Aspects of seven-dimensional relativity

Abstract

Two versions of a Kaluza-Klein model of seven-dimensional relativity are discussed. These models are characterized by having a fixed geometry for the internal parts of their manifolds. In version (1) the internal part is flat ($T_3$) and in version (2) it is curved ($S_3$). The physical interpretation of the internal coordinates is given in both versions. The main differences are that in version (2) the model features a cosmological constant given by the curvature constant of the sphere and the gauge fields depend in a definite way on the internal coordinates [they do not in version (1)]. These gauge fields appear as part of the metric tensor in seven dimensions. The result by Kerner and Cho which states that seven-dimensional relativity contains as a special case four-dimensional gravity coupled to Yang-Mills fields is rederived. The Dirac Lagrangian is given for both versions. It is defined to be the free-field Lagrangian in seven dimensions, i.e., it contains spinors coupled to seven-dimensional "gravity" only. Again as a special case it contains a gauge-invariant Lagrangian featuring a minimal coupling and a Fierz-Pauli term. The latter can be eliminated by choosing a particular way for the dimensional reduction procedure. Spinors carry an internal degree of freedom originating in the use of higher dimensions. For both versions this internal degree of freedom may be identified with the gauge degree of freedom. For version (1), scalar fields are also discussed and some restrictions concerning the inclusion of higher groups are given.