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Complete retarded point fields

1 · Prerequisites

2 · Summary

The completion of the retarded point-field layer of the classical electromagnetism mathematics track. The single item, Complete retarded point fields: unique retarded root, the delta substitution with denominator $1 - n \cdot β$ , Liénard–Wiechert potentials and fields, and a uniform far-zone remainder, fixes a $C^{3}$ worldline with a uniform subluminal bound $∣ z^{'} ∣ \leq v_{*} < c$ and assembles the full point-source computation in four clauses. Clause (a) states the event form of the retarded root: every event off the worldline has exactly one root $s = s (x, t) < t$ of $t - s = ∣ x - z (s) ∣ / c$ , the root map is $C^{3}$ with derivative formulas $\partial_{t} s = κ^{- 1}$ and $\nabla_{x} s = - n / (c κ)$ , the emission-time Jacobian $1 - n \cdot β = κ$ is uniformly bounded below by $1 - v_{*} / c$ , and $κ = 1 - n \cdot β$ is exactly the denominator produced by the delta substitution. Clause (b) records the delta substitution as the exact history-integral pairing identity with weight $(κ r)^{- 1}$ , under Countable Choice, and identifies the Liénard-Wiechert potentials $φ = C / (κ r)$ and $A = β φ / c$ as $C^{2}$ functions off the worldline whose zero extensions are locally integrable and define regular distributions. Clause (c) gives $E = - \nabla φ - \partial_{t} A$ and $B = curl A$ as $C^{1}$ fields off the worldline together with the velocity/acceleration split and the uniform bound $∣ E_{v e l} ∣ \leq ∣ C ∣ (1 + v_{*} / c) (1 - v_{*} / c)^{- 3} r^{- 2}$ . Clause (d) evaluates the field on the ray family $x_{r} = z (s_{0}) + r n$ , $t_{r} = s_{0} + r / c$ , where the root is exactly the emission time $s_{0}$ , and proves the uniform far-zone remainder with explicit constant and the equivalent $O (r^{- 1})$ limit of $r E$ .

The item is mathematics: it carries domain: mathematics, depends only on mathematical suppliers, and asserts no self-force law, radiation-reaction postulate, physical Maxwell coupling, finite-energy or radiation-flux conclusion. Its physical consumers are the classical electromagnetism pages that use the retarded fields; empirical qualifications of any physical use remain with the consuming physical item. The companion examples page records the three hypothesis/convention boundaries on which the item's hypotheses depend.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Complete retarded point fields: unique retarded root, the delta substitution with denominator $1 - n \cdot β$ , Liénard–Wiechert potentials and fields, and a uniform far-zone remainder

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ), used through the change-of-variables corollaries inside the proof of the retarded-convolution identity cited in clause (b) (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $c > 0$ , let $v_{*} < c$ , let $z : R \to R^{3}$ be of class $C^{3}$ with $∣ z^{'} (s) ∣ \leq v_{*}$ for every $s \in R$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, $C^{k}$ Euclidean maps and diffeomorphisms), let $q \in R$ and $ε_{0} > 0$ , put $μ_{0} : = (c^{2} ε_{0})^{- 1}$ and $C : = q / (4 π ε_{0})$ , and write $Ω : = R^{3} \times R$ and $Γ : = {(z (s), s) : s \in R}$ .

(a) [Root and emission-time Jacobian] For every $(x, t) \in Ω ∖ Γ$ (equivalently $x \neq z (t)$ ) there is exactly one $s = s (x, t) < t$ with $t - s = ∣ x - z (s) ∣ / c$ . The map $s : Ω ∖ Γ \to R$ is of class $C^{3}$ , and with $R : = x - z (s)$ , $r : = ∣ R ∣$ , $n : = R / r$ , $β : = z^{'} (s) / c$ and $κ : = 1 - n \cdot β$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ) one has $\partial_{t} s (x, t) = κ^{- 1}, \nabla_{x} s (x, t) = - \frac{n}{c κ}, κ \in [1 - v_{*} / c, 1 + v_{*} / c],$ so $κ \geq 1 - v_{*} / c > 0$ uniformly on $Ω ∖ Γ$ . Moreover, for the fixed $x$ and near the root, the function $s^{'} \mapsto s^{'} + ∣ x - z (s^{'}) ∣ / c$ is differentiable at $s^{'} = s$ with derivative $κ (x, t)$ : this emission-time Jacobian $1 - n \cdot β$ is the denominator produced by the delta substitution of clause (b).

