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The Dimensions of Space

Now for something stranger. The passage from dots to filaments is only part of the new framework introduced by string theory. In the early days of string theory research, physicists encountered pernicious mathematical flaws (called quantum anomalies), entailing unacceptable processes like the spontaneous creation or destruction of energy. Typically, when problems like this afflict a proposed theory, physicists respond swiftly and sharply. They discard the theory. Indeed, many in the 1970s thought this the best course of action regarding strings. But the few researchers who stayed the course came upon an alternative way of proceeding.

In a dazzling development, they discovered that the problematic features were entwined with the number of dimensions of space. Their calculations revealed that were the universe to have more than the three dimensions of everyday experience—more than the familiar left/right, back/forth, and up/down—then string theory’s equations could be purged of their problematic features. Specifically, in a universe with nine dimensions of space and one of time, for a total of ten spacetime dimensions, the equations of string theory become trouble-free.

I’d love to explain in purely nontechnical terms how this comes about, but I can’t, and I’ve never encountered anyone who can. I made an attempt in The Elegant Universe, but that treatment only describes, in general terms, how the number of dimensions affects aspects of string vibrations, and doesn’t explain where the specific number ten comes from. So, in one slightly technical line, here’s the mathematical skinny. There’s an equation in string theory that has a contribution of the form (D — 10) times (Trouble), where D represents the number of spacetime dimensions and Trouble is a mathematical expression resulting in troublesome physical phenomena, such as the violation of energy conservation mentioned above. As to why the equation takes this precise form, I can’t offer any intuitive, nontechnical explanation. But if you do the calculation, that’s where the math leads. Now, the simple but key observation is that if the number of spacetime dimensions is ten, not the four we expect, the contribution becomes 0 times Trouble. And since 0 times anything is 0, in a universe with ten spacetime dimensions the trouble gets wiped away. That’s how the math plays out. Really. And that’s why string theorists argue for a universe with more than four spacetime dimensions.

Even so, no matter how open you may be to following the trail blazed by mathematics, if you’ve never encountered the idea of extra dimensions, the possibility may nevertheless sound nutty. Dimensions of space don’t go missing like car keys or one member of your favorite pair of socks. If there were more to the universe than length, width, and height, surely someone would’ve noticed. Well, not necessarily. Even as far back as the early decades of the twentieth century, a prescient series of papers by the German mathematician Theodor Kaluza and by the Swedish physicist Oskar Klein suggested that there might be dimensions that are proficient at evading detection. Their work envisioned that unlike the familiar spatial dimensions that extend over great, possibly infinite, distances, there might be additional dimensions that are tiny and curled up, making them difficult to see.

To picture this, think of a common drinking straw. But for the purpose at hand, make it decidedly uncommon by imagining it as thin as usual but as tall as the Empire State Building. The surface of the tall straw (like that of any straw) has two dimensions. The long vertical dimension is one; the short circular dimension, which curls around the straw, is the other. Now imagine viewing the tall straw from across the Hudson River, as in Figure 4.4a. Because the straw is so thin, it looks like a vertical line stretching from ground to sky. At this distance, you don’t have the visual acuity to see the straw’s tiny circular dimension,