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Causal kernels, gauge and point fields after wave existence

1 · Prerequisites

2 · Summary

This page supplies the mathematical layer that sits after wave existence and before any physical adoption of the electromagnetic potentials: the vocabulary of scalar-vector potential pairs, the retarded kernel, gauge reachability, the forced Maxwell Cauchy system, and the point-history potentials and fields with their distributional Maxwell equations. Everything here carries domain: mathematics and depends only on mathematical suppliers; no postulate, observation or empirical claim is made, and physical use remains with the consuming electromagnetism pages.

The page opens with Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression, which fixes the conventions once and for all: a potential pair of class $C^{k}$ on an open $Ω \subseteq R^{3} \times R$ , the extension sense on closed time slabs, the fields $E = - \nabla φ - \partial_{t} A$ and $B = curl A$ , the Lorenz divergence $G = div A + c^{- 2} \partial_{t} φ$ , the gauge transform $T_{χ} (φ, A) = (φ - \partial_{t} χ, A + \nabla χ)$ and the normalization $W = c^{- 2} \partial_{t}^{2} - Δ$ . The definition asserts no existence or solvability statement; those come with their own hypotheses.

The retarded wave kernel is a distribution supported on the cone, solves the point-wave equation, and convolves to the retarded solution constructs the retarded kernel $K (ψ) = (4 π)^{- 1} \int ψ (y, ∣ y ∣ / c) ∣ y ∣^{- 1} d y$ as a distribution and proves that its support is exactly the retarded cone, that the convolution $K * h$ with a compactly supported smooth $h$ is a $C^{\infty}$ function solving $W (K * h) = h$ pointwise, and that $W K = δ_{(0, 0)}$ distributionally; no smoothness of $K$ itself on the cone is claimed. Gauge reachability of Lorenz gauge by the free-space wave solution, with field invariance then uses the free-space wave solution with zero Cauchy data and source $G$ to reach Lorenz gauge from an arbitrary smooth pair on a slab, preserving the fields, and records the homogeneous wave equation satisfied by differences; boundary-value gauge fixing on bounded spatial domains is outside it. The smooth forced Maxwell Cauchy system on a slab: existence, uniqueness, Gauss constraints and the cone of dependence treats the stipulated smooth first-order system $\partial_{t} E = c^{2} curl B - J / ε$ , $\partial_{t} B = - curl E$ with $ε μ c^{2} = 1$ : it has a unique smooth solution on a slab, the solution obeys both Gauss laws under the stated compatibility of the data, and it obeys the $c$ -cone of dependence.

The last two items pass to a point history. Retarded potentials of a point history as the retarded convolution of the point sources, and the off-worldline Lienard-Wiechert fields computes the retarded pairings of the point current and of the retarded history against the kernel, identifies them with the regular distributions of the locally integrable zero extensions of $φ = C / (κ r)$ and $A = β φ / c$ , and derives the off-worldline Liénard–Wiechert fields with the exact velocity/acceleration split, together with their higher-regularity branches. The retarded point-history pair solves wave, Lorenz and Maxwell equations distributionally, with locally integrable field representatives off the worldline is the distributional completion: the proper point-history pair satisfies the wave, Lorenz and Maxwell equations in $D^{'} (R^{3} \times R)$ with the exact point source, the fields have locally integrable representatives whose regular distributions are those vector distributions, and the off-worldline $C^{1}$ fields satisfy the source-free classical Maxwell equations pointwise. No singular products, field values on the worldline, finite self-energy or self-force law are asserted. The companion examples page em-causal-wave-kernel-prerequisites-examples records the explicit static, gauge and support checks of these conventions.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableprecheck passjudge pass (gpt-6.1-sol)Open item page →

Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression

Definition

Let $c > 0$ and let $Ω \subseteq R^{3} \times R$ be open. A scalar-vector potential pair of class $C^{k}$ on $Ω$ , for an integer $k \geq 1$ , is a pair $(φ, A)$ with $φ : Ω \to R$ and $A : Ω \to R^{3}$ each of class $C^{k}$ in the sense of $C^{k}$ Euclidean maps and diffeomorphisms, the components of $A$ being of class $C^{k}$ separately; the pair is smooth when both members are $C^{\infty}$ , and continuous, or of class $C^{0}$ , when both members are continuous. A pair defined on a set of the form $R^{3} \times [t_{0}, T]$ with $t_{0} < T$ is of class $C^{k}$ , $k \geq 1$ , in the extension sense when there are an open interval $J \supset [t_{0}, T]$ and a potential pair of class $C^{k}$ on the open set $R^{3} \times J$ whose restriction to $R^{3} \times [t_{0}, T]$ the given pair is; the same wording is used for $C^{\infty}$ and for the single functions $φ$ and $χ$ below.

All operators $\nabla$ , $div$ , $curl$ and $Δ$ below act on the three spatial coordinates with $t$ fixed: for $Ω_{t} : = {x \in R^{3} : (x, t) \in Ω}$ , apply the cited three-dimensional operators to the restrictions $x \mapsto φ (x, t)$ , $x \mapsto A (x, t)$ or $x \mapsto u (x, t)$ on the open set $Ω_{t}$ . In particular $\nabla φ = (\partial_{x_{1}} φ, \partial_{x_{2}} φ, \partial_{x_{3}} φ)$ and $Δ u = \sum_{i = 1}^{3} \partial_{x_{i}}^{2} u$ ; $\partial_{t}$ differentiates only the time coordinate.

For a potential pair of class $C^{k}$ with $k \geq 1$ its electric field and magnetic field are

$E (φ, A) : = - \nabla φ - \partial_{t} A, B (φ, A) : = curl A,$

where $\nabla$ is the gradient of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, $\partial_{t}$ is the partial derivative of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , and $div$ and $curl$ are those of Divergence and curl of a $C^{1}$ vector field. Both $E (φ, A)$ and $B (φ, A)$ are maps $Ω \to R^{3}$ of class $C^{k - 1}$ .

For $k \geq 1$ the Lorenz divergence of the pair is the function

$G (φ, A) : = div A + c^{- 2} \partial_{t} φ,$

also of class $C^{k - 1}$ , and the pair is in Lorenz gauge on $Ω$ when $G (φ, A) = 0$ throughout $Ω$ .

For $k \geq 2$ and a function $χ : Ω \to R$ of class $C^{k}$ , a gauge function, the gauge transform of the pair is

$T_{χ} (φ, A) : = (φ - \partial_{t} χ, A + \nabla χ),$

again a potential pair, of class $C^{k - 1}$ .

Finally, for a function $u : Ω \to R$ of class $C^{2}$ put

$W u : = c^{- 2} \partial_{t}^{2} u - Δ u,$

with $Δ$ the Laplacian of The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field and $\partial_{t}^{2}$ the second partial derivative in $t$ (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ); this is the normalization $W = c^{- 2} □_{c}$ of the wave operator $□_{c} = \partial_{t}^{2} - c^{2} Δ$ of speed $c$ of Wave equation, Cauchy data and wave speed. For a map $U : Ω \to R^{m}$ of class $C^{2}$ with components $U_{1}, \dots, U_{m}$ , one writes $W U : = (W U_{1}, \dots, W U_{m})$ componentwise.

Nothing in this definition asserts that a gauge function with prescribed auxiliary data exists, that every potential pair is gauge equivalent to a pair in Lorenz gauge, or that $W u = g$ is solvable: those are conclusions with their own hypotheses. The signs $- \nabla φ - \partial_{t} A$ and $A + \nabla χ$ , the factor $c^{- 2}$ in $G$ , and the spelling Lorenz are part of the convention.

Remarks

Regularity is consumed exactly once. If $φ$ and $A$ are $C^{k}$ with $k \geq 1$ , then $\nabla φ$ and the three partial derivatives $\partial_{t} A_{i}$ are $C^{k - 1}$ by the cited definitions of gradient and partial derivative, so the fields $E$ and $B$ and the divergence $G$ are $C^{k - 1}$ ; for $k = 1$ this uses the convention fixed above that class $C^{0}$ means continuity. Likewise a $C^{k}$ gauge function with $k \geq 2$ has $C^{k - 1}$ first partial derivatives, so $T_{χ} (φ, A)$ is of class $C^{k - 1}$ .
Why the pair is defined on an arbitrary open set. The definition is stated for any open $Ω \subseteq R^{3} \times R$ so that the full slab $R^{3} \times (t_{0}, T)$ and the complement of a worldline are both instances; the extension sense above handles closed time slabs.
The name. Lorenz here refers to Ludvig Lorenz, not to Hendrik Lorentz; in four-coordinate notation, set $x^{0} = c t$ , $x^{i} = x_{i}$ for $i = 1, 2, 3$ , $A^{0} = φ / c$ , $A^{i} = A_{i}$ and $\partial_{μ} = \partial / \partial x^{μ}$ . Then $\sum_{μ = 0}^{3} \partial_{μ} A^{μ} = c^{- 2} \partial_{t} φ + div A = G (φ, A)$ , so its vanishing is precisely the defined Lorenz condition. The sign and the factor $c^{- 2}$ belong to the convention and are not consequences of the definition.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Gauge reachability of Lorenz gauge by the free-space wave solution, with field invariance

