The high concentration of the graviton near the brane—let's call the brane where gravity is localized the Planck brane—leads to a natural solution to the hierarchy problem in a universe with two branes. For the particular geometry that solves Einstein's equations, when you go out some distance in an extra dimension, you see an exponentially suppressed gravitational force. This is remarkable because it means that a huge separation of mass scales—sixteen orders of magnitude—can result from a relatively modest separation of branes. If we are living on the second brane (not the Planck brane), we would find that gravity was very weak. Such a moderate distance between branes is not difficult to achieve and is many orders of magnitude smaller than that necessary for the large-extra-dimensions scenario just discussed. A localized graviton plus a second brane separated from the brane on which the standard model of particle physics is housed provides a natural solution to the hierarchy problem—the problem of why gravity is so incredibly weak. The strength of gravity depends on location, and away from the Planck brane it is exponentially suppressed.
This theory has exciting experimental implications, since it applies to a particle physics scale—namely, the TeV scale. In this theory's highly curved geometry, Kaluza-Klein particles—those particles with momentum in the extra dimensions—would have mass of about a TeV; thus there is a real possibility of producing them at colliders in the near future. They would be created like any other particle and they would decay in much the same way. Experiments could then look at their decay products and reconstruct the mass and spin that is their distinguishing property. The graviton is the only particle we know about that has spin 2. The many Kaluza-Klein particles associated with the graviton would also have spin 2 and could therefore be readily identified. Observation of these particles would be strong evidence of the existence of additional dimensions and would suggest that this theory is correct.
As exciting as this explanation of the existence of very different mass scales is, Raman and I discovered something perhaps even more surprising. Conventionally, it was thought that extra dimensions must be curled up or bounded between two branes, or else we would observe higher-dimensional gravity. The aforementioned second brane appeared to serve two purposes: It explained the hierarchy problem because of the small probability for the graviton to be there, and it was also responsible for bounding the extra dimension so that at long distances (bigger than the dimension's size) only three dimensions are seen.
The concentration of the graviton near the Planck brane can, however, have an entirely different implication. If we forget the hierarchy problem for the moment, the second brane is unnecessary! That is, even if there is an infinite extra dimension and we live on the Planck brane in this infinite dimension, we wouldn't know about it. In this "warped geometry," as the space with exponentially decreasing graviton amplitude is known, we would see things as if this dimension did not exist and the world were only three-dimensional.
Because the graviton has such a small probability of being located away from the Planck brane, anything going on far away from the Planck brane should be irrelevant to physics on or near it. The physics far away is in fact so entirely irrelevant that the extra dimension can be infinite, with absolutely no problem from a three-dimensional vantage point. Because the graviton makes only infrequent excursions into the bulk, a second brane or a curled-up dimension isn't necessary to get a theory that describes our three-dimensional world, as had previously been thought. We might live on the Planck brane and address the hierarchy problem in some other manner—or we might live on a second brane out in the bulk, but this brane would not be the boundary of the now infinite space. It doesn't matter that the graviton occasionally leaks away from the Planck brane; it's so highly localized there that the Planck brane essentially mimics a world of three dimensions, as though an extra dimension didn't exist at all. A four-spatial-dimensions world, say, would look almost identical to one with three spatial dimensions. Thus all the evidence we have for three spatial dimensions could equally well be evidence for a theory in which there are four spatial dimensions of infinite extent.
It's an exciting but frustrating game. We used to think the easiest thing to rule out would be large extra dimensions, because large extra dimensions would be associated with low energies, which are more readily accessible. Now, however, because of the curvature of space, there is a theory permitting an infinite fourth dimension of space in a configuration that so closely mimics three dimensions that the two worlds are virtually indistinguishable.
If there are differences, they will be subtle. It might turn out that black holes in the two worlds would behave differently. Energy can leak off the brane, so when a black hole decays it might spit out particles into the extra dimension and thus decay much more quickly. Physicists are now doing some interesting work on what black holes would look like if this extra-dimensional theory with the highly concentrated graviton on the brane is true; however, initial inquiries suggest that black holes, like everything else, would look too similar to distinguish the four- and three dimensional theories. With extra dimensions, there are an enormous number of possibilities for the overall structure of space. There can be different numbers of dimensions and there might be arbitrary numbers of branes contained within. Branes don't even all have to be three-plus-one-dimensional; maybe there are other dimensions of branes in addition to those that look like ours and are parallel to ours. This presents an interesting question about the global structure of space, since how space evolves with time would be different in the context of the presence of many branes. It's possible that there are all sorts of forces and particles we don't know about that are concentrated on branes and can affect cosmology.
In the above example, physics everywhere—on the brane and in the bulk—looks three-dimensional. Even away from the Planck brane, physics appears to be three dimensional, albeit with weaker gravitational coupling. Working with Andreas Karch (now at the University of Washington), I discovered an even more amazing possibility: Not only can there be an infinite extra dimension but physics in different locations can reflect different dimensionality. Gravity is localized near us in such a way that it's only the region near us that looks three-dimensional; regions far away reflect a higher-dimensional space. It may be that we see three spatial dimensions not because there really are only three spatial dimensions but because we're stuck to this brane and gravity is concentrated near it, while the surrounding space is oblivious to our lower-dimensional island. There are also some possibilities that matter can move in and out of this isolated four-dimensional region, seeming to appear and disappear as it enters and leaves our domain. These are very hard phenomena to detect in practice, but theoretically there are all sorts of interesting questions about how such a construct all fits together.
Whether or not these theories are right will not necessarily be answered experimentally but could be argued for theoretically, if one or more of them ties into a more fundamental theory. We've used the basic elements found in string theory—namely, the existence of branes and extra dimensions—but we would really like to know if there is a true brane construction. Could you take the very specific branes given by string theory and produce a universe with a brane that localizes gravity? Whether you can actually derive this from string theory or some more fundamental theory is important. The fact that we haven't done it yet isn't evidence that it's not true, and Andreas and I have made good headway into realizing our scenario in string theory. But it can be very, very hard to solve these complicated geometrical set-ups. In general, the problems that get solved, although they seem very complicated, are in many ways simple problems. There is much more work to be done; exciting discoveries await, and they will have implications for other fields.
In cosmology, for instance. Alan Guth's mechanism whereby exponential expansion smooths out the universe works very well, but another possibility has been suggested: a cyclic universe, Paul Steinhardt's idea, wherein a smaller amount of exponential expansion happens many times. Such a theory prompts you to ask questions. First of all, is it really consistent with what we see? The jury's out on that. Does it really have a new mechanism in it? In some sense, the cyclic idea still uses inflation to smooth out the universe. Sometimes it's almost too easy to come up with theories. What grounds your theories? What ties them down? What restricts you from just doing anything? Is there really a new idea there? Do we really have a new mechanism at work? Does it connect to some other, more fundamental theoretical idea? Does it help make that work? Recently I have been exploring the implications of extra dimensions for cosmology. It seems that inflation with extra dimensions works even better than without! What's so nice about this theory is that one can reliably calculate the effect of the extra dimension; no ad hoc assumptions are required. Furthermore, the theory has definite implications for cosmology experiments. All along, I've been emphasizing what we actually see. It's my hope that time and experiments will distinguish among the possibilities.