Observer congruences, their observer curves and adapted charts

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) and let $U\subseteq M$ be a nonempty open subset, carrying the restricted smooth structure of An open subset of a smooth manifold has a canonical restricted smooth structure.

(a) Congruences. An observer congruence on $U$ is a smooth vector field $n$ on $U$ (A smooth vector field is a smooth section of the tangent bundle) such that $n_q$ is an observer at $q$ for every $q\in U$ (Observers, rest spaces and the spatial projection at an event); explicitly,

$g_q(n_q,n_q)=-1\text{and}{g}_q(n_q,N_q)<0\text{for every }q\in U\mathrm{.}$

Since $g_q(n_q,n_q)=-1$, the field $n$ is nowhere vanishing. A single observer at a point is therefore not a congruence, and a congruence is not an ordered basis or frame of $T_{q}M$.

(b) Observer curves. Because $n$ is a smooth vector field on $U$, the theorem on maximal integral curves (Through each point there is a unique maximal integral curve, Integral curves of a vector field) assigns to every $q\in U$ a unique maximal integral curve ${\gamma }_q:I_q\to U$ of $n$ with ${\gamma }_q(0)=q$, defined on an open interval $I_q\subseteq \mathbb{R}$. We call these curves the observers (or observer curves) of the congruence. Each curve is regular: for every $s\in I_q$, ${\gamma }_q^{\mathrm{\prime }}(s)=n_{{\gamma }_q(s)}$ and

$g_{{\gamma }_q(s)}(n_{{\gamma }_q(s)},n_{{\gamma }_q(s)})=-1,$

so its tangent is nonzero, unit timelike and future-directed. Completeness of the maximal intervals $I_q$, and global product or foliation structure of $U$, are not asserted. The parameter $s$ is the flow parameter of $n$; it is not identified with any clock reading here.

(c) Adapted charts. An $n$-adapted chart is a smooth chart $(u^0,u^1,u^2,u^3):U^{\mathrm{\prime }}\to {\mathbb{R}}^4$ of $M$ whose domain $U^{\mathrm{\prime }}$ is a nonempty open subset of $U$ and whose spatial coordinates $u^1,u^2,u^3$ are constant along every integral curve $\gamma :J\to U^{\mathrm{\prime }}$ of the restricted field $n{\mathrm{\mid }}_{U^{\mathrm{\prime }}}$, where $J$ is an interval. Equivalently, for each maximal observer curve ${\gamma }_q:I_q\to U$, each connected component $K$ of ${\gamma }_q^{-1}(U^{\mathrm{\prime }})$ (a maximal nonempty interval contained in this open subset of $\mathbb{R}$), and $i=1,2,3$, the function $u^i\circ {\gamma }_q$ is constant on $K$. If $q\in U^{\mathrm{\prime }}$ and $0\in K$, this constant is $u^i(q)$. This condition does not identify labels on different components when a maximal curve leaves and re-enters $U^{\mathrm{\prime }}$. No global adapted chart, and no adapted chart on all of $U$, is asserted by this definition; one exists near every point of $U$ by the existence corollary proved on this page.

(d) Scope. A congruence is a field of observer directions on an open set; it is not a chart, not a tetrad field, and not a family of parametrized worldlines with specified domains. The physical reading of its observer curves as ideal-clock worldlines, and the identification of the flow parameter with a measurable time, belong to the physical postulates of the framework and are not asserted here.

Choice convention. The smooth structure used for the vector field $n$ is the tangent-bundle structure of A smooth vector field is a smooth section of the tangent bundle, which assumes the axiom ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$); this definition inherits that assumption and uses no further choice principle. The restricted structure on $U$ is the canonical one of the cited proposition.