Charts, frames, tetrads, congruences and observers are distinct objects

This page introduces several objects that are related but not interchangeable; the distinctions below are part of their definitions and are used throughout the general-relativity track.

• Observer. An observer at $p$ is one future-directed unit timelike vector $n\in T_{p}M$, $g_p(n,n)=-1$ (Observers, rest spaces and the spatial projection at an event). It carries no parametrized curve and no clock reading by itself.
• Congruence. An observer congruence on an open set $U$ is a smooth field $n$ whose value at every point is an observer; its maximal integral curves are its observer curves, and its flow parameter is not a physical time (Observer congruences, their observer curves and adapted charts).
• Frame and coordinate frame. A local frame field is a tuple of four pointwise independent smooth vector fields; the coordinate frame of a smooth chart is the special case of the fields $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^a$, whose members pairwise commute. A frame whose members do not commute is therefore not the coordinate frame of any chart (Local frame fields, coordinate frames and tetrad fields).
• Tetrad. A tetrad field is a pointwise $g$-orthonormal frame field; it is future-directed when its first member is future-directed, and it is adapted to a congruence $n$ when that first member equals $n$. A tetrad field need not be a coordinate frame. Local existence near every point is proved in Existence of a smooth future-directed tetrad field near every event; global existence on $M$ is not asserted and would be a nontrivial parallelizability property of the manifold.
• Adapted chart. A chart is $n$-adapted when its spatial coordinates are constant along each integral curve of $n$ contained in its domain, with an interval as parameter domain (Observer congruences, their observer curves and adapted charts, Definition (c)). Near every point of a congruence there is such a chart with $n=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{u}^0$ (Every congruence admits a local adapted chart). The definition does not require this normalization: replacing $u^0$ in that chart by $v^0=2u^0$ and keeping the spatial coordinates gives another adapted chart with $n=2\mathrm{\partial }\mathrm{/}\mathrm{\partial }{v}^0$. Thus adapted charts are not unique, and their zeroth coordinate field need not equal $n$.
• Rest spaces and their integrability. The rest spaces $n_q^{\perp }$ of a congruence always form a smooth rank-$3$ distribution, and they are tangent to local hypersurfaces exactly when $\alpha \wedge d\alpha =0$ for $\alpha =n^{\mathrm{♭}}$. Thus an adapted chart always exists, while an orthogonal rest-space foliation need not (Smoothness of the rest-space family and its Frobenius integrability criterion).
• Components. Coordinate components of a tensor field transform by products of chart Jacobians (The component transformation law of a smooth tensor field), while its components in two future-directed tetrads are related by a matrix $\mathrm{\Lambda }$ with ${\mathrm{\Lambda }}^{\mathsf{T}}\eta \mathrm{\Lambda }=\eta$ and ${\mathrm{\Lambda }}^0{}_0>0$ (The change between two future-directed tetrads is a Lorentz matrix). The two component systems are different data (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).
• Observer-relative quantities. For $P\in T_{p}M$ and a fixed real parameter $c>0$, observer energy $E_n(P)=-c{g}_p(P,n)$ and spatial momentum $q_n(P)=h_n(P)$ depend on the chosen observer $n$, where $h_n(P)=P+g_p(n,P)n$ is the spatial projection of Observers, rest spaces and the spatial projection at an event, Definition (c). Different observers generally assign different energies to the same tangent vector, and no observer-independent energy-momentum split is asserted. In contrast, rest mass $m(P)=\sqrt{-g_p(P,P)}\mathrm{/}c$, defined when $g_p(P,P)\le 0$, depends only on $P$, $g_p$ and $c$, and is independent of $n$ (Observer-relative energy, spatial momentum and rest mass of a tangent vector, Definition (a)-(c)).
• Physical readings. The interpretation of these geometric objects as observers, reference frames, clocks, particles and measured energies belongs to the physical postulates and unit conventions of the framework, recorded elsewhere on this track; the items on this page assert no measurement, dynamics or claim about nature.