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The Heart of Riemannian Geometry

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  • The Heart of Riemannian Geometry

    Table of Contents
    .......The Elegant Universe
    THE ELEGANT UNIVERSE, Brian Greene, 1999, 2003
    ```(annotated and with added bold highlights by Epsilon=One)
    Chapter 10 - Quantum Geometry
    The Heart of Riemannian Geometry
    If you jump on a trampoline, the weight of your body causes it to warp by stretching its elastic fibers. This stretching is most severe right under your body and becomes less noticeable toward the trampoline's edge. You can see this clearly if a familiar image such as the Mona Lisa is painted on the trampoline. When the trampoline is not supporting any weight, the Mona Lisa looks normal. But when you stand on the trampoline, the image of the Mona Lisa becomes distorted, especially the part directly under your body, as illustrated in Figure 10.1.

    Figure 10.1 When standing on the Mona Lisa trampoline, the image becomes most distorted under your weight.
    This example cuts to the heart of Riemann's mathematical framework for describing warped shapes. Riemann, building on earlier insights of the mathematicians Carl Friedrich Gauss, Nikolai Lobachevsky, Janos Bolyai, and others, showed that a careful analysis of the distances between all locations on or in an object provides a means of quantifying the extent of its curvature. Roughly speaking, the greater the (nonuniform) stretching—the greater the deviation from the distance relations on a flat shape—the greater the curvature of the object. For example, the trampoline is most significantly stretched right under your body and therefore the distance relations between points in this area are most severely distorted. This region of the trampoline, therefore, has the largest amount of curvature, in line with what you expect, since this is where the Mona Lisa suffers the greatest distortion, yielding the hint of a grimace at the corner of her customary enigmatic smile.

    Einstein adopted Riemann's mathematical discoveries by giving them a precise physical interpretation. He showed, as we discussed in Chapter 3, that the curvature of spacetime embodies the gravitational force. But let's now think about this interpretation a little more closely. Mathematically, the curvature of spacetime—like the curvature of the trampoline—reflects the distorted distance relations between its points. Physically, the gravitational force felt by an object is a direct reflection of this distortion. In fact, by making the object smaller and smaller, the physics and the mathematics align ever more precisely as we get closer and closer to physically realizing the abstract mathematical concept of a point. But string theory limits how precisely Riemann's geometrical formalism can be realized by the physics of gravity, because there is a limit to how small we can make any object. Once you get down to strings, you can't go any further. The traditional notion of a point particle does not exist in string theory—an essential element in its ability to give us a quantum theory of gravity. This concretely shows us that Riemann's geometrical framework, which relies fundamentally upon distances between points, is modified on ultramicroscopic scales by string theory.

    This observation has a very small effect on ordinary macroscopic applications of general relativity. In cosmological studies, for example, physicists routinely model whole galaxies as if they are points, since their size, in relation to the whole of the universe, is extremely tiny. For this reason, implementing Riemann's geometrical framework in this crude manner proves to be a very accurate approximation, as evidenced by the success of general relativity in a cosmological context. But in the ultramicroscopic realm, the extended nature of the string ensures that Riemann's geometry simply will not be the right mathematical formalism. Instead, as we will now see, it must be replaced by the quantum geometry of string theory, leading to dramatically new and unexpected properties.