Warped Passages - Lisa Randall [192]
In the particular geometry described in this chapter, we’ll see that space is so strongly warped in the presence of two flat boundary branes that the hierarchy problem of particle physics is automatically solved—without the need for a large dimension, or for any arbitrary large number at all. In this scenario, one brane experiences a large gravitational force but the other does not. Spacetime changes so rapidly along the fifth dimension that it parlays a modest number associated with the separation between the two branes into a huge number (about ten million billion) associated with the relative strength of the gravitational force.
We’ll first explain the weakness of gravity on the second brane in terms of the graviton probability function, which determines the graviton’s interactions at any particular location in the fifth dimension. But we’ll also explain gravity’s weakness in different terms, based on the warped geometry itself rather than the graviton interaction strength. We’ll see that one of the amazing consequences of warped geometry is that size, mass, and even time depend on position along the fifth dimension. The warping of space and time in this two-brane setup is like the warping of time near the horizon of a black hole. But in this case, time dilates, geometry expands, and on one of the branes particles have a small mass—so the hierarchy problem gets automatically solved.
After discussing the warped geometry and its implications for the hierarchy problem, we’ll conclude this chapter with a discussion of the distinctive implications of the theory for future experiments. One of the most exciting aspects of this theory, as with the large extra-dimension models of the previous chapter, is that if it is correct it will very soon have observable consequences at particle accelerators. In fact, we’ll see they will be even more dramatic than the missing energy signature we discussed. The KK partners of the gravitons, though visitors from higher-dimensional space, will be distinguishable, visible particles that will decay into familiar particles on our four-dimensional brane.
Warped Geometry and Its Surprising Implications
The geometry that we’ll consider in this chapter contains two branes that bound a fifth dimension of space, as illustrated in Figure 78. This setup is similar to the one considered in Chapter 17 in that there are two branes with a fifth dimension that extends between them. However, it really is quite a different theory. The particles and the distribution of energy are different, and the theory is not supersymmetric. Nevertheless, as with that theory, we assume that all of the Standard Model particles, along with a Higgs particle responsible for breaking electroweak symmetry, are confined to one of the two branes.
Also as before, in this setup we’ll assume that gravity is the only force that exists throughout the fifth dimension. This means that were it not for gravity, each of the branes would look like a conventional four-dimensional universe. Gauge bosons and particles confined to the branes would communicate forces and interact as if the fifth dimension didn’t exist. Standard Model particles would travel only in the three flat spatial brane dimensions, and forces would spread out only along the flat three-dimensional surface of the brane.35
Figure 78. The warped five-dimensional geometry with a single brane. The universe has five spacetime dimensions, but the Standard Model resides on a brane (the Weakbrane) that has four. Again, the total number of spacetime dimensions in this setup is five, whereas the number of spatial dimensions is four, three of which extend along the branes and one that extends between them.
Gravity, however, is different since it is not restricted to a brane, but instead exists in the full five-dimensional bulk. The force of gravity would be felt everywhere in the fifth dimension. But this does not necessarily mean that it is felt equally everywhere.