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THE FABRIC of the COSMOS, Brian Greene, 2004
```(annotated and with added bold highlights by Epsilon=One)
```(annotated and with added bold highlights by Epsilon=One)
Chapter 12 - The World on a String
The Hidden Dimensions
To understand Klein's idea, picture Philippe Petit walking on a long, rubber-coated tightrope stretched between Mount Everest and Lhotse. Viewed from a distance of many miles, as in Figure 12.5, the tightrope appears to be a one-dimensional object like a line — an object that has extension only along its length. If we were told that a tiny worm was slithering along the tightrope in front of Philippe, we'd wildly cheer it on because it needs to stay ahead of Philippe's step to avoid disaster. Of course, with a moment's reflection we all realize that there is more to the surface of the tightrope than the left/right dimension we can directly perceive. Although difficult to see with the naked eye from a great distance, the surface of the tightrope has a second dimension: the clockwise/counterclockwise dimension that is "wrapped" around it. With the aid of a modest telescope, this circular dimension becomes visible and we see that the worm can move not only in the long, unfurled left/right direction but also in the short, "curled-up" clockwise/counterclockwise direction. That is, at every point on the tightrope, the worm has two independent directions in which it can move (that's what we mean when we say the tightrope's surface is two-dimensional *), so it can safely stay out of Philippe's way either by slithering ahead of him, as we initially envisioned, or by crawling around the tiny circular dimension and letting Philippe pass above.
The tightrope illustrates that dimensions — the independent directions in which anything can move — come in two qualitatively distinct varieties. They can be big and easy to see, like the left/right dimension of the tightrope's surface, or they can be tiny and more difficult to see, like the clockwise/counterclockwise dimension that circles around the tightrope's surface. In this example, it was not a major challenge to see the small circular girth of the tightrope's surface. All we needed was a reasonable magnifying instrument. But as you can imagine, the smaller a curled-up dimension, the more difficult it is to detect. At a distance of a few miles, it's one thing to reveal the circular dimension of a tightrope's surface; it would be quite another to reveal the circular dimension of something as thin as dental floss or a narrow nerve fiber.
Figure 12.5 From a distance, a tightrope wire looks one-dimensional, although with a strong enough telescope, its second, curled-up dimension becomes visible.
Klein's contribution was to suggest that what's true for an object within the universe might be true for the fabric of the universe itself. Namely, just as the tightrope's surface has both large and small dimensions, so does the fabric of space. Maybe the three dimensions we all know about — left/right, back/forth, and up/down — are like the horizontal extent of the tightrope, dimensions of the big, easy-to-see variety. But just as the surface of the tightrope has an additional, small, curled-up, circular dimension, maybe the fabric of space also has a small, curled-up, circular dimension, one so small that no one has powerful enough magnifying equipment to reveal its existence. Because of its tiny size, Klein argued, the dimension would be hidden.
How small is small? Well, by incorporating certain features of quantum mechanics into Kaluza's original proposal, Klein's mathematical analysis revealed that the radius of an extra circular spatial dimension would likely be roughly the Planck length, 16 certainly way too small for experimental accessibility (current state-of-the-art equipment cannot resolve anything smaller than about a thousandth the size of an atomic nucleus, falling short of the Planck length by more than a factor of a million billion). Yet, to an imaginary, Planck-sized worm, this tiny, curled-up circular dimension would provide a new direction in which it could roam just as freely as an ordinary worm negotiates the circular dimension of the tightrope in Figure 12.5. Of course, just as an ordinary worm finds that there isn't much room to explore in the clockwise direction before it finds itself back at its starting point, a Planck-sized worm slithering along a curled-up dimension of space would also repeatedly circle back to its starting point. But aside from the length of the travel it permitted, a curled-up dimension would provide a direction in which the tiny worm could move just as easily as it does in the three familiar unfurled dimensions.
Figure 12.6
The surface of a tightrope has one long dimension with a circular dimension tacked on at each point.
