Warped Passages - Lisa Randall [19]
Figure 15. When you view a hose spread over a football field from above, it looks like it has one dimension. But when you view it up close, you see that the surface has two dimensions and the volume it encloses has three.
Figure 16. When one dimension is curled up, a two-dimensional universe looks one-dimensional.
Imagine again a garden hose, which can be thought of as a long sheet of rubber rolled up into a tube with a small circular cross-section. This time, we’ll think of the hose as the entire universe (not an object inside the universe). * If the universe were shaped like this garden hose, we would have one very long dimension and one very small, rolled-up dimension—exactly what we want.
For a little creature—a flat bug, say—that lived in the garden-hose universe, the universe would look two-dimensional. (In this scenario, our bug has to stick to the surface of the hose—the two-dimensional universe doesn’t include the interior, which is three-dimensional.) The bug could crawl in two directions: along the length of the hose or around it. Like the Dodo, who could run laps in its two-dimensional universe, a bug that started somewhere along the hose could crawl around and eventually return to where it started. Because the second dimension is small, the bug wouldn’t travel very far before it returned.
If a population of bugs living on the hose experienced forces, such as the electric force or gravity, those forces would be able to attract or repel bugs in any direction on the surface of the hose. Bugs could be separated from one another either along the length of the hose or around the hose’s circumference, and would experience any force that was present on the hose. Once there is sufficient resolution to distinguish distances as small as the diameter of the hose, forces and objects exhibit both of the dimensions they actually have.
However, if our bug could observe its surroundings, it would notice that the two dimensions were very different. The one along the length of the hose would be very big. It could even be infinitely long. The other dimension, on the other hand, would be very small. Two bugs could never get very far from each other in the direction around the hose. And a bug that tried to take a long trip in that direction would quickly end up back where it started. A thoughtful bug that liked to stretch its legs would know that its universe was two-dimensional, and that one dimension extended a long way while the other was very small and rolled up into a circle.
But the bug’s perspective is nothing like the one that creatures like us would have in Klein’s universe, in which the extra dimension is rolled up to an extremely small size, 10-33 cm. Unlike the bug, we are not small enough to detect—never mind travel in—a dimension of such a tiny size.
So to complete our analogy, suppose that something much bigger than a bug, capable only of much coarser resolution and therefore unable to detect small objects or structure, lived in the garden-hose universe. Since the lens through which this bigger being views the world blurs any details that are as small as the hose’s diameter, from the vantage point of this bigger being the extra dimensions would be invisible. It would see only a single dimension. Someone would see that the garden-hose universe had more than a single dimension only if he had sufficiently sharp vision to register something as small as the width of the hose. If his vision is too fuzzy to register that width, all he’ll ever notice is a line.
Moreover, physical