We will refer to any physical theory with a fixed four dimensional background a 4-sphere with radius as an S4 physics. In particular we leave open possible particle theories to ‘a gauge theory with compact symmetry group . We refer to hypersurfaces of as an ‘S4-physical universe’ or when ambiguity does not arise simply as a ‘physical universe’. These can arise as critical points of an Einstein-Hilbert action where ranges over smooth connected hypersurfaces of . Therefore, all S4 physics models restrict the geometry of possible physical universes. When time is added to these theories, we consider a constant space-independent time.

Justifications for S4 physics can be given directly empirically (where it can be shown that redshift slope is perfectly explained in a static S4 universe) or assuming (or using empirical evidence for) that the three-dimensional universe is compact and contains a pair of massive fermions satisfying and where is Planck constant and is a function so that the manifold can be isometrically immersed in with the mean curvature by a theorem of Lawn-Roth. Therefore restricting the three dimensional universe models to the hypersurfaces of the static is not a significant physical restriction.

Since the Euler-Lagrange equations of the Einstein-Hilbert action in this case is the same as the classical Einstein case, the gravitational field equations are not different and in principle explains the main empirical basis for relativity: the gravitational deflection of light and the mercury perihelion. But we can see immediately that so long as we can describe quantum phenomena from the geometry of a 4-sphere, we have perfect coherence of gravity and quantum phenomena without invoking a quantum field theory. For a three dimensional physical manifold we have discrete spectrum for the Dirac operator which represents the possible masses of fermions in that physical universe. Positive scalar curvature of the ambient manifold disallows massless fermions which in principle could be violated (massless fermions could exist) for a hypersurface but this is not the case whenever the length of the second fundamental form is sufficiently small. In that situation the work on the Chern conjecture shows rigidity of minimal hypersurfaces to Clifford tori and the equator both with positive scalar curvature. It is likely that harmonic spinors, i.e. massless fermions cannot exist for any physically realizable hypersurface.

Thus S4 physics could be thought of as a class of models of physics (gravity and gauge forces) that have empirical support but are much simpler than the efforts for ‘quantum gravity’. If the S4 picture is right, it is evident that ‘quantizing gravity’ is not a useful project as quantum phenomena are not merely ‘subatomic’ phenomana but by the bridge of universal radius they are always inherently global phenomena while gravity is inherently three dimensional while quantum phenomena are inherently four dimensional.

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