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HOW TO EXTEND UNIFICATION OF ELECTROMAGNETISM AND GRAVITY TO S4 UNIVERSE

One can extract from Jonah Miller’s Kaluza-Klein paper (jonah-miller-kaluza-klein-theory) the following result:

If the metric for a four (space) dimensional space can be written as:

$ds^2 = e^{2 \alpha \phi} g_{\mu\nu} dx^\mu dx^\nu + e^{2 \beta\phi}(dx^4 + A_{\mu} dx^{\mu})^2$

then the components of the curvature 2-form are:

$R^{zz} = \beta^2 e^{-2\alpha\phi} \box \phi + \frac{1}{4}e^{2(\beta - 2\alpha)\phi}F^2$

and other terms and the scalar curvature in the snapshot:

Now we can consider this a generic theorem for four space-dimensional manifolds and specialize to four-sphere hypersurfaces with the following special $A_\mu dx^\mu$: we take the second fundamental form 3×3 matrix and diagonalize it, setting $A_{\mu} = h_{\mu\mu}$.  Miller’s arguments for unification of electromagnetism and gravitation is unchanged given that the vector potential is given in this special form.