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

### Part IV. String Theory and the Fabric of Spacetime

### Chapter 11. 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.

**Figure 11.1** (a) In a "U-shaped" universe, the only way to get from one end to the other is by traversing the whole cosmos. (b) The fabric of space tears, and two ends of a wormhole start to grow. (c) The two wormhole ends merge together, forming a new bridge—a shortcut—from one end of the universe to the other.

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)).

**Figure 11.2** The highlighted region inside a Calabi-Yau shape contains a sphere.

**Figure 11.3** A sphere inside a Calabi-Yau space shrinks down to a point, pinching the fabric of space. We simplify this and subsequent figures by showing only part of the full Calabi-Yau shape.

**Figure 11.4** A pinched Calabi-Yau space tears open and grows a sphere that smoothes out its surface. The original sphere of Figure 11.3 is "flopped."

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.

But then, in 1991 the Norwegian physicist Andy Lütken together with Paul Aspinwall, a graduate-school classmate of mine from Oxford and now a professor at Duke University, asked themselves what proved to be a very interesting question: If the spatial fabric of the Calabi-Yau portion of our universe were to undergo a space-tearing flop transition, what would it look like from the perspective of the mirror Calabi-Yau space? To understand the motivation for this question, you must recall that the physics emerging from either member of a mirror pair of Calabi-Yau shapes (if selected for the extra dimensions) is identical, but the complexity of the mathematics that a physicist must employ to extract the physics can differ significantly between the two. Aspinwall and Lütken speculated that the mathematically complicated flop transition of Figures 11.3 and 11.4 might have a far simpler mirror description—one that might give a more transparent view on the associated physics.

At the time of their work, mirror symmetry was not understood at the depth required to answer the question they posed. However, Aspinwall and Lütken noted that there did not seem to be anything in the mirror description that would indicate a disastrous physical consequence associated with the spatial tears of flop transitions. Around the same time, the work Plesser and I had done in finding mirror pairs of Calabi-Yau shapes (see Chapter 10) unexpectedly led us to think about flop transitions as well. It is a well-known mathematical fact that gluing various points together as in Figure 10.4—the procedure we had used to construct mirror pairs—leads to geometrical situations that are identical to the pinch and puncture in Figures 11.3 and 11.4. Physically, though, Plesser and I could find no associated calamity. Moreover, inspired by the observations of Aspinwall and Lütken (as well as a previous paper of theirs with Graham Ross), Plesser and I realized that we could repair the pinch mathematically in two different ways. One way led to the Calabi-Yau shape in Figure 11.3(a) while the other led to that in Figure 11.4(d). This suggested to us that the evolution from Figure 11.3(a) through Figure 11.4(d) was something that could actually occur in nature.

By late 1991, then, at least a few string theorists had a strong feeling that the fabric of space *can* tear. But no one had the technical facility to definitively establish or refute this striking possibility.

**Inching Forward**

Off and on during 1992, Plesser and I tried to show that the fabric of space can undergo space-tearing flop transitions. Our calculations yielded bits and pieces of supporting circumstantial evidence, but we could not find definitive proof. Sometime during the spring, Plesser visited the Institute for Advanced Study in Princeton to give a talk, and privately told Witten about our recent attempts to realize the mathematics of space-tearing flop transitions within the physics of string theory. After summarizing our ideas, Plesser waited for Witten's response. Witten turned from the blackboard and stared out of his office window. After a minute of silence, maybe two, he turned back to Plesser and told him that if our ideas worked out, "it would be spectacular." This rekindled our efforts. But after a while, with our progress stalled, each of us turned to working on other string theory projects.

Even so, I found myself mulling over the possibility of space-tearing flop transitions. As the months went by, I felt increasingly sure that they had to be part and parcel of string theory. The preliminary calculations Plesser and I had done, together with insightful discussions with David Morrison, a mathematician from Duke University, made it seem that this was the only conclusion that mirror symmetry naturally supported. In fact, during a visit to Duke, Morrison and I, together with some helpful observations from Sheldon Katz of Oklahoma State University, who was also visiting Duke at the time, outlined a strategy for proving that flop transitions can occur in string theory. But when we sat down to do the required calculations, we found that they were extraordinarily intensive. Even on the world's fastest computer, they would take more than a century to complete. We had made progress, but we clearly needed a new idea, one that could greatly enhance the efficiency of our calculational method. Unwittingly, Victor Batyrev, a mathematician from the University of Essen, revealed such an idea through a pair of papers released in the spring and summer of 1992.