(b) [Delta substitution and Liénard-Wiechert potentials] Let $ρ, J \in D^{'} (Ω)$ be the point-source distributions of Point-support spacetime densities: distributions, continuity equation and endpoint fluxes attached to $q$ and the worldline $z$ , let $K$ be the retarded wave kernel of The retarded wave kernel is a distribution supported on the cone, solves the point-wave equation, and convolves to the retarded solution, and for $ψ \in D (Ω)$ put $(K ⊛ ψ) (y, τ) : = ⟨ K, ψ (y + \cdot, τ + \cdot) ⟩$ and define the retarded-history pairings by the explicit integrals $⟨ Φ, ψ ⟩ : = (q / ε_{0}) \int_{R} (K ⊛ ψ) (z (s), s) d s$ , $⟨ Υ, ψ ⟩ : = μ_{0} q \int_{R} z^{'} (s) (K ⊛ ψ) (z (s), s) d s$ , componentwise in the second case. These are the history-integral interpretation supplied by clause (b) of Retarded potentials of a point history as the retarded convolution of the point sources, and the off-worldline Lienard-Wiechert fields, not ordinary distribution action on a noncompact test. Then these pairings are finite for every $ψ$ and their exact retarded-convolution values are $⟨ Φ, ψ ⟩ = \int_{Ω ∖ Γ} \frac{C ψ}{κ r} d x d t, ⟨ Υ, ψ ⟩ = \int_{Ω ∖ Γ} \frac{β}{c} \frac{C ψ}{κ r} d x d t$ (the second identity componentwise); equivalently, $Φ$ and $Υ$ are the regular distributions of the locally integrable zero extensions to $Γ$ of the $C^{2}$ functions $φ : = C / (κ r)$ and $A : = β φ / c$ on $Ω ∖ Γ$ . The weight $(κ r)^{- 1}$ with $κ = 1 - n \cdot β$ is the exact form of the substitution with denominator $1 - n \cdot β$ (the rigorous statement is the displayed pairing identity; composition of $δ$ with a function is not part of the distribution calculus used here).

(c) [Fields and the velocity/acceleration split] With $E : = - \nabla φ - \partial_{t} A$ and $B : = curl A$ on $Ω ∖ Γ$ (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression), and writing $\dot{β} : = z^{''} (s) / c$ at the root, $E = E_{v e l} + E_{a c c}, E_{v e l} = \frac{C (1 - ∣ β ∣^{2}) (n - β)}{κ^{3} r^{2}}, E_{a c c} = \frac{C n \times ((n - β) \times \dot{β})}{c κ^{3} r}, B = \frac{n \times E}{c}$ at every event of $Ω ∖ Γ$ (The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors). Both $E$ and $B$ are $C^{1}$ on $Ω ∖ Γ$ , and the velocity field obeys the bound $∣ E_{v e l} (x, t) ∣ \leq ∣ C ∣ \frac{1 + v_{*} / c}{(1 - v_{*} / c)^{3}} r (x, t)^{- 2}$ at every event.

(d) [Uniform far-zone remainder] For every $s_{0} \in R$ and every $n \in R^{3}$ with $∣ n ∣ = 1$ , put $x_{r} : = z (s_{0}) + r n$ and $t_{r} : = s_{0} + r / c$ for $r > 0$ , and set $β_{0} : = z^{'} (s_{0}) / c$ , ${\dot{β}}_{0} : = z^{''} (s_{0}) / c$ , $κ_{0} : = 1 - n \cdot β_{0}$ . Then $(x_{r}, t_{r}) \in Ω ∖ Γ$ and $s (x_{r}, t_{r}) = s_{0}$ for every $r > 0$ , and $E (x_{r}, t_{r}) = \frac{C n \times ((n - β_{0}) \times {\dot{β}}_{0})}{c κ_{0}^{3} r} + ρ_{E} (s_{0}, n, r), B (x_{r}, t_{r}) = \frac{n \times E (x_{r}, t_{r})}{c},$ where the remainder is bounded uniformly in the direction, $∣ ρ_{E} (s_{0}, n, r) ∣ \leq ∣ C ∣ \frac{1 + v_{*} / c}{(1 - v_{*} / c)^{3}} r^{- 2} for every n \in S^{2} and every r > 0.$ Equivalently, $r E (x_{r}, t_{r}) \to C n \times ((n - β_{0}) \times {\dot{β}}_{0}) / (c κ_{0}^{3})$ as $r \to \infty$ , uniformly over $n \in S^{2}$ at the explicit rate $O (r^{- 1})$ given by the same bound multiplied by $r$ . The uniformity is in the direction $n$ at a fixed emission event $s_{0}$ ; no uniformity in $s_{0}$ is asserted ( $z^{''}$ is not assumed bounded), and no finite-energy, decay or radiation-flux conclusion is drawn.