To get an intuitive sense of what this looks like, notice that what we've been referring to as the tightrope's curled-up dimension — the clockwise/counterclockwise direction — exists at each point along its extended dimension. The earthworm can slither around the circular girth of the tightrope at any point along its outstretched length, and so the tightrope's surface can be described as having one long dimension, with a tiny, circular direction tacked on at each point, as in Figure 12.6. This is a useful image to have in mind because it also applies to Klein's proposal for hiding Kaluza's extra dimension of space.
Figure 12.7
The Kaluza-Klein proposal is that on very small scales, space has an extra circular dimension tacked on to each familiar point.
To see this, let's again examine the fabric of space by sequentially showing its structure on ever smaller distance scales, as in Figure 12.7. At the first few levels of magnification, nothing new is revealed: the fabric of space still appears three-dimensional (which, as usual, we schematically represent on the printed page by a two-dimensional grid). However, when we get down to the Planck scale, the highest level of magnification in the figure, Klein suggested that a new, curled-up dimension becomes visible.
Just as the circular dimension of the tightrope exists at each point along its big, extended dimension, the circular dimension in this proposal exists at each point in the familiar three extended dimensions of daily life. In Figure 12.7, we illustrate this by drawing the additional circular dimension at various points along the extended dimensions (since drawing the circle at every point would obscure the image) and you can immediately see the similarity with the tightrope in Figure 12.6. In Klein's proposal, therefore, space should be envisioned as having three unfurled dimensions (of which we show only two in the figure) with an additional circular dimension tacked on to each point. Notice that the extra dimension is not a bump or a loop within the usual three spatial dimensions, as the graphic limitations of the figure might lead you to think. Instead, the extra dimension is a new direction, completely distinct from the three we know about, which exists at every point in our ordinary three-dimensional space, but is so small that it escapes detection even with our most sophisticated instruments.
With this modification to Kaluza's original idea, Klein provided an answer to how the universe might have more than the three space dimensions of common experience that could remain hidden, a framework that has since become known as Kaluza-Klein theory. And since an extra dimension of space was all Kaluza needed to merge general relativity and electromagnetism, Kaluza-Klein theory would seem to be just what Einstein was looking for. Indeed, Einstein and many others became quite excited about unification through a new, hidden space dimension, and a vigorous effort was launched to see whether this approach would work in complete detail. But it was not long before Kaluza-Klein theory encountered its own problems. Perhaps most glaring of all, attempts to incorporate the electron into the extra-dimensional picture proved unworkable. 17 Einstein continued to dabble in the Kaluza-Klein framework until at least the early 1940s, but the initial promise of the approach failed to materialize, and interest gradually died out.
Within a few decades, though, Kaluza-Klein theory would make a spectacular comeback.
The tightrope illustrates that dimensions — the independent directions in which anything can move — come in two qualitatively distinct varieties. They can be big and easy to see, like the left/right dimension of the tightrope's surface, or they can be tiny and more difficult to see, like the clockwise/counterclockwise dimension that circles around the tightrope's surface. In this example, it was not a major challenge to see the small circular girth of the tightrope's surface. All we needed was a reasonable magnifying instrument. But as you can imagine, the smaller a curled-up dimension, the more difficult it is to detect. At a distance of a few miles, it's one thing to reveal the circular dimension of a tightrope's surface; it would be quite another to reveal the circular dimension of something as thin as dental floss or a narrow nerve fiber.
Klein's contribution was to suggest that what's true for an object within the universe might be true for the fabric of the universe itself. Namely, just as the tightrope's surface has both large and small dimensions, so does the fabric of space. Maybe the three dimensions we all know about — left/right, back/forth, and up/down — are like the horizontal extent of the tightrope, dimensions of the big, easy-to-see variety. But just as the surface of the tightrope has an additional, small, curled-up, circular dimension, maybe the fabric of space also has a small, curled-up, circular dimension, one so small that no one has powerful enough magnifying equipment to reveal its existence. Because of its tiny size, Klein argued, the dimension would be hidden.