Batyrev had become very interested in mirror symmetry, especially in the wake of the success of Candelas and his collaborators in using it to solve the sphere-counting problem described at the end of Chapter 10. With a mathematician's perspective, though, Batyrev was unsettled by the methods Plesser and I had invoked to find mirror pairs of Calabi-Yau spaces. Although our approach used tools familiar to string theorists, Batyrev later told me that our paper seemed to him to be "black magic." This reflects the large cultural divide between the disciplines of physics and mathematics, and as string theory blurs their borders, the vast differences in language, methods, and styles of each field become increasingly apparent. Physicists are more like avant-garde composers, willing to bend traditional rules and brush the edge of acceptability in the search for solutions. Mathematicians are more like classical composers, typically working within a much tighter framework, reluctant to go to the next step until all previous ones have been established with due rigor. Each approach has its advantages as well as drawbacks; each provides a unique outlet for creative discovery. Like modern and classical music, it's not that one approach is right and the other wrong—the methods one chooses to use are largely a matter of taste and training.

Batyrev set out to recast the construction of mirror manifolds in a more conventional mathematical framework, and he succeeded. Inspired by earlier work of Shi-Shyr Roan, a mathematician from Taiwan, he found a systematic mathematical procedure for producing pairs of Calabi-Yau spaces that are mirrors of one another. His construction reduces to the procedure Plesser and I had found in the examples we had considered, but offers a more general framework that is phrased in a manner more familiar to mathematicians.

The flip side is that Batyrev's papers invoked areas of mathematics that most physicists had never previously encountered. I, for example, could extract the gist of his arguments, but had significant difficulty in understanding many crucial details. One thing, however, was clear: The methods of his paper, if properly understood and applied, could very well open a new line of attack on the issue of space-tearing flop transitions.

By late summer, energized by these developments, I decided that I wanted to return to the problem of flops with full and undistracted intensity. I had learned from Morrison that he was going on leave from Duke to spend a year at the Institute for Advanced Study, and I knew that Aspinwall would also be there, as a postdoctoral fellow. After a few phone calls and e-mails, I arranged to take leave from Cornell University and spend the fall of 1992 at the Institute as well.

**A Strategy Emerges**

One would be hard pressed to think of a more ideal place for long hours of intense concentration than the Institute for Advanced Study. Founded in 1930, it is set within gently rolling fields on the border of an idyllic forest a few miles from the campus of Princeton University. It is said that you can't get distracted from your work at the Institute, because, well, there aren't any distractions.

After leaving Germany in 1933, Einstein joined the Institute and remained there for the duration of his life. It takes little imagination to picture him pondering unified field theory in the Institute's quiet, lonely, almost ascetic surroundings. The legacy of deep thought infuses the atmosphere, which, depending on your own immediate state of progress, can be either exciting or oppressive.

Shortly after arriving at the Institute, Aspinwall and I were walking down Nassau Street (the main commercial street in the town of Princeton) trying to agree on a place to have dinner. This was no small task since Paul is as devout a meat eater as I am a vegetarian. In the midst of catching up on each other's lives as we were walking along, he asked me if I had any ideas about new things to work on. I told him I did, and recounted my take on the importance of establishing that the universe, if truly described by string theory, can undergo space-tearing flop transitions. I also outlined the strategy I had been pursuing, as well as my newfound hope that Batyrev's work might allow us to fill in the missing pieces. I thought that I was preaching to the converted, and that Paul would be excited by this prospect. He wasn't. In retrospect, his reticence was due largely to our good-natured and long-standing intellectual joust in which we each play devil's advocate to the other's ideas. Within days, he came around and we turned our full attention to flops.

