Geometry of Kaluza-Klein theory. I. Basic setting

Kaluza-Klein space theory is derived from the hypothesis that the four-dimensional space-time is locally and isometrically embedded in a high-dimensional space which presumably originated at the big bang. For mathematical simplicity the high-dimensional space is taken to be a flat, Minkowski space with 14 dimensions assumed to be the ground state of the theory. The resulting metric is more general than the usual zero-mode metric ansatz but it reduces to the latter in the low-energy sector of the theory. The compactification of the internal space results from the existence of the second quadratic form of the embedded V₄. A simple model of spherical compact space is considered as a working example, where the spontaneous compactification is a hyperbolic function of the strength of the gravitational field. The symmetry group of the embedding is a combined symmetry which breaks into P₄×SO(10) in the flat limit of the space-time.