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 in V a chart chi that is v-adapted and "nice", i.e., such that the mapping bar chi: l |-> x is injective, have a generic 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 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; reference fluid; 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-01093350v2⟩