Warped Passages - Lisa Randall [24]
Figure 23. Water entering a garden hose through a pinhole at the end first spreads in three dimensions before traveling only along the single long dimension of the hose.
As before, we can imagine a fixed number of force lines emanating from a massive sphere. At a distance smaller than the extra dimensions’ size, these force lines will spread out equally in all directions. If you could measure gravity on that small scale, you would measure the consequences of higher-dimensional gravity. The force lines would spread the way water does as it enters the hose through the pinhole and spreads throughout the hose’s interior.
However, at distances greater than the extra dimensions’ sizes, the force lines can spread only in the infinite directions (see Figure 24). In the small, compact dimensions, the force lines will hit the edge of space, so they can’t spread out any farther that way. They have to bend, and the only way for them to go is in the direction of the large dimensions. Therefore, at distances greater than the sizes of the extra dimensions, it’s just as if the extra dimensions didn’t exist, and the force law reverts to Newton’s inverse square law—the one we observe. This means that even from a quantitative point of view, you won’t know there are extra dimensions if you measure the gravitational force only between objects with separations greater than the curled-up dimensions’ size. The distance dependence reflects extra dimensions only in the tiny region inside the compact space.
Figure 24. Gravitational force lines emitted from a massive object when a dimension is curled up. The force lines spread radially over short distances, but over long distances they extend only along the infinite dimension.
Other Ways to Bound Dimensions?
We’ve now established that when extra dimensions are sufficiently small, they are invisible and have no detectable consequences on the length scales we observe. For a long time, string theorists assumed that extra dimensions were Planck-length dimensions, but recently some of us have questioned this assumption.
No one understands string theory well enough to say definitively what the sizes of extra dimensions will turn out to be. Sizes comparable to the Planck length are possible, but any dimension too small to observe is also in the running. The Planck length is so tiny that even considerably larger curled-up dimensions might well escape notice. An important question for the study of extra dimensions is just how big these dimensions can be, given that we haven’t seen them yet.
The questions we’ll address in this book include how big extra dimensions can be, whether these dimensions have any discernible effect on elementary particles, and how experiments might probe them. We will see that the existence of extra dimensions can significantly change the rules by which we do particle physics and, furthermore, that some of these changes will have experimentally observable consequences.
An even more radical question we’ll investigate is whether additional dimensions have to be small. We don’t see tiny dimensions, but do dimensions have to be small to be invisible? Could an extra dimension possibly extend for ever without our seeing it? If so, extra dimensions would have to be very different from the dimensions we’ve looked at. So far I’ve presented only the simplest possibility. We’ll see later why even the radical possibility of an infinite extra dimension cannot be excluded if it is sufficiently different from the three familiar infinite dimensions.
The next chapter will address yet another question that might have occurred to you: why can’t small extra dimensions just be intervals, not curled up into a ball but instead bounded between two “walls”? This possibility didn’t occur to anyone right away—but why