The Elegant Universe (37 page)

It’s somewhat as if someone requires you to count exactly the number of oranges that are haphazardly jumbled together in an enormous bin, some 50 feet on each side and 10 feet deep. You start to count them one by one, but soon realize that the task is just too laborious. Luckily, though, a friend comes along who was present when the oranges were delivered. He tells you that they arrived neatly packed in smaller boxes (one of which he just happens to be holding) that when stacked were 20 boxes long, by 20 boxes deep, by 20 boxes high. You quickly calculate that they arrived in 8,000 boxes, and that all you need to do is figure out how many oranges are packed in each. This you easily do by borrowing your friend’s box and filling it with oranges, allowing you to finish your huge counting task with almost no effort. In essence, by cleverly reorganizing the calculation, you were able to make it substantially easier to accomplish.

The situation with numerous calculations in string theory is similar. From the perspective of one Calabi-Yau space, a calculation might involve an enormous number of difficult mathematical steps. By translating the calculation to its mirror, though, the calculation is reorganized in a far more efficient manner, allowing it to be completed with relative ease. This point was made by Plesser and me, and was impressively put into practice in subsequent work by Candelas with his collaborators Xenia de la Ossa and Linda Parkes, from the University of Texas, and Paul Green, from the University of Maryland. They showed that calculations of almost unimaginable difficulty could be accomplished by using the mirror perspective, with a few pages of algebra and a desktop computer.

This was an especially exciting development for mathematicians, because some of these calculations were precisely the ones they had been stuck on for many years. String theory—or so the physicists claimed—had beaten them to the solution.

Now you should bear in mind that there is a good deal of healthy and generally good-natured competition between mathematicians and physicists. And as it turns out, two Norwegian mathematicians—Geir Ellingsrud and Stein Arilde Strømme—happened to be working on one of numerous calculations that Candelas and his collaborators had successfully conquered with mirror symmetry. Roughly speaking, it amounted to calculating the number of spheres that could be “packed” inside a particular Calabi-Yau space, somewhat like our analogy of counting oranges in a large bin. At a meeting of physicists and mathematicians in Berkeley in 1991, Candelas announced the result reached by his group using string theory and mirror symmetry: 317,206,375. Ellingsrud and Strømme announced the result of their very difficult mathematical calculation: 2,682,549,425. For days, mathematicians and physicists debated: Who was right? The question turned into a real litmus test of the quantitative reliability of string theory. A number of people even commented—somewhat in jest—that this test was the next best thing to being able to compare string theory with experiment. Moreover, Candelas’s results went far beyond the single numerical result that Ellingsrud and Strømme claimed to have calculated. He and his collaborators claimed to have also answered many other questions that were tremendously more difficult—so difficult in fact, that no mathematician had ever even attempted to address them. But could the string theory results be trusted? The meeting ended with a great deal of fruitful exchange between mathematicians and physicists, but no resolution of the discrepancy.

About a month later, an e-mail message was widely circulated among participants in the Berkeley meeting with the subject heading Physics Wins! Ellingsrud and Strømme had found an error in their computer code that, when corrected, confirmed Candelas’s result. Since then, there have been many mathematical checks on the quantitative reliability of the mirror symmetry of string theory: It has passed all with flying colors. Even more recently, almost a decade after physicists discovered mirror symmetry, mathematicians have made great progress in revealing its inherent mathematical foundations. By utilizing substantial contributions of the mathematicians Maxim Kontsevich, Yuri Manin, Gang Tian, Jun Li, and Alexander Givental, Yau and his collaborators Bong Lian and Kefeng Liu have finally found a rigorous mathematical proof of the formulas used to count spheres inside Calabi-Yau spaces, thereby solving problems that have puzzled mathematicians for hundreds of years.

Beyond the particulars of this success, what these developments really highlight is the role that physics has begun to play in modern mathematics. For quite some time, physicists have “mined” mathematical archives in search of tools for constructing and analyzing models of the physical world. Now, through the discovery of string theory, physics is beginning to repay the debt and to provide mathematicians with powerful new approaches to their unsolved problems. String theory not only provides a unifying framework for physics, but it may well forge an equally deep union with mathematics as well.

