage and even a satirical caste system based upon one's geometrical shape. Whereas it's hard to imagine anything interesting happening in Lineland—there is just not enough room—life on a Garden-hose becomes replete with possibilities. The evolution from one to two observably large space dimensions is dramatic.

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Figure 8.5 One Linebeing can see directly into the interior of another's body when Lineland expands into the Garden-hose universe.

And now the refrain: Why stop there? The two-dimensional universe might itself have a curled-up dimension and therefore secretly be three-dimensional. We can illustrate this with Figure 8.4, so long as we recognize that we are now imagining that there are only two extended space dimensions (whereas when we first introduced this figure we were imagining the flat grid to represent three extended dimensions). If the circular dimension should expand, a two-dimensional being would find itself in a vastly new world in which movement is not limited just to left-right and back-forth along the extended dimensions. Now, a being can also move in a third dimension—the "up-down" direction along the circle. In fact, if the circular dimension were to grow to a large enough size, this could be our three-dimensional universe. We do not know at present whether any of our three spatial dimensions extends outward forever, or in fact curls back on itself in the shape of a giant circle, beyond the range of our most powerful telescopes. If the circular dimension in Figure 8.4 got big enough—billions of light-years in extent—the figure could very well be a drawing of our world.

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Figure 8.6 Flat, two-dimensional beings living in the Garden-hose universe.

But the refrain replays: Why stop there? This takes us to Kaluza's and Klein's vision: that our three-dimensional universe might have a previously unanticipated curled-up fourth spatial dimension. If this striking possibility, or its generalization to numerous curled-up dimensions (to be discussed shortly) is true, and if these curled-up dimensions were themselves to expand to a macroscopic size, the lower-dimensional examples discussed make it clear that life as we know it would change immensely.

Surprisingly, though, even if they should always stay curled up and small, the existence of extra curled-up dimensions has profound implications.

Unification in Higher Dimensions

Although Kaluza's 1919 suggestion that our universe might have more spatial dimensions than those of which we are directly aware was a remarkable possibility in its own right, something else really made it compelling. Einstein had formulated general relativity in the familiar setting of a universe with three spatial dimensions and one time dimension. The mathematical formalism of his theory, however, could be extended fairly directly to write down analogous equations for a universe with additional space dimensions. Under the "modest" assumption of one extra space dimension, Kaluza carried out the mathematical analysis and explicitly derived the new equations.

He found that in the revised formulation the equations pertaining to the three ordinary dimensions were essentially identical to Einstein's. But because he included an extra space dimension, not surprisingly Kaluza found extra equations beyond those Einstein originally derived. After studying the extra equations associated with the new dimension, Kaluza realized that something amazing was going on. The extra equations were none other than those Maxwell had written down in the 1880s for describing the electromagnetic force! By adding another space dimension, Kaluza had united Einstein's theory of gravity with Maxwell's theory of light.

Before Kaluza's suggestion, gravity and electromagnetism were thought of as two unrelated forces; nothing had even hinted that there might be a relation between them. By having the bold creativity to imagine that our universe has an additional space dimension, Kaluza suggested that there was a deep connection, indeed. His theory argued that both gravity and electromagnetism are associated with ripples in the fabric of space. Gravity is carried by ripples in the familiar three space dimensions, while electromagnetism is carried by ripples involving the new, curled-up dimension.

Kaluza sent his paper to Einstein, and at first Einstein was quite intrigued. On April 21, 1919, Einstein wrote back to Kaluza and told him that it had never occurred to him that unification might be achieved "through a five-dimensional [four space and one time] cylinder-world." He added, "At first glance, I like your idea enormously."4 About a week later, though, Einstein wrote Kaluza again, this time with some skepticism: "I have read through your paper and find it really interesting. Nowhere, so far, can I see an impossibility. On the other hand, I have to admit that the arguments brought forward so far do not appear convincing enough."5 But then, on October 14, 1921, more than two years later, Einstein wrote to Kaluza again, having had time to digest Kaluza's novel approach more fully: "I am having second thoughts about having restrained you from publishing your idea on a unification of gravitation and electricity two years ago. . .. If you wish, I shall present your paper to the academy after all."6 Belatedly, Kaluza had received the master's stamp of approval.

Although it was a beautiful idea, subsequent detailed study of Kaluza's proposal, augmented by Klein's contributions, showed that it was in serious conflict with experimental data. The simplest attempts to incorporate the electron into the theory predicted relations between its mass and its charge that were vastly different from their measured values. Because there did not seem to be any obvious way of getting around this problem, many of the physicists who had taken notice of Kaluza's idea lost interest. Einstein and others continued, now and then, to dabble with the possibility of extra curled-up dimensions, but it quickly came to be an enterprise on the outskirts of theoretical physics.

