Warped Passages - Lisa Randall [23]
Let’s now apply this reasoning to gravity, and derive the precise distance dependence of the gravitational force in three dimensions. Newton’s gravitational force law follows from two facts: that gravity acts equally in all directions, and that there are three dimensions of space. Let’s now imagine a planet, which attracts any mass in its vicinity. Because the gravitational force is the same in all directions, the strength of the gravitational attraction that the planet exerts on another massive object—a moon, for example—will depend not on direction, but on the distance between them.
To pictorially represent the strength of the gravitational force, the left of Figure 22 shows radial lines extending outwards from the planet’s center, resembling water spreading out from a sprinkler. The density of these lines determines the strength of gravitational attraction that the planet exerts on anything in its vicinity. More force lines passing through an object would mean a greater gravitational attraction, and fewer force lines would mean a smaller gravitational attraction.
Figure 22. Gravitational force lines emitted from a massive object, such as a planet. The same number of lines intersect a sphere of any radius; therefore, the force lines are more diffuse and gravity is weaker the farther you are from the massive object at the center.
Notice that the same number of force lines intersect a spherical shell drawn any distance away, no matter how far or near (center and right of Figure 22). The number of force lines never changes. But because the force lines are spread out among all the points on the sphere’s surface, the force at a greater distance is necessarily weaker. The precise dilution factor is determined by the quantitative measure of how widely distributed the force lines are at any given distance.
A fixed number of force lines passes through a sphere’s surface, whatever its distance from the mass. The area of that sphere’s surface is proportional to its radius squared: the surface area is equal to a number multiplied by the square of the radius. Because the fixed number of gravitational force lines is spread out over the sphere’s surface, the gravitational force has to decrease as the square of the radius. This spreading out of the gravitational field is the origin of the inverse square law for gravity.
Newton’s Law with Compact Dimensions
So we now know that in three dimensions, gravity should obey an inverse square law. Notice that the argument seems to depend critically on the fact that there are three spatial dimensions. Had there been only two dimensions, gravity would have been spread out only over a circle, and the force of gravity would have decreased with distance at a slower rate. Had there been more than three dimensions, the surface area of a hypersphere would have grown far more rapidly with the separation between the planet and its moon, and the force would have fallen off that much more quickly. It seems that only three spatial dimensions yields the inverse square distance dependence. But if that is the case, how can theories with extra dimensions yield Newton’s inverse square law for gravity?
It is very interesting to see how compactified dimensions resolve this potential conflict. The essence of the logic is that force lines cannot spread arbitrarily far into the compact dimensions because those compact dimensions have finite size. Although force lines initially spread out in all dimensions, when they have spread beyond the extra dimensions’ sizes they have no choice but to spread out solely in the directions of the infinite dimensions.
This can be illustrated once again with our hose example. Imagine that water enters the hose through a small pinhole in a cap covering the end of the hose (see Figure 23). Water directed through the puncture will not immediately travel directly down the hose, but will first spread throughout the tube’s cross-section. Nonetheless, it should be clear that if you were at the other end of the hose, watering your flower, the way the water entered would make no difference