Warped Passages - Lisa Randall [183]
However, ADD were then left with the question of why gravity should be so weak. After all, the reason that the Planck scale mass is so big is that gravity is weak—gravity’s strength is inversely proportional to this scale. A much smaller fundamental mass scale for gravity would make gravitational interactions far too strong.
But this problem wasn’t insurmountable. ADD pointed out that it was only higher-dimensional gravity that was necessarily strong. They reasoned that large extra dimensions could dilute the strength of gravity so much that although the gravitational force would be very strong in higher dimensions, gravity in the lower-dimensional effective theory would be very feeble. In their picture, gravity appears feeble to us because it gets diluted in a very large extra-dimensional space. The electromagnetic, strong, and weak forces, on the other hand, would not be feeble because those forces would be confined to a brane and would not be diluted at all. Large dimensions and a brane could therefore conceivably explain why gravity is so much feebler than the other forces.
Nima told me that the turning point in their research was when he and his collaborators understood the precise relationship between the strengths of higher-and lower-dimensional gravity. This relationship was not new. String theorists, for example, always used it to relate the four-dimensional gravitational scale to the ten-dimensional one. And, as I briefly explained in Chapter 16, Hořava and Witten used the relationship between the strengths of ten-and eleven-dimensional gravity when they observed that gravity can be unified with other forces: a large eleventh dimension permits the higher-dimensional gravitational scale, and hence the string scale, to be as low as the GUT scale. But no one before had recognized that higher-dimensional gravity could be sufficiently strong to address the hierarchy problem so long as extra dimensions are large enough to adequately dilute it. After Nima, Savas, and Gia had thought about extra dimensions for a while and learned how to relate higher-and lower-dimensional gravity, they understood this extraordinary implication.
Relating Higher-and Lower-Dimensional Gravity
In Chapter 2 we saw that when you explore only those distances that are larger than the size of curled-up extra dimensions, the extra dimensions are imperceptible. However, that doesn’t necessarily mean that additional dimensions don’t have physical consequences; even though we don’t see them, they can still influence the values of quantities we do see. Chapter 17 gave an example of this phenomenon. In the sequestering model of supersymmetry breaking, in which supersymmetry breaking occurred on a distant brane and the graviton communicated the breaking to the supersymmetric partners of Standard Model particles, the values of superpartner masses reflected the extra-dimensional origin of supersymmetry breaking and its communication via gravity.
We’ll now consider another example in which extra dimensions influence the values of measurable quantities. The sizes of the compactified dimensions determine the relationship between the strength of four-dimensional gravity (that is, the one we observe) and the strength of the higher-dimensional gravity from which it derives. Gravity is diluted in extra dimensions and is weaker when curled-up extra dimensions enclose a larger volume.
To see how this works, let’s return to the example of Chapter 2, where we considered the three-dimensional garden-hose universe as an analogy for a bulk three-dimensional space bounded by branes. If water were to enter the hose through a pinhole (see Figure 23, Chapter 2), it would initially spurt out from the hole and spread in all three dimensions. However, once the water reached the width of the hose, it would