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Retarded point fields off the worldline: Lienard-Wiechert potentials, the velocity/acceleration split and the uniform far-zone remainder

Physical scope

Retarded potentials and fields of a prescribed point charge with charge parameter $q \in R$ on a uniformly subluminal $C^{3}$ worldline, off the worldline: exact Lienard-Wiechert split, $B = n \times E / c$ , the velocity-field bound and the uniform-in-direction far-zone remainder at a fixed emission event; and the constructed proper point-source weak Maxwell solution with $L_{l o c}^{1}$ field representatives and distributional Lorenz-gauge potentials. Excluded: field values on the worldline and any nonlinear self-interaction (no products of distributions, no self-force, no radiation reaction); energy, flux or power (deferred to the emitted-power items of this page); uniform bounds in the emission event; and every empirical or measurement claim.

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) as carried by the point-source change-of-variables suppliers below (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let the laboratory region, chart and 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 ( $c > 0$ , $μ_{0} > 0$ , $ε_{0} = (μ_{0} c^{2})^{- 1}$ ), let $q \in R$ and let $z$ , $v_{*}$ be a retarded point-geometry datum of Uniformly subluminal prescribed worldline, retarded root and retarded geometric data off the worldline (a prescribed $C^{3}$ uniformly subluminal worldline of a point charge, Worldlines, four-velocity, chart readings and the frame/observer distinction, with the constant $C : = q / (4 π ε_{0})$ ), write $Ω : = R^{3} \times R$ and $Γ : = {(z (s), s) : s \in R}$ , and let $ρ, J$ be the point-source distributions of that charge on the worldline in the declared point-source class of Distributional Maxwell equations with point and stationary-sheet sources(b) (the $C^{3}$ branch of the point-current construction). Let $(φ, A)$ be the retarded history pairings of Retarded potentials of a point history as the retarded convolution of the point sources, and the off-worldline Lienard-Wiechert fields(b) -- the framework's retarded construction for this source under the adopted selection Adopted retarded solution selection: the physical branch carries no incoming free field -- and define $E : = - \nabla φ - \partial_{t} A$ , $B : = \nabla \times A$ on the off-worldline domain $Ω ∖ Γ$ (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression, The electric and magnetic fields as fixed-time spatial vector fields). Then:

(a) [Potentials] $φ = C / (κ r)$ and $A = β φ / c$ on $Ω ∖ Γ$ , where $r, κ, β$ are the retarded data of Uniformly subluminal prescribed worldline, retarded root and retarded geometric data off the worldline(b); the zero extensions of $φ, A$ to $Γ$ are locally integrable on $Ω$ .

(b) [Exact fields] On $Ω ∖ Γ$ , $E = E_{v e l} + E_{a c c}, E_{v e l} = \frac{C (1 - ∣ β ∣^{2}) (n - β)}{κ^{3} r^{2}}, E_{a c c} = \frac{C n \times ((n - β) \times \dot{β})}{c κ^{3} r}, B = \frac{n \times E}{c},$ and $E, B$ are of class $C^{1}$ there; the velocity part obeys $∣ E_{v e l} ∣ \leq ∣ C ∣ (1 + v_{*} / c) (1 - v_{*} / c)^{- 3} r^{- 2}$ .

(c) [Uniform far-zone remainder] For every emission event $s_{0}$ and direction $n \in S^{2}$ , with $x_{r} : = z (s_{0}) + r n$ , $t_{r} : = s_{0} + r / c$ , $β_{0} : = z^{'} (s_{0}) / c$ , ${\dot{β}}_{0} : = z^{''} (s_{0}) / c$ and $κ_{0} : = 1 - n \cdot β_{0}$ : $(x_{r}, t_{r}) \in Ω ∖ Γ$ , $s (x_{r}, t_{r}) = s_{0}$ , and $E (x_{r}, t_{r}) = \frac{C n \times ((n - β_{0}) \times {\dot{β}}_{0})}{c κ_{0}^{3} r} + ρ_{E} (s_{0}, n, r), ∣ ρ_{E} (s_{0}, n, r) ∣ \leq ∣ C ∣ \frac{1 + v_{*} / c}{(1 - v_{*} / c)^{3}} r^{- 2}, B (x_{r}, t_{r}) = \frac{n \times E (x_{r}, t_{r})}{c};$ the bound is uniform in $n$ at fixed $s_{0}$ .