This is a mathematical model computation for the stipulated $C^{3}$ uniformly subluminal worldline and the retarded choice; no self-force law, no radiation-reaction postulate and no physical Maxwell coupling is asserted or proved, and the uniform speed bound and the exclusion of $Γ$ are essential hypotheses (see the companion example page).

Facts & Assumptions

Given: A speed $c > 0$ , a real bound $v_{*} < c$ , a worldline $z : R \to R^{3}$ of class $C^{3}$ with $∣ z^{'} (s) ∣ \leq v_{*}$ for every $s \in R$ , a real $q$ and a positive $ε_{0}$ , together with the constants $μ_{0} = (c^{2} ε_{0})^{- 1}$ , $C = q / (4 π ε_{0})$ and the sets $Ω = R^{3} \times R$ , $Γ = {(z (s), s) : s \in R}$ .

[F1]

Let $c > 0$ , $0 \leq v_{*} < c$ and $z : R \to R^{3}$ of class $C^{1}$ with $∣ z^{'} (s) ∣ \leq v_{*}$ for every $s$ . Then for every $(x, t) \in R^{3} \times R$ there is exactly one real root $s = s (x, t)$ of $t - s = ∣ x - z (s) ∣ / c$ , and $s \leq t$ with equality exactly when $x = z (t)$ ; on the open off-instantaneous-worldline domain $D = {(x, t) : x \neq z (t)}$ one has $s < t$ , $r = ∣ x - z (s) ∣ = c (t - s) > 0$ , the root is $C^{1}$ , and with $n = (x - z (s)) / r$ , $β = z^{'} (s) / c$ , $κ = 1 - n \cdot β$ its derivatives are $\partial_{t} s = 1 / κ$ and $\nabla_{x} s = - n / (c κ)$ , where $1 - v_{*} / c \leq κ \leq 1 + v_{*} / c$ . If $z$ is $C^{k}$ for an integer $k \geq 1$ , the root is $C^{k}$ on $D$ ; in particular a $C^{3}$ trajectory gives a $C^{3}$ root (Global existence, uniqueness and differentiability of the retarded root).

[F2]

For $f : A \to R^{m}$ with $A \subseteq R$ and a limit point $c \in A$ , $f$ is differentiable at $c$ when the difference quotients $q_{f, c} (x) = \frac{1}{x - c} (f (x) - f (c))$ converge as $x \to c$ , and then the derivative is the limit $f^{'} (c) : = \lim_{x \to c} q_{f, c} (x) \in R^{m}$ ; the notion is intrinsic and is computed componentwise (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

[F3]

A map $f : U \to R^{q}$ on an open $U \subseteq R^{m}$ is of class $C^{k}$ when every word of at most $k$ coordinate partial derivatives of each component exists and is continuous, including the word of length zero; it is smooth when it is $C^{k}$ for every $k$ ( $C^{k}$ Euclidean maps and diffeomorphisms).

[F4]

On $R^{n}$ the Euclidean inner product is $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ , it is bilinear and symmetric, $⟨ z, z ⟩ \geq 0$ with $⟨ z, z ⟩ = 0$ only for $z = 0$ , and the Euclidean norm is $∥ x ∥_{2} = \sqrt{⟨ x, x ⟩}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F5]

For all $x, y \in R^{n}$ one has $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ ; $∥ \cdot ∥_{2}$ is a norm, so $∥ u + v ∥_{2} \leq ∥ u ∥_{2} + ∥ v ∥_{2}$ and $∥ λ u ∥_{2} = ∣ λ ∣ ∥ u ∥_{2}$ , and consequently the reverse triangle inequality $∣ ∥ u ∥_{2} - ∥ v ∥_{2} ∣ \leq ∥ u - v ∥_{2}$ holds (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[F6]

Let $I \subseteq R$ be an open interval, $z : I \to R^{3}$ a $C^{1}$ trajectory, $q \in R$ , and $Ω \supseteq Γ$ open on which the graph map is proper. Then $⟨ ρ, ψ ⟩ = q \int_{I} ψ (z (t), t) d t$ and $⟨ J, ψ ⟩ = q \int_{I} z^{'} (t) ψ (z (t), t) d t$ define distributions with $\partial_{t} ρ + div J = 0$ ; properness is automatic for $Ω = R^{3} \times I$ , in particular for the all-time history $I = R$ (Point-support spacetime densities: distributions, continuity equation and endpoint fluxes).

