The geometry of space–time–matter

We formulate a global, differential geometric structure for the space–time–matter theory introduced by Wesson and coworkers. In addition to giving a coordinate-free, intrinsic approach to the theory, we extend the discussion from 5-dimensions to arbitrary dimensions.Our model for space–time–matter is a Ricci flat, semi-Riemannian manifold (E,g¯), where EE is a fiber bundle over MM (the spacetime) and g¯ is a Kaluza–Klein metric on EE. Each space–time–matter manifold (E,g¯) generates spacetimes (M,g˜), one for each embedding of MM in EE, with stress–energy tensor for MM determined by the geometry of EE and the nature of the embedding.The use of a fiber bundle EE (with fibers isomorphic to the gauge groups) affords a natural way of incorporating the gauge-field potentials into the metric g¯ (perhaps the only global way to do so). The gauge field potentials determine a horizontal bundle orthogonal to the vertical bundle VEVE of TETE and when the spacetime MM is embedded horizontally, the gauge fields FF vanish when restricted to MM. Thus, the fiber bundle approach clarifies how F=0F=0 arises from the geometry rather than from the usual assumptions on the metric in traditional space–time–matter theory.