Tearing the Fabric of Space

I

f you relentlessly stretch a rubber membrane, sooner or later it will tear. This simple fact has inspired numerous physicists over the years to ask whether the same might be true of the spatial fabric making up the universe. That is, can the fabric of space rip apart, or is this merely a misguided notion that arises from taking the rubber membrane analogy too seriously?

Einstein’s general relativity says no, the fabric of space cannot tear.1 The equations of general relativity are firmly rooted in Riemannian geometry and, as we noted in the preceding chapter, this is a framework that analyzes distortions in the distance relations between nearby locations in space. In order to speak meaningfully about these distance relations, the underlying mathematical formalism requires that the substrate of space is smooth—a term with a technical mathematical meaning, but whose everyday usage captures its essence: no creases, no punctures, no separate pieces “stuck” together, and no tears. Were the fabric of space to develop such irregularities, the equations of general relativity would break down, signaling some or other variety of cosmic catastrophe—a disastrous outcome that our apparently well-behaved universe avoids.

This has not kept imaginative theorists over the years from pondering the possibility that a new formulation of physics that goes beyond Einstein’s classical theory and incorporates quantum physics might show that rips, tears, and mergers of the spatial fabric can occur. In fact, the realization that quantum physics leads to violent short-distance undulations led some to speculate that rips and tears might be a commonplace microscopic feature of the spatial fabric. The concept of wormholes (a notion with which any fan of Star Trek: Deep Space Nine is familiar) makes use of such musings. The idea is simple: Imagine you’re the CEO of a major corporation with headquarters on the ninetieth floor of one of New York City’s World Trade Center towers. Through the vagaries of corporate history, an arm of your company with which you need to have ever increasing contact is ensconced on the ninetieth floor of the other tower. As it is impractical to move either office, you come up with a natural suggestion: Build a bridge from one office to the other, connecting the two towers. This allows employees to move freely between the offices without having to go down and then up ninety floors.

A wormhole plays a similar role: It is a bridge or tunnel that provides a shortcut from one region of the universe to another. Using a two-dimensional model, imagine that a universe is shaped as in Figure 11.1.

If your corporate headquarters are located near the lower circle in 11.1 (a), you can get to your field office, located near the upper circle, only by traversing the entire U-shaped path, taking you from one end of the universe to another. But if the fabric of space can tear, developing punctures as in 11.1(b), and if these punctures can “grow” tentacles that merge together as in 11.1(c), a spatial bridge would connect the previously remote regions. This is a wormhole. You should note that the wormhole has some similarity to the World Trade Center bridge, but there is one essential difference: The World Trade Center bridge would traverse a region of existing space—the space between the two towers. On the contrary, the wormhole creates a new region of space, since the curved two-dimensional space in Figure 11.1(a) is all there is (in the setting of our two-dimensional analogy). Regions lying off of the membrane merely reflect the inadequacy of the illustration, which depicts the U-shaped universe as if it were an object within our higher-dimensional universe. The wormhole creates new space and therefore blazes new spatial territory

Do wormholes exist in the universe? No one knows. And if they do, it is far from clear whether they would take on only a microscopic form or if they could span vast regions of the universe (as in Deep Space Nine). But one essential element in assessing whether they are fact or fiction is determining whether or not the fabric of space can tear.

Black holes provide another compelling example in which the fabric of space is stretched to its limits. In Figure 3.7, we saw that the enormous gravitational field of a black hole results in such extreme curvature that the fabric of space appears to be pinched or punctured at the black hole’s center. Unlike in the case of wormholes, there is strong experimental evidence supporting the existence of black holes, so the question of what really happens at their central point is one of science, not speculation. Once again, the equations of general relativity break down under such extreme conditions. Some physicists have suggested that there really is a puncture, but that we are protected from this cosmic “singularity” by the event horizon of the black hole, which prevents anything from escaping its gravitational grip. This reasoning led Roger Penrose of Oxford University to speculate on a “cosmic censorship hypothesis” that allows these kinds of spatial irregularities to occur only if they are deeply hidden from our view behind the shroud of an event horizon. On the other hand, prior to the discovery of string theory, some physicists surmised that a proper merger of quantum mechanics and general relativity would show that the apparent puncture of space is actually smoothed out—“sewn up,” so to speak—by quantum considerations.