(d) [Scope] These are the stipulated retarded model fields of the prescribed charge: the item identifies, through the same adopted causal selection used by Retarded potentials, Lorenz gauge and Maxwell fields of a compactly supported conserved smooth source for the compact branch, the point-source potentials and fields of the model, and proves their exact form and bounds. It asserts no value of the fields on the worldline $Γ$ (where the formulas are undefined and the self-field is singular), no integral or energy statement, no self-force and no radiation reaction, and derives no mass. The uniform bound $v_{*} < c$ and the exclusion of $Γ$ are essential hypotheses; the companion example page shows that a merely pointwise subluminal curve can fail to have a retarded root, and no uniformity in the emission event $s_{0}$ is claimed ( $z^{''}$ is not assumed bounded). The signed constant $C$ is retained in every displayed formula while absolute-value bounds use $∣ C ∣$ .

(e) [Constructed weak solution] Let $Φ, Υ$ be the global regular potential distributions in (a), and let $E = - \nabla Φ - \partial_{t} Υ$ , $B = curl Υ$ . By The retarded point-history pair solves wave, Lorenz and Maxwell equations distributionally, with locally integrable field representatives off the worldline, their representatives are exactly the locally integrable zero extensions of the displayed off-worldline fields. They satisfy all four identities of Distributional Maxwell equations with point and stationary-sheet sources(e) with the proper all-time point pair $ρ, J$ ; the volume source parts are zero. Thus they are an admitted point-source model of the explicit physical adoption Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources, under the same retarded selection. The potentials satisfy $W Φ = ρ / ε_{0}$ , $W Υ = μ_{0} J$ and the distributional Lorenz identity. This existence and solution property is supplied by the mathematical theorem, not by the selection postulate or by the compact-source theorem. The zero extensions on the null graph give no physically defined self-field and no nonlinear products there. All original exact field, regularity and uniform far-zone branches remain as stated. On each product neighborhood disjoint from $Γ$ the fields satisfy the smooth source-free Maxwell equations. Consequently any use of Local and fixed-volume electromagnetic energy balance by the radiation items is local on such neighborhoods, with $ρ = J = 0$ ; it neither applies that smooth theorem across the point source nor forms self-work or a global finite-energy identity.

Facts & Assumptions

Given: ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )); the laboratory chart, units and constants 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; a retarded point-geometry datum: $q \in R$ , $z : R \to R^{3}$ of class $C^{3}$ with $∣ z^{'} (s) ∣ \leq v_{*} < c$ , and the off-worldline domain $Ω ∖ Γ$ with the retarded data $s, r, n, β, κ, \dot{β}$ of Uniformly subluminal prescribed worldline, retarded root and retarded geometric data off the worldline; the constant $C = q / (4 π ε_{0})$ .

[F1]

Retarded point geometry: on $Ω ∖ Γ$ the root $s = s (x, t) < t$ solves $t - s = ∣ x - z (s) ∣ / c$ , is $C^{3}$ , satisfies $\partial_{t} s = 1 / κ$ and $\nabla_{x} s = - n / (c κ)$ , and $κ = 1 - n \cdot β \in [1 - v_{*} / c, 1 + v_{*} / c]$ ; at an emission event $s_{0}$ and direction $n \in S^{2}$ , the event $(x_{r}, t_{r}) = (z (s_{0}) + r n, s_{0} + r / c)$ lies in $Ω ∖ Γ$ and has root $s (x_{r}, t_{r}) = s_{0}$ . (Uniformly subluminal prescribed worldline, retarded root and retarded geometric data off the worldline (a)-(c))

[F2]

Point-source history pairings: for the point-source distributions $ρ, J$ attached to $q$ and $z$ , the retarded-history pairings $Φ, Υ$ define distributions on $Ω$ , are finite for every test, and satisfy $⟨ Φ, ψ ⟩ = \int_{Ω ∖ Γ} C ψ / (κ r)$ and $⟨ Υ, ψ ⟩ = \int_{Ω ∖ Γ} (β / c) C ψ / (κ r)$ componentwise; consequently $φ = C / (κ r)$ and $A = β φ / c$ are $C^{2}$ functions off the worldline whose locally integrable zero extensions have $Φ, Υ$ as their regular distributions. (Retarded potentials of a point history as the retarded convolution of the point sources, and the off-worldline Lienard-Wiechert fields (b))

[F3]

Exact fields: with $E = - \nabla φ - \partial_{t} A$ and $B = curl A$ on $Ω ∖ Γ$ , one has $E = E_{v e l} + E_{a c c}$ with the displayed velocity and acceleration parts, $B = n \times E / c$ , both fields $C^{1}$ off the worldline, and $∣ E_{v e l} ∣ \leq ∣ C ∣ (1 + v_{*} / c) (1 - v_{*} / c)^{- 3} r^{- 2}$ . (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 (c))

