The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [50]
For another visualization, think of a huge carpet blanketing Utah’s salt flats. From an airplane, the carpet looks like a flat surface with two dimensions that extend north/south and east/west. But after you parachute down and view the carpet up close, you realize that its surface is composed of a tight pile: tiny cotton loops attached to each point on the flat carpet backing. The carpet has two large, easy-to-see dimensions (north/south and east/west), but also one small dimension (the circular loops) that is harder to detect (Figure 4.4b).
The Kaluza-Klein proposal suggested that a similar distinction, between dimensions that are big and easily seen, and others that are tiny and thus more difficult to reveal, might apply to the fabric of space itself. The reason we are all aware of the familiar three dimensions of space would be that their extent, like the vertical dimension of the straw and the north/south and east/west dimensions of the carpet, is huge (possibly infinite). However, if an extra dimension of space were curled up like the circular part of the straw or carpet, but to an extraordinarily small size—millions or even billions of times smaller than a single atom—it could be as ubiquitous as the familiar unfurled dimensions and yet remain beyond our ability to detect even with today’s most powerful magnifying equipment. The dimension would indeed go missing. Such was the beginning of Kaluza-Klein theory, the proposition that our universe has spatial dimensions beyond the three of everyday experience (Figure 4.5).
This line of thought establishes that the suggestion of “extra” spatial dimensions, however unfamiliar, is not absurd. That’s a good start, but it invites an essential question: Why, back in the 1920s, would someone invoke such an exotic idea? Kaluza’s motivation came from an insight he had shortly after Einstein published the general theory of relativity. He found that with a single stroke of the pen—literally—he could modify Einstein’s equations to make them apply to a universe with one additional dimension of space. And when he analyzed those modified equations, the results were so thrilling that, as his son has recounted, Kaluza discarded his normally reserved demeanor, pounded his desk with both hands, shot to his feet, and erupted into an aria from The Marriage of Figaro.10 Within the modified equations, Kaluza found the ones Einstein had already used successfully to describe gravity in the familiar three dimensions of space and one of time. But because his new formulation included an additional dimension of space, Kaluza found an additional equation. Lo and behold, when Kaluza derived this equation he recognized it as the very one Maxwell had discovered half a century earlier to describe the electromagnetic field.
Figure 4.4 (a) The surface of a tall straw has two dimensions; the vertical dimension is long and easy to see, while the circular dimension is small and harder to detect. (b) A gigantic carpet has three dimensions; the north/south and east/west dimensions are big and easy to see, while the circular part, the carpet’s pile, is small and therefore harder to detect.
Figure 4.5 Kaluza-Klein theory posits tiny extra spatial dimensions attached to every point in the familiar three large spatial dimensions. If we could magnify the spatial fabric sufficiently, the hypothesized extra dimensions would become visible. (For the sake of visual clarity, extra dimensions are attached only on grid points in the illustration.)
Kaluza revealed that in a universe with an additional dimension of space, gravity and electromagnetism can both be described in terms of spatial ripples. Gravity ripples through the familiar three spatial dimensions, while electromagnetism ripples through the fourth. An outstanding problem with Kaluza’s proposal was to explain why we don’t see this fourth spatial dimension. It was here that Klein made