By then, Morrison had also arrived, and the three of us met in the Institute's tea-room to formulate a strategy. We agreed that the central goal was to determine whether the evolution from Figure 11.3(a) to Figure 11.4(d) can actually occur in our universe. But a direct attack on the question was forbidding, because the equations describing this evolution are extremely difficult, especially when the spatial tear occurs. Instead, we chose to rephrase the issue using the mirror description, hoping that the equations involved might be more manageable. This is schematically illustrated in Figure 11.5, in which the top row is the original evolution from Figure 11.3(a) to Figure 11.4(d), and the bottom row is the same evolution from the perspective of the mirror Calabi-Yau shapes. As a number of us had already realized, it turns out that in the mirror rephrasing it appears that string physics is perfectly well behaved and encounters no catastrophes. As you can see, there does not seem to be any pinching or tearing in the bottom row in Figure 11.5. However, the real question this observation raised for us was this: Were we pushing mirror symmetry beyond the bounds of its applicability? Although the upper and lower Calabi-Yau shapes drawn on the far left-hand side of Figure 11.5 yield identical physics, is it true that at every step in the evolution to the right-hand side of Figure 11.5—necessarily passing through the pinch-tear-repair stage in the middle—the physical properties of the original and mirror perspective are identical?

Although we had solid reason to believe that the powerful mirror relationship holds for the shape progression leading to the tear in the upper Calabi-Yau shape in Figure 11.5, we realized that no one knew whether the upper and lower Calabi-Yau shapes in Figure 11.5 continue to be mirrors after the tear has occurred. This is a crucial question, because if they are, then the absence of a catastrophe in the mirror perspective would mean an absence in the original, and we would have demonstrated that space can tear in string theory. We realized that this question could be reduced to a calculation: Extract the physical properties of the universe for the upper Calabi-Yau shape after the tear (using, say, the upper-right Calabi-Yau shape in Figure 11.5) and for its supposed mirror (the lower-right Calabi-Yau shape in Figure 11.5), and see if they are identical.

**Figure 11.5** A space-tearing flop transition (top row) and its purported mirror rephrasing (bottom row).

It was this calculation to which Aspinwall, Morrison, and I devoted ourselves in the fall of 1992.

**Late Nights at Einstein's Final Stomping Ground**

Edward Witten's razor-sharp intellect is clothed in a soft-spoken demeanor that often has a wry, almost ironic, edge. He is widely regarded as Einstein's successor in the role of the world's greatest living physicist. Some would go even further and describe him as the greatest physicist of all time. He has an insatiable appetite for cutting-edge physics problems and he wields tremendous influence in setting the direction of research in string theory.

The breadth and depth of Witten's productivity is legendary. His wife, Chiara Nappi, who is also a physicist at the Institute, paints a picture of Witten sitting at their kitchen table, mentally probing the edge of string theory knowledge, and only now and then returning to pick up pen and paper to verify an elusive detail or two.^{3} Another story is told by a postdoctoral fellow who, one summer, had an office next to Witten's. He describes the unsettling juxtaposition of laboriously struggling with complex string theory calculations at his desk while hearing the incessant rhythmic patter of Witten's keyboard, as paper after groundbreaking paper poured forth directly from mind to computer file.

A week or so after I arrived, Witten and I were chatting in the Institute's courtyard, and he asked about my research plans. I told him about the space-tearing flops and the strategy we were planning to pursue. He lit up upon hearing the ideas, but cautioned that he thought the calculations would be horrendously difficult. He also pointed out a potential weak link in the strategy I described, having to do with some work I had done a few years earlier with Vafa and Warner. The issue he raised turned out to be only tangential to our approach for understanding flops, but it started him thinking about what ultimately turned out to be related and complementary issues.

Aspinwall, Morrison, and I decided to split our calculation in two pieces. At first a natural division might have seemed to involve first extracting the physics associated with the final Calabi-Yau shape from the upper row of Figure 11.5, and then doing the same for the final Calabi-Yau shape from the lower row of Figure 11.5. If the mirror relationship is not shattered by the tear in the upper Calabi-Yau, then these two final Calabi-Yau shapes should yield identical physics, just like the two initial Calabi-Yau shapes from which they evolved. (This way of phrasing the problem avoids doing any of the very difficult calculations involving the upper Calabi-Yau shape just when it tears.) It turns out, though, that calculating the physics associated with the final Calabi-Yau shape in the upper row is pretty straightforward. The real difficulty in carrying out this program lies in first figuring out the *precise shape* of the final Calabi-Yau space in the lower row of Figure 11.5—the putative mirror of the upper Calabi-Yau—and then in extracting the associated physics.

