Defining the space in a general spacetime

Abstract

A global vector field $v$ on a "spacetime" differentiable manifold $\mathrm{V}$, of dimension $N+1$, defines a congruence of world lines: the maximal integral curves of $v$, or orbits. The associated global space $\mathrm{N}\_v$ is the set of these orbits. A "$v$-adapted" chart on $\mathrm{V}$ is one for which the $\mathbb{R}^N$ vector ${\bf x}\equiv (x^j)\ (j=1,...,N)$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$X\in \mathrm{V}$a chart$\chi $that is$v$-adapted and "nice", i.e., such that the mapping$\bar{\chi }: l\mapsto {\bf x}$is injective --- unless$v$has some "pathological" character. This leads us to define a notion of "normal" vector field. For any such vector field, the mappings$\bar{\chi }$build an atlas of charts, thus providing$\mathrm{N}\_v$with a canonical structure of differentiable manifold (when the topology defined on$\mathrm{N}\_v$is Hausdorff, for which we give a sufficient condition met in important physical situations). Previously, a local space manifold$\mathrm{M}\_\mathrm{F}$had been associated with any "reference frame"$\mathrm{F}$, defined as an equivalence class of charts. We show that, if$\mathrm{F}$is made of nice$v$-adapted charts,$\mathrm{M}\_\mathrm{F}$is naturally identified with an open subset of the global space manifold$\mathrm{N}\_v$.Comment: 38 pages. v3: version accepted for publication in Int. J. Geom. Meth. Mod. Phys.: stronger statements in Prop. 0 and Prop. 8, and precisions in the abstract, following from referee's suggestions; stronger form of Theorem 5; new example