[F4]

Uniform far-zone remainder: at the emission sphere $E (x_{r}, t_{r}) = C n \times ((n - β_{0}) \times {\dot{β}}_{0}) / (c κ_{0}^{3} r) + ρ_{E}$ with $∣ ρ_{E} ∣ \leq ∣ C ∣ (1 + v_{*} / c) (1 - v_{*} / c)^{- 3} r^{- 2}$ uniformly over $n \in S^{2}$ at fixed $s_{0}$ , and $B (x_{r}, t_{r}) = n \times E (x_{r}, t_{r}) / c$ . (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 (d))

[F5]

Distributional point-source Maxwell theorem: $W Φ = ρ / ε_{0}$ , $W Υ = μ_{0} J$ and $div Υ + c^{- 2} \partial_{t} Φ = 0$ in $D^{'} (Ω)$ ; the distributions $E = - \nabla Φ - \partial_{t} Υ$ , $B = curl Υ$ satisfy the four distributional Maxwell identities with the point pair; and off $Γ$ they restrict to the exact $C^{1}$ fields of [F3], the locally integrable zero extensions of the field formulas being their regular representatives. (The retarded point-history pair solves wave, Lorenz and Maxwell equations distributionally, with locally integrable field representatives off the worldline (a)-(c))

[F6]

Weak source class and solution predicate: the all-time $C^{3}$ uniformly subluminal worldline with its proper graph is an admitted point source, the predicate consists of the four vanishing distributional pairings with the source pairings of the declared point class, and no product of two distributions is formed anywhere. (Distributional Maxwell equations with point and stationary-sheet sources (b),(e),(f))

[F7]

Adopted singular-source physics: point-source models satisfying the weak predicate are the adopted physical models of the separate postulate, which the earlier smooth-source postulate neither covers nor implies. (Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources)

[F8]

Adopted selection: the retarded branch with no free addition is the framework's selected branch, and for the point-source branch the same selection is adopted in the weak equations of the singular-source model. (Adopted retarded solution selection: the physical branch carries no incoming free field (b),(c))

[F9]

Conventions: $E = - \nabla φ - \partial_{t} A$ and $B = \nabla \times A$ are those of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression with field classes of The electric and magnetic fields as fixed-time spatial vector fields; $ε_{0} = (μ_{0} c^{2})^{- 1}$ and $C = q / (4 π ε_{0})$ are the SI conventions of Framework-local SI units, the vacuum constants and the exact/measured status of their values; the curve $s \mapsto (z (s), s)$ is the laboratory reading of a prescribed massive future timelike worldline by Worldlines, four-velocity, chart readings and the frame/observer distinction. (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression, The electric and magnetic fields as fixed-time spatial vector fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values, Worldlines, four-velocity, chart readings and the frame/observer distinction)

[F10]

The same adopted causal selection is used by the compact-source theorem Retarded potentials, Lorenz gauge and Maxwell fields of a compactly supported conserved smooth source for its branch, and that theorem explicitly leaves point sources to the point-source items of its page; the uniform speed bound and the exclusion of $Γ$ are essential hypotheses of the point construction. (Retarded potentials, Lorenz gauge and Maxwell fields of a compactly supported conserved smooth source (b),(g))

[F11]

Local Poynting balance: for jointly $C^{1}$ fields solving the smooth source-free vacuum Maxwell equations on a region, the local identity $\partial_{t} u + div S = - J \cdot E$ of Local and fixed-volume electromagnetic energy balance holds there, and on a product neighborhood disjoint from $Γ$ the point pair restricts to zero. (Local and fixed-volume electromagnetic energy balance (a),(c))

Proof

technique · direct identification; clauses (a)-(d) instantiate the retarded point-geometry datum and the two point-source suppliers, and clause (e) instantiates the distributional point-source theorem and the weak solution predicate, with the physical adoption supplied separately by the postulates

1.1F1F2F9algebra

Potentials (a). By [F1] the retarded data are defined, with $s, r, n$ of class $C^{3}$ and $β, κ$ of class $C^{2}$ on $Ω ∖ Γ$ with $κ \geq 1 - v_{*} / c > 0$ , and $C = q / (4 π ε_{0})$ is the stated signed constant [F9]. Clause (b) of [F2] identifies the retarded-history pairings with the exact integrals $\int_{Ω ∖ Γ} C ψ / (κ r)$ and $\int_{Ω ∖ Γ} (β / c) C ψ / (κ r)$ , and states that the potentials are the $C^{2}$ functions $φ = C / (κ r)$ and $A = β φ / c$ off the worldline whose zero extensions to $Γ$ are locally integrable. This is (a).