A procedure for accomplishing the second task—extracting the physical features of the final Calabi-Yau space in the lower row, once its shape was precisely known—had been worked out a few years earlier by Candelas. His approach, however, was calculationally intensive and we realized that it would require a clever computer program to carry it out in our explicit example. Aspinwall, who in addition to being a renowned physicist is a crackerjack programmer, took on this task. Morrison and I set out to accomplish the first task, namely, to identify the precise shape of the candidate mirror Calabi-Yau space.

It was here that we felt Batyrev's work could provide us some important clues. Once again, though, the cultural divide between mathematics and physics—in this case, between Morrison and me—started to impede progress. We needed to join the power of the two fields to find the *mathematical* form of the lower Calabi-Yau shapes that should correspond to the same *physical* universe as the upper Calabi-Yau shapes, if flop tears are within nature's repertoire. But neither of us was sufficiently conversant in the other's language to see clear to reaching this end. It became obvious to both of us that we needed to bite the bullet: Each of us needed to take a crash course in the other's field of expertise. And so, we decided to spend our days pushing forward as best we could on the calculation, while spending evenings being both professor and student in a class of one: I would lecture to Morrison for an hour or two on the relevant physics; he would then lecture to me for an hour or two on the relevant mathematics. School would typically let out at about 11 P.M.

We stuck to the program, day in and day out. Progress was slow, but we could sense that things were starting to fall into place. Meanwhile, Witten was making significant headway on reformulating the weak link he had earlier identified. His work was establishing a new and more powerful method of translation between the physics of string theory and the mathematics of the Calabi-Yau spaces. Aspinwall, Morrison, and I had almost daily impromptu meetings with Witten at which he would show us new insights following from his approach. As the weeks went by, it gradually became clear that unexpectedly, his work, from a vantage point completely different from our own, was converging on the issue of flop transitions. Aspinwall, Morrison, and I realized that if we didn't complete our calculation soon, Witten would beat us to the punch.

**Of Six-Packs and Working Weekends**

Nothing focuses the mind of a physicist like a healthy dose of competition. Aspinwall, Morrison, and I went into high gear. It's important to note that this meant one thing to Morrison and me, and quite another to Aspinwall. Aspinwall is a curious mixture of upper-class British sensibility, largely a reflection of the decade he spent at Oxford as both an undergraduate and a graduate student, infused ever so slightly with a prankster's roguishness. As far as work habits go, he is perhaps the most civilized physicist I know. While many of us work deep into the evening, he never works past 5 P.M. While many of us work weekends, Aspinwall does not. He gets away with this because he is both sharp and efficient. Going into high gear for him merely amounts to notching up his efficiency level to even greater heights.

By this time, it was early December. Morrison and I had been lecturing to one another for several months and it was starting to pay off. We were very close to being able to identify the precise shape of the Calabi-Yau space we were seeking. Moreover, Aspinwall had just about finished his computer code, and he now awaited our result, which would be the required input for his program. It was a Thursday night when Morrison and I finally had confidence that we knew how to identify the sought-after Calabi-Yau shape. That, too, boiled down to a procedure that required its own, fairly simple, computer code. By Friday afternoon we had written the program and debugged it; by late Friday night we had our result.

But it was after 5 P.M. and it was Friday. Aspinwall had gone home and would not return until Monday. There was nothing we could do without his computer code. Neither Morrison nor I could imagine waiting out the whole weekend. We were on the verge of answering the long-pondered question of spatial tears in the fabric of the cosmos, and the suspense was too much to bear. We called Aspinwall at home. At first he refused to come to work the next morning as we asked. But then, after much groaning, he consented to join us, as long as we bought him a six-pack of beer. We agreed.

