Free Hyperspace by Michio Kaku, Robert O'Keefe Page A

unification of all forces in the next several chapters.

    Figure 2.2. A plane has zero curvature. In Euclidean geometry, the interior angles of a triangle sum to 180 degrees, and parallel lines never meet. In non-Euclidean geometry, a sphere has positive curvature. A triangle’s interior angles sum to greater than 180 degrees and parallel lines always meet. (Parallel lines include arcs whose centers coincide with the center of the sphere. This rules out latitudinal lines.) A saddle has negative curvature. The interior angles sum to less than 180 degrees. There are an infinite number of lines parallel to a given line that go through a fixed point
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    Figure 2.3. Riemann’s metric tensor contains all the information necessary to describe mathematically a curved space in
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dimensions. It takes 16 numbers to describe the metric tensor for each point in four-dimensional space. These numbers can be arranged in a square array (six of these numbers are actually redundant; so the metric tensor has ten independent numbers)
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    (The secret of unification, we will see, lies in expanding Riemann’s metric to
N-
dimensional space and then chopping it up into rectangular pieces. Each rectangular piece corresponds to a different force. In this way, we can describe the various forces of nature by slotting them into the metric tensor like pieces of a puzzle. This is the mathematical expression of the principle that higher-dimensional space unifies the laws of nature, that there is “enough room” to unite them in
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-dimensional space. More precisely, there is “enough room” in Riemann’s metric to unite the forces of nature.)
    Riemann anticipated another development in physics; he was one of the first to discuss multiply connected spaces, or wormholes. To visualize this concept, take two sheets of paper and place one on top of the other. Make a short cut on each sheet with scissors. Then glue the two sheets together along the two cuts ( Figure 2.4 ). (This is topologically the same as Figure 1.1 , except that the neck of the wormhole has length zero.)
    If a bug lives on the top sheet, he may one day accidentally walk into the cut and find himself on the bottom sheet. He will be puzzled because everything is in the wrong place. After much experimentation, the bug will find that he can re-emerge in his usual world by re-entering the cut. If he walks around the cut, then his world looks normal; but when he tries to take a short-cut through the cut, he has a problem.

    Figure 2.4. Riemann ’s cut, with two sheets are connected together along a line. If we walk around the cut, we stay within the same space. But if we walk through the cut, we pass from one sheet to the next. This is a multiply connected surface
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    Riemann’s cuts are an example of a wormhole (except that it has zero length) connecting two spaces. Riemann’s cuts were used with great effect by the mathematician Lewis Carroll in his book
Through the Looking-Glass
. Riemann’s cut, connecting England with Wonderland, is the looking glass. Today, Riemann’s cuts survive in two forms. First, they are cited in every graduate mathematics course in the world when applied to the theory of electrostatics or conformal mapping. Second, Riemann’s cuts can be found in episodes of “The Twilight Zone.” (It should be stressed that Riemann himself did not view his cuts as a mode of travel between universes.)
Riemann’s Legacy
     
    Riemann persisted with his work in physics. In 1858, he even announced that he had finally succeeded in a unified description of light and electricity. He wrote, “I am fully convinced that my theory is the correct one, and that in a few years it will be recognized as such.” 8 Although his metric tensor gave him a powerful way to describe any curved space in any dimension, he did not know the precise equations that the metric tensor obeyed; that is, he did not know what made the sheet crumple.
    Unfortunately, Riemann’s efforts to solve