2.1step 1.1F3F9algebra

Exact fields (b). Clause (c) of [F3] gives, at every event of $Ω ∖ Γ$ , the split $E = E_{v e l} + E_{a c c}$ with the displayed velocity and acceleration parts, the relation $B = n \times E / c$ , the $C^{1}$ regularity of both fields, and the velocity bound $∣ E_{v e l} ∣ \leq ∣ C ∣ (1 + v_{*} / c) (1 - v_{*} / c)^{- 3} r^{- 2}$ ; the fields are defined as $E = - \nabla φ - \partial_{t} A$ and $B = curl A$ with the conventions of [F9], and the constant $C$ is retained with its sign in the field formulas while the bound uses $∣ C ∣$ . This is (b).

2.2step 1.1F1F4algebra

Uniform far-zone remainder (c). Fix $s_{0} \in R$ and $n \in S^{2}$ ; by F1 the event $(x_{r}, t_{r}) = (z (s_{0}) + r n, s_{0} + r / c)$ lies in $Ω ∖ Γ$ and its root is $s (x_{r}, t_{r}) = s_{0}$ , so the retarded data at that event are exactly $β_{0}, {\dot{β}}_{0}, κ_{0}$ and $r$ . Clause (d) of [F4] then gives the displayed decomposition of $E (x_{r}, t_{r})$ with the uniform bound $∣ ρ_{E} (s_{0}, n, r) ∣ \leq ∣ C ∣ (1 + v_{*} / c) (1 - v_{*} / c)^{- 3} r^{- 2}$ over $n \in S^{2}$ at fixed $s_{0}$ , together with $B (x_{r}, t_{r}) = n \times E (x_{r}, t_{r}) / c$ . This is (c).

3.1step 1.1step 2.2F8F10algebra

Scope (d). The identification of the retarded construction of clause (a)-(b) with the model fields of the prescribed charge is the adopted causal selection of [F8], the same selection used by the compact-source theorem [F10]; nothing in steps 1.1-2.2 assigns a field value on $Γ$ (the formulas require $r > 0$ ), forms a product of the singular source with a field, or proves a self-force, radiation-reaction, energy or mass statement, and [F10] records that the compact-source branch explicitly leaves point sources to this page. The bound in (c) is uniform in $n$ at fixed $s_{0}$ only, since ${\dot{β}}_{0}$ and $κ_{0}^{- 3}$ depend on the emission event and $z^{''}$ is not assumed bounded. This is (d).

4.1step 1.1step 2.1step 2.2step 3.1F5F6F7F8F11algebra∎

Constructed weak solution (e) and conclusion. By F5 the potential distributions of [F2] satisfy $W Φ = ρ / ε_{0}$ , $W Υ = μ_{0} J$ and the distributional Lorenz identity, and by F5 the field distributions $E, B$ satisfy all four distributional Maxwell identities with the point pair $ρ, J$ ; F5 identifies their representatives with the locally integrable zero extensions of the off-worldline fields of [F3] and records that the off-worldline $C^{1}$ fields satisfy the source-free classical Maxwell equations. The all-time $C^{3}$ uniformly subluminal worldline with its proper graph is an admitted point source of the declared class [F6], the source pairings are those of the predicate, and the volume source parts vanish; hence the four identities are exactly the solution predicate of F6 with $(ρ, J)$ , and by the explicit physical adoption [F7] under the same retarded selection [F8] these constructed weak fields are the model fields of the point charge. This is the solution property claimed in (e); it is supplied by the mathematical theorem and not by the selection postulate. No product of distributions is formed anywhere (F6), and the zero extension on the null graph gives no physically defined self-field there. Finally, on a product neighborhood disjoint from $Γ$ the point pair restricts to the zero source so that the smooth source-free Maxwell equations of F5 hold and the local Poynting identity [F11] applies there with $ρ = J = 0$ ; it is never applied across the point source and forms no self-work or global finite-energy identity. This completes (e) and, with steps 1.1-3.1, all clauses (a)-(e).

Retarded point fields off the worldline: Lienard-Wiechert potentials, the velocity/acceleration split and the uniform far-zone remainder

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources