Multi-Fibers Bundles as a new model for high-dimensional Spacetimes

Abstract

Around 1920, Kaluza and Klein had the idea to add a fifth dimension to the classical 4-dimensional spacetime of general relativity to create a geometric theory of gravitation and electromagnetism. Today, theoretical evidences, like string theory, suggest the need for a spacetime with more than five dimensions. The mathematical translation of the heuristic idea of a 4-dimensional classical spacetime equipped with extra "small" dimensions, is a fiber bundle structure π : M → M on a (4 + k)dimensional manifold M , with fiber a compact manifold F of dimension k, more shortly a F -fibration.Kaluza and Klein used a fibration with fiber the standard circle S 1 , this fiber carrying the electromagnetic potential. Inclusion of other physical interaction would therefore require a F -fibration with F of the form F = S 1 ×W , W being a compact manifold.However, with such a F -fibration, they is no naturally defined fiber diffeomorphic to S 1 at each point of the manifold, and therefore one looses the possibility to define simply the electromagnetic potential.We want to present in this paper a mathematical structure generalizing the fiber bundle, that enable the possible definition of multiple naturally defined fibers at each point of the manifold, on which therefore one can define objects that depend only on one of the components, S 1 or W , of the global (S 1 × W ) fiber.Although we do not pretend here to model precisely other known physical interactions, we present this geometric structure as a possible way to model or encode deviations from standard 4-dimensional General Relativity, or "dark" effects such as dark matter or energy ; (we refer to the authors' article [3] from which this paper is extracted, but whose purpose is different). Also this geometry was a starting point for the second author's new approach to a geometric unification of general relativity and quantum physics ( see [19]). 1