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suggested that this latter role for the second brane was a red herring, and that the second brane was not essential for gravity to mimic that of a genuinely four-dimensional universe: the four-dimensional graviton’s interactions were virtually independent of the fifth dimension’s size. A calculation showed that gravity would have the same strength if the second brane stayed where it was, or if it were twice as far from the Gravitybrane, or if it were ten times as far out into the bulk, further away from the first brane. In fact, four-dimensional gravity persisted even if our model put the second brane at infinity—which is to say, eliminated it altogether. This should not be true if the second brane and a finite dimension were essential for reproducing four-dimensional gravity.

This was our first clue that the intuition that we need a second brane was based on flat dimensions and wasn’t necessarily true for warped spacetime. With a flat extra dimension, the second brane is compulsory for four-dimensional gravity. We can see this with the aid of the sprinkler analogy from Chapter 20. A flat extra dimension would correspond to water being distributed equally everywhere along a long straight sprinkler (see Figure 81, Chapter 20).* The longer the sprinkler, the less water would be sprinkled on any given garden. If we were to extend this reasoning to an infinitely long sprinkler, we would see that the water would be spread so thinly that essentially no water would be sprinkled on any finite-sized garden.

Similarly, if gravity were spread throughout an infinite uniform dimension, the gravitational force would be so attenuated along the extra dimension that it would be reduced to nothing. A geometry with an infinite extra dimension would have to contain some subtlety that goes beyond this simple intuitive picture if gravity is to behave four-dimensionally. And indeed, warped spacetime provides the requisite added ingredient.

To see how this works, let’s once again use our sprinkler analogy to identify the loophole in the argument above. Suppose that you have an infinitely long sprinkler, but you don’t distribute water equally everywhere. Instead, you have control over how the water is allocated, giving you the option of ensuring that your own garden is well watered. One way of accomplishing this would be to deliver half the water to your plot of land and the remaining half of the water everywhere else. In that case, although the gardens far away would be badly treated, your garden would be guaranteed to receive all the water it needs. Your garden would always receive half the water, even though the sprinkler continues delivering water indefinitely far away. With an inequitable distribution of water, you would get all the water you need. The sprinkler could be infinite but you wouldn’t know the distance.

Similarly, the graviton’s probability function in our warped geometry is always very big near the Gravitybrane, despite the infinite fifth dimension. As in the previous chapter, the probability function for the graviton peaks on this brane (see Figure 87), and falls off exponentially as the graviton moves away from the Gravitybrane into the fifth dimension. In this theory, however, the graviton’s probability function continues indefinitely far but it is inconsequential to the size of the graviton probability function near the brane.

Figure 87. The graviton’s probability function in infinite warped spacetime with a single brane.

A plummeting probability function of this sort tells us that the likelihood of finding the graviton far from the Gravitybrane is extremely tiny—so tiny that we can generally ignore the distant regions of the fifth dimension. Although in principle the graviton can be anywhere along the fifth dimension, the exponential decrease makes the graviton’s probability function very concentrated in the vicinity of the Gravitybrane. The situation is almost, but not quite, as if a second brane confined the graviton to a limited region.

The high probability for the graviton to be found near the Gravitybrane, and