Free Hyperspace by Michio Kaku, Robert O'Keefe

more difficult than manipulating three-dimensional space. It is nothing short of amazing that on a plain sheet of paper, you can mathematically describe the properties of higher-dimensional objects that cannot be visualized by our brains.)

    Figure 2.1. The length of a diagonal of a cube is given by a three-dimensional version of the Pythagorean Theorem:
a 2 + b 2 + c 2 = d 2 .
By simply adding more terms to the Pythagorean Theorem, this equation easily generalizes to the diagonal of a hypercube in
N
dimensions. Thus although higher dimensions cannot be visualized, it is easy to represent
N
dimensions mathematically
.
     
    Riemann then generalized these equations for spaces of arbitrary dimension. These spaces can be either flat or curved. If flat, then the usual axioms of Euclid apply: The shortest distance between two points is a straight line, parallel lines never meet, and the sum of the interior angles of a triangle add to 180 degrees. But Riemann also found that surfaces can have “positive curvature,” as in the surface of a sphere, where parallel lines always meet and where the sum of the angles of a triangle can exceed 180 degrees. Surfaces can also have “negative curvature,”as in a saddle-shaped or a trumpet-shaped surface. On these surfaces, the sum of the interior angles of a triangle add to less than 180 degrees. Given a line and a point off that line, there are an infinite number of parallel lines one can draw through that point ( Figure 2.2 ).
    Riemann’s aim was to introduce a new object in mathematics that would enable him to describe all surfaces, no matter how complicated. This inevitably led him to reintroduce Faraday’s concept of the field.
    Faraday’s field, we recall, was like a farmer’s field, which occupies a region of two-dimensional space. Faraday’s field occupies a region of three-dimensional space; at any point in space, we assign a collection of numbers that describes the magnetic or electric force at that point. Riemann’s idea was to introduce a collection of numbers at every point in space that would describe how much it was bent or curved.
    For example, for an ordinary two-dimensional surface, Riemann introduced a collection of three numbers at every point that completely describe the bending of that surface. Riemann found that in four spatial dimensions, one needs a collection of ten numbers at each point to describe its properties. No matter how crumpled or distorted the space, this collection of ten numbers at each point is sufficient to encode all the information about that space. Let us label these ten numbers by the symbols g 11 , g 12 , g 13 ,…. (When analyzing a four-dimensional space, the lower index can range from one to four.) Then Riemann’s collection of ten numbers can be symmetrically arranged as in Figure 2.3 . 7 (It appears as though there are 16 components. However, g 12 = g 21 , g 13 = g 31 , and so on, so there are actually only ten independent components.) Today, this collection of numbers is called the Riemann
metric tensor
. Roughly speaking, the greater the value of the metric tensor, the greater the crumpling of the sheet. No matter how crumpled the sheet of paper, the metric tensor gives us a simple means of measuring its curvature at any point. If we flattened the crumpled sheet completely, then we would retrieve the formula of Pythagoras.
    Riemann’s metric tensor allowed him to erect a powerful apparatus for describing spaces of any dimension with arbitrary curvature. To his surprise, he found that all these spaces are well defined and self-consistent. Previously, it was thought that terrible contradictions would arise when investigating the forbidden world of higher dimensions. To his surprise, Riemann found none. In fact, it was almost trivial to extend his work to
N
-dimensional space. The metric tensor would now resemble the squares of a checker board that was N × N in size. This will have profound physical implications when we discuss the