With the discovery of string theory and the harmonious merger of quantum mechanics and gravity, we are finally poised to study these issues. As yet, string theorists have not been able to answer them fully, but during the last few years closely related issues have been solved. In this chapter we discuss how string theory, for the first time, definitively shows that there are physical circumstances—differing from wormholes and black holes in certain ways—in which the fabric of space can tear.

A Tantalizing Possibility

In 1987, Shing-Tung Yau and his student Gang Tian, now at the Massachusetts Institute of Technology, made an interesting mathematical observation. They found, using a well-known mathematical procedure, that certain Calabi-Yau shapes could be transformed into others by puncturing their surface and then sewing up the resulting hole according to a precise mathematical pattern.2 Roughly speaking, they identified a particular kind of two-dimensional sphere—like the surface of a beach ball—sitting inside an initial Calabi-Yau space, as in Figure 11.2. (A beach ball, like all familiar objects, is three-dimensional. Here, however, we are referring solely to its surface; we are ignoring the thickness of the material from which it is made as well as the interior space it encloses. Points on the beach ball’s surface can be located by giving two numbers—“latitude” and “longitude”—much as we locate points on the earth’s surface. This is why the surface of the beach ball, like the surface of the garden hose discussed in preceding chapters, is two-dimensional.) They then considered shrinking the sphere until it is pinched down to a single point, as we illustrate with the sequence of shapes in Figure 11.3. This figure, and subsequent ones in this chapter, have been simplified by focusing in on the most relevant “piece” of the Calabi-Yau shape, but in the back of your mind you should note that these shape transformations are occuring within a somewhat larger Calabi-Yau space, as in Figure 11.2. And finally, Tian and Yau imagined slightly tearing the Calabi-Yau space at the pinch (Figure 11.4(a)), opening it up and gluing in another beach ball-like shape (Figure 11.4(b)), which they could then reinflate to a nice plump form (Figures 11.4(c) and 11.4(d)).

Mathematicians call this sequence of manipulations a flop-transition. It’s as if the original beach ball shape is “flopped” over into a new orientation within the overall Calabi-Yau shape. Yau, Tian, and others noted that under certain circumstances, the new Calabi-Yau shape produced by a flop, as in Figure 11.4(d), is topologically distinct from the initial Calabi-Yau shape in Figure 11.3(a). This is a fancy way of saying that there is absolutely no way to deform the initial Calabi-Yau space in Figure 11.3(a) into the final Calabi-Yau space shown in Figure 11.4(d) without tearing the fabric of the Calabi-Yau space at some intermediate stage.

From a mathematical standpoint, this procedure of Yau and Tian is of interest because it provides a way to produce new Calabi-Yau spaces from ones that are known. But its real potential lies in the realm of physics, where it raises a tantalizing question: Could it be that, beyond its being an abstract mathematical procedure, the sequence displayed from Figure 11.3(a) through Figure 11.4(d) might actually occur in nature? Might it be that, contrary to Einstein’s expectations, the fabric of space can tear apart and subsequently be repaired in the manner described?

The Mirror Perspective

For a couple of years after their 1987 observation, Yau would, every so often, encourage me to think about the possible physical incarnation of these flop transitions. I didn’t. To me it seemed that flop transitions were merely a piece of abstract mathematics without any bearing on the physics of string theory. In fact, based on the discussion in Chapter 10 in which we found that circular dimensions have a minimum radius, one might be tempted to say that string theory does not allow the sphere in Figure 11.3 to shrink all the way down to a pinched point. But remember, as also noted in Chapter 10, that if a chunk of space collapses—in this case, a spherical piece of a Calabi-Yau shape—as opposed to the collapse of a complete spatial dimension, the argument identifying small and large radii is not directly applicable. Nevertheless, even though this idea for ruling out flop transitions does not stand up to scrutiny, the possibility that the fabric of space could tear still seemed rather unlikely.