In LawnRothIsometricImmersionS4Spinors Lawn and Roth gave a sufficient condition for a three manifold to be isometrically immersed into a scaled 4-sphere. Their results are slightly more general but we are interested only in the case of isometric immersion into a 4-sphere. The condition is that if there exist two spinor fields and that satisfy the Dirac equations

and

where can be a function on the manifold rather than simply a constant, then the three-manifold can be isometrically embedded in a 4-sphere of curvature in such a way that will appear as the mean curvature of the embedding.

This theorem can be used to show that the physical universe can be isometrically immersed in S4(1/h) is there exists in the physical universe a massive fermion and it antiparticle. In general it is useful to analyze the static eigenspinors noting that the time-dependent Dirac equation will be satisfied by multiplying these eigenspinors by just as is usually done to solve wave equations using Laplace eigenfunctions. So if we assume the physical universe is a compact three-dimensional manifold and that a massive fermion and its antiparticle exists, then we have eigenspinors with eigenvalue , the mass and its antiparticle of opposite signed mass. Since energy is quantized in units of we can assume and we see that we have fulfilled the Lawn-Roth necessary conditions with and therefore can be isometrically immersed in S4(1/h) with mean curvature .

The above application requires the assumption that the physical universe is compact, which is satisfied for our universe by evidence I had presented before using the CBR uniform lower bound and therefore is physically meaningful. This application of the Lawn-Roth theorem shows that S4 physics is not synthetic; there are rigorous ways of obtaining the 4-sphere ambient universe from physical phenomena even with the assumption of a three-dimensional compact universe. This application of the Lawn-Roth theorem only provides a mathematical 4-sphere construction but we can look at other evidence to argue that in fact the actual universe has four macroscopic spatial dimensions.

## Leave a Reply