The Elegant Universe - Brian Greene [104]
In fact, if the circular dimension were to grow, "inflating" Lineland into the Garden-hose universe, your life would change in profound ways. Take your body, for example. As a Linebeing, anything between your two eyes constitutes the interior of your body. Your eyes, therefore, play the same role for your linebody as skin plays for an ordinary human body: They constitute the barrier between the inside of your body and the outside world. A doctor in Lineland can access the interior of your linebody only by puncturing its surface—in other words, "surgery" in Lineland takes place through the eyes.
But now imagine what happens if Lineland does, a la Kaluza K. Line, have a secret, curled-up dimension, and if this dimension expands to an observably large size. Now one Linebeing can view your body at an angle and thereby directly see into its interior, as we illustrate in Figure 8.5. Using this second dimension, a doctor can operate on your body by reaching directly inside your exposed interior. Weird! In time, Linebeings, no doubt, would develop a skinlike cover to shield the newly exposed interior of their bodies from contact with the outside world. And moreover, they would undoubtedly evolve into beings with length as well as breadth: Flatbeings sliding along the two-dimensional Garden-hose universe as illustrated in Figure 8.6. If the circular dimension were to grow very large this two-dimensional universe would be closely akin to Abbott's Flatland—an imaginary two-dimensional world Abbott suffused with a rich cultural heritage and even a satirical caste system based upon one's geometrical shape. Whereas it's hard to imagine anything interesting happening in Lineland—there is just not enough room—life on a Garden-hose becomes replete with possibilities. The evolution from one to two observably large space dimensions is dramatic.
Figure 8.5 One Linebeing can see directly into the interior of another's body when Lineland expands into the Garden-hose universe.
And now the refrain: Why stop there? The two-dimensional universe might itself have a curled-up dimension and therefore secretly be three-dimensional. We can illustrate this with Figure 8.4, so long as we recognize that we are now imagining that there are only two extended space dimensions (whereas when we first introduced this figure we were imagining the flat grid to represent three extended dimensions). If the circular dimension should expand, a two-dimensional being would find itself in a vastly new world in which movement is not limited just to left-right and back-forth along the extended dimensions. Now, a being can also move in a third dimension—the "up-down" direction along the circle. In fact, if the circular dimension were to grow to a large enough size, this could be our three-dimensional universe. We do not know at present whether any of our three spatial dimensions extends outward forever, or in fact curls back on itself in the shape of a giant circle, beyond the range of our most powerful telescopes. If the circular dimension in Figure 8.4 got big enough—billions of light-years in extent—the figure could very well be a drawing of our world.
Figure 8.6 Flat, two-dimensional beings living in the Garden-hose universe.
But the refrain replays: Why stop there? This takes us to Kaluza's and Klein's vision: that our three-dimensional universe might have a previously unanticipated curled-up fourth spatial dimension. If this striking possibility, or its generalization to numerous curled-up dimensions (to be discussed shortly) is true, and if these curled-up dimensions were themselves to expand to a macroscopic size, the lower-dimensional examples discussed make it clear that life as we know it would change immensely.
Surprisingly, though, even if they should always stay curled up and small, the existence of extra curled-up dimensions has profound implications.
Unification in Higher