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string developments of the time: both our work and the research of string theorists involved a similarly warped geometry. In fact, perhaps because our research didn’t directly challenge string theory models, the string theory community actually accepted and recognized the significance of our work sooner than the model building community. Although it had initially seemed coincidental, maybe this was some indication that we were all on the right track. And happily, Raman had no trouble getting job offers afterward. (He’s now a professor at Johns Hopkins.)

However, some skeptics remained. The precise model Raman and I considered led to interesting questions that no one could answer right away. Did localization depend on the form of spacetime very far away? When people tried to find examples of the type of geometry that Raman and I had suggested in supergravity theories, the form of gravity far from the localizing brane seemed to be the stumbling block. But were those conditions essential? Another question we wanted to answer was, did spacetime necessarily look four-dimensional everywhere? Localized gravity made the entire five-dimensional universe behave as if there were four-dimensional gravity. Did this always happen, or could some regions look four-dimensional and some regions behave differently? And what happens when the Gravitybrane isn’t completely flat? Does localization work the same way for a brane with a different geometry? These are some of the questions that locally localized gravity, the theory that Andreas and I developed, could address.

Locally Localized Gravity

How many dimensions of space are there? Do we really know? By now, I hope you will agree that it would be overreaching to claim that we know for certain that extra dimensions do not exist. We see three dimensions of space, but there could be more that we haven’t yet detected.

You now know that extra dimensions can be hidden either because they are curled up and small, or because spacetime is warped and gravity so concentrated in a small region that even an infinite dimension is invisible. Either way, whether dimensions are compact or localized, spacetime would appear to be four-dimensional everywhere, no matter where you are.

This might be a little less obvious in the localized gravity scenario, in which the graviton’s probability function becomes smaller and smaller as you go out into the fifth dimension. Gravity acts as it does in four dimensions if you’re near the brane. But what about everywhere else?

The answer is that in RS2, the influence of four-dimensional gravity is inescapable, no matter where you are in the fifth dimension. Although the graviton’s probability function is largest on the Gravitybrane, objects everywhere can interact with one another by exchanging a graviton, and therefore all objects would experience four-dimensional gravity, independently of location. Gravity everywhere looks four-dimensional because the graviton’s probability function is never actually zero—it continues on for ever. In the localized scenario, objects far from a brane would have extremely weak gravitational interactions, but weak gravity would nonetheless behave in a four-dimensional manner. So, for example, Newton’s inverse square law would hold, no matter where you were along the fifth dimension.

The small but nonzero graviton probability function away from the Gravitybrane was essential to the solution to the hierarchy problem I presented in Chapter 20. The Weakbrane, located away from the Gravitybrane in the bulk, experiences gravity that appears to be four-dimensional, even if it experiences that gravity only extremely feebly. Like water far from your own garden in the sprinkler analogy, there is always some water, just not a lot.

But suppose we reflect even further and ask what we really know with certainty about the dimensions of space. We do not know that space everywhere looks three-dimensional, only that space near us looks three-dimensional. Space appears to have three dimensions (and spacetime to have four) at the distances that we can