Warped Passages - Lisa Randall [22]
It is nice to have a pictorial, descriptive explanation for why the extra dimensions are hidden when compactified, or rolled up, to a very minute size. But it’s a good idea to check that the laws of physics accord with this intuition.
Let’s take a look at Newton’s gravitational force law, the well-established form of the gravitational force law that Newton proposed in the seventeenth century. This law tells us how the gravitational force depends on the distance between two massive objects. * It’s known as an inverse square law, which means that the strength of gravity decreases with distance proportionally to the distance squared. For example, if you double the distance between two objects, the strength of their gravitational attraction goes down by a factor of four. If the separation is increased to three times its original value, gravitational attraction decreases by a factor of nine. The inverse square law of gravity is one of the oldest and most important laws of physics. Among other things, it is the reason that planets have the type of orbits they do. Any viable physical theory of gravity must reproduce the inverse square law or it would be bound to fail.
The way in which the gravitational force law depends on distance, which is encoded in Newton’s inverse square law, is intimately connected to the number of spatial dimensions. This is because the number of dimensions determines how quickly gravity diffuses as it spreads out in space.
Let’s reflect on the connection, which will be very relevant to us later on when we consider extra dimensions. We’ll do this by imagining a water supply whose water can be directed through either a hose or a sprinkler. We’ll assume that both the hose and the sprinkler have the same amount of water running through them, and that they can each water a certain flower in a garden (see Figure 20). When the water goes through the hose, which is directed at the flower, the flower will get all the water. The distance from the base of the hose to the nozzle directed at the flower is irrelevant, because all the water must end up on the flower, no matter how far away the hose happened to be.
However, suppose instead that the same water is directed through a sprinkler that simultaneously waters many flowers. That is, the sprinkler sends out water in a circle, reaching all the flowers a certain distance away. Because the water will now be distributed among everything at that distance, the original flower will no longer get all the water. Moreover, the farther away the flower is from the source, the more greenery the sprinkler will water, and the more widely distributed the water will be (see Figure 21). That’s because you can fit more plants on a circle three meters in circumference, say, than a circle just one meter in circumference. Because the water is more widely spread out, a farther away flower receives less water.
Figure 20. The amount of water delivered to a flower by a sprinkler that sprinkles water around a circle is less than the amount delivered directly by a hose.
Figure 21. When a sprinkler delivers water around a circle of larger radius, the water is spaced out more and the flower receives less water.
Similarly, anything that is shared uniformly in more than one direction will have a smaller impact on any particular thing that is farther away—whether that thing is a flower or, as we will soon see, an object experiencing the force of gravity. Gravity, like water, is more widely distributed when it is farther away.
With this example, we can also see why the distribution depends so strongly on the number of dimensions in which water (or gravity) is spread. The water from the two-dimensional sprinkler is spread out with distance, unlike the water from the one-dimensional hose, which is not spread out at all. Now imagine a sprinkler that spreads its water over the surface of a sphere, and not just around a circle. (Such a sprinkler would look something like a dandelion gone to seed.) Here, the water will spread out with distance much more quickly.