[F7]

The retarded wave kernel is the distribution $K$ on $R^{3} \times R$ given by $⟨ K, ψ ⟩ = \frac{1}{4 π} \int_{R^{3}} \frac{ψ (y, ∣ y ∣ / c)}{∣ y ∣} d y$ , and for $(y, τ) \in R^{3} \times R$ the shifted pairing $(K ⊛ ψ) (y, τ) : = ⟨ K, ψ (y + \cdot, τ + \cdot) ⟩$ equals the absolutely convergent integral $\frac{1}{4 π} \int \frac{ψ (y + z, τ + ∣ z ∣ / c)}{∣ z ∣} d z$ (The retarded wave kernel is a distribution supported on the cone, solves the point-wave equation, and convolves to the retarded solution).

[F8]

Assume ${A C}_{ω}$ . Let $c > 0$ , $q \in R$ , $ε_{0} > 0$ and let $z : R \to R^{3}$ be $C^{3}$ with $∣ z^{'} (s) ∣ \leq v_{*} < c$ for all $s$ . With $K$ the retarded kernel of [F7] and, for tests $ψ$ , $⟨ Φ, ψ ⟩ = (q / ε_{0}) \int_{R} (K ⊛ ψ) (z (s), s) d s$ , $⟨ Υ, ψ ⟩ = μ_{0} q \int_{R} z^{'} (s) (K ⊛ ψ) (z (s), s) d s$ componentwise, these pairings are finite for every $ψ$ and satisfy the exact retarded-convolution identities $⟨ Φ, ψ ⟩ = \int_{Ω ∖ Γ} \frac{C ψ}{κ r} d x d t$ and $⟨ Υ, ψ ⟩ = \int_{Ω ∖ Γ} \frac{β}{c} \frac{C ψ}{κ r} d x d t$ ; consequently $Φ$ and $Υ$ are the regular distributions of the locally integrable zero extensions of the $C^{2}$ functions $φ = C / (κ r)$ and $A = β φ / c$ on $Ω ∖ Γ$ (Retarded potentials of a point history as the retarded convolution of the point sources, and the off-worldline Lienard-Wiechert fields).

[F9]

In the situation of [F8], the fields $E = - \nabla φ - \partial_{t} A$ and $B = curl A$ on $Ω ∖ Γ$ satisfy $E = C [\frac{(1 - ∣ β ∣^{2}) (n - β)}{κ^{3} r^{2}} + \frac{n \times ((n - β) \times \dot{β})}{c κ^{3} r}], B = \frac{n \times E}{c}$ and are of class $C^{1}$ there, where $\dot{β} = z^{''} (s) / c$ is evaluated at the retarded root (Retarded potentials of a point history as the retarded convolution of the point sources, and the off-worldline Lienard-Wiechert fields).

[F10]

For $c > 0$ and an open $Ω \subseteq R^{3} \times R$ , a $C^{k}$ scalar-vector potential pair $(φ, A)$ on $Ω$ with $k \geq 1$ has fields $E = - \nabla φ - \partial_{t} A$ and $B = curl A$ , both of class $C^{k - 1}$ (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression).

[F11]

The right-handed cross product on $R^{3}$ is $u \times v = (u_{y} v_{z} - u_{z} v_{y}, u_{z} v_{x} - u_{x} v_{z}, u_{x} v_{y} - u_{y} v_{x})$ (The cross product in $R^{3}$ ); it is bilinear and alternating, $⟨ u \times v, w ⟩ = \det [u v w]$ , and $u \times v$ is orthogonal to both $u$ and $v$ (The cross product is bilinear, alternating, and orthogonal to both factors).

