by Brian Greene
By contrast, if the universe had two or even just one space dimension, how would Newton's formula change? Well, Figure 13.5a shows a two-dimensional version of the sun and its orbiting satellite. As you can see, at any given distance the sun's gravitational field lines uniformly spread out on a circle, the analog of a sphere in one lower dimension. Since the circle's circumference is proportional to its radius (not to the square of its radius), if you double the sun-satellite separation, the density of field lines will decrease by a factor of 2 (not 4) and so the strength of the sun's gravitational pull will drop only by a factor of 2 (not 4). If the universe had only two space dimensions, then, gravitational pull would be inversely proportional to separation, not the square of separation.
If the universe had only one space dimension, as in Figure 13.5b, the law of gravity would be simpler still. Gravitational field lines would have no room to spread out, and so the force of gravity would not decrease with separation. If you were to double the distance between the sun and the
Figure 13.5 ( a ) In a universe with only two spatial dimensions, the gravitational force drops in proportion to separation, because gravitational field lines uniformly spread on a circle whose circumference is proportional to its radius. ( b ) In a universe with one space dimension, gravitational field lines do not have any room to spread, so the gravitational force is constant, regardless of separation.
satellite (assuming that versions of such objects could exist in such a universe), the same number of field lines would penetrate the satellite and hence the force of gravity acting between them would not change at all.
Although it is impossible to draw, the pattern illustrated by Figures 13.4 and 13.5 extends directly to a universe with four or five or six or any number of space dimensions. The more space dimensions there are, the more room gravitational lines of force have to spread out. And the more they spread out, the more precipitously the force of gravity drops with increasing separation. In four space dimensions, Newton's law would be an inverse cube law (double the separation, force drops by a factor of 8); in five space dimensions, it would be an inverse fourth-power law (double the separation, force drops by a factor of 16); in six space dimensions, it would be an inverse fifth-power law (double the separation, force drops by a factor of 32); and so on for ever higher-dimensional universes.
You might think that the success of the inverse square version of Newton's law in explaining a wealth of data—from the motion of planets to the paths of comets—confirms that we live in a universe with precisely three space dimensions. But that conclusion would be hasty. We know that the inverse square law works on astronomical scales, 6 and we know that it works on terrestrial scales, and that jibes well with the fact that on such scales we see three space dimensions. But do we know that it works on smaller scales? How far down into the microcosmos has gravity's inverse square law been tested? As it turns out, experimenters have confirmed it down to only about a tenth of a millimeter; if two objects are brought to within a separation of a tenth of a millimeter, the data verify that the strength of their gravitational attraction follows the predictions of the inverse square law. But so far, it has proven a significant technical challenge to test the inverse square law on shorter scales (quantum effects and the weakness of gravity complicate the experiments). This is a critical issue, because deviations from the inverse square law would be a convincing signal of extra dimensions.
To see this explicitly, let's work with a lower-dimensional toy example that we can easily draw and analyze. Imagine we lived in a universe with one space dimension—or so we thought, because only one space dimension was visible and, moreover, centuries of experiments had shown that the force of gravity does not vary with the separation between objects. But also imagine that in all those years experimenters had been able only to test the law of gravity down to distances of about a tenth of a millimeter. For distances shorter than that, no one had any data. Now, imagine further that, unbeknownst to everyone but a handful of fringe theoretical physicists, the universe actually had a second, curled-up space dimension making its shape like the surface of Philippe Petit's tightrope, as in Figure 12.5. How would this affect future, more refined gravitational tests? We can deduce the answer by looking at Figure 13.6. As two tiny objects are brought close enough together—much closer than the circumference of the curled-up dimension—the two-dimensional character of space would become apparent immediately, because on those scales gravitational field lines would have room to spread out (Figure 13.6a). Rather than being independent of distance, the force of gravity would vary inversely with separation when objects were close enough together.
Figure 13.6 ( a ) When objects are close, the gravitational pull varies as it does in two space dimensions. ( b ) When objects are farther apart, the gravitational pull behaves as it does in one space dimension—it is constant.
Thus, if you were an experimenter in this universe, and you developed exquisitely accurate methods for measuring gravitational attraction, here's what you would find. When two objects were extremely close, much closer than the size of the curled-up dimension, their gravitational attraction would diminish in proportion to their separation, just as you expect for a universe with two space dimensions. But then, when the objects were about as far apart as the circumference of the curled-up dimension, things would change. Beyond this distance, the gravitational field lines would be unable to spread any further. They would have spilled out as far as they could into the second curled-up dimension— they would have saturated that dimension—and so from this distance onward the gravitational force would no longer diminish, as illustrated in Figure 13.6b. You can compare this saturation with the plumbing in an old house. If someone opens the faucet in the kitchen sink when you're just about to rinse the shampoo out of your hair, the water pressure can drop because the water spreads between the two outlets. The pressure will diminish yet again should someone open the faucet in the laundry room, since the water will spread even more. But once all the faucets in the house are open, the pressure will remain constant. Although it might not provide the relaxing, high-water-pressure experience you'd anticipated, the pressure in the shower will not drop any further once the water has completely spread throughout all "extra" outlets. Similarly, once the gravitational field has completely spread throughout the extra curled-up dimension, it will not diminish with further separation.
