The geometry of space–time–matter

Abstract

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 E is a fiber bundle over M (the spacetime) and g¯ is a Kaluza–Klein metric on E. Each space–time–matter manifold (E,g¯) generates spacetimes (M,g˜), one for each embedding of M in E, with stress–energy tensor for M determined by the geometry of E and the nature of the embedding.The use of a fiber bundle E (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 VE of TE and when the spacetime M is embedded horizontally, the gauge fields F vanish when restricted to M. Thus, the fiber bundle approach clarifies how F=0 arises from the geometry rather than from the usual assumptions on the metric in traditional space–time–matter theory.