[F12]

${A C}_{ω}$ : every family $(X_{n})_{n \in N}$ of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

Given: The hypotheses of the Statement: $c > 0$ , $v_{*} < c$ , a $C^{3}$ worldline $z$ with $∣ z^{'} (s) ∣ \leq v_{*}$ for every $s$ , $q \in R$ , $ε_{0} > 0$ , and the derived data $μ_{0}$ , $C$ , $Ω$ , $Γ$ .

Proof technique: direct: clause (a) is the root lemma together with the one-variable computation of the emission-time Jacobian; clause (b) is the supplier's pairing identity read in the stated history-integral interpretation; clause (c) is the supplier's field formula with an elementary uniform bound; clause (d) evaluates the field on the ray family and reads off the uniform limit.

1.1F1F2F3given

Hypothesis audit and the root. The hypothesis gives $0 \leq ∣ z^{'} (0) ∣ \leq v_{*}$ , so $0 \leq v_{*} < c$ , and $z$ is of class $C^{3}$ , hence $C^{1}$ ; the root lemma [F1] therefore applies with these $c$ , $v_{*}$ , $z$ , and since $z$ is $C^{3}$ its $C^{k}$ branch with $k = 3$ applies. Hence for every $(x, t) \in Ω ∖ Γ$ there is exactly one $s = s (x, t) < t$ with $t - s = ∣ x - z (s) ∣ / c$ , the root map is of class $C^{3}$ on $Ω ∖ Γ$ , and with $R = x - z (s)$ , $r = ∣ R ∣ > 0$ , $n = R / r$ , $β = z^{'} (s) / c$ , $κ = 1 - n \cdot β$ one has $\partial_{t} s = κ^{- 1}$ , $\nabla_{x} s = - n / (c κ)$ and $1 - v_{*} / c \leq κ \leq 1 + v_{*} / c$ , so $κ \geq 1 - v_{*} / c > 0$ . The differentiability vocabulary and the $C^{k}$ vocabulary used throughout are [F2] and [F3].

1.2F2F4F5algebra

The norm term. Fix $(x, t) \in Ω ∖ Γ$ with root $s$ and put $R = x - z (s)$ , $r = ∥ R ∥_{2} > 0$ , $n = R / r$ ; I show that $ψ (s^{'}) : = ∥ x - z (s^{'}) ∥_{2}$ is differentiable at $s^{'} = s$ with $ψ^{'} (s) = - ⟨ z^{'} (s), R ⟩ / r$ . For $h \neq 0$ put $V_{h} : = \frac{z (s + h) - z (s)}{h}$ ; by [F2] one has $V_{h} \to z^{'} (s)$ as $h \to 0$ , so with $M : = ∥ z^{'} (s) ∥_{2} + 1$ there is $δ > 0$ with $∥ V_{h} ∥_{2} \leq M$ for $0 < ∣ h ∣ < δ$ ; shrinking $δ$ if necessary so that $M δ < r / 2$ , the vector $w_{2} : = x - z (s + h)$ satisfies $∥ w_{2} - R ∥_{2} = ∥ z (s + h) - z (s) ∥_{2} = ∣ h ∣ ∥ V_{h} ∥_{2} \leq M ∣ h ∣$ and $∥ w_{2} ∥_{2} \geq r - M ∣ h ∣ > r / 2$ . Writing $w_{1} : = R$ , the identity $∥ w_{2} ∥_{2} - ∥ w_{1} ∥_{2} = \frac{⟨ w_{2} - w_{1}, w_{2} + w_{1} ⟩}{∥ w_{2} ∥_{2} + ∥ w_{1} ∥_{2}}$ holds, because $∥ w_{2} ∥_{2}^{2} - ∥ w_{1} ∥_{2}^{2} = ⟨ w_{2} + w_{1}, w_{2} - w_{1} ⟩$ by bilinearity and symmetry and the denominator is positive [F4]; since $w_{2} - w_{1} = - h V_{h}$ , dividing by $h$ gives $\frac{ψ (s + h) - ψ (s)}{h} = - ⟨ V_{h}, U_{h} ⟩$ with $U_{h} : = \frac{w_{1} + w_{2}}{∥ w_{1} ∥_{2} + ∥ w_{2} ∥_{2}}$ . The triangle inequality gives $∥ U_{h} ∥_{2} \leq 1$ , and $∥ U_{h} - n ∥_{2} = \frac{∥ r w_{2} - ∥ w_{2} ∥_{2} w_{1} ∥_{2}}{r (r + ∥ w_{2} ∥_{2})} \leq \frac{2 ∥ w_{2} - w_{1} ∥_{2}}{r + ∥ w_{2} ∥_{2}} \leq \frac{2 M ∣ h ∣}{r}$ , where the middle step is the reverse triangle inequality [F5]. Cauchy-Schwarz then gives $∣ ⟨ V_{h}, U_{h} ⟩ - ⟨ z^{'} (s), n ⟩ ∣ \leq ∥ V_{h} - z^{'} (s) ∥_{2} ∥ U_{h} ∥_{2} + ∥ z^{'} (s) ∥_{2} ∥ U_{h} - n ∥_{2} \leq ∥ V_{h} - z^{'} (s) ∥_{2} + ∥ z^{'} (s) ∥_{2} \frac{2 M ∣ h ∣}{r}$ , which tends to $0$ as $h \to 0$ ; hence the difference quotients converge and $ψ^{'} (s) = - ⟨ z^{'} (s), n ⟩ = - n \cdot z^{'} (s)$ [F5].

