Knocking on Heaven's Door - Lisa Randall [156]
[ FIGURE 67 ] The Randall-Sundrum setup contains two branes that bound a fourth dimension of space (a fifth dimension of spacetime). In this space, the graviton wavefunction (which tells the probability of finding the graviton at any point in space) decreases exponentially from the Gravitybrane to the Weakbrane.
If the idea that Raman and I had is right, the LHC will soon teach us fascinating properties about the nature of space. It turns out that the universe we suggested is dramatically curved, in accordance with what Einstein taught us about spacetime in the presence of matter and energy. In technical terminology, the geometry we derived from Einstein’s equations is “warped” (that really was the pre-existing technical term). What that means is that space and time vary along the single additional dimension of interest. It does so in such a way that space and time, as well as masses and energy, are all rescaled as you move from one place in extra-dimensional space to another, as we will soon get to and is illustrated in Figure 68.
One important consequence of this warped spacetime geometry is that whereas the Higgs particle would have been heavy in some other location in extra-dimensional space, it will have weak scale mass—exactly as should be the case—in the location where we reside. This might sound somewhat arbitrary, but it is not. According to our scenario, there is a brane on which we live—the Weakbrane—and a second brane where gravity is concentrated, known as the Gravitybrane—or among physicists, the Planck brane. This brane would contain another universe that is separated from us in an extra dimension. (See Figure 67.) In this scenario, the second brane would in fact be right next door—separated by an infinitesimal distance, a million trillion trillion times smaller than a centimeter.
The remarkable property that follows from the warped geometry (illustrated in the Figure 67), is that the graviton, the particle that communicates the force of gravity, is far more heavily weighted on the other brane than on ours. That would make gravity strong elsewhere in the other dimension, but very weak where we live. In fact, Raman and I found that gravity should be exponentially weaker in our vicinity than on the other brane, thereby giving a natural explanation for the weakness of gravity.
An alternative way of interpreting the consequences of this setup is through the geometry of spacetime, schematically illustrated in Figure 68. The scale of spacetime depends on location in the fourth spatial dimension. Masses get exponentially rescaled too—and they do so in a way that the Higgs boson mass is what it needs to be. Although one can debate the assumptions our model relies on—namely, two large flat branes bounding an extra-dimensional universe—the geometry itself follows directly from Einstein’s theory of gravity once you postulate the energy carried by the branes and by the extra-dimensional space known as the bulk. Raman and I solved the equations of general relativity. And when we did, we found the geometry I just described—namely, the curved warped space in which masses get rescaled in the way required to solve the hierarchy problem.
[ FIGURE 68 ] Another way to understand why warped geometry solves the hierarchy problem is in terms of the geometry itself. Space, time, energy, and mass all are rescaled exponentially as you go from one brane to the next. In this scenario, it would be very natural to find that the Higgs mass is exponentially smaller than the Planck mass.
Unlike the large extra-dimensional models, the models based on the warped geometry don’t replace the old enigma of the hierarchy problem by a new one (why are the extra dimensions so large?). In the warped geometry, the extra dimension is not large. The large numbers arise from an exponential rescaling of space and time. The exponential rescaling makes the ratio of sizes—and masses—of objects enormous, even when those objects are separated only modestly in extra-dimensional