Knocking on Heaven's Door (49 page)

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 these branes, even when gravity and space extend through more dimensions. (Figure 64 presents a schematic of a braneworld showing a person and a magnet on a brane, with gravity spreading both on and off it.)

String theory’s extra dimensions might have physical import for the observable world and so too might three-dimensional branes. Perhaps the most important reason to consider extra dimensions is that they might affect visible phenomena, and, in particular, address outstanding puzzles such as the hierarchy problem of particle physics. Extra dimensions and branes could be the key to resolving this question—addressing the issue of why gravity is so weak.

[
FIGURE 64
]
Standard Model particles and forces can be stuck on a braneworld that lives in higher-dimensional space. In that case, my cousin Matt, the matter and stars we know, forces such as electromagnetism, and our galaxy and universe all live in its three spatial dimensions. Gravity, on the other hand, can always spread throughout all of space. (Photo courtesy of Marty Rosenberg)

Which brings us to what is perhaps the best reason right now to think about extra dimensions of space. They can have consequences for phenomena we are now trying to understand, and if so, we might see evidence in the imminent future.

Recall that we can phrase the hierarchy problem in two different ways. We can say it is the question of why the Higgs mass—and hence the weak scale—is so much smaller than the Planck mass. This is the question we considered when thinking about supersymmetry and technicolor. But we can also ask an equivalent question: Why is gravity so weak compared to the other known fundamental forces? The strength of gravity depends on the Planck mass scale, the enormous mass ten thousand trillion times greater than the weak scale. The bigger the Planck mass, the weaker the force of gravity. Only when masses are at or near the Planck scale is gravity strong. As long as particles are a good deal lighter than the scale set by the Planck mass, as they are in our world, the force of gravity is extremely weak.

The puzzle of why gravity is so weak is in fact equivalent to the hierarchy problem—the solution of one solves the other. But even though the problems are equivalent, phrasing the hierarchy problem in terms of gravity helps guide our thinking toward extra-dimensional solutions. We’ll now delve into a couple of the leading suggestions.

LARGE EXTRA DIMENSIONS AND THE HIERARCHY

Ever since people first started thinking about the hierarchy problem, physicists thought the resolution must involve modified particle interactions at the weak energy scale of about a TeV. With only Standard Model particles, the quantum contributions to the Higgs particle mass are simply too enormous. Something has to kick in to tame the large quantum mechanical contributions to the Higgs particle mass.

Supersymmetry and technicolor are two examples in which new heavy particles might participate in high-energy interactions and cancel the contributions or prevent them from arising in the first place. Until the 1990s, all proposed solutions to the hierarchy problem could be categorized similarly, with new particles and forces and even new symmetries emerging at the weak energy scale.

In 1998, Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali
⁶⁵
proposed an alternative way of addressing the problem. They pointed out that since the problem involves not just the weak energy scale alone, but its ratio to the Planck energy scale associated with gravity, perhaps the problem lay in an incorrect understanding of the basic nature of gravity itself.

They suggested that there is in fact no hierarchy in masses at all—at least with respect to the fundamental scale of gravity compared to the weak scale. Maybe gravity is instead much stronger in the extra-dimensional universe, but is only measured to be so feeble in our three-plus-one-dimensional world because it is diluted throughout all the dimensions that we don’t see. Their hypothesis was that the mass scale at which gravity becomes strong in the extra-dimensional universe is in fact the weak mass scale. In that case, we measure gravity to be minuscule in strength not because it is fundamentally weak but rather because it spreads throughout large unseen dimensions.

One way to understand this is to imagine an analogous situation with a water sprinkler. Think about the water that emerges from this sprinkler. If the water spread only in our dimensions, its impact would depend on the amount of water emerging from the hose and how far it had to travel. But if there were additional dimensions to space, the water would spread throughout those dimensions as well after emerging from the end of the hose. We would experience much less water than we would otherwise at a given distance from the source because water would also spread throughout the dimensions we don’t observe. (This is illustrated schematically in Figure 65.)

[
FIGURE 65
]
The strengths of forces weaken more quickly with distance in a higher-dimensional space than in a lower-dimensional one. This is analogous to a higher-dimensional water sprinkler for which the water dilutes much more quickly with distance. The water spreads more in three dimensions than it spreads in two—in the picture, only the flower receiving water from the lower-dimensional sprinkler is adequately maintained.

If the extra dimensions were of finite size, the water would reach the boundaries of the extra dimensions and no longer spread out. But the amount of water anything would receive at any given place in the extra-dimensional space would be far less than if it had never spread out in those dimensions in the first place.

Similarly, gravity could spread into other dimensions. Even though the force wouldn’t spread out forever if the dimensions have finite size, large dimensions would dilute the gravitational force we would experience in our three-dimensional world. If the dimensions were sufficiently large, we would experience very weak gravity, even though the fundamental strength of higher-dimensional gravity could be quite big. Keep in mind, however, that for this idea to work, the extra dimensions have to be enormous compared to what theoretical considerations lead us to expect, since gravity indeed appears so weakly in a three-dimensional world.

Nonetheless, the LHC will subject this idea to experimental tests. Even though the idea now seems improbable, reality and not our ease in finding models is the final arbiter of what is right. If realized in the world, these models would lead to a distinctive characteristic signature. Because higher-dimensional gravity is strong at energies of about the weak scale—the energies that the LHC will generate—p articles would collide together and produce a higher-dimensional graviton—the particle that communicates the force of higher-dimensional gravity. But this graviton travels into the extra dimensions. The gravity we are familiar with is extremely weak—far too weak to produce a graviton if there are only three dimensions of space. But in this new scenario, higher-dimensional gravity would be sufficiently strong to produce a graviton at the energies reached by the LHC.