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ), carried through The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support and its sphere integrations (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $c > 0$ , $t_{0} < T$ , and let $(φ, A)$ be a potential pair of class $C^{\infty}$ on the slab $Q : = R^{3} \times [t_{0}, T]$ in the extension sense of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression: $φ$ and $A$ are restrictions of $C^{\infty}$ functions on $R^{3} \times J$ for some open interval $J \supset [t_{0}, T]$ ( $C^{k}$ Euclidean maps and diffeomorphisms). Put $G : = G (φ, A) = div A + c^{- 2} \partial_{t} φ$ and $W = c^{- 2} \partial_{t}^{2} - Δ$ . Then:

(a) there is a gauge function $χ : R^{3} \times [t_{0}, T] \to R$ of class $C^{\infty}$ in the extension sense with $χ (\cdot, t_{0}) = 0$ , $\partial_{t} χ (\cdot, t_{0}) = 0$ and $W χ = G$ on $R^{3} \times (t_{0}, T)$ ;

(b) the gauge-transformed pair $T_{χ} (φ, A) = (φ - \partial_{t} χ, A + \nabla χ)$ is again of class $C^{\infty}$ in the extension sense and its fields are unchanged:

$E (T_{χ} (φ, A)) = E (φ, A), B (T_{χ} (φ, A)) = B (φ, A) on Q;$

(d) if $χ_{1}, χ_{2}$ are of class $C^{2}$ on $Q$ in the extension sense and $W χ_{1} = W χ_{2} = G$ on $R^{3} \times (t_{0}, T)$ , then $W (χ_{1} - χ_{2}) = 0$ on $R^{3} \times (t_{0}, T)$ .

Nothing is asserted about prescribed boundary values of $χ$ on a bounded spatial domain, about spatial decay of $χ$ , or about Lorenz gauge fixing for a cavity problem with prescribed boundary data; the older global gauge promise on bounded domains is outside this result.

Facts & Assumptions

Given: The constant $c > 0$ , the times $t_{0} < T$ , the potential pair $(φ, A)$ of class $C^{\infty}$ on $Q$ in the extension sense, and the countable-choice hypothesis.

[F1]

For a potential pair, $E (φ, A) = - \nabla φ - \partial_{t} A$ , $B (φ, A) = curl A$ , $G (φ, A) = div A + c^{- 2} \partial_{t} φ$ , the gauge transform is $T_{χ} (φ, A) = (φ - \partial_{t} χ, A + \nabla χ)$ and $W u = c^{- 2} \partial_{t}^{2} u - Δ u$ (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression).

[F2]

For $f$ of class $C^{2}$ , its Laplacian is $Δ f = div \nabla f = \sum_{i < 3} \partial_{i} \partial_{i} f$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field).

[F3]

Let $a, b \in C^{\infty} (R^{3})$ and let $h$ be of class $C^{\infty}$ on $R^{3} \times [t_{0}, T]$ in the extension sense, and put $W u = c^{- 2} \partial_{t}^{2} u - Δ u$ . Then the function $u = χ$ supplied by the theorem with $a = b = 0$ and $h$ is of class $C^{\infty}$ on $R^{3} \times [t_{0}, T]$ in the extension sense, satisfies $W χ = h$ on $R^{3} \times (t_{0}, T)$ and satisfies $χ (\cdot, t_{0}) = 0$ , $\partial_{t} χ (\cdot, t_{0}) = 0$ ; the theorem assumes the Axiom of Countable Choice, used through its sphere and parameter integrations (The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support).

[F4]

If $ϕ$ is of class $C^{2}$ on an open subset of $R^{3}$ , then $curl \nabla ϕ = 0$ there (The curl of the gradient of a $C^{2}$ function vanishes).

[F5]

If $ϕ$ is of class $C^{2}$ on an open subset of $R^{m}$ , then $\partial_{i} \partial_{j} ϕ = \partial_{j} \partial_{i} ϕ$ for all coordinate indices (Clairaut--Schwarz theorem for continuous second partial derivatives).

[F6]

For $C^{1}$ fields and scalars, $div$ and $curl$ are linear: $div (F + G) = div F + div G$ , $curl (F + G) = curl F + curl G$ , and likewise for scalar multiples (Divergence and curl are linear and satisfy the scalar product rules).

[F7]

The divergence of a $C^{1}$ field and the curl of a $C^{1}$ field in three coordinates are computed pointwise from first partial derivatives, so they gain the regularity of those derivatives (Divergence and curl of a $C^{1}$ vector field).

[F8]

The gradient collects coordinate partial derivatives, $\nabla ϕ = (\partial_{0} ϕ, \partial_{1} ϕ, \partial_{2} ϕ)$ , and a partial derivative is the one-variable derivative along a coordinate line; sums and scalar multiples differentiate termwise (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[A1]

The Axiom of Countable Choice ${A C}_{ω}$ : every at most countable family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1givenF1F7F8

Since $A$ and $φ$ are of class $C^{\infty}$ on $Q$ in the extension sense, the field $div A$ and the partial derivative $\partial_{t} φ$ are of class $C^{\infty}$ on $Q$ in the extension sense: they are finite sums of first partial derivatives of $C^{\infty}$ functions, and first partial derivatives of $C^{\infty}$ functions are again $C^{\infty}$ . Hence $G = div A + c^{- 2} \partial_{t} φ$ is of class $C^{\infty}$ on $Q$ in the extension sense.

2.1step 1.1F3A1

Apply [F3] with $a = b = 0$ and $h = G$ , which is legitimate because $G$ is of class $C^{\infty}$ on the slab in the extension sense; the assumption [A1] is carried by that theorem. This supplies a function $χ$ of class $C^{\infty}$ on $Q$ in the extension sense with $W χ = G$ on $R^{3} \times (t_{0}, T)$ , $χ (\cdot, t_{0}) = 0$ and $\partial_{t} χ (\cdot, t_{0}) = 0$ , proving (a).

3.1step 2.1F1F8

Since $χ$ is of class $C^{\infty}$ in the extension sense, its partial derivatives $\partial_{t} χ$ and $\nabla χ$ are of class $C^{\infty}$ in the extension sense; adding these to the $C^{\infty}$ pair members, $T_{χ} (φ, A) = (φ - \partial_{t} χ, A + \nabla χ)$ is a potential pair of class $C^{\infty}$ on $Q$ in the extension sense.

4.1step 3.1F1F4F6

The magnetic field is unchanged: $curl (A + \nabla χ) = curl A + curl \nabla χ = curl A$ , because $curl$ is linear and $curl \nabla χ = 0$ for the $C^{\infty}$ , hence $C^{2}$ , function $χ$ .

4.2step 3.1F1F5F8

The electric field is unchanged: expanding with the linearity of $\nabla$ and $\partial_{t}$ , $- \nabla (φ - \partial_{t} χ) - \partial_{t} (A + \nabla χ) = - \nabla φ - \partial_{t} A + (\nabla \partial_{t} χ - \partial_{t} \nabla χ)$ , and the bracket vanishes because each of its coordinates is a difference $\partial_{i} \partial_{t} χ - \partial_{t} \partial_{i} χ = 0$ of mixed second partials of the $C^{2}$ function $χ$ .

4.3step 2.1step 3.1F1F2F6F8

The Lorenz divergence of the transformed pair vanishes: $div (A + \nabla χ) + c^{- 2} \partial_{t} (φ - \partial_{t} χ) = div A + div \nabla χ + c^{- 2} \partial_{t} φ - c^{- 2} \partial_{t}^{2} χ = G + Δ χ - c^{- 2} \partial_{t}^{2} χ = G - W χ = 0$ on $R^{3} \times (t_{0}, T)$ , using $Δ χ = div \nabla χ$ and the linearity of $div$ and $\partial_{t}$ .

5.1givenF1F8∎

For (d), if $χ_{1}, χ_{2}$ are $C^{2}$ on $Q$ in the extension sense, then each second partial derivative is additive, $\partial_{i} \partial_{j} (χ_{1} - χ_{2}) = \partial_{i} \partial_{j} χ_{1} - \partial_{i} \partial_{j} χ_{2}$ , by applying the one-variable linearity of differentiation along coordinate lines twice; summing over $i = j$ and applying the definition of $W$ gives $W (χ_{1} - χ_{2}) = W χ_{1} - W χ_{2} = G - G = 0$ on $R^{3} \times (t_{0}, T)$ .

Remarks

The countable-choice hypothesis travels with the conclusion. Parts (a)–(c) consume the existence clause of the in-run wave theorem The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support, which is stated under ${A C}_{ω}$ ; part (d) is a purely local algebraic statement and does not itself use it. No arbitrary-index choice and no dependent choice is used.
What is not proved. The construction is the free-space wave solution with zero Cauchy data and source $G$ ; it produces no control of $χ$ on a bounded spatial domain, no prescribed boundary values and no decay at spatial infinity. In particular it does not supply a gauge in the presence of a cavity with prescribed boundary data.

