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Geometry in Translation

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  • Geometry in Translation

    THE FABRIC of the COSMOS, Brian Greene, 2004
    ```(annotated and with added bold highlights by Epsilon=One)
    Chapter 16 - The Future of an Allusion
    Geometry in Translation
    Another consideration, one physicists call geometrical duality, also suggests that spacetime may not be fundamental, but suggests it from a very different viewpoint. Its description is a little more technical than quantum averaging, so feel free to go into skim mode if at any point this section gets too heavy. But because many researchers consider this material to be among string theory's most emblematic features it's worth trying to get the gist of the ideas.

    In Chapter 13 (we saw how the five supposedly distinct string theories are actually different translations of one and the same theory. Among other things, we emphasized that this is a powerful realization because, when translated, supremely difficult questions sometimes become far simpler to answer. But there is a feature of the translation dictionary unifying the five theories that I've so far neglected to mention. Just as a question's degree of difficulty can be changed radically by the translation from one string formulation to another, so, too, can the description of the geometrical form of spacetime. Here's what I mean.

    Because string theory requires more than the three space dimensions and one time dimension of common experience, we were motivated in Chapters 12 and 13 to take up the question of where the extra dimensions might be hiding. The answer we found is that they may be curled up into a size that, so far, has eluded detection because it's smaller than we are able to probe experimentally. We also found that physics in our familiar big dimensions is dependent on the precise size and shape of the extra dimensions because their geometrical properties, affect the vibrational patterns strings can execute. Good. Now for the part I left out.

    The dictionary that translates questions posed in one string theory into different questions posed in another string theory also translates the geometry of the extra dimensions in the first theory into a different extra-dimensional geometry in the second theory. If, for example, you are studying the physical implications of, say, the Type IIA string theory with extra dimensions curled up into a particular size and shape, then every conclusion you reach can, at least in principle, be deduced by considering appropriately translated questions in, say, the Type IIB string theory. But the dictionary for carrying out the translation demands that the extra dimensions in the Type IIB string theory be curled up into a precise geometrical form that depends on—but generally differs from—the form given by the Type IIA theory. In short, a given string theory with curled-up dimensions in one geometrical form is equivalent to—is a translation of—another string theory with curled-up dimensions in a different geometrical form.

    And the differences in spacetime geometry need not be minor. For example, if one of the extra dimensions of, say, the Type IIA string theory should be curled up into a circle, as in Figure 12.7, the translation dictionary shows that this is absolutely equivalent to the Type IIB string theory with one of its extra dimensions also curled up into a circle, but one whose radius is inversely proportional to the original. If one circle is tiny, the other is big, and vice versa—and yet there is absolutely no way to distinguish between the two geometries. (Expressing lengths as multiples of the Planck length, if one circle has radius R, the mathematical dictionary shows that the other circle has radius 1/R). You might think that you could easily and immediately distinguish between a big and a small dimension, but in string theory this is not always the case. All observations derive from the interactions of strings, and these two theories, the Type IIA with a big circular dimension and the Type IIB with a small circular dimension, are merely different translations of— different ways of expressing—the same physics. Every observation you describe within one string theory has an alternative and equally viable description within the other string theory, even though the language of each theory and the interpretation it gives may differ. (This is possible because there are two qualitatively different configurations for strings moving on a circular dimension: those in which the string is wrapped around the circle like a rubber band around a tin can, and those in which the string resides on a portion of the circle but does not wrap around it. The former have energies that are proportional to the radius of the circle [the larger the radius, the longer the wrapped strings are stretched, so the more energy they embody], while the latter have energies that are inversely proportional to the radius [the smaller the radius, the more hemmed in the strings are, so the more energetically they move because of quantum uncertainty]. Notice that if we were to replace the original circle by one of inverted radius, while also exchanging "wrapped" and "not wrapped" strings, physical energies—and, it turns out, physics more generally—would remain unaffected. This is exactly what the dictionary translating from the Type IIA theory to the Type IIB theory requires, and why two seemingly different geometries—a big and a small circular dimension—can be equivalent.)

    A similar idea also holds when circular dimensions are replaced with the more complicated Calabi-Yau shapes introduced in Chapter 12. A given string theory with extra dimensions curled up into a particular Calabi-Yau shape gets translated by the dictionary into a different string theory with extra dimensions curled up into a different Calabi-Yau shape (one that is called the mirror or dual of the original). In these cases, not only can the sizes of the Calabi-Yaus differ, but so can their shapes, including the number and variety of their holes. But the translation dictionary ensures that they differ in just the right way, so that even though the extra dimensions have different sizes and shapes, the physics following from each theory is absolutely identical. (There are two types of holes in a given Calabi-Yau shape, but it turns out that string vibrational patterns—and hence physical implications—are sensitive only to the difference between the number of holes of each type. So if one Calabi-Yau has, say, two holes of the first kind and five of the second, while another Calabi-Yau has five holes of the first kind and two of the second, then even though they differ as geometrical shapes; they can give rise to identical physics.*)

    From another perspective, then, this bolsters the suspicion that space is not a foundational concept. Someone describing the universe using one of the five string theories would claim that space, including the extra dimensions, has a particular size and shape, while someone else using one of the other string theories would claim that space, including the extra dimensions, has a different size and shape. Because the two observers would simply be using alternative mathematical descriptions of the same physical universe, it is not that one would be right and the other wrong. They would both be right, even though their conclusions about space—its size and shape—would differ. Note too, that it's not that they would be slicing up spacetime in different, equally valid ways, as in special relativity. These two observers would fail to agree on the overall structure of spacetime itself. And that's the point. If spacetime were really fundamental, most physicists expect that everyone, regardless of perspective—regardless of the language or theory used—would agree on its geometrical properties. But the fact that, at least within string theory, this need not be the case, suggests that spacetime may be a secondary phenomenon.

    We are thus led to ask: if the clues described in the last two sections are pointing us in the right direction, and familiar spacetime is but a large-scale manifestation of some more fundamental entity, what is that entity and what are its essential properties? As of today, no one knows. But in the search for answers, researchers have found yet further clues, and the most important have come from thinking about black holes.

    *For details on geometrical duality involving both circles and Calabi-Yau shapes, see The Elegant Universe, Chapter 10.
    Last edited by al-Brunardot; 04-26-2014, 04:38 AM.