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of the gravitational force lines emitted from a massive object, which were diluted because they were spread throughout large dimensions. If we so chose, we could have described this dilution as a consequence of the graviton’s probability function. Remember that the graviton’s probability function tells us how gravity is spread out over space. Because gravity in the large extra-dimensions scenario is equally strong everywhere in the extra dimensions, the graviton’s probability function in this case is flat. Such a flat graviton probability function would tell us that the graviton, the particle communicating gravity, is spread out over the large region enclosed by the extra dimensions. This flat probability function, that is equally distributed over all of extra-dimensional space, tells us that gravity’s influence in four dimensions is vastly diluted.

In the warped five-dimensional spacetime we are now considering, there is an interesting twist. The graviton no longer has equal probability of being in all places in five-dimensional space that lie between its two boundaries, the Gravitybrane and the Weakbrane. The distribution of the graviton is in fact far from democratic as an automatic consequence of the energy carried by the branes and in the bulk. The graviton’s probability function varies: it is big in one region and small in all others, and it is this variation that provides the dilution factor responsible for making gravity so weak in our world. Gravity is feeble on the Weakbrane because the graviton’s probability function there is so minuscule.

Let us momentarily return to the sprinkler analogy that we used earlier to explain how the strength of gravity decreases with distance. The larger the region over which the sprinkler distributes water (as illustrated in the upper part of Figure 81), the more the water gets diluted. When there are large extra dimensions, gravity is spread over a very big region, and it too gets diluted. Gravity therefore appears to be feeble in the low-energy, effective four-dimensional theory.

Figure 81. Three different sprinklers. By comparing the first and second sprinklers, we see that a longer sprinkler delivers less water over any particular region than a shorter one does. The third sprinkler demonstrates that water can be inequitably distributed so that the first garden always gets half the water, the second garden one-quarter, and so on. In that case, the amount of water delivered to the first garden is independent of the sprinkler’s length; it always gets half the water.

The warped geometry, on the other hand, resembles a sprinkler that does not distribute water equally in all directions, but instead delivers it preferentially to one particular region, the region around the Gravitybrane (see the lower part of Figure 81). With an undemocratic sprinkler, it is obvious that less water will be delivered everywhere aside from the favored region. And, if the amount of water delivered to other regions falls off exponentially from the highly favored location, the water fraction delivered to those other areas will be very tiny indeed, even if they are only a modest distance away. Clearly, the water delivered by the “warped” sprinkler is “diluted” far more than water that is equally distributed to all regions.

The upshot is that if all the Standard Model particles are housed on the Weakbrane, then gravity is so weak compared with the other three forces that the hierarchy problem of particle physics, the question of why gravity is so weak relative to the other forces, is solved. Feeble gravity is a natural consequence of the small amplitude of the graviton’s probability function on the Weakbrane, even when it is only a relatively modest distance (about ten times larger than the string-theory-favored Planck scale length) away from the Gravitybrane.

Growing and Shrinking in a Warped Dimension

The previous explanation of the hierarchy in terms of the exponentially falling probability function is entirely adequate for understanding warped spacetime. The intuitive explanation for the weakness