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The retarded wave kernel is a distribution supported on the cone, solves the point-wave equation, and convolves to the retarded solution

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ), used through the sphere and polar integrations below (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The polar surface set function on the unit sphere, Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma). Let $c > 0$ , write $Q : = R^{3} \times R$ , $D : = D (Q)$ for the test-function space of Test function space d of an open set, and $C : = {(y, t) \in Q : t = ∣ y ∣ / c}$ for the retarded cone. Define the functional $K$ on test functions by

$⟨ K, ψ ⟩ : = \frac{1}{4 π} \int_{R^{3}} \frac{ψ (y, ∣ y ∣ / c)}{∣ y ∣} d y (ψ \in D) .$

Then: (a) the integral converges absolutely for every $ψ \in D$ , the functional $K$ is a distribution on $Q$ (Distribution), $(K, ψ) = 0$ for every $ψ$ with $supp ψ \cap C = \emptyset$ , and for every $(y_{0}, t_{0}) \in C$ and every neighbourhood $U$ of $(y_{0}, t_{0})$ there is $ψ \in D$ with $supp ψ \subseteq U$ and $⟨ K, ψ ⟩ \neq 0$ ; consequently $supp K = C$ . (b) For every $h \in C_{c}^{\infty} (Q)$ the function

$(K * h) (x, t) : = \int_{R^{3}} \frac{h (y, t - ∣ x - y ∣ / c)}{4 π ∣ x - y ∣} d y$

is well defined (the integral converges absolutely), of class $C^{\infty}$ on $Q$ , and for every $ψ \in D$

$⟨ K * h, ψ ⟩ = ⟨ h, K ⊛ ψ ⟩, (K ⊛ ψ) (y, s) : = ⟨ K, ψ (y + \cdot, s + \cdot) ⟩ = \frac{1}{4 π} \int \frac{ψ (y + z, s + ∣ z ∣ / c)}{∣ z ∣} d z;$

moreover $W (K * h) = h$ on $Q$ in the classical pointwise sense, where $W = c^{- 2} \partial_{t}^{2} - Δ$ is the wave expression of Wave equation, Cauchy data and wave speed and The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field. (c) Read $W = c^{- 2} \partial_{t}^{2} - Δ$ as the distributional operator built from Distributional derivative; then $W K = δ_{(0, 0)}$ in the sense of Dirac delta and its derivatives, that is, $(W K) (ψ) = ψ (0, 0)$ for every $ψ \in D$ . This identifies $K$ as the retarded fundamental solution of $W$ ; no smoothness of $K$ itself on the cone is asserted, and the advanced fundamental solution is not treated.

Facts & Assumptions

Given: The speed $c > 0$ , the open set $Q = R^{3} \times R$ , its test space $D$ , the retarded cone $C$ , the functional $K$ of the Statement, and, for parts (b) and (c), a function $h \in C_{c}^{\infty} (Q)$ .

[F1]

$D (Q) = C_{c}^{\infty} (Q)$ is the test-function space with the LF topology, $D_{K}$ denotes the tests supported in a compact $K$ with its Fréchet seminorms $p_{m}$ , and a distribution is a continuous complex-linear functional on $D (Q)$ (Test function space d of an open set, Test function topology, Fixed support test function frechet space, Distribution).

[F2]

The support of a distribution is $supp u = Q ∖ Z$ , where $Z$ is the largest open set on which $u$ vanishes, and $u$ vanishes on an open $V$ exactly when $u (φ) = 0$ for every test with support in $V$ (Support of a distribution).

[F3]

Under ${A C}_{ω}$ the polar measure $σ$ on $S^{n - 1}$ is a finite Borel measure with $\int_{R^{n}} f d λ_{n} = \int_{0}^{\infty} \int_{S^{n - 1}} f (r ω) r^{n - 1} d σ (ω) d r$ for nonnegative Borel $f$ ; in particular $\int_{B_{R}} ∣ z ∣^{- 1} d z = σ (S^{2}) R^{2} / 2$ and $σ (S^{2}) = 4 π$ (The polar surface set function on the unit sphere, Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

[F4]

The chart surface measure on $S^{2}$ equals the polar measure $σ$ (Agreement with the existing polar sphere measure).

[F5]

Let $(X, A, μ)$ and $(Y, B, ν)$ be sigma-finite and $f \in L^{1} (μ \times ν)$ ; then the iterated integrals in either order exist and equal the product integral (Fubini's theorem for L^1 functions on a sigma-finite product).

[F6]

If $φ$ is $C^{1}$ and injective with $φ^{'} \neq 0$ on a neighborhood of $[a, b]$ and $f$ is continuous on an interval containing $φ ([a, b])$ , then $\int_{\min}^{} f = \int_{a}^{b} f (φ) ∣ φ^{'} ∣$ (In one dimension the compact-Jordan formula is substitution over the unoriented image interval with the absolute derivative).

[F7]

Under ${A C}_{ω}$ , for a $C^{1}$ diffeomorphism $T : U \to V$ of open subsets of $R^{m}$ and nonnegative Borel $g : V \to [0, \infty]$ , $\int_{V} g = \int_{U} g (T) ∣ \det D T ∣$ (Borel change of variables from the compact-support formula and Radon uniqueness).

[F8]

Under ${A C}_{ω}$ , for a $C^{1}$ diffeomorphism $T : U \to V$ of open subsets of $R^{m}$ and $f \in L^{1} (V)$ , $\int_{V} f = \int_{U} f (T) ∣ \det D T ∣$ ; translations of $R^{n}$ preserve Lebesgue outer measure, measurability and measure (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions, Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).

[F9]

Dominated convergence: if $f_{n} \to f$ pointwise a.e. and $∣ f_{n} ∣ \leq g$ a.e. with $\int g < \infty$ , then $\int ∣ f_{n} - f ∣ \to 0$ (Dominated convergence).

[F10]

$∣ \int f ∣ \leq \int ∣ f ∣$ for $f \in L^{1}$ (The modulus of an integral is bounded by the integral of the modulus).

[F11]

For measurable $f \geq 0$ , $\int f = 0$ if and only if $f = 0$ almost everywhere (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[F12]

Euclidean balls have positive finite Lebesgue measure, under ${A C}_{ω}$ (Euclidean balls have positive finite Lebesgue measure).

[F13]

Nonnegative integrals are monotone and scale, and $0 \cdot \infty$ is not formed (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F14]

Compact subsets of $R^{n}$ are closed and bounded, closed balls and spheres are compact, and closed boxes are compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, For $n \geq 1$ , every Euclidean closed ball and every Euclidean sphere of positive radius is compact).

[F15]

A continuous real function on a nonempty compact metric space is bounded and attains its maximum and minimum (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

[F16]

A unit-mass smooth bump $η \in C_{c}^{\infty} (R^{4})$ generates the mollifier family $η_{ε} (u) = ε^{- 4} η (u / ε)$ (The mollifier family generated by a unit-mass smooth bump).

[F17]

Under ${A C}_{ω}$ a mollifier family generated by a unit-mass smooth bump is an $L^{1}$ approximate identity (An $L^{1}$ approximate identity on $R^{n}$ , A unit-mass smooth bump generates an $L^{1}$ approximate identity).

[F18]

If $(K_{ε})$ is an $L^{1}$ approximate identity and $f$ is bounded and continuous, then $f * K_{ε} \to f$ uniformly on compacta ( $L^{1}$ approximate identities converge uniformly on compacta for bounded continuous functions).

[F19]

Let $c > 0$ , $t_{0} < T$ , $a, b \in C^{\infty} (R^{3})$ and let $h$ be of class $C^{\infty}$ on $R^{3} \times [t_{0}, T]$ in the extension sense, and put $W = c^{- 2} \partial_{t}^{2} - Δ$ . The function $u (x, t) = \partial_{s} H_{a} (x, t - t_{0}) + H_{b} (x, t - t_{0}) + c^{2} \int_{t_{0}}^{t} H_{h (\cdot, v)} (x, t - v) d v$ , with $H_{φ} (x, s) = s M_{c s} φ (x)$ and $M_{r} φ (x) = \frac{1}{4 π} \int_{S^{2}} φ (x + r n) d Ω (n)$ , is of class $C^{\infty}$ on $R^{3} \times [t_{0}, T]$ in the extension sense, satisfies $W u = h$ , $u (\cdot, t_{0}) = a$ , $\partial_{t} u (\cdot, t_{0}) = b$ , and is the unique such $C^{2}$ function; the theorem assumes ${A C}_{ω}$ (The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support).

[F20]

A locally integrable function $f$ defines the regular functional $u_{f} (φ) = \int f φ$ , well defined with $∣ u_{f} (φ) ∣ \leq \sup_{K} ∣ φ ∣ \int_{K} ∣ f ∣$ for $K = supp φ$ (Regular distribution from a locally integrable function).

[F21]

Distributional derivatives are $⟨ \partial^{α} u, φ ⟩ = (- 1)^{∣ α ∣} ⟨ u, \partial^{α} φ ⟩$ , are continuous linear on $D^{'}$ for the weak and strong topologies, and satisfy $\partial^{α} u_{f} = u_{\partial^{α} f}$ for $f \in C^{k}$ , $∣ α ∣ \leq k$ (the last clause requires ${A C}_{ω}$ ) (Distributional derivative, Distributional differentiation is continuous and commutes).

[F22]

The weak distribution topology is generated by the seminorms $u \mapsto ∣ u (φ) ∣$ , so weak convergence means $u_{i} (φ) \to u (φ)$ for every test (Weak and strong topologies on distributions).

[F23]

$δ_{a} (φ) = φ (a)$ is the Dirac distribution, continuous on each fixed-support stage (Dirac delta and its derivatives).