In a real sense, Kaluza's idea was way ahead of its time. The 1920s marked the start of a bull market for theoretical and experimental physics concerned with understanding the basic laws of the microworld. Theorists had their hands full as they sought to develop the structure of quantum mechanics and quantum field theory. Experimentalists had the detailed properties of the atom as well as numerous other elementary material constituents to discover. Theory guided experiment and experiment refined theory as physicists pushed forward for half a century, ultimately to reveal the standard model. It is no wonder that speculations on extra dimensions took a distant backseat during these productive and heady times. With physicists exploring powerful quantum methods, the implications of which gave rise to experimentally testable predictions, there was little interest in the mere possibility that the universe might be a vastly different place on length scales far too small to be probed by even the most powerful of instruments.

But sooner or later, bull markets lose steam. By the late 1960s and early 1970s the theoretical structure of the standard model was in place. By the late 1970s and early 1980s many of its predictions had been verified experimentally, and most particle physicists concluded that it was just a matter of time before the rest were confirmed as well. Although a few important details remained unresolved, many felt that the major questions concerning the strong, weak, and electromagnetic forces had been answered.

The time was finally ripe to return to the grandest question of all: the enigmatic conflict between general relativity and quantum mechanics. The success in formulating a quantum theory of three of nature's forces emboldened physicists to try to bring the fourth, gravity, into the fold. Having pursued numerous ideas that all ultimately failed, the mind-set of the community became more open to comparatively radical approaches. After being left for dead in the late 1920s, Kaluza-Klein theory was resuscitated.

Modern Kaluza-Klein Theory

The understanding of physics had significantly changed and substantially deepened in the six decades since Kaluza's original proposal. Quantum mechanics had been fully formulated and experimentally verified. The strong and the weak forces, unknown in the 1920s, had been discovered and were largely understood. Some physicists suggested that Kaluza's original proposal had failed because he was unaware of these other forces and had therefore been too conservative in his revamping of space. More forces meant the need for even more dimensions. It was argued that a single new, circular dimension, although able to show hints of a connection between general relativity and electromagnetism, was just not enough.

By the mid-1970s, an intense research effort was underway, focusing on higher-dimensional theories with numerous curled-up spatial directions. Figure 8.7 illustrates an example with two extra dimensions that are curled up into the surface of a ball—that is, a sphere. As in the case of the single circular dimension, these extra dimensions are tacked on to every point of the familiar extended dimensions. (For visual clarity we again have drawn only an illustrative sample of the spherical dimensions at regularly spaced grid points in the extended dimensions.) Beyond proposing a different number of extra dimensions, one can also imagine other shapes for the extra dimensions. For instance, in Figure 8.8 we illustrate a possibility in which there are again two extra dimensions, now in the shape of a hollow doughnut—that is, a torus. Although they are beyond our ability to draw, more complicated possibilities can be imagined in which there are three, four, five, essentially any number of extra spatial dimensions, curled up into a wide spectrum of exotic shapes. The essential requirement, again, is that all of these dimensions have a spatial extent smaller than the smallest length scales we can probe, since no experiment has yet revealed their existence.

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Figure 8.7 Two extra dimensions curled up into the shape of a sphere.

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Figure 8.8 Two extra dimensions curled up in the shape of a hollow doughnut, or torus.

The most promising of the higher-dimensional proposals were those that also incorporated supersymmetry. Physicists hoped that the partial cancelling of the most severe quantum fluctuations, arising from the pairing of superpartner particles, would help to soften the hostilities between gravity and quantum mechanics. They coined the name higher-dimensional supergravity to describe those theories encompassing gravity, extra dimensions, and supersymmetry.

As had been the case with Kaluza's original attempt, various versions of higher-dimensional supergravity looked quite promising at first. The new equations resulting from the extra dimensions were strikingly reminiscent of those used in the description of electromagnetism, and the strong and the weak forces. But detailed scrutiny showed that the old conundrums persisted. Most importantly, the pernicious short-distance quantum undulations of space were lessened by supersymmetry, but not sufficiently to yield a sensible theory. Physicists also found it difficult to find a single, sensible, higher-dimensional theory incorporating all features of forces and matter.7

It gradually became clear that bits and pieces of a unified theory were surfacing, but that a crucial element capable of tying them all together in a quantum-mechanically consistent manner was missing. In 1984 this missing piece—string theory—dramatically entered the story and took center stage.