1.3F1givenalgebra

The ray family. Fix $s_{0} \in R$ and $n \in R^{3}$ with $∥ n ∥_{2} = 1$ , and let $r > 0$ ; put $x_{r} : = z (s_{0}) + r n$ and $t_{r} : = s_{0} + r / c$ . Then $t_{r} - s_{0} = r / c = ∥ x_{r} - z (s_{0}) ∥_{2} / c$ , so $s_{0}$ solves the retarded equation at the event $(x_{r}, t_{r})$ ; by the uniqueness in [F1] applied at that event the root is $s (x_{r}, t_{r}) = s_{0}$ , and since $s_{0} < t_{r}$ , the equality clause of [F1], which says that the root equals $t$ exactly when the observation point is $z (t)$ , gives $x_{r} \neq z (t_{r})$ , that is $(x_{r}, t_{r}) \in Ω ∖ Γ$ .

2.1step 1.2F1F2algebra

Emission-time Jacobian. With the fixed $x$ of step 1.2, the function $g_{x} (s^{'}) : = s^{'} + \frac{1}{c} ∥ x - z (s^{'}) ∥_{2}$ is differentiable at $s^{'} = s$ , as the sum of the identity map, whose derivative is $1$ , and the constant multiple $\frac{1}{c} ψ$ of the differentiable map $ψ$ of step 1.2, and its derivative is $g_{x}^{'} (s) = 1 - \frac{1}{c} n \cdot z^{'} (s) = 1 - n \cdot β = κ (x, t)$ , using $β = z^{'} (s) / c$ and $κ = 1 - n \cdot β$ from step 1.1; this is exactly the emission-time Jacobian $1 - n \cdot β$ asserted in clause (a).

2.2F6F7F8F12step 1.1given

Clause (b). The definitions of the retarded-history pairings in clause (b) coincide with [F8], with the same signed constant $C = q / (4 π ε_{0})$ , the same $β = z^{'} (s) / c$ and $κ = 1 - n \cdot β$ of step 1.1, and the same shifted-kernel pairing $(K ⊛ ψ)$ of [F7]; the point-source distributions and their continuity equation are [F6], whose proper-graph hypothesis is automatic for the all-time history $I = R$ . The supplier [F8] proves, under ${A C}_{ω}$ [F12], that the pairings are finite for every test, that the displayed exact identities hold, and that $Φ$ and $Υ$ are the regular distributions of the locally integrable zero extensions of $φ = C / (κ r)$ and $A = β φ / c$ , which are $C^{2}$ on $Ω ∖ Γ$ by the supplier's regularity branch fed by the $C^{3}$ root of step 1.1; this is the whole content of clause (b).