[F24]

For compact $K \subseteq Ω$ open there is $χ \in C_{c}^{\infty} (Ω)$ with $0 \leq χ \leq 1$ and $χ = 1$ on a neighbourhood of $K$ , in ZF (Test function cutoffs and euclidean localization).

[F25]

The Lebesgue integral is complex-linear on $L^{1}$ (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[A1]

The Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1givenF1F3F10F13F14F15A1

If $ψ = 0$ , the defining integral is zero; otherwise fix $ψ \in D$ with nonempty support. The support of $ψ$ is compact in $Q$ , hence closed and bounded; choose $R > 0$ with $supp ψ \subseteq B_{R} (0) \times [- R, R]$ and put $M : = \sup ∣ ψ ∣ < \infty$ , the supremum taken over the nonempty compact support and finite by the extreme value theorem. Since $∣ ψ (y, ∣ y ∣ / c) ∣ \leq M$ for every $y$ and $∣ y ∣^{- 1}$ is nonnegative, the triangle inequality and the order rules for nonnegative integrals bound $∣ ⟨ K, ψ ⟩ ∣ \leq \frac{M}{4 π} \int_{B_{R}} \frac{d y}{∣ y ∣}$ . The polar formula applied to the nonnegative Borel function $y \mapsto ∣ y ∣^{- 1} 1_{B_{R}} (y)$ gives $\int_{B_{R}} ∣ y ∣^{- 1} d y = \int_{0}^{R} r^{- 1} σ (S^{2}) r^{2} d r = σ (S^{2}) R^{2} / 2 < \infty$ , using $σ (S^{2}) = 4 π$ . Hence the defining integral converges absolutely, and the bound is a fixed-support seminorm, so $K$ is continuous for the LF topology and $K \in D^{'}$ .

1.2givenF3F10F13F14F15

Fix $h \in C_{c}^{\infty} (Q)$ and $(x, t) \in Q$ . Since $supp h$ is compact, there is $ρ > 0$ with $supp h \subseteq {\overline{B}}_{ρ} (0) \times [- ρ, ρ]$ , and $∥ h ∥_{\infty} < \infty$ by the extreme value theorem. If $h (y, t - ∣ x - y ∣ / c) \neq 0$ , then $y \in {\overline{B}}_{ρ} (0)$ and $∣ t - ∣ x - y ∣ / c ∣ \leq ρ$ , so $∣ x - y ∣ \leq c (∣ t ∣ + ρ)$ ; therefore the absolute integral is at most $\frac{∥ h ∥_{\infty}}{4 π} \int_{B (x, c (∣ t ∣ + ρ))} ∣ x - y ∣^{- 1} d y = \frac{∥ h ∥_{\infty}}{4 π} \cdot \frac{σ (S^{2}) c^{2} (∣ t ∣ + ρ)^{2}}{2} = \frac{∥ h ∥_{\infty} c^{2} (∣ t ∣ + ρ)^{2}}{2} < \infty$ by the polar formula and the order rules. Hence $(K * h) (x, t)$ is well defined by an absolutely convergent integral for every $(x, t) \in Q$ .

1.3givenF3F13F14

Fix $ψ \in D$ and put $F : = K ⊛ ψ$ , that is, $F (y, s) = \frac{1}{4 π} \int ψ (y + z, s + ∣ z ∣ / c) ∣ z ∣^{- 1} d z$ . Choose $M > 0$ with $supp ψ \subseteq {\overline{B}}_{M} (0) \times [- M, M]$ and let $A : = ∥ ψ ∥_{\infty} < \infty$ . If $∣ y ∣ \leq 2 M$ , every nonzero contribution has $∣ z ∣ \leq M + ∣ y ∣ \leq 3 M$ , so $∣ F (y, s) ∣ \leq \frac{A}{4 π} \int_{B_{3 M}} ∣ z ∣^{- 1} d z = \frac{9}{2} M^{2} A$ ; if $∣ y ∣ > 2 M$ , then $z \in B (- y, M)$ forces $∣ z ∣ \geq ∣ y ∣ - M > M$ , so $∣ F (y, s) ∣ \leq \frac{A}{4 π} \int_{B (- y, M)} ∣ z ∣^{- 1} d z \leq \frac{A}{4 π} \cdot \frac{1}{M} \cdot \frac{4 π}{3} M^{3} = \frac{M^{2} A}{3}$ . Hence $F$ is bounded on all of $Q$ by $\frac{9}{2} M^{2} A$ .

2.1givenstep 1.1F3F11F12F13F24

Fix $(y_{0}, t_{0}) \in C$ and a neighbourhood $U$ of it. Since $C \subseteq Q$ is closed, take an open ball $W$ with $(y_{0}, t_{0}) \in W \subseteq U$ . If $y_{0} \neq 0$ : choose $0 \leq ψ \leq 1$ in $D (Q)$ with $ψ = 1$ on a neighbourhood of $(y_{0}, t_{0})$ and $supp ψ \subseteq W$ . Then $ψ (y, ∣ y ∣ / c) = 1$ for all $y$ in a neighbourhood $V$ of $y_{0}$ (the map $y \mapsto (y, ∣ y ∣ / c)$ is continuous at $y_{0} \neq 0$ ), and $y \mapsto ψ (y, ∣ y ∣ / c) / ∣ y ∣$ is nonnegative, continuous on $V$ , and bounded below by $1 / (2 ∣ y_{0} ∣)$ on a smaller ball around $y_{0}$ of positive measure; hence $⟨ K, ψ ⟩ > 0$ by the criterion that a nonnegative measurable function with positive values on a set of positive measure has positive integral, together with the finiteness of step 1.1 and the positivity of ball measure. If $y_{0} = 0$ , then $t_{0} = 0$ ; choose $0 \leq ψ \leq 1$ in $D (Q)$ with $ψ = 1$ on a neighbourhood of the origin contained in $W$ . Then $ψ (y, ∣ y ∣ / c) = 1$ for $∣ y ∣$ small, and the same argument, using the finiteness of step 1.1 again, gives $⟨ K, ψ ⟩ \geq \frac{1}{4 π} \int_{0 < ∣ y ∣ < δ} ∣ y ∣^{- 1} d y > 0$ for some $δ > 0$ . In both cases $supp ψ \subseteq U$ and $⟨ K, ψ ⟩ \neq 0$ .

2.2step 1.3F3F9F13F14

With $F = K ⊛ ψ$ as in step 1.3, $F$ is continuous at the origin and $F (0, 0) = ⟨ K, ψ ⟩$ . Indeed, for a sequence $(y_{n}, s_{n}) \to (0, 0)$ with $∣ y_{n} ∣ \leq M$ eventually, every nonzero contribution has $z \in B_{2 M}$ , and $ψ (y_{n} + z, s_{n} + ∣ z ∣ / c) ∣ z ∣^{- 1} \to ψ (z, ∣ z ∣ / c) ∣ z ∣^{- 1}$ for every $z \neq 0$ by continuity of $ψ$ ; the dominating function $A ∣ z ∣^{- 1} 1_{B_{2 M}} (z)$ is integrable by the polar formula, so dominated convergence gives $F (y_{n}, s_{n}) \to \frac{1}{4 π} \int ψ (z, ∣ z ∣ / c) ∣ z ∣^{- 1} d z = ⟨ K, ψ ⟩$ ; a sequence criterion for continuity at a point gives the claim.

2.3step 1.1F10F11

Let $ψ \in D$ satisfy $supp ψ \cap C = \emptyset$ . For every $y \neq 0$ the point $(y, ∣ y ∣ / c)$ lies in $C$ , so $ψ (y, ∣ y ∣ / c) = 0$ ; the exceptional set ${0}$ has Lebesgue measure zero. Hence the nonnegative measurable function $y \mapsto ∣ ψ (y, ∣ y ∣ / c) ∣ ∣ y ∣^{- 1}$ vanishes almost everywhere, and its integral is zero by the almost-everywhere criterion; the triangle inequality with the absolute convergence of step 1.1 then gives $⟨ K, ψ ⟩ = 0$ .

2.4step 1.2F3F4F5F6F7F8F13F19F25

Let $t_{0}$ lie below and $T$ above the time support of $h$ , so that $h$ vanishes on $R^{3} \times (- \infty, t_{0}]$ and on $R^{3} \times [T, \infty)$ and its restriction to $R^{3} \times [t_{0}, T]$ is of class $C^{\infty}$ in the extension sense. Apply the wave theorem with $a = b = 0$ and source $h$ : the function $u (x, t) = c^{2} \int_{t_{0}}^{t} (t - v) M_{c (t - v)} h (\cdot, v) (x) d v$ is $C^{\infty}$ on the slab in the extension sense and $W u = h$ on $R^{3} \times (t_{0}, T)$ . Writing the spherical mean as an integral over $S^{2}$ , using Fubini on the compact rectangle $[t_{0}, t] \times S^{2}$ , and substituting $r = c (t - v)$ , with the absolute-derivative one-dimensional change of variables, gives $u (x, t) = \int_{0}^{c (t - t_{0})} r \frac{1}{4 π} \int_{S^{2}} h (x + r n, t - r / c) d Ω (n) d r$ . Since the chart measure on $S^{2}$ is $σ$ , the polar formula applied to the nonnegative Borel function $z \mapsto ∣ h (x + z, t - ∣ z ∣ / c) ∣ (4 π ∣ z ∣)^{- 1} 1_{∣ z ∣ \leq c (t - t_{0})}$ , followed by linearity for the signed integrand, gives $u (x, t) = \int_{∣ z ∣ \leq c (t - t_{0})} \frac{h (x + z, t - ∣ z ∣ / c)}{4 π ∣ z ∣} d z$ ; the omitted range $∣ z ∣ > c (t - t_{0})$ has $t - ∣ z ∣ / c < t_{0}$ , where $h = 0$ , so the integration extends to all $z$ . Finally the translation $y = x + z$ , a $C^{1}$ diffeomorphism of Jacobian determinant one, changes this into $(K * h) (x, t)$ as defined in step 1.2. Thus $u = K * h$ on the slab, and consequently $K * h$ agrees, near each of its points, with a function of class $C^{\infty}$ in the extension sense.

