The paper is in the following UniverseIsS4
An updated version with match of the S4 model of redshift to actual Caltech data is here.UniverseIsS4
We now know that the redshift is an artifact of treating waves on a four-sphere as though they were waves in a flat space. Thus the fundamental support of Big Bang no longer exists. But a long rivalry between Big Bang and the Steady State cosmology of Hoyle, Gold and Bondi ended with the ascendancy of Big Bang because of the distribution of radio galaxies and quasars.
I will suggest that this distribution is affected by the same misunderstanding that produced expansionary models of the universe. The ‘discrepancy’ function for this mistakes puts more weight to higher wavelengths than lower. The discrepancy itself has the form f(x) = 1/x – 1/sqrt(x(x+3)). It is thus interesting to see what happens when we take the radio sources and correct their redshift. Part of the problem stems from people using the ‘Hubble law’ to determine distances.
Modern theory of electromagnetism is described as a U(1) gauge theory, which is an extremely general construction for any background. But I believe that the fundamental reason why electromagnetism is possible is that the universe has exactly four macroscopic spatial dimensions and corresponds to the fact that the double cover of SO(4), Spin(4) decomposes as SU(2) x SU(2).
Taking a step back, if the Maxwell system in three dimensions is described by curl E – ik H = 0 and curl H + ik E = 0. One defines the D = d + d*, and D_k = D + k e_4, and u = H – i e_4 E. Then the Maxwell equations are D_k u = 0 when E = E1 e1 + E2 e2 + E3 e3 and H = H1 e2 e3 + H2 e1 e3 + H3 e1 e2. Here e1, e2, e3, e4 are Clifford algebra generators. Thus the Maxwell’s equations are described in terms of the Dirac operator.
One can use the above equivalence construction to understand electromagnetism on a four-manifold where the splitting of Spin(4) divides electric and magnetic fields as components of a spinor.
First, recall the S4 theory position regarding the ‘gravitational’ field equations, that they describe the instantaneous coordinates of a three dimensional submanifold of a four dimensional scaled sphere. We can identify the ambient sphere as the metaphysical universe. The field equations then are simply the Ricci curvature equation for a submanifold with two restrictions of and that the divergence of T vanishes.
Second, we identify will to power, the Nietzschean concept for the fundamental principle of life, to always be accompanied by a certain sort of metaphysical movement. Abstractly, we are assuming that regardless of how material world manifestation of will to power occurs for human beings, they are always accompanied by a change in the metaphysical environment where the agent expands the metaphysical reach or clears metaphysical paths in four dimensions.
This is an intuitive sketch of what physics of metaphysical travel might look like. The Einstein gravitational field equations tell us how the physical universe, which is a three-dimensional submanifold of a four dimensional sphere of fixed radius (so the cosmological constant is h^2 or a constant multiple). Now metaphysical travel cannot break any laws of the correct four dimensional physical laws of the universe. It is definitely possible that a galaxy that is far distant in three dimensional distance is not so distant in the four dimensional sphere. Now metaphysical travel amounts to having conscious presence in some part of the physical universe without having physical presence. Metaphysical travel works often by taking advantage of flow of energy in some direction caused by entities and objects to ‘expand attention’ in particular directions. So it is analogous to sailing. Thus the physical body does not move but consciousness does. In effect we are literally ‘folding space’ as we travel metaphysically.
Twentieth century physics has expended enormous effort on the issue of the wave-particle duality of substances. In the longer view, we see light beginning with Newton’s corpuscles to the wave theories of nineteenth century return to the photon particle of Einstein’s 1905 paper. Now the central modern focus in physics had been on the theories of particles and their interactions leaving the background physical space as a variable. This explains why effort had been expended on conformal field theory in two dimensions on Riemann surfaces.
I want to point out that on stationary sphere models of the universe, both wave and particle characteristics are inherent features of all objects. This is because on spheres the spherical harmonics provide decomposition of arbitrary objects in terms of waves, regardless of their spatial localization.
Suppose M is the three dimensional physical universe, and for physical reasons solutions must exist to the Dirac-Einstein equation which are eigenspinors of the Dirac operator which also satisfy an Einstein equation with an energy-momentum tensor. Take the solutions and check that the energy momentum tensor satisfies a Codazzi equation. Then apply the Lawn-Roth theorem that tells us that these can be used to isometrically immerse M in an S4 of appropriate radius.