Nonlinear connections and 1+3 splittings of spacetime

Abstract

The analogy between 1+3 splittings of the spacetime tangent bundle and the splitting of the tangent bundle to the bundle of linear frames into vertical and horizontal sub-bundles is described from the unifying standpoint of the geometry of foliations. The physical nature of the line field on spacetime that plays the role of vertical sub-bundle is discussed in some detail. The notion that the complementary spatial bundle is most fundamentally a representation of the normal bundle to the foliation of spacetime by the integral curves of the line field is proposed, such that the geometry of space becomes the transverse geometry of the flow and the integration of the spatial bundle into proper time simultaneity hypersurfaces becomes a secondary issue to the geometry of space as the geometry of the leaf space of the foliation. The concept of an adapted convected frame field is introduced as a means of locally representing the transversal geometry of a flow in terms of the information that is contained in the flow and its derivatives, and some discussion is given to the role of the Bott connection.