The Apophenion: A Chaos Magick Paradigm - Peter J. Carroll 2008
The Shape of the Universe
Appendix
If you live in a hyperspherical universe with a positive space-time curvature but you assume that you live in a flat universe instead, then you will run into strange problems. You will basically end up with worse versions of the problems of horizons and edges that arise if you persist in believing in a flat earth.
A non-infinite universe must have a definite shape and size, but the finite and unbounded hypersphere or 3-sphere which the universe probably consists of does not easily submit to visualisation unless we remove one of the spatial dimensions for illustrative purposes.
The polar type projection mentioned in chapter 6 results from cutting the hypersphere into two hemi-hyperspheres which we can represent as spheres shown by circles in Figure 1.
These two circles represent spheres whose perimeters contact each other at every point on their surfaces. We can imagine this by allowing the spheres to roll freely around each other.
Position A represents an observer in a hypersphere where we have chosen slice it into two hemi-hyperspheres to position the observer in the centre of one of them. We could have cut it anywhere for illustrative purposes, a hypersphere contains no special positions in reality.
Now an observer at position A can set off in any direction and eventually reach position B, an antipode point which represents the furthest distance you can travel from A without starting to return towards it. All straight-line routes from A lead to B, in much the same way that all straight line trips from the North Pole of the earth lead to the South Pole. See figure 2.
In a hypersphere a straight line route, the shortest distance between two points in 3-dimensional space, has to follow the gravitationally induced curvature of the universe itself. Light also has to follow such routes, which we call geodesics.
Now we always construct an image as though light had travelled to us in a straight line. A lens or mirror actually bends the path of light, but because we construct images on the basis of the direction in which light approaches us, objects appear magnified or diminished by lenses or repositioned by mirrors.
When we look out into the cosmos we assume that light has come towards us in straight lines and that the apparent position of objects represents their actual positions.
This works reasonably well for short distances but at cosmic distances the curvature of space-time itself acts like a gigantic lens.
If we assume flat un-curved space then we can represent that by un-rolling the whole of one of the hemi-hyperspheres around the other. See figure 3. Here the antipode point of an observer at A has become spread out right round the horizon. This corresponds to the South Pole of the earth lying in every possible direction from the North Pole. If this planet had such an enormous density that it bent the paths of light around its surface, we would see something like this.
Figure 4 shows what happens to lines of sight in a hypersphere, they curve inwards towards the halfway to antipode distance, and then diverge towards the antipode, from the perspective of an observer who assumes flat space.
Thus, as Figure 5 shows, objects around the halfway to antipode distance will appear magnified whilst objects further away than that will appear diminished, because observers assume that they see in straight lines in un-curved space.
Now light travelling down those geodesics towards an observer will become redshifted to lower energies, and if the observer assumes a flat spacetime, this redshift will become interpreted as an expansion of the universe. However because hypespherical spacetime acts as a giant lens, the observer will notice a mismatch between the apparent magnitudes of objects at various distances and their apparent recession velocities calculated from redshift. High redshift objects will appear fainter, and thus apparently further away than they ought to. Thus our befuddled observer may conclude that not only does the universe expand, but that its expansion rate has speeded up during the expansion.
Of course neither of these things has actually occurred. It just looks like that because we inhabit a finite and unbounded universe of constant size whose curvature distorts what we can see.