**A Moment of Truth**

We all met at the Institute Saturday morning as planned. It was a bright sunny morning, and the atmosphere was jokingly relaxed. I, for one, half expected that Aspinwall would not show up; once he did, I spent 15 minutes extolling the import of this first weekend he had come into the office. He assured me it wouldn't happen again.

We all huddled around Morrison's computer in the office he and I shared. Aspinwall told Morrison how to bring his program up on the screen and showed us the precise form for the required input. Morrison appropriately formatted the results we had generated the previous night, and we were set to go.

The particular calculation we were performing amounts, roughly speaking, to determining the mass of a certain particle species—a specific vibrational pattern of a string—when moving through a universe whose Calabi-Yau component we had spent all fall identifying. We hoped, in line with the strategy discussed earlier, that this mass would agree identically with a similar calculation done on the Calabi-Yau shape emerging from the space-tearing flop transition. The latter was the relatively easy calculation, and we had completed it weeks before; the answer turned out to be 3, in the particular units we were using. Since we were now doing the purported mirror calculation numerically on a computer, we expected to get something extremely close to but not exactly 3, something like 3.000001 or 2.999999, with the tiny difference arising from rounding errors.

Morrison sat at the computer with his finger hovering over the enter button. With the tension mounting he said, "Here goes," and set the calculation in motion. In a couple of seconds the computer returned its answer: 8.999999. My heart sank. Could it be that space-tearing flop transitions shatter the mirror relation, likely indicating that they cannot actually occur? Almost immediately, though, we all realized that something funny must be going on. If there was a real mismatch in the physics following from the two shapes, it was extremely unlikely that the computer calculation should yield an answer so close to a whole number. If our ideas were wrong, there was no reason in the world to expect anything but a random collection of digits. We had gotten a wrong answer, but one that suggested, perhaps, that we had just made some simple arithmetic error. Aspinwall and I went to the blackboard, and in a moment we found our mistake: we had dropped a factor of 3 in the "simpler" calculation we had done weeks before; the true result was 9. The computer answer was therefore just what we wanted.

Of course, the after-the-fact agreement was only marginally convincing. When you know the answer you want, it is often all too easy to figure out a way of getting it. We needed to do another example. Having already written all of the necessary computer code, this was not hard to do. We calculated another particle mass on the upper Calabi-Yau shape, being careful this time to make no errors. We found the answer: 12. Once again, we huddled around the computer and set it on its way. Seconds later it returned 11.999999. *Agreement.* We had shown that the supposed mirror *is* the mirror, and hence space-tearing flop transitions are part of the physics of string theory.

At this I jumped out of my chair and ran an unrestrained victory lap around the office. Morrison beamed from behind the computer. Aspinwall's reaction, though, was rather different. "That's great, but I knew it would work," he calmly said. "And where's my beer?"

**Witten's Approach**

That Monday, we triumphantly went to Witten and told him of our success. He was very pleased with our result. And, as it turned out, he too had just found a way of establishing that flop transitions occur in string theory. His argument was quite different from ours, and it significantly illuminates the microscopic understanding of why the spatial tears do not have any catastrophic consequences.

His approach highlights the difference between a point-particle theory and string theory when such tears occur. The key distinction is that there are two types of string motion near the tear, but only one kind of point-particle motion. Namely, a string can travel adjacent to the tear, like a point particle does, but it can also encircle the tear as it moves forward, as illustrated in Figure 11.6. In essence, Witten's analysis reveals that strings which encircle the tear, something that cannot happen in a point-particle theory, shield the surrounding universe from the catastrophic effects that would otherwise be encountered. It's as if the world-sheet of the string—recall from Chapter 6 that this is a two-dimensional surface that a string sweeps out as it moves through space—provides a protective barrier that precisely cancels out the calamitous aspects of the geometrical degeneration of the spatial fabric.

