Knocking on Heaven's Door - Lisa Randall [152]
But why would we even think extra dimensions could be out there if we have never seen them? The history of physics holds many examples of finding things no one could see. No one could “see” atoms and no one could “see” quarks. Yet we now have strong experimental evidence of the existence of both.
No law of physics tells us that only three dimensions of space can exist. Einstein’s theory of general relativity works for any number of dimensions. In fact, soon after Einstein completed his theory of gravity, Theodor Kaluza extended Einstein’s ideas to suggest the existence of a fourth spatial dimension, and, five years later, Oskar Klein suggested how it might be curled up and differ from the familiar three.
String theory, a leading proposal for a theory combining quantum mechanics and gravity, is another reason physicists currently entertain the notion of extra dimensions. String theory does not obviously lead to the theory of gravity we are familiar with. String theory necessarily involves additional dimensions of space.
People often ask me the number of dimensions that exist in the universe. We don’t know. String theory suggests six or seven extra ones. But model builders keep an open mind. It’s conceivable that different versions of string theory will lead to other possibilities. In any case, dimensions model builders care about in the following discussions are only the ones that are sufficiently warped or so large that they can affect physical predictions. Other dimensions even smaller than the ones relevant to particle physics phenomena might exist, but we will ignore anything so super-tiny. We again take the effective theory approach and ignore anything too small or invisible to ever make any measurable differences.
String theory also introduces other elements—notably branes—that make for richer possibilities for the geometry of the universe, if indeed it contains extra dimensions. In the 1990s, the string theorist Joe Polchinski established that string theory was not just a theory of one-dimensional objects called strings. He, along with many others, demonstrated that higher-dimensional objects known as branes were also essential to the theory.
The word “brane” derives from “membrane.” Like membranes, which are two-dimensional surfaces in three-dimensional space, branes are lower-dimensional surfaces in higher-dimensional space. These branes can trap particles and forces so that they don’t travel through the full higher-dimensional space. Branes in higher-dimensional space are like a shower curtain in your bathroom, which is a two-dimensional surface in a three-dimensional room. (See Figure 62.) Water droplets might travel only over the two-dimensional surface of the curtain, much as particles and forces might be stuck on the lower-dimensional “surface” of a brane.
[ FIGURE 62 ] A brane traps particles and forces, which can move along it but not off—much like water droplets that can move on a shower curtain but don’t travel away.
Broadly speaking, two types of strings exist: open strings that have ends and closed strings that form loops like rubber bands. (See Figure 63.) String theorists in the 1990s realized that the ends of open strings can’t be just anywhere—they have to end on branes. When particles arise from the oscillations of the open strings that are anchored to a brane, they too are confined there. Particles, the oscillations of those strings, are then stuck. As with water drops on a shower curtain, they can travel along the dimensions of the branes, but they can’t travel off them.
[ FIGURE 63 ] An open string with two ends, and closed string with none.
String theory suggests the existence of many types of branes, but the ones that will be of most interest for models addressing the hierarchy problem involve those that extend over three dimensions—the three physical dimensions of space that we know. Particles and forces can be trapped on