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1.1.2 Curved Volume and the Hyper-Sphere

A circle may be described, geometrically, as a closed, symmetrically curved, line. Similarly, a sphere may be described as a closed, symmetrically curved, surface, and by extrapolation, a hyper-sphere is a closed, symmetrically curved volume. While circles and spheres are easily understood, the difficulty in visualizing the universe as an energy continuum shaped like a hyper-sphere is due entirely to our inability to even imagine curved volume. The reason for this is simply that human vision evolved on planet Earth where curvature of volume is too small to affect our senses. However, many of the important properties of a hyper-sphere also exist with an ordinary sphere in two dimensional form and in circles in one dimension.

The concept that the universe is shaped like a hyper-sphere was first introduced by Einstein in his 1917 paper. He wrote - "We start from a Euclidean space of four dimensions, \(\xi_1, \xi_2, \xi_3, \xi_4\),... In this space we consider the hyper-surface, \(R^2 = \xi_1^2+\xi_2^2+\xi_3^2+\xi_4^2\), where \(R\) denotes a constant. The points of this hyper-surface form a three dimensional continuum, a spherical space of radius of curvature R."

”Einstein’s definition of a three dimensional continuum views it as a subset of points in four dimensional space so that space itself is ’spherical’. In the following, it is the volume of an energy continuum that is spherical, even though it may take a four dimensional coordinate system to represent it.

It is important not to confuse the curved volume of a hyper-sphere with the volume of a ball, or curved surface area of a sphere with the area of a disc. The volume of a ball is not curved and neither is the surface area of a disc.

Since curved volume is impossible to visualize, the strategy used in this work is to describe the universe in terms of an expanding circle or an ordinary sphere. This allows us to think in terms of curved lines and surfaces which we can easily visualize, rather than the unimaginable shape of curved volume. It will only be necessary to take into account the extra dimension of a hyper-sphere when its curved volume is significant.

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