More Dimensions and String Theory

By now you should be convinced that our universe may have additional curled-up spatial dimensions; certainly, so long as they are small enough, nothing rules them out. But extra dimensions may strike you as an artifice. Our inability to probe distances smaller than a billionth of a billionth of a meter permits not only extra tiny dimensions but all manner of whimsical possibilities as well—even a microscopic civilization populated by even tinier green people. While the former certainly seems more rationally motivated than the latter, the act of postulating either of these experimentally untested—and, at present, untestable—possibilities might seem equally arbitrary.

Such was the case until string theory. Here is a theory that resolves the central dilemma confronting contemporary physics—the incompatibility between quantum mechanics and general relativity—and that unifies our understanding of all of nature's fundamental material constituents and forces. But to accomplish these feats, it turns out that string theory requires that the universe have extra space dimensions.

Here's why. One of the main insights of quantum mechanics is that our predictive power is fundamentally limited to asserting that such-and-such outcome will occur with such-and-such probability. Although Einstein felt that this was a distasteful feature of our modern understanding, and you may agree, it certainly appears to be fact. Let's accept it. Now, we all know that probabilities are always numbers between 0 and 1—equivalently, when expressed as percentages, probabilities are numbers between 0 and 100. Physicists have found that a key signal that a quantum-mechanical theory has gone haywire is that particular calculations yield "probabilities" that are not within this acceptable range. For instance, we mentioned earlier that a sign of the grinding incompatibility between general relativity and quantum mechanics in a point-particle framework is that calculations result in infinite probabilities. As we have discussed, string theory cures these infinities. But what we have not as yet mentioned is that a residual, somewhat more subtle problem still remains. In the early days of string theory physicists found that certain calculations yielded negative probabilities, which are also outside of the acceptable range. So, at first sight, string theory appeared to be awash in its own quantum-mechanical hot water.

With stubborn determination, physicists sought and found the cause of this unacceptable feature. The explanation begins with a simple observation. If a string is constrained to lie on a two-dimensional surface—such as the surface of a table or a garden hose—the number of independent directions in which it can vibrate is reduced to two: the left-right and back-forth dimensions along the surface. Any vibrational pattern that remains on the surface involves some combination of vibrations in these two directions. Correspondingly, we see that this also means that a string in Flatland, the Garden-hose universe, or in any other two-dimensional universe, is also constrained to vibrate in a total of two independent spatial directions. If, however, the string is allowed to leave the surface, the number of independent vibrational directions increases to three, since the string then can also oscillate in the up-down direction. Equivalently, in a universe with three spatial dimensions, a string can vibrate in three independent directions. Although it gets harder to envision, the pattern continues: In a universe with ever more spatial dimensions, there are ever more independent directions in which it can vibrate.

We emphasize this fact of string vibrations because physicists found that the troublesome calculations were highly sensitive to the number of independent directions in which a string can vibrate. The negative probabilities arose from a mismatch between what the theory required and what reality seemed to impose: The calculations showed that if strings could vibrate in nine independent spatial directions, all of the negative probabilities would cancel out. Well, that's great in theory, but so what? If string theory is meant to describe our world with three spatial dimensions, we still seem to be in trouble.

But are we? Taking a more than half-century-old lead, we see that Kaluza and Klein provide a loophole. Since strings are so small, not only can they vibrate in large, extended dimensions, they can also vibrate in ones that are tiny and curled up. And so we can meet the nine-space-dimension requirement of string theory in our universe, by assuming—a la Kaluza and Klein—that in addition to our familiar three extended spatial dimensions there are six other curled-up spatial dimensions. In this manner, string theory, which appeared to be on the brink of elimination from the realm of physical relevance, is saved. Moreover, rather than just postulating the existence of extra dimensions, as had been done by Kaluza, Klein, and their followers, string theory requires them. For string theory to make sense, the universe should have nine space dimensions and one time dimension, for a total of ten dimensions. In this way, Kaluza's 1919 proposal finds its most convincing and powerful forum.

