Defining the space in a general spacetime

Abstract

A global vector field [Formula: see text] on a “spacetime” differentiable manifold [Formula: see text], of dimension [Formula: see text], defines a congruence of world lines: the maximal integral curves of [Formula: see text], or orbits. The associated global space [Formula: see text] is the set of these orbits. A “[Formula: see text]-adapted” chart on [Formula: see text] is one for which the [Formula: see text] vector [Formula: see text] of the “spatial” coordinates remains constant on any orbit [Formula: see text]. We consider non-vanishing vector fields [Formula: see text] 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 [Formula: see text] a chart [Formula: see text] that is [Formula: see text]-adapted and “nice”, i.e. such that the mapping [Formula: see text] is injective — unless [Formula: see text] has some “pathological” character. This leads us to define a notion of “normal” vector field. For any such vector field, the mappings [Formula: see text] build an atlas of charts, thus providing [Formula: see text] with a canonical structure of differentiable manifold (when the topology defined on [Formula: see text] is Hausdorff, for which we give a sufficient condition met in important physical situations). Previously, a local space manifold [Formula: see text] had been associated with any “reference frame” [Formula: see text], defined as an equivalence class of charts. We show that, if [Formula: see text] is made of nice [Formula: see text]-adapted charts, [Formula: see text] is naturally identified with an open subset of the global space manifold [Formula: see text].