The Elegant Universe - Brian Greene [140]
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