The Fabric of the Cosmos: Space, Time, and the Texture of Reality (51 page)

Back in 1687, when Newton proposed his universal law of gravity, he was actually making a strong statement about the number of space dimensions. Newton didn't just say that the force of attraction between two objects gets weaker as the distance between them gets larger. He proposed a formula, the
inverse square law,
which describes precisely how the gravitational attraction will diminish as two objects are separated. According to this formula, if you double the distance between the objects, their gravitational attraction will fall by a factor of 4 (2
²
); if you triple the distance, it will fall by a factor of 9 (3
²
); if you quadruple the distance, it will fall by a factor of 16 (4
²
); and more generally, the gravitational force drops in proportion to the square of the separation. As has become abundantly evident over the last few hundred years, this formula works.

But
why
does the force depend on the square of the distance? Why doesn't the force drop like the cube of the separation (so that if you double the distance, the force diminishes by a factor of 8) or the fourth power (so that if you double the distance, the force diminishes by a factor of 16), or perhaps, even more simply, why doesn't the gravitational force between two objects drop in direct proportion to the separation (so that if you double the distance, the force diminishes by a factor of 2)? The answer is tied directly to the number of dimensions of space.

One way to see this is to think about how the number of gravitons emitted and absorbed by the two objects depends on their separation, or by thinking about how the curvature of spacetime that each object experiences diminishes as the distance between them increases. But let's take a simpler, more old-fashioned approach, which gets us quickly and intuitively to the correct answer. Let's draw a figure (Figure 13.4a) that schematically illustrates the gravitational field produced by a massive object—let's say the sun—much as Figure 3.1 schematically illustrates the magnetic field produced by a bar magnet. Whereas magnetic field lines sweep around from the magnet's north pole to its south pole, notice that gravitational field lines emanate radially outward in all directions and just keep on going. The strength of the gravitational pull another object— imagine it's an orbiting satellite—would feel at a given distance is proportional to the density of field lines at that location. The more field lines penetrate the satellite, as in Figure 13.4b, the greater the gravitational pull to which it is subject.

We can now explain the origin of Newton's inverse square law. An imaginary sphere centered on the sun and passing through the satellite's location, as in Figure 13.4c, has a surface area that—like the surface of any sphere in three-dimensional space—is proportional to the
square
of its radius, which in this case is the
square
of the distance between the sun and the satellite. This means that the density of field lines passing through the sphere—the total number of field lines divided by the sphere's area— decreases as the square of sun-satellite separation. If you double the distance, the same number of field lines are now uniformly spread out on a sphere with four times the surface area, and hence the gravitational pull at that distance will drop by a factor of four. Newton's inverse square law for gravity is thus a reflection of a geometrical property of spheres in three space dimensions.

Figure 13.4
(
a
)
The gravitational force exerted by the sun on an object, such as a satellite, is inversely proportional to the square of the distance between them. The reason is that the sun's gravitational field lines spread out uniformly as in
(
b
)
and hence have a density at a distance
d
that is inversely proportional to the area of an imaginary sphere of radius
d—
schematically drawn in
(
c
)—
an area which basic geometry shows to be proportional to d
²
.

By contrast, if the universe had two or even just one space dimension, how would Newton's formula change? Well, Figure 13.5a shows a two-dimensional version of the sun and its orbiting satellite. As you can see, at any given distance the sun's gravitational field lines uniformly spread out on a circle, the analog of a sphere in one lower dimension. Since the circle's circumference is proportional to its radius (not to the square of its radius), if you double the sun-satellite separation, the density of field lines will decrease by a factor of 2 (not 4) and so the strength of the sun's gravitational pull will drop only by a factor of 2 (not 4). If the universe had only two space dimensions, then, gravitational pull would be inversely proportional to separation, not the square of separation.

If the universe had only one space dimension, as in Figure 13.5b, the law of gravity would be simpler still. Gravitational field lines would have no room to spread out, and so the force of gravity would not decrease with separation. If you were to double the distance between the sun and the

Figure 13.5
(
a
)
In a universe with only two spatial dimensions, the gravitational force drops in proportion to separation, because gravitational field lines uniformly spread on a circle whose circumference is proportional to its radius.
(
b
)
In a universe with one space dimension, gravitational field lines do not have any room to spread, so the gravitational force is constant, regardless of separation.

satellite (assuming that versions of such objects could exist in such a universe), the same number of field lines would penetrate the satellite and hence the force of gravity acting between them would not change at all.