How small is small? Well, by incorporating certain features of quantum mechanics into Kaluza's original proposal, Klein's mathematical analysis revealed that the radius of an extra circular spatial dimension would likely be roughly the Planck length, 16 certainly way too small for experimental accessibility (current state-of-the-art equipment cannot resolve anything smaller than about a thousandth the size of an atomic nucleus, falling short of the Planck length by more than a factor of a million billion). Yet, to an imaginary, Planck-sized worm, this tiny, curled-up circular dimension would provide a new direction in which it could roam just as freely as an ordinary worm negotiates the circular dimension of the tightrope in Figure 12.5. Of course, just as an ordinary worm finds that there isn't much room to explore in the clockwise direction before it finds itself back at its starting point, a Planck-sized worm slithering along a curled-up dimension of space would also repeatedly circle back to its starting point. But aside from the length of the travel it permitted, a curled-up dimension would provide a direction in which the tiny worm could move just as easily as it does in the three familiar unfurled dimensions.
The surface of a tightrope has one long dimension with a circular dimension tacked on at each point.
To get an intuitive sense of what this looks like, notice that what we've been referring to as the tightrope's curled-up dimension — the clockwise/counterclockwise direction — exists at each point along its extended dimension. The earthworm can slither around the circular girth of the tightrope at any point along its outstretched length, and so the tightrope's surface can be described as having one long dimension, with a tiny, circular direction tacked on at each point, as in Figure 12.6. This is a useful image to have in mind because it also applies to Klein's proposal for hiding Kaluza's extra dimension of space.
The Kaluza-Klein proposal is that on very small scales, space has an extra circular dimension tacked on to each familiar point.
To see this, let's again examine the fabric of space by sequentially showing its structure on ever smaller distance scales, as in Figure 12.7. At the first few levels of magnification, nothing new is revealed: the fabric of space still appears three-dimensional (which, as usual, we schematically represent on the printed page by a two-dimensional grid). However, when we get down to the Planck scale, the highest level of magnification in the figure, Klein suggested that a new, curled-up dimension becomes visible.
Just as the circular dimension of the tightrope exists at each point along its big, extended dimension, the circular dimension in this proposal exists at each point in the familiar three extended dimensions of daily life. In Figure 12.7, we illustrate this by drawing the additional circular dimension at various points along the extended dimensions (since drawing the circle at every point would obscure the image) and you can immediately see the similarity with the tightrope in Figure 12.6. In Klein's proposal, therefore, space should be envisioned as having three unfurled dimensions (of which we show only two in the figure) with an additional circular dimension tacked on to each point. Notice that the extra dimension is not a bump or a loop within the usual three spatial dimensions, as the graphic limitations of the figure might lead you to think. Instead, the extra dimension is a new direction, completely distinct from the three we know about, which exists at every point in our ordinary three-dimensional space, but is so small that it escapes detection even with our most sophisticated instruments.
With this modification to Kaluza's original idea, Klein provided an answer to how the universe might have more than the three space dimensions of common experience that could remain hidden, a framework that has since become known as Kaluza-Klein theory. And since an extra dimension of space was all Kaluza needed to merge general relativity and electromagnetism, Kaluza-Klein theory would seem to be just what Einstein was looking for. Indeed, Einstein and many others became quite excited about unification through a new, hidden space dimension, and a vigorous effort was launched to see whether this approach would work in complete detail. But it was not long before Kaluza-Klein theory encountered its own problems. Perhaps most glaring of all, attempts to incorporate the electron into the extra-dimensional picture proved unworkable. 17 Einstein continued to dabble in the Kaluza-Klein framework until at least the early 1940s, but the initial promise of the approach failed to materialize, and interest gradually died out.
Within a few decades, though, Kaluza-Klein theory would make a spectacular comeback.
* Were you to count left, right, clockwise, and counterclockwise all separately, you'd conclude that the worm can move in four directions. But when we speak of "independent" directions, we always group those that lie along the same geometrical axis — like left and right, and also clockwise and counterclockwise.