3.1step 2.4F19

Since every point $(x_{*}, t_{*}) \in Q$ lies in the interior of a slab $R^{3} \times [t_{0}, T]$ with $t_{0}$ below and $T$ above the time support of $h$ , step 2.4 shows that $K * h$ agrees with a $C^{\infty}$ function on a neighbourhood of each point; hence $K * h$ is of class $C^{\infty}$ on $Q$ . For such a point, $W (K * h) (x_{*}, t_{*}) = W u (x_{*}, t_{*}) = h (x_{*}, t_{*})$ by the wave theorem applied on that slab, where $W = c^{- 2} \partial_{t}^{2} - Δ$ is evaluated classically; hence $W (K * h) = h$ on all of $Q$ .

3.2step 1.1step 2.1step 2.3F2

The cone $C = {(y, t) : t = ∣ y ∣ / c}$ is closed in $Q$ , since $∣ y ∣$ is continuous, so $Q ∖ C$ is open, and every test supported there has $supp ψ \cap C = \emptyset$ and pairs to zero with $K$ by step 2.3; thus $K$ vanishes on $Q ∖ C$ in the sense of the support definition, and $supp K \subseteq C$ . Conversely let $(y_{0}, t_{0}) \in C$ and suppose it were outside $supp K$ ; then $Q ∖ supp K$ would be an open neighbourhood of $(y_{0}, t_{0})$ on which $K$ vanishes, and step 2.1 would produce a test supported there with $⟨ K, ψ ⟩ \neq 0$ , contradicting the vanishing. Hence $C \subseteq supp K$ , and $supp K = C$ .

4.1step 1.2step 3.1F5F7F8F13F20F25

Let $ψ \in D (Q)$ . By step 3.1, $K * h$ is continuous, hence locally integrable, so $⟨ K * h, ψ ⟩$ is the regular-functional pairing $\int_{(x, t)} (K * h) ψ$ of the formula in step 1.2. For fixed $(x, t)$ the inner integral defining $K * h$ is absolutely convergent by step 1.2, and inserting it gives a triple integral of $h (y, t - ∣ x - y ∣ / c) ψ (x, t) (4 π ∣ x - y ∣)^{- 1}$ over $(y, x, t)$ . This triple integral is absolutely convergent: with $supp h \subseteq {\overline{B}}_{ρ} (0) \times [- ρ, ρ]$ and $supp ψ \subseteq {\overline{B}}_{M} (0) \times [- M, M]$ it is bounded by $\frac{∥ h ∥_{\infty} ∥ ψ ∥_{\infty}}{4 π} \int_{[- M, M]} \int_{{\overline{B}}_{ρ}} \int_{B_{M}} ∣ x - y ∣^{- 1} d x d y d t < \infty$ , the inner double integral being finite by the polar formula. Fubini therefore lets us integrate first in $(x, t)$ for fixed $y$ ; the substitution $(x, t) \mapsto (z, s) = (x - y, t - ∣ x - y ∣ / c)$ is a $C^{1}$ diffeomorphism from ${(x, t) : x \neq y}$ onto $(R^{3} ∖ {0}) \times R$ with determinant one, the omitted set ${x = y}$ being null, so the $L^{1}$ change-of-variables identity gives $\int_{(x, t)} \frac{h (y, t - ∣ x - y ∣ / c) ψ (x, t)}{4 π ∣ x - y ∣} d x d t = \int_{(z, s)} \frac{h (y, s) ψ (y + z, s + ∣ z ∣ / c)}{4 π ∣ z ∣} d z d s$ . Fubini once more and the definition of $K ⊛ ψ$ yield $⟨ K * h, ψ ⟩ = \int_{(y, s)} h (y, s) (K ⊛ ψ) (y, s) d y d s = ⟨ h, K ⊛ ψ ⟩$ .

5.1step 2.2step 3.1step 4.1F16F17F18F20F21F22F23F24A1∎

Let $η \in C_{c}^{\infty} (Q)$ satisfy $η \geq 0$ , $\int η = 1$ and $supp η \subseteq {\overline{B}}_{1} (0)$ : start from [F24] with compact set ${0}$ inside $B_{1} (0)$ , giving a cutoff equal to one on a neighbourhood of the origin and supported in $B_{1} (0)$ , which is nonnegative with positive integral over a ball of positive measure, and normalize it. Let $η_{ε} (u) = ε^{- 4} η (u / ε)$ be the mollifier family, an $L^{1}$ approximate identity under [A1]. Fix $ψ \in D$ . Since $W (K * η_{ε}) = η_{ε}$ pointwise on $Q$ by step 3.1 and $K * η_{ε}$ is $C^{\infty}$ , the classical-compatibility clause of the distributional calculus identifies the distributional $W$ of the regular functional $u_{K * η_{ε}}$ with $u_{W (K * η_{ε})} = u_{η_{ε}}$ ; and the pairing identity of step 4.1 gives $⟨ K * η_{ε}, ψ ⟩ = ⟨ η_{ε}, K ⊛ ψ ⟩$ . With $F = K ⊛ ψ$ , bounded and continuous at the origin with $F (0, 0) = ⟨ K, ψ ⟩$ by steps 1.3 and 2.2, the unit mass and support $supp η_{ε} \subseteq {\overline{B}}_{ε}$ give $∣ ⟨ η_{ε}, F ⟩ - F (0, 0) ∣ \leq \sup_{∣ u ∣ \leq ε} ∣ F (u) - F (0, 0) ∣ \to 0$ by continuity at the origin; thus $⟨ K * η_{ε}, ψ ⟩ \to F (0, 0) = ⟨ K, ψ ⟩$ without requiring global continuity of $F$ . Thus $K * η_{ε} \to K$ in the weak distribution topology; since $W$ is continuous linear for that topology and $⟨ η_{ε}, ψ ⟩ \to ψ (0, 0) = ⟨ δ_{(0, 0)}, ψ ⟩$ by the same approximate-identity argument applied to the bounded continuous $ψ$ , we obtain $⟨ W K, ψ ⟩ = \lim_{ε} ⟨ W (K * η_{ε}), ψ ⟩ = \lim_{ε} ⟨ η_{ε}, ψ ⟩ = ψ (0, 0)$ . As $ψ$ was arbitrary, $W K = δ_{(0, 0)}$ .

Remarks

What is and is not asserted about $K$ . The support computation says exactly that $supp K = C$ ; it asserts no regularity of $K$ as a function on the cone and gives no values there. Only the identity $W K = δ_{(0, 0)}$ and the convolution formula are used later on this page; the advanced kernel is not treated.
Choice accounting. ${A C}_{ω}$ enters through the polar measure and polar formula, through the two $C^{1}$ change-of-variables theorems, and through the classical-compatibility and injectivity clauses of the distributional calculus. The algebraic identities are choice-free, and no arbitrary-index choice or dependent choice is used.
No distribution product. The notation $K * h$ is defined by the explicit absolutely convergent integral of the Statement; nothing of the form $W K \cdot h$ or a product of two singular distributions is formed. The expression $K = δ (t - ∣ y ∣ / c) / (4 π ∣ y ∣)$ , if used elsewhere, abbreviates the pairing of the Statement.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

The smooth forced Maxwell Cauchy system on a slab: existence, uniqueness, Gauss constraints and the cone of dependence

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) through The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $c, ε, μ > 0$ with $ε μ c^{2} = 1$ , let $t_{0} < T$ , and let $ρ : R^{3} \times [t_{0}, T] \to R$ and $J : R^{3} \times [t_{0}, T] \to R^{3}$ be of class $C^{\infty}$ in the extension sense (restrictions of $C^{\infty}$ functions on $R^{3} \times I_{e x t}$ for some open interval $I_{e x t} \supset [t_{0}, T]$ , $C^{k}$ Euclidean maps and diffeomorphisms) satisfying the continuity equation

$\partial_{t} ρ + div J = 0 on R^{3} \times (t_{0}, T) .$

Let $E_{0}, B_{0} \in C^{\infty} (R^{3}; R^{3})$ satisfy $div E_{0} = ρ (\cdot, t_{0}) / ε$ and $div B_{0} = 0$ on $R^{3}$ . Then:

(a) the first-order system

$\partial_{t} E = c^{2} curl B - \frac{J}{ε}, \partial_{t} B = - curl E on R^{3} \times (t_{0}, T),$

with $E (\cdot, t_{0}) = E_{0}$ , $B (\cdot, t_{0}) = B_{0}$ , has exactly one solution $(E, B)$ with $E, B$ of class $C^{\infty}$ on $R^{3} \times [t_{0}, T]$ in the extension sense;

(b) this solution obeys both Gauss laws on $R^{3} \times (t_{0}, T)$ : $div E = ρ / ε$ and $div B = 0$ ;

(c) (domain of dependence) if $(\tilde{ρ}, \tilde{J}, {\tilde{E}}_{0}, {\tilde{B}}_{0})$ is second data of the same kind and, for some $x_{*} \in R^{3}$ and $t_{*} \in (t_{0}, T]$ , one has ${\tilde{E}}_{0} = E_{0}$ , ${\tilde{B}}_{0} = B_{0}$ on the closed base ball ${\overline{B}}_{c (t_{*} - t_{0})} (x_{*})$ and $\tilde{ρ} = ρ$ , $\tilde{J} = J$ on the closed backward cone $K (x_{*}, t_{*}) = {(y, v) : ∣ y - x_{*} ∣ \leq c (t_{*} - v), t_{0} \leq v \leq t_{*}}$ (Forward and backward wave cones, domain of dependence and influence), then $\tilde{E} (x_{*}, t_{*}) = E (x_{*}, t_{*})$ and $\tilde{B} (x_{*}, t_{*}) = B (x_{*}, t_{*})$ .

