Warped Passages Read Online Free PDF

a meter stick we all agree on.
    But that is not all a metric specifies. It also tells us whether space bends or curls around, like the surface of a balloon when it is blown up into a sphere. The metric contains all the information about the shape of space. A metric for curved space tells us about both distances and angles. Just as an inch can represent different distances, an angle can correspond to different shapes. I’ll go into this later on when we explore the connection between curved space and gravity. For now, let’s just say that the surface of a sphere is not the same as the surface of a flat piece of paper. Triangles on one don’t look like triangles on the other, and the difference between these two-dimensional spaces can be seen in their metrics. 2
    As physics has evolved, so has the amount of information stored in the metric. When Einstein developed relativity, he recognized that a fourth dimension—time—is inseparable from the three dimensions of space. Time, too, needs a scale, so Einstein formulated gravity by using a metric for four-dimensional spacetime , adding the dimension of time to the three dimensions of space.
    And more recent developments have shown that additional spatial dimensions might also exist. In that case, the true spacetime metric will involve more than three dimensions of space. The number of dimensions and the metric for those dimensions is how one describes such a multidimensional space. But before we explore metrics and metrics for multidimensional spaces any further, let’s think more about the meaning of the term “multidimensional space.”
    Playful Passages Through Extra Dimensions
    In Roald Dahl’s Charlie and the Chocolate Factory , Willy Wonka introduced visitors to his “Wonkavator.” In his words, “An elevator can only go up and down, but a Wonkavator goes sideways and slantways and longways and backways and frontways and squarewaysand any other ways that you can think of…” * Really, what he had was a device that moved in any direction, so long as it was a direction in the three dimensions we know. It was a nice, imaginative idea.
    However, the Wonkavator didn’t really go any way “you can think of.” Willy Wonka was remiss in that he neglected extra-dimensional passages. Extra dimensions are other directions entirely. They are hard to describe, but they may be easier to understand by analogy.
    In 1884, to explain the notion of extra dimensions, the English mathematician Edwin A. Abbott wrote a novel called Flatland . † It takes place in a fictitious two-dimensional universe—the Flatland of the title—where two-dimensional beings (of various geometric shapes) reside. Abbott shows us why Flatlanders, who live their whole lives in two dimensions—on a table top, for example—are as mystified by three dimensions as people in our world are by the idea of four.
    For us, more than three dimensions requires a stretch of the imagination, but in Flatland three dimensions are beyond its inhabitants’ comprehension. Everyone thinks it is obvious that the universe holds no more than their two perceived dimensions. Flatlanders are as insistent about this as most people here are about three.
    The book’s narrator, A. Square (the namesake of the author, Edwin A 2 ), is introduced to the reality of a third dimension. In the first stage of his education, while he is still confined to Flatland, he watches a three-dimensional sphere travel vertically through his two-dimensional world. Because A. Square is confined to Flatland, he sees a series of disks that increase and then decrease in size, which are slices of the sphere as it passes through A. Square’s plane (see Figure 6).
    This is initially perplexing to the two-dimensional narrator, who has never imagined more than two dimensions and has never contemplated a three-dimensional object like a sphere. It is not until A. Square has been lifted out of Flatland into the surrounding three-dimensional world that he can