The Elegant Universe - Brian Greene [172]
Figure 13.2 Spheres of dimensions that can be easily visualized—those of (a) two, (b) one, and (c) zero dimensions.
First of all, what are one-and zero-dimensional spheres? Well, let's reason by analogy. A two-dimensional sphere is the collection of points in three-dimensional space that are the same distance from a chosen center, as shown in Figure 13.2(a). By following the same idea, a one-dimensional sphere is the collection of points in two-dimensional space (the surface of this page, for example) that are the same distance from a chosen center. As shown in Figure 13.2(b), this is nothing but a circle. Finally, following the pattern, a zero-dimensional sphere is the collection of points in a one-dimensional space (a line) that are the same distance from a chosen center. As shown in Figure 13.2(c), this amounts to two points, with the "radius" of the zero-dimensional sphere equal to the distance each point is from their common center. And so, the lower-dimensional analogy alluded to in the preceding paragraph involves a circle (a one-dimensional sphere) pinching down, followed by space tearing, and then being replaced by a zero-dimensional sphere (two points). Figure 13.3 puts this abstract idea into practice.
We imagine beginning with the surface of a doughnut, in which a one-dimensional sphere (a circle) is embedded, as highlighted in Figure 13.3. Now, let's imagine that as time goes by, the highlighted circle collapses, causing the fabric of space to pinch. We can repair the pinch by allowing the fabric to momentarily tear, and then replacing the pinched one-dimensional sphere—the collapsed circle—with a zero-dimensional sphere—two points—plugging the holes in the upper and lower portions of the shape arising from the tear. As shown in Figure 13.3, the resulting shape looks like a warped banana, which through gentle deformation (non–space tearing) can be reshaped smoothly into the surface of a beach ball. We see, therefore, that when a one-dimensional sphere collapses and is replaced by a zero-dimensional sphere, the topology of the original doughnut, that is, its fundamental shape, is drastically altered. In the context of the curled-up spatial dimensions, the space-tearing progression of Figure 13.3 would result in the universe depicted in Figure 8.8 evolving into that depicted in Figure 8.7.
Figure 13.3 A circular piece of a doughnut (a torus) collapses to a point. The surface tears open, yielding two puncture holes. A zero-dimensional sphere (two points) is "glued in," replacing the original one-dimensional sphere (the circle) and repairing the torn surface. This allows a transformation to a completely different shape—a beach ball.
Although this is a lower-dimensional analogy, it captures the essential features of what Morrison and I foresaw for the second half of Strominger's story. After the collapse of a three-dimensional sphere inside a Calabi-Yau space, it seemed to us that space could tear and subsequently repair itself by growing a two-dimensional sphere, leading to far more drastic changes in topology than Witten and we had found in our earlier work (discussed in Chapter 11). In this way, one Calabi-Yau shape could, in essence, transform itself into a completely different Calabi-Yau shape—much like the doughnut transforming into the beach ball in Figure 13.3—while string physics remained perfectly well behaved. Although a picture was starting to emerge, we knew that there were significant aspects that we would need to work out before we could establish that our second half of the story did not introduce any singularities—that is, pernicious and physically