From your data you would deduce two things. First, from the fact that the gravitational force diminished in proportion to distance when objects are very close, you'd realize that the universe has two space dimensions, not one. Second, from the crossover to a gravitational force that is constant—the result known from hundreds of years of previous experiments—you'd conclude that one of these dimensions is curled up, with a size about equal to the distance at which the crossover takes place. And with this result, you'd overturn centuries, if not millennia, of belief regarding something so basic, the number of space dimensions, that it seemed almost beyond questioning.
Although I set this story in a lower-dimensional universe, for visual ease, our situation could be much the same. Hundreds of years of experiments have confirmed that gravity varies inversely with the square of distance, giving strong evidence that there are three space dimensions. But as of 1998, no experiment had ever probed gravity's strength on separations smaller than a millimeter (today, as mentioned, this has been pushed to a tenth of a millimeter). This led Savas Dimopoulos, of Stanford, Nima Arkani-Hamed, now of Harvard, and Gia Dvali, of New York University, to propose that in the braneworld scenario extra dimensions could be as large as a millimeter and would still have been undetected. This radical suggestion inspired a number of experimental groups to initiate a study of gravity at submillimeter distances in hopes of finding violations of the inverse square law; so far, none have been found, down to a tenth of a millimeter. Thus, even with today's state-of-the-art gravity experiments, if we are living within a three-brane, the extra dimensions could be as large as a tenth of a millim
eter, and yet we wouldn't know it.
This is one of the most striking realizations of the last decade. Using the three nongravitational forces, we can probe down to about a billionth of a billionth (10 -18 ) of a meter, and no one has found any evidence of extra dimensions. But in the braneworld scenario, the nongravitational forces are impotent in searching for extra dimensions since they are trapped on the brane itself. Only gravity can give insight into the nature of the extra dimensions, and, as of today, the extra dimensions could be as thick as a human hair and yet they'd be completely invisible to our most sophisticated instruments. Right now, right next to you, right next to me, and right next to everyone else, there could be another spatial dimension—a dimension beyond left/right, back/forth, and up/down, a dimension that's curled up but still large enough to swallow something as thick as this page—that remains beyond our grasp. 38
Large Extra Dimensions and Large Strings
By trapping three of the four forces, the braneworld scenario significantly relaxes experimental constraints on how big the extra dimensions can be, but the extra dimensions aren't the only thing this approach allows to get bigger. Drawing on insights of Witten, Joe Lykken, Constantin Bachas, and others, Ignatios Antoniadis, together with Arkani-Hamed, Dimopoulos, and Dvali, realized that in the braneworld scenario even unexcited, low-energy strings can be much larger than previously thought. In fact, the two scales—the size of extra dimensions and the size of strings—are closely related.
Remember from the previous chapter that the basic size of string is determined by requiring that its graviton vibrational pattern communicate a gravitational force of the observed strength. The weakness of gravity translates into the string's being very short, about the Planck length (10 -33 centimeters). But this conclusion is highly dependent on the size of the extra dimensions. The reason is that in string/M-theory, the strength of the gravitational force we observe in our three extended dimensions represents an interplay between two factors. One factor is the intrinsic, fundamental strength of the gravitational force. The second factor is the size of the extra dimensions. The larger the extra dimensions, the more gravity can spill into them and the weaker its force will appear in the familiar dimensions. Just as larger pipes yield weaker water pressure because they allow water more room to spread out, so larger extra dimensions yield weaker gravity, because they give gravity more room to spread out.
The original calculations that determined the string's length assumed that the extra dimensions were so tiny, on the order of the Planck length, that gravity couldn't spill into them at all. Under this assumption, gravity appears weak because it is weak. But now, if we work in the braneworld scenario and allow the extra dimensions to be much larger than had previously been considered, the observed feebleness of gravity no longer means that it's intrinsically weak. Instead, gravity could be a relatively powerful force that appears weak only because the relatively large extra dimensions, like large pipes, dilute its intrinsic strength. Following this line of reasoning, if gravity is much stronger than once thought, the strings can be much longer than once thought, too.
As of today, the question of exactly how long doesn't have a unique, definite answer. With the newfound freedom to vary both the size of strings and the size of the extra dimensions over a much wider range than previously envisioned, there are a number of possibilities. Dimopoulos and his collaborators have argued that existing experimental results, both from particle physics and from astrophysics, show that unexcited strings can't be larger than about a billionth of a billionth of a meter (10 -18 meters). While small by everyday standards, this is about a hundred million billion (10 17 ) times larger than the Planck length—nearly a hundred million billion times larger than previously thought. As we'll now see, that would be large enough that signs of strings could be detected by the next generation of particle accelerators.
String Theory Confronts Experiment?
