Warped Passages - Lisa Randall [209]
You can also see why the fifth dimension doesn’t affect gravity very much by considering the gravitational field surrounding an object on the Gravitybrane. We have seen that in flat spatial dimensions, the force lines emanating from an object spread equally in all directions. And when there were finite extra dimensions, the field lines extended in all directions until some reached a boundary and bent around. For this reason, gravitational field lines that are further from an object than the size of the extra dimensions spread only along the three infinite dimensions of the lower-dimensional world.
Figure 88. If ducks are concentrated near the shore, you can count almost all of them by counting the ones nearby.
In the warped scenario, on the other hand, field lines do not distribute themselves equally in all directions; it’s only along the brane that they extend equally in all directions. Perpendicular to the brane, they extend very little (see Figure 89). Because the gravitational field lines spread out primarily along the brane, the gravitational field looks almost identical to the field associated with an object in four dimensions. The spread into the fifth dimension is so small (not much more than the Planck scale length, 10-33 cm) that we can ignore it. Although the extra dimension is infinite, it is irrelevant to the gravitational field of a brane-bound object.
You can also see how Raman and I resolved the initial puzzle we faced: why the size of the fifth dimension is irrelevant to the strength of gravity. Returning to the sprinkler analogy above, suppose we now specify the distribution of water over the entire sprinkler, so that it resembles the distribution of gravity from the plummeting graviton’s probability function: after you take half of the water for your garden, you then send half of the remaining water to the adjacent garden, half of that amount to the next one, and so on, with everyone in successive gardens receiving half as much water as their neighbor. To mimic a second brane in the fifth dimension, we’ll assume that we stop delivering water beyond a certain point, just as a second brane in the fifth dimension would cut off the graviton’s probability function at some point along the fifth dimension. And to mimic an infinite fifth dimension, we’ll assume that the water delivers water indefinitely along its length.
Figure 89. In the warped scenario, the field lines are equally distributed in all the directions on the brane. However, off the brane the field lines bend back around so that they become essentially parallel to the brane, almost as if the fifth dimension were finite. Even with an infinite dimension, the gravitational field is localized near the brane, and field lines spread essentially as if there were only four (spacetime) dimensions.
To show that the size of the fifth dimension is irrelevant to the strength of gravity near the brane, we would want to show that the first few gardens get very nearly the same amount of water, regardless of whether we stop delivering water when we get to the fifth garden or the tenth garden or we don’t stop delivering water at all. So let’s consider what happens if the sprinkler ended after the first five gardens. Because the sixth garden and beyond would have received so little water, the total amount of water that the sprinkler would send to the first several gardens would differ from that of an infinite sprinkler by only a few percent. And if you stopped the sprinkler after the seventh garden, it would differ by even less. With our