Some Questions

This raises a number of questions. First, why does string theory require the particular number of nine space dimensions to avoid nonsensical probability values? This is probably the hardest question in string theory to answer without appealing to mathematical formalism. A straightforward string theory calculation reveals this answer, but no one has an intuitive, nontechnical explanation for the particular number that emerges. The physicist Ernest Rutherford once said, in essence, that if you can't explain a result in simple, nontechnical terms, then you don't really understand it. He wasn't saying that this means your result is wrong; rather, he was saying that it means you do not fully understand its origin, meaning, or implications. Perhaps this is true regarding the extradimensional character of string theory. (In fact, let's take this opportunity to brace—parenthetically—for a central aspect of the second superstring revolution that we will discuss in Chapter 12. The calculation underlying the conclusion that there are ten spacetime dimensions—nine space and one time—turns out to be approximate. In the mid-1990s, Witten, based on his own insights and previous work by Michael Duff from Texas A&M University and Chris Hull and Paul Townsend from Cambridge University, gave convincing evidence that the approximate calculation actually misses one space dimension: String theory, he argued to most string theorists' amazement, actually requires ten space dimensions and one time dimension, for a total of eleven dimensions. We will ignore this important result until Chapter 12, as it will have little direct bearing on the material we develop before then.)

Second, if the equations of string theory (or, more precisely, the approximate equations guiding our pre–Chapter 12 discussion) show that the universe has nine space dimensions and one time dimension, why is it that three space (and one time) dimensions are large and extended while all of the others are tiny and curled up? Why aren't they all extended, or all curled up, or some other possibility in between? At present no one knows the answer to this question. If string theory is right, we should eventually be able to extract the answer, but as yet our understanding of the theory is not refined enough to reach this goal. That's not to say that there haven't been valiant attempts to explain it. For instance, from a cosmological perspective, we can imagine that all of the dimensions start out being tightly curled up and then, in a big bang–like explosion, three spatial dimensions and one time dimension unfurl and expand to their present large extent while the other spatial dimensions remain small. Rough arguments have been put forward as to why only three space dimensions grow large, as we will discuss in Chapter 14, but it's fair to say that these explanations are only in the formative stages. In what follows, we will assume that all but three space dimensions are curled up, in accordance with what we see around us. A primary goal of modern research is to establish that this assumption emerges from the theory itself.

Third, given the requirement of numerous extra dimensions, is it possible that some are additional time dimensions, as opposed to additional space dimensions? If you think about this for a moment, you will see that it's a truly bizarre possibility. We all have a visceral understanding of what it means for the universe to have multiple space dimensions, since we live in a world in which we constantly deal with a plurality—three. But what would it mean to have multiple times? Would one align with time as we presently experience it psychologically while the other would somehow be "different?"

It gets even stranger when you think about a curled-up time dimension. For instance, if a tiny ant walks around an extra space dimension that is curled up like a circle, it will find itself returning to the same position over and over again as it traverses complete circuits. This holds little mystery as we are familiar with the ability to return, should we so choose, to the same location in space as often as we like. But, if a curled-up dimension is a time dimension, traversing it means returning, after a temporal lapse, to a prior instant in time. This, of course, is well beyond the realm of our experience. Time, as we know it, is a dimension we can traverse in only one direction with absolute inevitability, never being able to return to an instant after it has passed. Of course, it might be that curled-up time dimensions have different properties from the familiar, vast time dimension that we imagine reaching back to the creation of the universe and forward to the present moment. But, in contrast to extra spatial dimensions, new and previously unknown time dimensions would clearly require an even more monumental restructuring of our intuition. Some theorists have been exploring the possibility of incorporating extra time dimensions into string theory, but as yet the situation is inconclusive. In our discussion of string theory, we will stick to the more "conventional" approach in which all of the curled-up dimensions are space dimensions, but the intriguing possibility of new time dimensions could well play a role in future developments.

The Physical Implications of Extra Dimensions

Years of research, dating back to Kaluza's original paper, have shown that even though any extra dimensions that physicists propose must be smaller than we or our equipment can directly "see" (since we haven't seen them), they do have important indirect effects on the physics that we observe. In string theory, this connection between the microscopic properties of space and the physics we observe is particularly transparent.

