Warped Passages - Lisa Randall [184]
But even though the water travels only along the single dimension of the hose, its pressure depends on the size of the cross-section. One way to understand this is by imagining what would happen if the width of the hose increased. The water that entered through the pinhole would then spread out over a larger region, and the pressure of the water exiting the hose would be weaker.
If the pressure of water represents gravitational force lines, and the water entering the hose through the pinhole represents the field lines emerging from a massive object, then the force lines from this massive object would initially spread in all three directions, just like the water in the previous example. And when the force lines reach the walls of the universe (the branes), they would bend and run solely along the single large dimension. With the hose, we found that the wider the nozzle, the weaker the water pressure. Similarly, the area of extra dimensions in our toy garden-hose universe would determine how dilute the field lines will be in the lower-dimensional world. The larger the area of the extra dimensions, the weaker the gravitational field strength in the effective lower-dimensional universe would be.
The same argument applies to rolled-up dimensions in a universe with any number of curled-up dimensions. The larger the volume of the extra dimensions, the more dilute the gravitational force and the weaker the strength of gravity. We can see this with a higher-dimensional hose analogous to the one we just considered. Gravitational force lines in a higher-dimensional hose would first spread out in all dimensions, including the extra curled-up dimensions. The force lines would reach the boundary of the curled-up dimensions, after which they would spread out only along the infinite dimensions of the lower-dimensional space. The initial spreading out in the extra dimensions would reduce the density of force lines in the lower-dimensional space, so the strength of gravity experienced there would be weaker.34
Back to the Hierarchy Problem
Because of the dilution of gravity in extra dimensions, lower-dimensional gravity is weaker when the volume of the extra-dimensional compactified space is bigger. ADD observed that this dilution of gravity into extra dimensions could conceivably be so large that it could account for the observed weakness of four-dimensional gravity in our world.
They reasoned as follows. Suppose that gravity in a higher-dimensional theory does not depend on the enormous Planck scale mass of 1019 GeV, but instead on a much smaller energy, about a TeV, sixteen orders of magnitude smaller. They chose a TeV to eliminate the hierarchy problem: if a TeV or some nearby energy were the energy at which gravity became strong, there would be no hierarchy of masses in particle physics. Everything, both particle physics and gravity, would be characterized by the TeV scale. So maintaining a reasonably light Higgs particle with mass of about a TeV would not be a problem in their model.
According to their assumption, at energies of about a TeV higher-dimensional gravity would be a reasonably strong force, comparable in strength to the other known forces. To have a sensible theory that agrees with what we see, ADD therefore needed to explain why four-dimensional gravity looks so weak. The added ingredient in their model was the assumption that the extra dimensions are extremely large. Ultimately we would want to explain this large size. But according to their proposal, the curled-up dimensions enclose such a large volume. And, in keeping with the logic of the previous section, four-dimensional gravity would be extremely feeble. Gravity in our world would be weak because extra dimensions are large, not because there is fundamentally a big mass responsible for the tiny gravitational force. The Planck scale mass that we measure in four dimensions is