This is a statement about the stipulated smooth system with coefficients and sources given; no physical identification of $(E, B)$ with an electromagnetic field, no finite-domain boundary problem and no decay at spatial infinity is asserted.

Facts & Assumptions

Given: The constants $c, ε, μ > 0$ with $ε μ c^{2} = 1$ , the times $t_{0} < T$ , the extension-sense smooth sources $ρ, J$ with $\partial_{t} ρ + div J = 0$ , and the data $E_{0}, B_{0}$ with the two constraint identities.

[F1]

Put $W = c^{- 2} \partial_{t}^{2} - Δ$ . For every $a, b \in C^{\infty} (R^{3})$ and every extension-sense smooth source $h$ on the slab there is a unique extension-sense $C^{2}$ function $u$ with $W u = h$ on $R^{3} \times (t_{0}, T)$ , $u (\cdot, t_{0}) = a$ , $\partial_{t} u (\cdot, t_{0}) = b$ ; it is $C^{\infty}$ in the extension sense. Moreover, if $\tilde{a}, \tilde{b}, \tilde{h}$ are second data with $\tilde{a} = a$ , $\tilde{b} = b$ on a closed ball ${\overset{ˉ}{B}}_{c (t_{*} - t_{0})} (x_{*})$ and $\tilde{h} = h$ on the closed backward cone $K (x_{*}, t_{*})$ , then the corresponding solutions agree at $(x_{*}, t_{*})$ . The theorem assumes ${A C}_{ω}$ (The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support).

[F2]

If $ϕ$ is $C^{2}$ then $curl \nabla ϕ = 0$ wherever defined (The curl of the gradient of a $C^{2}$ function vanishes).

[F3]

If $F$ is a $C^{2}$ field then $div (curl F) = 0$ (The divergence of the curl of a $C^{2}$ field vanishes).

[F4]

If $F$ is a $C^{2}$ field on an open subset of $R^{3}$ , then $curl curl F = \nabla div F - Δ F$ (The curl of a curl is the gradient of the divergence minus the Laplacian).

[F5]

Divergence and curl are linear on $C^{1}$ fields, and $div (f F) = ⟨ \nabla f, F ⟩ + f div F$ (Divergence and curl are linear and satisfy the scalar product rules).

[F6]

If $ϕ$ is $C^{2}$ on an open subset of $R^{m}$ then $\partial_{i} \partial_{j} ϕ = \partial_{j} \partial_{i} ϕ$ (Clairaut--Schwarz theorem for continuous second partial derivatives).

[F7]

One-variable derivatives are linear and obey the sum rule, and total derivatives are additive and homogeneous (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives).

[F8]

$div F = \sum_{i} \partial_{i} F_{i}$ for $C^{1}$ $F$ , $curl F = (\partial_{y} F_{z} - \partial_{z} F_{y}, \partial_{z} F_{x} - \partial_{x} F_{z}, \partial_{x} F_{y} - \partial_{y} F_{x})$ for $C^{1}$ $F$ on open subsets of $R^{3}$ , $Δ f = div \nabla f$ for $C^{2}$ scalars, partial derivatives are one-variable derivatives along coordinate lines, and $\nabla f = (\partial_{0} f, \partial_{1} f, \partial_{2} f)$ (Divergence and curl of a $C^{1}$ vector field, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

[F9]

A map is of class $C^{k}$ when each component is, in the extension sense on a closed time slab when it is the restriction of a $C^{k}$ map on an open time neighbourhood ( $C^{k}$ Euclidean maps and diffeomorphisms).

[F10]

The closed backward cone with vertex $(x_{*}, t_{*})$ is $K (x_{*}, t_{*}) = {(y, v) : ∣ y - x_{*} ∣ \leq c (t_{*} - v), t_{0} \leq v \leq t_{*}}$ , with base ball ${\overline{B}}_{c (t_{*} - t_{0})} (x_{*})$ (Forward and backward wave cones, domain of dependence and influence).

[F11]

A continuous function on an interval with zero derivative at every interior point is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

[A1]

The Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1givenF1F8F9A1

Existence. Define $E$ componentwise by [F1] on the slab with data $E (\cdot, t_{0}) = E_{0}$ , $\partial_{t} E (\cdot, t_{0}) = c^{2} curl B_{0} - J (\cdot, t_{0}) / ε$ and source $- \nabla ρ / ε - μ \partial_{t} J$ , and $B$ componentwise by [F1] with data $B (\cdot, t_{0}) = B_{0}$ , $\partial_{t} B (\cdot, t_{0}) = - curl E_{0}$ and source $μ curl J$ . All data and sources are smooth in the extension sense because $ρ, J, E_{0}, B_{0}$ are, and $curl$ and $\nabla$ are finite combinations of first partial derivatives; hence $E$ and $B$ are of class $C^{\infty}$ on the slab in the extension sense.

2.1step 1.1F1F2F4F5F6F7

The first-order equations. Put $R : = E_{t} - c^{2} curl B + J / ε$ and $Q : = B_{t} + curl E$ . Since $\partial_{t}$ commutes with $W$ on $C^{\infty}$ functions and $curl$ commutes with $W$ on $C^{\infty}$ fields (both are finite combinations of constant-coefficient first and second partial derivatives, and mixed partials commute), the wave equations of step 1.1 give $W (E_{t}) = (W E)_{t} = - \nabla ρ_{t} / ε - μ J_{t t}$ , $W (curl B) = curl (W B) = μ curl curl J$ and $W (J / ε) = c^{- 2} J_{t t} / ε - Δ J / ε = μ J_{t t} - c^{2} μ Δ J$ . Adding, with $ε μ c^{2} = 1$ and so $c^{2} μ = 1 / ε$ , the $J_{t t}$ and $Δ J$ terms cancel and $W R = - \nabla ρ_{t} / ε - c^{2} μ \nabla div J = - \nabla (ρ_{t} + div J) / ε = 0$ by the continuity equation. Likewise $W Q = (W B)_{t} + curl (W E) = μ curl J_{t} + curl (- \nabla ρ / ε - μ J_{t}) = 0$ , using $curl \nabla ρ = 0$ and the interchange of $curl$ with $\partial_{t}$ . At $t_{0}$ : $R (t_{0}) = E_{t} (t_{0}) - c^{2} curl B_{0} + J (\cdot, t_{0}) / ε = 0$ and $Q (t_{0}) = B_{t} (t_{0}) + curl E_{0} = 0$ by the prescribed data. For the time derivatives at $t_{0}$ : $E_{t t} (t_{0}) = c^{2} Δ E_{0} + c^{2} (W E) (t_{0}) = c^{2} Δ E_{0} - c^{2} \nabla ρ (\cdot, t_{0}) / ε - c^{2} μ J_{t} (\cdot, t_{0})$ and $c^{2} curl curl E_{0} = c^{2} \nabla div E_{0} - c^{2} Δ E_{0} = c^{2} \nabla (ρ (\cdot, t_{0}) / ε) - c^{2} Δ E_{0}$ , so $R_{t} (t_{0}) = E_{t t} (t_{0}) + c^{2} curl curl E_{0} + J_{t} (\cdot, t_{0}) / ε = (- c^{2} μ + 1 / ε) J_{t} (\cdot, t_{0}) = 0$ ; similarly $B_{t t} (t_{0}) = c^{2} Δ B_{0} + c^{2} (W B) (t_{0}) = c^{2} Δ B_{0} + c^{2} μ curl J (\cdot, t_{0})$ and $Q_{t} (t_{0}) = B_{t t} (t_{0}) + curl E_{t} (t_{0}) = c^{2} Δ B_{0} + c^{2} μ curl J (\cdot, t_{0}) + c^{2} curl curl B_{0} - curl J (\cdot, t_{0}) / ε = c^{2} \nabla div B_{0} = 0$ . Thus $R$ and $Q$ are extension-sense smooth fields with $W R = W Q = 0$ on $R^{3} \times (t_{0}, T)$ and zero initial values and time derivatives; the uniqueness clause of [F1], applied componentwise, gives $R = Q = 0$ , that is, $E_{t} = c^{2} curl B - J / ε$ and $B_{t} = - curl E$ on the slab.

