Warped Passages - Lisa Randall [185]
How large would these extra dimensions have to be? The answer depends on the number of extra dimensions. ADD considered different possible numbers of dimensions for their model, since experiments haven’t yet decided how many dimensions there are. Notice that we are interested only in the large dimensions at this point. So if you think that you and your local string theorist know that the number of spatial dimensions is nine or ten, you can still consider different possibilities for the number of large dimensions and assume that all the other dimensions are small enough to ignore.
The size of the dimensions in the ADD proposal depends on how many there are, because volume depends on the number of dimensions. If all dimensions were the same size, a higher-dimensional region would enclose more volume than a lower-dimensional one and would therefore dilute gravity more. You can see this easily enough from the fact that lower-dimensional objects fit inside higher-dimensional ones. Or, returning to our sprinkler analogy in Chapter 2, you can see that a plant receives more water from a sprinkler that spreads water only over a line segment of a particular size (one dimension) than one that spreads water over the surface area enclosed by a circle (two dimensions) whose diameter is the same size. When water spreads over a higher-dimensional region, it becomes more diluted.
If there were only one large extra dimension, it would have to be enormous to satisfy the ADD proposal. It would have to be as large as the distance from the Earth to the Sun in order to dilute gravity enough. That’s not allowed. If the extra dimension were that big, the universe would behave as if it were five-dimensional at measurable distances. We already know that Newton’s gravitational force law applies at these distances; a large extra dimension that would modify gravity at such large distances is clearly ruled out.
However, with as few as two additional dimensions, the size of the dimensions is almost acceptably small. If there were just two additional dimensions, they could be as small as a millimeter and still adequately dilute gravity. That is the reason ADD paid so much attention to the millimeter scale. Not only was it on the verge of experimental probes, but two additional dimensions of this size could be relevant to the hierarchy problem. Gravity would spread throughout these two millimeter-size dimensions and yield the weak gravitational force we know. Of course, a millimeter is still pretty big, but as we said earlier, gravity tests are not nearly as restrictive as you might think. Spurred on by the ADD scenario, people thought harder about looking for rolled-up dimensions of this size.
With more than two additional dimensions, gravity is modified only at a very small distance. With more additional dimensions, it can be sufficiently diluted even if those extra dimensions are relatively small. For example, with six extra dimensions the size need only be about 10-13 cm, one ten thousandth of a billionth of a centimeter.
Even with such small dimensions we could, if we’re lucky, find evidence of one of these examples some time very soon—not in the direct gravitational tests we’ll discuss in the next section, but in the experiments at high-energy particle colliders that we’ll consider afterwards.
Looking for Large Dimensions
How would one go about finding differences in gravity at small distances? What should one look for? We know that if there are curled-up dimensions, the strength of gravity at distances less than the size of the extra dimensions would decrease more rapidly with distance than Newton had predicted, because gravity would spread out in more than three spatial dimensions. Whenever objects were separated by less than the extra dimensions’ sizes, higher-dimensional gravity would apply. A bug sufficiently small to circle a curled-up dimension would experience the extra dimension, both because it could travel in it and because the gravitational