2.3step 1.1F5F8F9F10F11algebra

Clause (c). By step 1.1 and [F8] the pair $(φ, A)$ is a $C^{2}$ potential pair on $Ω ∖ Γ$ , so [F10] identifies its fields as $E = - \nabla φ - \partial_{t} A$ and $B = curl A$ , both $C^{1}$ ; the supplier's formula [F9] gives the displayed $E$ and $B$ , whose acceleration numerator is the cross product of [F11], and splitting the bracket defines $E_{v e l}$ and $E_{a c c}$ as in the Statement. For the bound, at every event $∥ n ∥_{2} = 1$ and $∥ β ∥_{2} = ∥ z^{'} (s) ∥_{2} / c \leq v_{*} / c$ , so $∣ 1 - ∥ β ∥_{2}^{2} ∣ \leq 1$ and the triangle inequality [F5] gives $∥ n - β ∥_{2} \leq 1 + ∥ β ∥_{2} \leq 1 + v_{*} / c$ ; with $κ \geq 1 - v_{*} / c > 0$ from step 1.1 this yields $κ^{3} \geq (1 - v_{*} / c)^{3}$ and hence $∥ E_{v e l} (x, t) ∥_{2} \leq ∣ C ∣ \frac{1 + v_{*} / c}{(1 - v_{*} / c)^{3}} r^{- 2}$ , the signed constant $C$ being kept in the formula and $∣ C ∣$ used in the bound.

3.1F5F11step 1.3step 2.3algebra

The far-zone expansion. At the event $(x_{r}, t_{r})$ of step 1.3 the retarded data are the root $s_{0}$ , the separation $x_{r} - z (s_{0}) = r n$ , the retarded distance $∥ x_{r} - z (s_{0}) ∥_{2} = r$ , which is the ray parameter, the unit vector $n$ , and $β = β_{0} = z^{'} (s_{0}) / c$ , $\dot{β} = {\dot{β}}_{0} = z^{''} (s_{0}) / c$ , $κ = κ_{0} = 1 - n \cdot β_{0}$ ; hence step 2.3 gives $E (x_{r}, t_{r}) = \frac{C n \times ((n - β_{0}) \times {\dot{β}}_{0})}{c κ_{0}^{3} r} + ρ_{E} (s_{0}, n, r)$ with $ρ_{E} (s_{0}, n, r) : = E_{v e l} (x_{r}, t_{r})$ and $B (x_{r}, t_{r}) = \frac{n \times E (x_{r}, t_{r})}{c}$ , while the uniform bound of step 2.3 gives $∣ ρ_{E} (s_{0}, n, r) ∣ \leq ∣ C ∣ \frac{1 + v_{*} / c}{(1 - v_{*} / c)^{3}} r^{- 2}$ for every unit $n$ and every $r > 0$ . Multiplying by $r$ yields $∥ r E (x_{r}, t_{r}) - \frac{C n \times ((n - β_{0}) \times {\dot{β}}_{0})}{c κ_{0}^{3}} ∥_{2} \leq ∣ C ∣ \frac{1 + v_{*} / c}{(1 - v_{*} / c)^{3}} \frac{1}{r}$ , which tends to $0$ as $r \to \infty$ with a constant depending only on $C$ , $v_{*}$ and $c$ ; hence $r E (x_{r}, t_{r}) \to \frac{C n \times ((n - β_{0}) \times {\dot{β}}_{0})}{c κ_{0}^{3}}$ uniformly over the unit directions $n$ at the fixed emission event $s_{0}$ , at the explicit rate $O (r^{- 1})$ , and no bound on $z^{''}$ enters the remainder.

4.1givenstep 1.1step 1.3step 2.1step 2.2step 2.3step 3.1∎

Assembly. Steps 1.1 and 2.1 give clause (a): at every event off the worldline there is a unique retarded root strictly below the observation time, the root map is $C^{3}$ , $\partial_{t} s = κ^{- 1}$ , $\nabla_{x} s = - n / (c κ)$ and $1 - v_{*} / c \leq κ \leq 1 + v_{*} / c$ , and the emission-time Jacobian is $1 - n \cdot β = κ$ ; step 2.2 gives clause (b) with the finiteness of the history pairings, the exact pairing identities and the identification of $Φ, Υ$ as the regular distributions of the zero extensions of the $C^{2}$ potentials $φ = C / (κ r)$ and $A = β φ / c$ ; step 2.3 gives clause (c) with the velocity/acceleration split, the $C^{1}$ regularity and the uniform velocity-field bound; and steps 1.3 and 3.1 give clause (d) with $s (x_{r}, t_{r}) = s_{0}$ , the expansion, the direction-uniform remainder bound and the equivalent $O (r^{- 1})$ limit of $r E$ . All hypotheses are those of the Statement, ${A C}_{ω}$ is assumed and used only through clause (b), the signed constant $C$ is retained in the field formulas while the bounds use $∣ C ∣$ , and no self-force, radiation-reaction, Maxwell-coupling, finite-energy, decay or flux conclusion is asserted.

5 · Examples, experiments and counterexamples

None yet.