Inertial and piecewise inertial worldlines of the affine Minkowski model

Let $(A,V,\eta ,C^{+})$ be the oriented time-oriented affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives)), let $I$ be an order-convex subset of $\mathbb{R}$ with at least two elements (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and let curves $x:I\to A$ and their derivatives be as in Curves, regular causal curves and future timelike worldlines in affine Minkowski space: the displacement $x(s+h)-x(s)$ is taken in the affine structure of Affine spaces over a real vector space and displacement vectors, and $x^{\mathrm{\prime }}(s)\in V$, when it exists, is the limit of the difference quotients, read in the coordinate calculus of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral in any inertial chart (Inertial frames and inertial charts of the affine Minkowski model, The affine atlas makes the Minkowski model a smooth 4-manifold diffeomorphic to R^4).

(a) Inertial worldlines

A curve $x:I\to A$ is inertial (also called uniform or straight) when there are a parameter value $s_0\in I$, an event $p\in A$ and a vector $d\in V$, with $d$ future timelike (Time orientation, future and past timelike/null/causal vectors, and the closed cones), such that

$x(s)=p+(s-s_0)d\text{for every }s\in I\mathrm{.}$

The affine action is defined for every real parameter and every vector, so the display defines a map $I\to A$ (Affine spaces over a real vector space and displacement vectors); its image is $x[I]=\{p+\lambda d:\lambda =s-s_0,s\in I\}$, and a restriction of an inertial worldline to a subinterval $J\subseteq I$ with at least two elements is inertial with the same $d$: for any $t_0\in J$ its formula is $x(s)=x(t_0)+(s-t_0)d$ on $J$. A singleton restriction is a map but is not a curve or worldline under the cited definition. Its tangent on $I$ is the constant vector $d$: in the chart $x_A$ of an inertial frame the coordinate path is the affine function $s\mapsto x_A(p)+(s-s_0)E^{-1}(d)$, whose difference quotients equal the constant $E^{-1}(d)$ (A finite list $v:n\to V$ is an ordered basis if and only if every $x\in V$ equals ${\sum }_{i<n}{\lambda }_i{v}_i$ for exactly one $\lambda :n\to F$; those scalars are the coordinates of $x$ in that ordered basis), so $x^{\mathrm{\prime }}(s)=d$ at every $s\in I$ by the difference-quotient definition in Curves, regular causal curves and future timelike worldlines in affine Minkowski space; the same computation gives $C^k$ regularity for every $k\ge 1$. Since the constant tangent $d$ is future timelike by hypothesis, an inertial curve is a future timelike worldline of the model (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Curves, regular causal curves and future timelike worldlines in affine Minkowski space (Worldlines)). If two inertial worldlines $x(s)=p+(s-s_0)d$ and $y(t)=p^{\mathrm{\prime }}+(t-t_0)d^{\mathrm{\prime }}$ have the same image and the same direction of traversal, that is $d^{\mathrm{\prime }}=\alpha d$ with $\alpha >0$, then $p^{\mathrm{\prime }}$ lies in the image of $x$, so $p^{\mathrm{\prime }}=p+\mu d$ for some real $\mu$, and

$y(t)=x(s_0+\mu +\alpha (t-t_0))\text{for every }t\text{ in the domain of }y,$

an increasing affine reparametrization. Here an increasing affine reparametrization means composition with a bijection of parameter domains of the form $t\mapsto \alpha t+\beta$ with $\alpha >0$. Conversely, composing an inertial worldline with such a bijection gives an inertial worldline with the same image. This computation uses only the affine axioms and $d\mathrm{\ne }0$.

(b) Piecewise inertial worldlines

Let $a<b$ be real numbers with $[a,b]\subseteq I$. A curve $x:[a,b]\to A$ is piecewise inertial when there are a finite partition $a=s_0<s_1<\cdots <s_m=b$, events $p_1,\dots ,p_m\in A$ and future timelike vectors $d_1,\dots ,d_m\in V$ such that

$x(s)=p_i+(s-s_{i-1})d_i\text{for every }s\in [s_{i-1},s_i]\text{ and every }i=1,\dots ,m,$

and consecutive pieces agree at the partition points, that is $p_{i+1}=p_i+(s_i-s_{i-1})d_i=x(s_i)$ for $i=1,\dots ,m-1$. The joining condition makes the piecewise formulas define a single function on $[a,b]$, whose tangent on each open piece $(s_{i-1},s_i)$ is the constant $d_i$ and whose finite one-sided difference-quotient tangents at an interior partition point $s_i$ are $d_i$ from the left and $d_{i+1}$ from the right; at the endpoints $a$ and $b$ the one-sided quotients are $d_1$ and $d_m$. Hence a piecewise inertial curve is a piecewise $C^1$ future timelike worldline on $[a,b]$ in the weak sense of Curves, regular causal curves and future timelike worldlines in affine Minkowski space — no continuous extension of the derivative to the closed pieces is required, and the tangents are future timelike at every smooth point and on both sides of every corner — and it is $C^1$ on $[a,b]$ if and only if $d_1=d_2=\cdots =d_m$. The events $x(s_i)$ with $0<i<m$ are the corners of the itinerary, and the itinerary is a two-leg itinerary when $m=2$.

(c) Scope

The definition asserts the existence of no worldlines beyond the explicitly parametrized straight and piecewise straight ones, and it attaches no clock reading to any curve: readings of ideal clocks are the subject of the model's postulates, not of this definition. No choice principle is used, since every object is finitely parametrized and every operation is finite-dimensional.