Worldlines, four-velocity, chart readings and the frame/observer distinction

The particle and light objects of the framework are the following curves and vector fields of the affine Minkowski model (Oriented time-oriented affine Minkowski space (the spacetime primitives), Inertial frames and inertial charts of the affine Minkowski model), with the physical interpretation of Adopted Minkowski physical model of the electromagnetic framework and the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

(a) [Worldlines] A worldline is a regular causal curve $x:J\to A$ in one of the regularity classes of Curves, regular causal curves and future timelike worldlines in affine Minkowski space ($J$ an order-convex subset of $\mathbb{R}$ with at least two elements). A massive worldline is a $C^k$ future timelike worldline, $k\ge 1$, with everywhere nonzero future timelike tangent. A future null curve is a $C^1$ curve with everywhere nonzero future null tangent. An ideal vacuum light ray of the adopted physical model follows a null straight line: on its parameter interval it has the form $x(s)=p+\lambda (s)v$, where $p\in A$, $v\in V$ is nonzero future null, and $\lambda$ is $C^1$ with ${\lambda }^{\mathrm{\prime }}(s)>0$. This is the straight-ray adoption of Adopted Minkowski physical model of the electromagnetic framework; arbitrary future null curves are retained as kinematic objects without asserting that they describe vacuum light propagation. Along a null curve the integrand $-\eta (x^{\mathrm{\prime }},x^{\mathrm{\prime }})$ of the proper-time construction (Proper time along a future timelike worldline) vanishes identically, so that construction yields no proper-time parametrization of it, and a light ray is never treated as a massive worldline. A worldline is a curve in spacetime; its reading $r\circ x$ in a named laboratory chart is a coordinate description of it, not an object of another observer.

(b) [Four-velocity and four-acceleration] For a massive worldline the four-velocity $U$ and, when it is $C^2$, the four-acceleration $A$ are those of Four-velocity and four-acceleration of a future timelike worldline; wherever defined they satisfy $\eta (U,U)=-c^2$ and $\eta (U,A)=0$, by Norms of four-velocity and four-acceleration, and the observer rest-space decomposition.

(c) [Chart reading] If a massive $C^2$ worldline is described in a laboratory chart of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields by $t\mapsto (ct,r(t))$ on an open interval of laboratory times, then $\mathrm{\mid }{r}^{\mathrm{\prime }}(t)\mathrm{\mid }<c$ for every $t$, with four-velocity $U=\gamma (c,r^{\mathrm{\prime }})$ and Lorentz factor $\gamma =(1-\mathrm{\mid }{r}^{\mathrm{\prime }}{\mathrm{\mid }}^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}$, by clause (e) of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition.

(d) [Frames and observers] An inertial frame in this framework is the origin-with-basis object of Inertial frames and inertial charts of the affine Minkowski model; a point observer is a future unit timelike vector $n\in V$ with rest space $n^{\perp }$ and rest projection $h$, as in clauses (c)-(d) of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition. On an open spacetime region $W\subseteq A$, an observer congruence field means a $C^1$ map $n:W\to V$ with $n(p)$ future timelike and $\eta (n(p),n(p))=-1$ at every $p$; its observer histories, when given, are integral curves satisfying $dx\mathrm{/}d\tau =cn(x)$. A tetrad field is a tuple of $C^1$ maps $(f_0,f_1,f_2,f_3):W\to V^4$ whose values form a positively oriented $\eta$-orthonormal basis at every event, with $f_0$ future timelike. Regularity here means componentwise $C^1$ in inertial coordinates. These fields need not be constant and are distinct from an inertial origin-with-basis frame, its coordinate chart and a single point observer; no global congruence or tetrad is asserted to exist.

(e) [Scope] No force law, no self-field and no extended-body structure is supplied here; an accelerating massive worldline requires a force model stated separately, and an ideal light ray carries no dispersive, quantum or polarisation structure in this definition.