The Apophenion: A Chaos Magick Paradigm - Peter J. Carroll 2008

Hypersphere from Radius Excess
Appendix

Positively curved space has the strange property of having a greater internal radius than an observer would suspect from looking at it from the outside. Thus in a sense a massive object has more space inside it than its outward appearance suggests, rather like those Tardis vehicles of the mythical Time Lords.

To visualise how this can happen, consider a curved space of just two dimensions like the surface of the earth. A small circle drawn on the surface will have a radius r, given by the Euclidian formula

r = where C equals the circumference.

However a vast circle drawn on the surface of the earth will have a radius longer than this because it will have to go over the hump created by the curvature of the earth.

A circle around the equator will have a radius of a quarter of the entire circumference.

Now the three dimensional version of curved space does not submit to easy visualisation but a hypersphere or 3-sphere has a similar property an ordinary sphere or 2-sphere. Whereas a 2-sphere has a diameter equal to half of its circumference (in 2-dimensional terms), a 3-sphere also has a diameter equal to half of its circumference (in 3-dimensional terms). This occurs because in 2-dimensional terms we have to measure over the curvature of the earth, and in 3-dimensional terms we have to measure over the curvature of space. This arises because the 2-sphere surface lies embedded in 3-dimensional space, and the 3-sphere lies embedded in 4-dimensional space.

Now Schwarzschild derived a formula from the equations of General Relativity that shows how the mass of any object curves space and leads to a radius excess inside of it. The radius excess depends only on the mass m, of the object and takes the form

Radius excess =

Where G = the gravitational constant, and where c = lightspeed.

The earth incidentally has a radius excess of only about 1.5 mm, whilst the much more massive sun has a radius excess of about 0.5 km.

The phenomenon of radius excess allows a cheeky little proof that at some state of density, a sphere must become a hypersphere as its radius excess increases its diameter to half of the circumference and beyond.

In the following proof, C = circumference, to which we add radius excess to see at what ratio of mass to diameter, the diameter becomes half of the circumference.

Thus only has to exceed about 85% ofto achieve hyperspherical geometry and topology, and in the H6D model of the universe, equals 100% of if we equate L, antipode distance, with d, diameter.

Thus it seems unlikely that spacetime singularities can feature in the universe, either as an initial condition or as the result of gravitational collapse, because hyperspheres will form instead.

Appendix iii shows that hyperspheres naturally vorticitate, thus preventing further collapse and creating three-dimensional time.