Defining the space(s) in a general spacetime

Mayeul Arminjon (1)

1 3SR - Laboratoire sols, solides, structures - risques [Grenoble]

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 N_v is the set of these orbits. A "v-adapted" chart on V is one for which the R^N vector x ≡ (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 show by transversality arguments that, among those vector fields, those for which there exists in the neighborhood of any point X ∈ V a chart that is v-adapted and "nice", i.e., such that the mapping ¯ χ : l → x is injective, have a generic character. For any "normal" vector field, the mappings ¯ χ build an atlas of charts, thus providing N_v with a canonical structure of differentiable manifold. Previously, a local space manifold M_F 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, M_F is naturally identified with an open subset of the global space manifold N_v .

Mots clés: Physical space; global vector field; orbit space; adapted chart; differentiable manifold

Domaines: Topologie géométrique [math.GT]; Systèmes dynamiques [math.DS]; Relativité Générale et Cosmologie Quantique [gr-qc]

Mayeul Arminjon. Defining the space(s) in a general spacetime. 2014. ⟨hal-01093350v1⟩