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 $χ$ that is $v$-adapted and "nice", i.e., such that the mapping $\bar{χ}: l\mapsto {\bf x}$ 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 $\bar{χ}$ 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 $\mathrm{N}\_v$.