Although it is impossible to draw, the pattern illustrated by Figures 13.4 and 13.5 extends directly to a universe with four or five or six or any number of space dimensions. The more space dimensions there are, the more room gravitational lines of force have to spread out. And the more they spread out, the more precipitously the force of gravity drops with increasing separation. In four space dimensions, Newton's law would be an inverse cube law (double the separation, force drops by a factor of 8); in five space dimensions, it would be an inverse fourth-power law (double the separation, force drops by a factor of 16); in six space dimensions, it would be an inverse fifth-power law (double the separation, force drops by a factor of 32); and so on for ever higher-dimensional universes.

You might think that the success of the inverse square version of Newton's law in explaining a wealth of data—from the motion of planets to the paths of comets—confirms that we live in a universe with precisely three space dimensions. But that conclusion would be hasty. We know that the inverse square law works on astronomical scales,
⁶
and we know that it works on terrestrial scales, and that jibes well with the fact that on such scales we see three space dimensions. But do we know that it works on smaller scales? How far down into the microcosmos has gravity's inverse square law been tested? As it turns out, experimenters have confirmed it down to only about a tenth of a millimeter; if two objects are brought to within a separation of a tenth of a millimeter, the data verify that the strength of their gravitational attraction follows the predictions of the inverse square law. But so far, it has proven a significant technical challenge to test the inverse square law on shorter scales (quantum effects and the weakness of gravity complicate the experiments). This is a critical issue, because deviations from the inverse square law would be a convincing signal of extra dimensions.

To see this explicitly, let's work with a lower-dimensional toy example that we can easily draw and analyze. Imagine we lived in a universe with one space dimension—or so we thought, because only one space dimension was visible and, moreover, centuries of experiments had shown that the force of gravity does not vary with the separation between objects. But also imagine that in all those years experimenters had been able only to test the law of gravity down to distances of about a tenth of a millimeter. For distances shorter than that, no one had any data. Now, imagine further that, unbeknownst to everyone but a handful of fringe theoretical physicists, the universe actually had a second, curled-up space dimension making its shape like the surface of Philippe Petit's tightrope, as in Figure 12.5. How would this affect future, more refined gravitational tests? We can deduce the answer by looking at Figure 13.6. As two tiny objects are brought close enough together—much closer than the circumference of the curled-up dimension—the two-dimensional character of space would become apparent immediately, because on those scales gravitational field lines
would
have room to spread out (Figure 13.6a). Rather than being independent of distance, the force of gravity would vary
inversely
with separation when objects were close enough together.

Figure 13.6
(
a
)
When objects are close, the gravitational pull varies as it does in two space dimensions.
(
b
)
When objects are farther apart, the gravitational pull behaves as it does in one space dimension—it is constant.

Thus, if you were an experimenter in this universe, and you developed exquisitely accurate methods for measuring gravitational attraction, here's what you would find. When two objects were extremely close, much closer than the size of the curled-up dimension, their gravitational attraction would diminish in proportion to their separation, just as you expect for a universe with two space dimensions. But then, when the objects were about as far apart as the circumference of the curled-up dimension, things would change. Beyond this distance, the gravitational field lines would be unable to spread any further. They would have spilled out as far as they could into the second curled-up dimension— they would have saturated that dimension—and so from this distance onward the gravitational force would no longer diminish, as illustrated in Figure 13.6b. You can compare this saturation with the plumbing in an old house. If someone opens the faucet in the kitchen sink when you're just about to rinse the shampoo out of your hair, the water pressure can drop because the water spreads between the two outlets. The pressure will diminish yet again should someone open the faucet in the laundry room, since the water will spread even more. But once all the faucets in the house are open, the pressure will remain constant. Although it might not provide the relaxing, high-water-pressure experience you'd anticipated, the pressure in the shower will not drop any further once the water has completely spread throughout all "extra" outlets. Similarly, once the gravitational field has completely spread throughout the extra curled-up dimension, it will not diminish with further separation.