Uniformly subluminal prescribed worldline, retarded root and retarded geometric data off the worldline

Let the laboratory region, inertial chart, spatial Euclidean structure and SI units be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values, in particular $c>0$, ${\mu }_0>0$, ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$. A retarded point-geometry datum consists of a real charge parameter $q\in \mathbb{R}$ in coulombs (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current(a)) and a prescribed map $z:\mathbb{R}\to {\mathbb{R}}^3$ of class $C^3$ ($C^k$ Euclidean maps and diffeomorphisms) together with a constant $0\le v_{\ast }<c$ such that $\mathrm{\mid }{z}^{\mathrm{\prime }}(s)\mathrm{\mid }\le v_{\ast }$ for every $s\in \mathbb{R}$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$); the spacetime curve $s\mapsto (z(s),s)$ is the laboratory reading of a prescribed massive future timelike worldline of the physical model (Worldlines, four-velocity, chart readings and the frame/observer distinction), and $v_{\ast }$ is a declared bound on its coordinate speed, not a measured quantity. Write $\mathrm{\Gamma }:=\{(z(s),s):s\in \mathbb{R}\}$ and $D:=\{(x,t)\in {\mathbb{R}}^3\times \mathbb{R}:x\mathrm{\ne }z(t)\}$ for the open off-worldline observation domain. Then:

(a) [Retarded root] For every observation event $(x,t)\in D$ there is exactly one $s=s(x,t)<t$ with $t-s=\mathrm{\mid }x-z(s)\mathrm{\mid }\mathrm{/}c$; the map $s:D\to \mathbb{R}$ is of class $C^3$, and with the data of (b) below one has ${\mathrm{\partial }}_{t}s=1\mathrm{/}\kappa$ and ${\mathrm{\nabla }}_{x}s=-n\mathrm{/}(c\kappa )$ (Global existence, uniqueness and differentiability of the retarded root, whose $C^3$ branch applies because $z$ is $C^3$ and uniformly subluminal).

(b) [Retarded data] For $(x,t)\in D$ define $R:=x-z(s(x,t))$, $r:=\mathrm{\mid }R\mathrm{\mid }>0$, $n:=R\mathrm{/}r$, $\beta :=z^{\mathrm{\prime }}(s(x,t))\mathrm{/}c$, $\kappa :=1-n\cdot \beta$, $\dot{\beta }:=z^{\mathrm{\prime }\mathrm{\prime }}(s(x,t))\mathrm{/}c$; then $r=c(t-s)$ and $\kappa \in [1-v_{\ast }\mathrm{/}c,1+v_{\ast }\mathrm{/}c]\subseteq (0,\mathrm{\infty })$. On $D$ the objects $r$ and $n$ are of class $C^3$, $\beta$ and $\kappa$ are of class $C^2$, and $\dot{\beta }$ is of class $C^1$ (each is a composition of the $C^3$ root with $z$ or its derivatives).

(c) [Emission-event data and emitted spheres] At an emission event $s_0\in \mathbb{R}$ and a direction $n\in S^2$ put $x_r:=z(s_0)+rn$, $t_r:=s_0+r\mathrm{/}c$ for $r>0$, and ${\beta }_0:=z^{\mathrm{\prime }}(s_0)\mathrm{/}c$, ${\kappa }_0:=1-n\cdot {\beta }_0$. Then $(x_r,t_r)\in D$ and $s(x_r,t_r)=s_0$ for every $r>0$: the sphere $\{x_r\}$ at observation time $t_r$ is emitted at the single event $s_0$.

(d) [Branches and scope] Only the all-time uniformly subluminal branch (a)-(c) is defined by this datum. For a curve prescribed on a finite interval $J$ the same formulas define retarded data only at observation events for which the root equation has a solution $s\in J$; each consumer must assume that cone intersection explicitly. Pointwise subluminality $\mathrm{\mid }{z}^{\mathrm{\prime }}(s)\mathrm{\mid }<c$ for every finite $s$ does not imply (a): the companion example on the examples page exhibits a smooth curve with $\mathrm{\mid }{z}^{\mathrm{\prime }}\mathrm{\mid }<c$ pointwise for which the retarded equation has no root at some event, so the uniform bound $v_{\ast }<c$, or an explicit cone-intersection hypothesis, is part of the datum. This definition is kinematic: it fixes the root, the retarded data and the notation, and asserts no field, no dynamics, no force law, no self-force and no empirical content. The identification of these data with the emission data of the physical field is made by the selection postulate Adopted retarded solution selection: the physical branch carries no incoming free field together with the theorems of this page.

Clauses (a)-(b) are exactly the all-time uniformly subluminal $C^3$ branch of Global existence, uniqueness and differentiability of the retarded root: that lemma supplies, for every event of ${\mathbb{R}}^3\times \mathbb{R}$, existence and uniqueness of the root, the identity of the equality case $s=t$ with $x=z(t)$, the strict inequality $s<t$ on $D$, the derivative formulas with $1-v_{\ast }\mathrm{/}c\le \kappa \le 1+v_{\ast }\mathrm{/}c$, and the $C^k$ regularity of the root on $D$ for a $C^k$ trajectory; no new analysis is introduced here. Clause (c) is the elementary root check $t_r-s_0=\mathrm{\mid }{x}_r-z(s_0)\mathrm{\mid }\mathrm{/}c=r\mathrm{/}c$ combined with the uniqueness in clause (a); no boundedness of $z^{\mathrm{\prime }\mathrm{\prime }}$ is used. Clause (d) records the branch boundaries honestly: the finite-history and nonuniform cases are left to consumers, and the failure of existence under mere pointwise subluminality is exhibited on the examples page rather than asserted here. The phrase 'prescribed worldline' is licensed by Worldlines, four-velocity, chart readings and the frame/observer distinction(a) and (c), which relate chart readings $t\mapsto r(t)$ of massive worldlines to the condition $\mathrm{\mid }{r}^{\mathrm{\prime }}(t)\mathrm{\mid }<c$. The definition introduces no physical field, so no Maxwell, self-force or empirical assertion is made; the cross-reference to Adopted retarded solution selection: the physical branch carries no incoming free field records that the retarded root is the kinematic object that the adopted selection turns into an emission time. No choice principle is used in the root lemma or in these kinematic constructions; its higher-regularity branch is explicitly choice-free.