Particle spacetime, observers and worldlines

Use the signature $(-,+,+,+)$ throughout. In the flat case, Minkowski spacetime means the affine model $(A,V,\eta ,C^{+})$ of Oriented time-oriented affine Minkowski space (the spacetime primitives): $A$ is an affine space, $\eta$ is a nondegenerate Lorentz form on its four-dimensional displacement space, and $C^{+}$ selects the future cone. In the curved case, a background spacetime means a prescribed smooth time-oriented Lorentzian four-manifold $(M,g,N)$ of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model; the metric and time orientation are fixed data, not dynamical unknowns here. No field equation or completeness assumption is included.

A massive worldline on an open interval $I\subseteq \mathbb{R}$ is a $C^2$ curve $x:I\to M$ with future-directed timelike tangent $T=dx\mathrm{/}ds$ at every parameter value. This is the $C^2$ subclass of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold. A null worldline is a $C^2$ curve with nonzero future-directed null tangent everywhere. Its tangent has no proper-time normalization: $g(T,T)=0$, so the massive proper-time formula does not apply. In the flat model these are $C^2$ curves $x:I\to A$, with tangent defined by differentiating the displacement $x(s)-o$ for any fixed $o\in A$, lying in the selected future timelike or nonzero null cone of Oriented time-oriented affine Minkowski space (the spacetime primitives). The derivative is independent of $o$ because a change of origin adds a constant displacement. No affine-line or geodesic condition is imposed.

An observer at an event $p\in M$ is a future unit timelike vector $n\in T_{p}M$ with $g_p(n,n)=-1$, as in Observers, rest spaces and the spatial projection at an event. Its instantaneous rest space is $n^{\perp }=\{v:g_p(v,n)=0\}$. This is a pointwise vector-space construction; the definition does not assign a clock reading or a moving frame to a lone observer.

An observer congruence on an open set $U\subseteq M$ is a smooth field of future unit timelike vectors on $U$, as in Observer congruences, their observer curves and adapted charts; its observer curves are its integral curves. A tetrad on a nonempty open set $U$ means a smooth ordered frame field $(E_0,E_1,E_2,E_3)$ with $g(E_a,E_b)=\mathrm{diag}(-1,1,1,1{)}_{ab}$ for $a,b=0,1,2,3$, as in Local frame fields, coordinate frames and tetrad fields. A tetrad is future-directed when additionally $E_0$ is future-directed everywhere: $g(E_0,N)<0$ in the curved case, or $E_0(p)\in C^{+}$ in the flat case with $g=\eta$. In flat spacetime this future-directed subclass is the tetrad-field convention of Charts, tetrads, observer congruences and adapted charts in affine Minkowski space; the unqualified tetrad here retains the general GR convention. A chart is a coordinate map; an adapted chart is a chart whose spatial coordinates are constant along the congruence's integral curves. These objects are distinct: a chart need not be inertial, a tetrad need not be coordinate induced, and a congruence is a field of observers rather than a chart or a frame. This distinction agrees with Charts, tetrads, observer congruences and adapted charts in affine Minkowski space.

The coordinates and units used later are SI: in an inertial chart $x^0=ct$ and $x^i$ are metres, $t$ is seconds, $c>0$ is metres per second, and proper time is measured in seconds. In a general chart, coordinate components inherit their coordinate units; invariant equations are tensorial. No global inertial chart, global tetrad, or global observer congruence is asserted to exist.