Kaluza theory with zero-length extra dimensions

Abstract

A new approach to the Kaluza theory and its relation to the gauge theory is presented. Two degenerate metrics on the $4+d$-dimensional total manifold are used, one corresponding to the spacetime metric and giving the fiber of the gauge bundle, and the other one to the metric of the fiber and giving the horizontal bundle of the connection. When combined, the two metrics give the Kaluza metric and its generalization to the non-abelian case, justifying thus his choice. Considering the two metrics as fundamental rather than the Kaluza metric explains why Kaluza's theory should not be regarded as five-dimensional vacuum gravity. This approach suggests that the only evidence of extra dimensions is given by the existence of the gauge forces, explaining thus why other kinds of evidence are not available. In addition, because the degenerate metric corresponding to the spacetime metric vanishes along the extra dimensions, the lengths in the extra dimensions is zero, preventing us to directly probe them. Therefore this approach suggests that it is not justified to search for experimental evidence of the extra dimensions as if they are merely extra spacetime dimensions. On the other hand the new approach uses a very general formalism, which can be applied to known and new generalizations of the Kaluza theory aiming to achieve more and make different experimental predictions.