Mathematically the difference between any S4 theory, by which we mean any physics where the background is the de Sitter space with fixed radius, and general relativity is small. Whereas the classical theory has an Einstein-Hilbert action on a fixed spacetime with a Lorentzian metric in any S4 theory an Einstein-Hilbert action of similar form with similar Euler-Lagrange equations (field equations) is taken over hypersurfaces of but automatically with the cosmological constant . Therefore the empirical basis of general relativity can serve as one for any S4 theory. But here we have available a clear extrinsic geometry for the hypersurface so that we can identify mass of fermions (spinor fields) in the physical universe with mean curvature of the hypersurface by the Lawn-Roth theorem and constructions. In fact, we can even understand the equivalence of the stress-energy tensor with the second fundamental form of an isometric immersion. But one of the real merits of S4 theories is that quantization of energy and other quantum phenomena are tied to S4 geometry such as the exact spectrum of the Dirac operator. In particular in S4 theories there is a unity between quantum phenomena, gravity, electromagnetism and particle theories (classical Yang-Mills theories well studied for 4-sphere). The mass gap problem is resolved in S4 theories relatively easily and there cannot exist massless fermions. There is no need for a theory of quantum gravity here as there is a geometric unity of gravity, gauge forces. Further thought shows that in fact electromagnetism is not really a gauge theory in the S4 setting where a peculiar new geometric phenomenon serves to understand electromagnetism: for any hypersurface , the normal geodesic at every point is a perfect circle of length . The natural conjecture is that it is these that could be mistaken for fibers of a circle-bundle and lead to a -gauge theory for electromagnetism while something else is happening in reality.

## Difference of any S4 physics from classical general relativity

May 5, 2014 by zulfahmed

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