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Precise problem of objective metaphysics

The S4 picture even in its current form gives a clear recipe for what to expect vis-a-vis an objective metaphysics.  The universe has four macroscopic electromagnetic spatial dimensions; the physical universe at every moment in time is a hypersurface of $S^4(1/h)$ where electromagnetism propagates in all four dimensions and gravity restricts the physical universe to three space dimensions (which continually evolve within the ambient universe).  Electromagnetism is not at any time a $U(1)$ gauge theory but rather can be defined via a 1-form on the ambient space.  This means that it will be a defect of $U(1)$ gauge invariance that can allow us to attempt examining four space dimensions indirectly.

For an idea of how this can be done concretely, although in three dimensions we may not be able to observe four dimensional phenomena, the S4 picture would posit that at each point in the physical three dimensions, some information about the integrals along the normal circles might be available through observed electromagnetic field.  Thus we have a situation familiar from tomography such as Radon transform.  There is a useful inversion formula for the Radon transform that can give us desired information.  The answer is vague for the moment and pending a concrete theory including four dimensional electromagnetism.  Abstractly there is no trouble describing four dimensional electromagnetism.  One takes the potential $A = (A_0, A_1, A_2, A_3)$ and puts the $A_0$ component, i.e. the scalar electric potential in the normal direction to the physical hypersurface.

In terms of interpretation, the theory would suggest that although we cannot directly examine four spatial dimensions, we can still gain information about it by indirect means.  A more aggressive suggestion would be to examine whether human beings can or do have direct perception of four spatial dimensions.  Although the procedure sketched in the last paragraph are quite vague, experience from Radon transform in tomography can be used to model this problem.  The engineering required for this is an open problem and awaiting more concrete computations.  But here we have sketched something fundamentally new: a rational objective method for treating objective metaphysical reality.