3.1step 1.1step 2.1F1F3F5F7

The Gauss laws. Put $D : = div E - ρ / ε$ . Since $W$ commutes with $div$ on extension-sense $C^{3}$ fields, $W D = div (W E) - (c^{- 2} ρ_{t t} - Δ ρ) / ε = - Δ ρ / ε - μ div J_{t} - (c^{- 2} ρ_{t t} - Δ ρ) / ε = - μ (div J_{t} + ρ_{t t}) = 0$ , the last equality being the time derivative of the continuity equation. Moreover $D (t_{0}) = div E_{0} - ρ (\cdot, t_{0}) / ε = 0$ and, using the first-order equation from step 2.1 and $div curl = 0$ , $D_{t} (t_{0}) = div (c^{2} curl B_{0} - J (\cdot, t_{0}) / ε) - ρ_{t} (\cdot, t_{0}) / ε = - (div J + ρ_{t}) (\cdot, t_{0}) / ε = 0$ . By the uniqueness clause of [F1] componentwise, $D = 0$ . Likewise $W (div B) = div (W B) = μ div curl J = 0$ , $div B (t_{0}) = div B_{0} = 0$ and $\partial_{t} div B (t_{0}) = - div curl E_{0} = 0$ , so $div B = 0$ on the slab.

4.1step 1.1step 3.1F1F2F3F4F5F6F11

Uniqueness and the forced wave problems. Let $(\tilde{E}, \tilde{B})$ be any second $C^{\infty}$ solution of the first-order system on the slab in the extension sense with the same data. Taking divergence directly in its first-order equations gives $\partial_{t} (div \tilde{E} - ρ / ε) = - (div J + ρ_{t}) / ε = 0$ and $\partial_{t} div \tilde{B} = - div curl \tilde{E} = 0$ by [F3] and the continuity equation. For each fixed spatial point these defects have zero derivative, hence are constant in time by [F11], and their initial values vanish; thus $div \tilde{E} = ρ / ε$ and $div \tilde{B} = 0$ . Differentiating ${\tilde{E}}_{t} = c^{2} curl \tilde{B} - J / ε$ in time and using $curl curl = \nabla div - Δ$ gives $c^{- 2} {\tilde{E}}_{t t} = - curl curl \tilde{E} - μ J_{t}$ , hence $W \tilde{E} = - \nabla div \tilde{E} - μ J_{t} = - \nabla ρ / ε - μ J_{t}$ , while $W \tilde{B} = μ curl J$ . Its initial data and initial time derivatives coincide with those used in step 1.1 (the derivative values are determined by the system and the data: ${\tilde{E}}_{t} (\cdot, t_{0}) = c^{2} curl B_{0} - J (\cdot, t_{0}) / ε$ and ${\tilde{B}}_{t} (\cdot, t_{0}) = - curl E_{0}$ ). By the uniqueness clause of [F1] applied componentwise, $\tilde{E} = E$ and $\tilde{B} = B$ .

5.1step 1.1step 4.1F1F10∎

Domain of dependence. Let $(\tilde{ρ}, \tilde{J}, {\tilde{E}}_{0}, {\tilde{B}}_{0})$ be second data as in the Statement, with corresponding solution $(\tilde{E}, \tilde{B})$ furnished by step 1.1, and fix $x_{*} \in R^{3}$ , $t_{*} \in (t_{0}, T]$ . By step 4.1's reduction, $E$ solves the wave problem with source $- \nabla ρ / ε - μ \partial_{t} J$ and initial data $E_{0}$ , $c^{2} curl B_{0} - J (\cdot, t_{0}) / ε$ , and $\tilde{E}$ solves the corresponding wave problem with hatted data. The data agree on the closed base ball ${\overline{B}}_{c (t_{*} - t_{0})} (x_{*})$ : the displacement data agree by hypothesis, and the time-derivative data $c^{2} curl B_{0} - J (\cdot, t_{0}) / ε$ agree because $B_{0} = {\tilde{B}}_{0}$ and $J (\cdot, t_{0}) = \tilde{J} (\cdot, t_{0})$ there (the latter is the base of the cone in [F10]). The sources agree on the closed backward cone: $ρ, J$ and their first partial derivatives agree on the cone, since the identity holds on the open cone (for $ρ, J$ by hypothesis, for the derivatives by the same local differential expressions) and all these functions are continuous on the closed cone, on which they agree by continuity. Applying the cone clause of [F1] to each component of the wave problems for $E$ and $\tilde{E}$ gives $\tilde{E} (x_{*}, t_{*}) = E (x_{*}, t_{*})$ ; the same argument, with source $μ curl J$ and initial data $B_{0}$ , $- curl E_{0}$ , gives $\tilde{B} (x_{*}, t_{*}) = B (x_{*}, t_{*})$ .

Remarks

Choice accounting. The countable-choice assumption enters only through the existence, uniqueness and cone clauses of the wave theorem [F1]; all differential-algebraic steps are choice-free. The statement records this by assuming ${A C}_{ω}$ globally.
Role of the constraint hypotheses. $div E_{0} = ρ (\cdot, t_{0}) / ε$ and $div B_{0} = 0$ are used in step 2.1 to propagate the Gauss laws; without them the first-order system would still have a unique solution but the first-order divergence identities would keep the corresponding initial Gauss defects constant in time at each spatial point. The continuity equation is used in steps 2.1, 3.1 and 4.1.
No physical claim. The constants $c, ε, μ$ are parameters of the mathematical system; no identification with an electromagnetic field or with SI quantities is made here.

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

Retarded potentials of a point history as the retarded convolution of the point sources, and the off-worldline Lienard-Wiechert fields

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $c > 0$ , $q \in R$ , $ε_{0} > 0$ , put $μ_{0} : = (c^{2} ε_{0})^{- 1}$ and $C : = q / (4 π ε_{0})$ , and let $z : R \to R^{3}$ be of class $C^{3}$ with $∣ z^{'} (s) ∣ \leq v_{*} < c$ for every $s \in R$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral). Write $Ω : = R^{3} \times R$ and $Γ : = {(z (s), s) : s \in R} \subseteq Ω$ . For $(x, t) \in Ω ∖ Γ$ let $s = s (x, t)$ be the unique retarded root of Global existence, uniqueness and differentiability of the retarded root with $t - s = ∣ x - z (s) ∣ / c$ , and put $R : = x - z (s)$ , $r : = ∣ R ∣ > 0$ , $n : = R / r$ , $β : = z^{'} (s) / c$ , $κ : = 1 - n \cdot β$ , $N : = n \cdot β$ , $D : = κ r$ , $\dot{β} : = z^{''} (s) / c$ and $d : = n \cdot \dot{β}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ). Then:

(a) the scalar distribution $ρ \in D^{'} (Ω)$ and the vector distribution $J \in (D^{'} (Ω))^{3}$ of Point-support spacetime densities: distributions, continuity equation and endpoint fluxes,

$⟨ ρ, ψ ⟩ = q \int_{R} ψ (z (s), s) d s, ⟨ J, ψ ⟩ = q \int_{R} z^{'} (s) ψ (z (s), s) d s,$

satisfy $\partial_{t} ρ + div J = 0$ .

(b) 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 $ψ \in D (Ω)$ and $τ \in R$ , $y \in R^{3}$ , put $(K ⊛ ψ) (y, τ) : = ⟨ K, ψ (y + \cdot, τ + \cdot) ⟩$ . Define the retarded-history pairings by the explicit integrals

$⟨ Φ, ψ ⟩ : = \frac{q}{ε_{0}} \int_{R} (K ⊛ ψ) (z (s), s) d s, ⟨ Υ, ψ ⟩ : = μ_{0} q \int_{R} z^{'} (s) (K ⊛ ψ) (z (s), s) d s .$

Here the second integral is componentwise. The notation $ε_{0}^{- 1} ⟨ ρ, K ⊛ ψ ⟩$ and $μ_{0} ⟨ J, K ⊛ ψ ⟩$ , if used, means precisely these explicit history integrals, rather than ordinary distribution action on a compactly supported test. These pairings define distributions on $Ω$ , are finite for every $ψ \in D (Ω)$ and satisfy the exact retarded-convolution identities

$⟨ Φ, ψ ⟩ = \int_{Ω ∖ Γ} \frac{C ψ (x, t)}{κ (x, t) r (x, t)} d x d t, ⟨ Υ, ψ ⟩ = \int_{Ω ∖ Γ} \frac{β (x, t)}{c} \frac{C ψ (x, t)}{κ (x, t) r (x, t)} d x d t$

(for $Υ$ , componentwise against $ψ \in D (Ω)$ ; both integrands are continuous on the open set $Ω ∖ Γ$ and integrable there against $ψ$ ). Consequently the potentials are the $C^{2}$ functions

$φ : = \frac{C}{D} = \frac{C}{κ r} : Ω ∖ Γ \to R, A : = \frac{β φ}{c} = \frac{β C}{c κ r} : Ω ∖ Γ \to R^{3}$

off the worldline, and the zero extensions $\tilde{φ}, \tilde{A}$ on $Γ$ belong to $L_{l o c}^{1} (Ω)$ ; $Φ$ is the regular distribution of $\tilde{φ}$ , and $Υ$ that of $\tilde{A}$ (Regular distribution from a locally integrable function).