You might well ask, What if such a tear should occur, and it just so happens that there are no strings in the vicinity to shield it? Moreover, you might also be concerned that at the instant in time that a tear occurs, a string—an infinitely thin loop—would provide as effective a barrier as shielding yourself from a cluster bomb by hiding behind a hula hoop. The resolution to both of these issues relies on a central feature of quantum mechanics that we discussed in Chapter 4. There we saw that in Feynman's formulation of quantum mechanics, an object, be it a particle or a string, travels from one location to another by "sniffing out" all possible trajectories. The resulting motion that is observed is a combination of *all* possibilities, with the relative contributions of each possible trajectory precisely determined by the mathematics of quantum mechanics. Should a tear in the fabric of space occur, then among the possible trajectories of travelling strings are those that encircle the tear—trajectories such as those in Figure 11.6. Even if no strings seem to be near the tear when it occurs, quantum mechanics takes account of physical effects from all possible string trajectories and among these are numerous (infinite, in fact) protective paths that encircle the tear. It is these contributions that Witten showed precisely to cancel out the cosmic calamity that the tear would otherwise create.

**Figure 11.6** The world-sheet swept out by a string provides a shield that cancels the potentially cataclysmic effects associated with a tear in the fabric of space.

In January 1993, Witten and the three of us released our papers simultaneously to the electronic Internet archive through which physics papers are immediately made available worldwide. The two papers described, from our widely different perspectives, the first examples of *topology-changing transitions*—the technical name for the space-tearing processes we had found. The long-standing question about whether the fabric of space can tear had been settled quantitatively by string theory.

**Consequences**

We have made much of the realization that spatial tears can occur without physical calamity. But what *does* happen when the spatial fabric rips? What are the observable consequences? We have seen that many properties of the world around us depend upon the detailed structure of the curled-up dimensions. And so, you would think that the fairly drastic transformation from one Calabi-Yau to another as shown in Figure 11.5, would have a significant physical impact. In fact, though, the lower-dimensional drawings that we use to visualize the spaces make the transformation appear to be somewhat more complicated than it actually is. If we could visualize six-dimensional geometry, we would see that, yes, the fabric is tearing, but it does so in a fairly mild way. It's more like the handiwork of a moth on wool than that of a deep knee bend on shrunken trousers.

Our work and that of Witten show that physical characteristics such as the number of families of string vibrations and the types of particles within each family are unaffected by these processes. As the Calabi-Yau space evolves through a tear, what can be affected are the precise values of the masses of the individual particles—the energies of the possible patterns of string vibrations. Our papers showed that these masses will vary continuously in response to the changing geometrical form of the Calabi-Yau component of space, some going up while others go down. Of primary importance, though, is the fact that there is no catastrophic jump, spike, or any unusual feature of these varying masses as the tear actually occurs. From the point of view of physics, the moment of tearing has no distinguishing characteristics.

This point raises two issues. First, we have focused on tears in the spatial fabric that occur in the extra six-dimensional Calabi-Yau component of the universe. Can such tears also occur in the more familiar three extended spatial dimensions? The answer, almost certainly, is yes. After all, space is space—regardless of whether it is tightly curled up into a Calabi-Yau shape or is unfurled into the grand expanse of the universe we perceive on a clear, starry night. In fact, we have seen earlier that the familiar spatial dimensions might themselves actually be curled up into the form of a giant shape that curves back on itself, way on the other side of the universe, and that therefore even the distinction between which dimensions are curled up and which are unfurled is somewhat artificial. Although our and Witten's analyses did rely on special mathematical features of Calabi-Yau shapes, the result—that the fabric of space can tear—is certainly of wider applicability.

Second, could such a topology-changing tear happen today or tomorrow? Could it have happened in the past? Yes. Experimental measurements of elementary particle masses show their values to be quite stable over time. But if we head back to the earliest epochs following the big bang, even non-string-based theories invoke important periods during which elementary particle masses do change over time. These periods, from a string-theoretic perspective, could certainly have involved the topology-changing tears discussed in this chapter. Closer to the present, the observed stability of elementary particle masses implies that if the universe is currently undergoing a topology-changing spatial tear, it must be doing it exceedingly slowly—so slowly that its effect on elementary particle masses is smaller than our present experimental sensitivity. Remarkably, so long as this condition is met, the universe could currently be in the midst of a spatial rupture. If it were occurring slowly enough, we would not even know it was happening. This is one of those rare instances in physics in which the lack of a striking observable phenomenon is cause for great excitement. The absence of an observable calamitous consequence from such an exotic geometrical evolution is testament to how far beyond Einstein's expectations string theory has gone.