Defining the space in a general spacetime

Abstract

A global vector field v on a "spacetime" differentiable manifold V, of dimension N + 1, defines a congruence of world lines: The maximal integral curves of v, or orbits. The associated global space Nv is the set of these orbits. A "v-Adapted" chart on V is one for which the ℝN vector x = (xj) (j = 1,..,N) of the "spatial" coordinates remains constant on any orbit l. We consider non-vanishing vector fields v that have non-periodic orbits, each of which is a closed set. We prove transversality theorems relevant to such vector fields. Due to these results, it can be considered plausible that, for such a vector field, there exists in the neighborhood of any point X V a chart ? that is v-Adapted and "nice", i.e. such that the mapping l→x is injective-unless v has some "pathological" character. This leads us to define a notion of "normal" vector field. For any such vector field, the mappings build an atlas of charts, thus providing Nv with a canonical structure of differentiable manifold (when the topology defined on Nv is Hausdorff, for which we give a sufficient condition met in important physical situations). Previously, a local space manifold MF had been associated with any "reference frame" F, defined as an equivalence class of charts. We show that, if F is made of nice v-Adapted charts, MF is naturally identified with an open subset of the global space manifold Nv.