The possibility that we are living within a large three-brane is, of course, just that: a possibility. And, within the braneworld scenario, the possibility that the extra dimensions could be much larger than once thought—and the related possibility that strings could also be much larger than once thought—are also just that: possibilities. But they are tremendously exciting possibilities. True, even if the braneworld scenario is right, the extra dimensions and the string size could still be Planckian. But the possibility within string/M-theory for strings and the extra dimensions to be much larger—to be just beyond the reach of today's technology—is fantastic. It means that there is at least a chance that in the next few years, string/ M-theory will make contact with observable physics and become an experimental science.
How big a chance? I don't know, and nor does anyone else. My intuition tells me it's unlikely, but my intuition is informed by a decade and a half of working within the conventional framework of Planck-sized strings and Planck-sized extra dimensions. Perhaps my instincts are old-fashioned. Thankfully, the question will be settled without the slightest concern for anyone's intuition. If the strings are big, or if some of the extra dimensions are big, the implications for upcoming experiments are spectacular.
In the next chapter, we'll consider a variety of experiments that will test, among other things, the possibilities of comparatively large strings and large extra dimensions, so here I will just whet your appetite. If strings are as large as a billionth of a billionth (10 -18 ) of a meter, the particles corresponding to the higher harmonic vibrations in Figure 12.4 will not have enormous masses, in excess of the Planck mass, as in the standard scenario. Instead, their masses will be only a thousand to a few thousand times that of a proton, and that's low enough to be within reach of the Large Hadron Collider now being built at CERN. If these string vibrations were to be excited through high-energy collisions, the accelerator's detectors would light up like the Times Square crystal ball on New Year's Eve. A whole host of never-before-seen particles would be produced, and their masses would be related to one another's much as the various harmonics are related on a cello. String theory's signature would be etched across the data with a flourish that would have impressed John Hancock. Researchers wouldn't be able to miss it, even without their glasses.
Moreover, in the braneworld scenario, high-energy collisions might even produce—get this—miniature black holes. Although we normally think of black holes as gargantuan structures out in deep space, it's been known since the early days of general relativity that if you crammed enough matter together in the palm of your hand, you'd create a tiny black hole. This doesn't happen because no one's grip—and no mechanical device—is even remotely strong enough to exert a sufficient compression force. Instead, the only accepted mechanism for black hole production involves the gravitational pull of an enormously massive star's overcoming the outward pressure normally exerted by the star's nuclear fusion processes, causing the star to collapse in on itself. But if gravity's intrinsic strength on small scales is far greater than previously thought, tiny black holes could be produced with significantly less compression force than previously believed. Calculations show that the Large Hadron Collider may have just enough squeezing power to create a cornucopia of microscopic black holes through high-energy collisions between protons. 7 Think about how amazing that would be. The Large Hadron Collider might turn out to be a factory for producing microscopic black holes! These black holes would be so small and would last for such a short time that they wouldn't pose us the slightest threat (years ago, Stephen Hawking showed that all black holes disintegrate via quantum processes—big ones very slowly, tiny ones very quickly), but their production would provide confirmation of some of the most exotic ideas ever contemplated.
Braneworld Cosmology
A primary goal of current research, one that is being hotly pursued by scientists worldwide (including me), is to formulate an understanding of cosmology that incorporates the new insights of string/M-theory. The reason is clear: not only does cosmology grapple with the big, gulp-in-the-throat questio
ns, and not only have we come to realize that aspects of familiar experience—such as the arrow of time—are bound up with conditions at the universe's birth, but cosmology also provides a theorist with what New York provided Sinatra: a proving ground par excellence. If a theory can make it in the extreme conditions characteristic of the universe's earliest moments, it can make it anywhere.
As of today, cosmology according to string/M-theory is a work in progress, with researchers heading down two main pathways. The first and more conventional approach imagines that just as inflation provided a brief but profound front end to the standard big bang theory, string/M-THEORY provides a yet earlier and perhaps yet more profound front end to inflation. The vision is that string/M-theory will unfuzz the fuzzy patch we've used to denote our ignorance of the universe's earliest moments, and after that, the cosmological drama will unfold according to inflationary theory's remarkably successful script, recounted in earlier chapters.
While there has been progress on specific details required by this vision (such as trying to understand why only three of the universe's spatial dimensions underwent expansion, as well as developing mathematical methods that may prove relevant to analyzing the spaceless/timeless realm that may precede inflation), the eureka moments have yet to occur. The intuition is that whereas inflationary cosmology imagines the observable universe getting ever smaller at ever earlier times—and hence being ever hotter, denser, and energetic—string/M-theory tames this unruly (in physics-speak, "singular") behavior by introducing a minimal size (as in our discussion on pages 350-351) below which new and less singular physical quantities become relevant. This reasoning lies at the heart of string/M-theory's successful merger of general relativity and quantum mechanics, and many researchers expect that we will shortly determine how to apply the same reasoning in the context of cosmology. But, as of now, the fuzzy patch still looks fuzzy, and it's anybody's guess when clarity will be achieved.