To understand this, you need to recall that masses and charges of particles in string theory are determined by the possible resonant vibrational string patterns. Picture a tiny string as it moves and oscillates, and you will realize that the resonant patterns are influenced by its spatial surroundings. Think, for example, of ocean waves. Out in the grand expanse of the open ocean, isolated wave patterns are relatively free to form and travel this way or that. This is much like the vibrational patterns of a string as it moves through large, extended spatial dimensions. As in Chapter 6, such a string is equally free to oscillate in any of the extended directions at any moment. But if an ocean wave passes through a more cramped spatial environment, the detailed form of its wave motion will surely be affected by, for example, the depth of the water, the placement and shape of the rocks encountered, the canals through which the water is channeled, and so on. Or, think of an organ pipe or a French horn. The sounds that each of these instruments can produce are a direct consequence of the resonant patterns of vibrating air streams in their interior; these are determined by the precise size and shape of the spatial surroundings within the instrument through which the air streams are channeled. Curled-up spatial dimensions have a similar impact on the possible vibrational patterns of a string. Since tiny strings vibrate through all of the spatial dimensions, the precise way in which the extra dimensions are twisted up and curled back on each other strongly influences and tightly constrains the possible resonant vibrational patterns. These patterns, largely determined by the extradimensional geometry, constitute the array of possible particle properties observed in the familiar extended dimensions. This means that extradimensional geometry determines fundamental physical attributes like particle masses and charges that we observe in the usual three large space dimensions of common experience.

This is such a deep and important point that we say it once again, with feeling. According to string theory, the universe is made up of tiny strings whose resonant patterns of vibration are the microscopic origin of particle masses and force charges. String theory also requires extra space dimensions that must be curled up to a very small size to be consistent with our never having seen them. But a tiny string can probe a tiny space. As a string moves about, oscillating as it travels, the geometrical form of the extra dimensions plays a critical role in determining resonant patterns of vibration. Because the patterns of string vibrations appear to us as the masses and charges of the elementary particles, we conclude that these fundamental properties of the universe are determined, in large measure, by the geometrical size and shape of the extra dimensions. That's one of the most far-reaching insights of string theory.

Since the extra dimensions so profoundly influence basic physical properties of the universe, we should now seek—with unbridled vigor—an understanding of what these curled-up dimensions look like.

What Do the Curled-Up Dimensions Look Like?

The extra spatial dimensions of string theory cannot be "crumpled" up any which way; the equations that emerge from the theory severely restrict the geometrical form that they can take. In 1984, Philip Candelas of the University of Texas at Austin, Gary Horowitz and Andrew Strominger of the University of California at Santa Barbara, and Edward Witten showed that a particular class of six-dimensional geometrical shapes can meet these conditions. They are known as Calabi-Yau spaces (or Calabi-Yau shapes) in honor of two mathematicians, Eugenio Calabi from the University of Pennsylvania and Shing-Tung Yau from Harvard University, whose research in a related context, but prior to string theory, plays a central role in understanding these spaces. Although the mathematics describing Calabi-Yau spaces is intricate and subtle, we can get an idea of what they look like with a picture.8

In Figure 8.9 we show an example of a Calabi-Yau space.9 As you view this figure, you must bear in mind that the image has built-in limitations. We are trying to represent a six-dimensional shape on a two-dimensional piece of paper, and this introduces significant distortions. Nevertheless, the image does convey the rough idea of what a Calabi-Yau space looks like.10 The shape in Figure 8.9 is but one of many tens of thousands of examples of Calabi-Yau shapes that meet the stringent requirements for the extra dimensions that emerge from string theory. Although belonging to a club with tens of thousands of members might not sound very exclusive, you must compare it with the infinite number of shapes that are mathematically possible; by this measure Calabi-Yau spaces are rare indeed.

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Figure 8.9 One example of a Calabi-Yau space.

To put it all together, you should now imagine replacing each of the spheres in Figure 8.7—which represented two curled-up dimensions—with a Calabi-Yau space. That is, at every point in the three familiar extended dimensions, string theory claims that there are six hitherto unanticipated dimensions, tightly curled up into one of these rather complicated-looking shapes, as illustrated in Figure 8.10. These dimensions are an integral and ubiquitous part of the spatial fabric; they exist everywhere. For instance, if you sweep your hand in a large arc, you are moving not only through the three extended dimensions, but also through these curled-up dimensions. Of course, because the curled-up dimensions are so small, as you move your hand you circumnavigate them an enormous number of times, repeatedly returning to your starting point. Their tiny extent means that there is not much room for a large object like your hand to move—it all averages out so that after sweeping your arm, you are completely unaware of the journey you took through the curled-up Calabi-Yau dimensions.

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Figure 8.10 According to string theory, the universe has extra dimensions curled up into a Calabi-Yau shape.

This is a stunning feature of string theory. But if you are practically minded, you are bound to bring the discussion back to an essential and concrete issue. Now that we have a better sense of what the extra dimensions look like, what are the physical properties that emerge from strings that vibrate through them, and how do these properties compare with experimental observations? This is string theory's $64,000 question.