(c) Let $E : = - \nabla φ - \partial_{t} A$ and $B : = curl A$ on $Ω ∖ Γ$ (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression). Then, off the worldline,

$E = C [\frac{(1 - ∣ β ∣^{2}) (n - β)}{κ^{3} r^{2}} + \frac{n \times ((n - β) \times \dot{β})}{c κ^{3} r}], B = \frac{n \times E}{c} .$

The fields $E, B$ are $C^{1}$ off the worldline. More generally, if $z \in C^{k}$ , $k \geq 3$ , the retarded root is $C^{k}$ , the potentials are $C^{k - 1}$ and the fields are $C^{k - 2}$ there; if $z \in C^{\infty}$ , all these objects are smooth off the worldline. The result is a mathematical model computation for the stipulated uniformly subluminal smooth worldline and the retarded choice; no self-force law, no radiation-reaction postulate and no physical Maxwell coupling is asserted or proved.

Facts & Assumptions

Given: The constants $c, q, ε_{0}, μ_{0}, C$ , the all-time uniformly subluminal $C^{3}$ trajectory $z$ , the domain $Ω = R^{3} \times R$ with its worldline graph $Γ$ , and the retarded data $s, R, r, n, β, κ, N, D, \dot{β}, d$ of the Statement.

[F1]

For an open interval $I$ , a $C^{1}$ trajectory $z : I \to R^{3}$ whose graph map into an open $Ω$ is proper, and $ψ \in D (Ω)$ , the functionals $⟨ ρ, ψ ⟩ = q \int_{I} ψ (z (s), s) d s$ and $⟨ J, ψ ⟩ = q \int_{I} z^{'} (s) ψ (z (s), s) d s$ are distributions and satisfy $\partial_{t} ρ + div J = 0$ ; properness is automatic for $I = R$ (Point-support spacetime densities: distributions, continuity equation and endpoint fluxes).

[F2]

For $c > 0$ and $C^{1}$ $z$ with $∣ z^{'} ∣ \leq v_{*} < c$ there is exactly one retarded root $s$ of $t - s = ∣ x - z (s) ∣ / c$ , with $s \leq t$ and equality exactly on the worldline; off it $s < t$ , $r = ∣ x - z (s) ∣$ satisfies $r = c (t - s) > 0$ , $κ = 1 - n \cdot z^{'} (s) / c$ obeys $1 - v_{*} / c \leq κ \leq 1 + v_{*} / c$ , and $\partial_{t} s = 1 / κ$ , $\nabla s = - n / (c κ)$ ; a $C^{k}$ trajectory gives a $C^{k}$ root on the off-worldline domain, in particular $C^{3}$ for $C^{3}$ $z$ (Global existence, uniqueness and differentiability of the retarded root).

[F3]

For the retarded kernel $K$ one has $(K ⊛ ψ) (y, τ) = (4 π)^{- 1} \int ψ (y + w, τ + ∣ w ∣ / c) ∣ w ∣^{- 1} d w$ for $ψ \in D (Ω)$ (The retarded wave kernel is a distribution supported on the cone, solves the point-wave equation, and convolves to the retarded solution).

[F4]

For a potential pair, $E = - \nabla φ - \partial_{t} A$ , $B = curl A$ (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression).

[F5]

A locally integrable function defines the regular functional $u_{f} (φ) = \int f φ$ ; distributions are the continuous complex-linear functionals on the LF test space, whose fixed-support seminorms bound absolute pairings (Regular distribution from a locally integrable function, Distribution, Test function topology).

[F6]

Under ${A C}_{ω}$ , for nonnegative Borel $g$ on $R^{n}$ , $\int_{R^{n}} g d λ_{n} = \int_{0}^{\infty} \int_{S^{n - 1}} g (r ω) r^{n - 1} d σ (ω) d r$ ; in particular $\int_{B_{L}} ∣ w ∣^{- 1} d w = σ (S^{2}) L^{2} / 2$ (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

[F7]

Tonelli and Fubini on sigma-finite products: nonnegative measurable functions may be integrated in either order, and for $f \in L^{1} (μ \times ν)$ the three integrals agree (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product).

[F8]

Euclidean Borel sets are the Borel sets of the product structure, and the Lebesgue product measure agrees with Euclidean Lebesgue measure on Borel sets (The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}).

[F9]

Under ${A C}_{ω}$ , for a $C^{1}$ diffeomorphism $T : U \to V$ of open sets, $\int_{V} g = \int_{U} g (T) ∣ \det D T ∣$ for nonnegative Borel $g$ , and $\int_{V} f = \int_{U} f (T) ∣ \det D T ∣$ for $f \in L^{1} (V)$ (Borel change of variables from the compact-support formula and Radon uniqueness, A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions).

[F10]

The integral is linear on $L^{1}$ , satisfies $∣ \int f ∣ \leq \int ∣ f ∣$ , nonnegative integrals are monotone and scale, and a nonnegative measurable function has zero integral exactly when it vanishes almost everywhere (The Lebesgue integral is linear on $L^{1} (μ)$ , The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral, A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[F11]

Compact subsets of $R^{n}$ are bounded and closed, and continuous real functions on compacta are bounded and attain extremes (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

[F12]

The gradient, the coordinate partial derivatives, and the divergence and curl on $C^{1}$ fields are those fixed by the cited definitions; vector-valued one-variable derivatives and integrals are defined componentwise (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Divergence and curl of a $C^{1}$ vector field, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

[F13]

The chain rule for total derivatives, the algebra of one-variable derivatives, equality of mixed second partials for $C^{2}$ functions, and the $C^{k}$ vocabulary hold as cited (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Clairaut--Schwarz theorem for continuous second partial derivatives, $C^{k}$ Euclidean maps and diffeomorphisms).

[F14]

The Euclidean inner product is bilinear and symmetric and the cross product is bilinear and alternating; in particular $n \times (n - β) = - n \times β$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors).

[F15]

The divergence and curl obey $curl (f F) = \nabla f \times F + f curl F$ (Divergence and curl are linear and satisfy the scalar product rules).

[A1]

The Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1givenF1F11

Since $z$ is $C^{3}$ , it is $C^{1}$ , and $I : = R$ is open; the graph map $s \mapsto (z (s), s)$ into $Ω = R^{3} \times R$ is proper because a compact subset of $Ω$ has bounded time coordinate, so its inverse image in $R$ is closed and bounded. Hence [F1] applies with $q, ε_{0}$ as given: the functionals $ρ, J$ of the Statement are distributions on $Ω$ and $\partial_{t} ρ + div J = 0$ , which is (a).

1.2givenF3F5F6F10F11A1

Finiteness and bounds for the history pairings. Fix $ψ \in D (Ω)$ ; by compactness of its support choose reals $M \geq 0$ and $t_{0} \leq t_{1}$ with $supp ψ \subseteq {(x, t) : ∣ x ∣ \leq M, t_{0} \leq t \leq t_{1}}$ , and put $A_{ψ} : = \sup ∣ ψ ∣ < \infty$ , $Z : = ∣ z (0) ∣$ , $S : = \max {0, (M + Z - c t_{0}) / (c - v_{*})}$ , $H : = \max {S, ∣ t_{1} ∣}$ , $L : = 1 + M + Z + v_{*} H$ , $T : = \max {0, t_{1} + S}$ and $M_{Q} : = T σ (S^{2}) L^{2} / (8 π)$ . If $g (s, w) : = (z (s) + w, s + ∣ w ∣ / c)$ lies in $supp ψ$ , then $s \leq t_{1}$ ; and for $s \leq 0$ the bound $∣ z (s) ∣ \leq Z + v_{*} ∣ s ∣$ gives $t_{0} \leq s + ∣ w ∣ / c \leq - (1 - v_{*} / c) ∣ s ∣ + (M + Z) / c$ , so $s \geq - S$ . Also $∣ w ∣ \leq M + ∣ z (s) ∣ \leq M + Z + v_{*} H < L$ . Therefore $∣ ⟨ Φ, ψ ⟩ ∣ \leq \frac{∣ q ∣}{ε_{0}} \int_{- S}^{t_{1}} \frac{1}{4 π} \int_{B_{L}} \frac{A_{ψ}}{∣ w ∣} d w d s \leq \frac{∣ q ∣ M_{Q} A_{ψ}}{ε_{0}}$ , and componentwise $∣ ⟨ Υ_{j}, ψ ⟩ ∣ \leq μ_{0} ∣ q ∣ v_{*} M_{Q} A_{ψ}$ . These bounds are fixed-support seminorms, so $Φ$ and the components of $Υ$ are distributions.

1.3givenF2F9F13F14

The retarded substitution map. Put $U : = R \times (R^{3} ∖ {0})$ , $V : = Ω ∖ Γ$ and $g (s, w) : = (z (s) + w, s + ∣ w ∣ / c)$ . Then $V$ is open: $Γ$ is closed, since a convergent sequence $(z (s_{j}), s_{j}) \to (x, t)$ has $s_{j} \to t$ , hence $x = z (t)$ by continuity. The map $g : U \to V$ is a bijection with inverse $(x, t) \mapsto (s (x, t), x - z (s (x, t)))$ : the root $s (x, t)$ of [F2] exists, satisfies $x - z (s) \neq 0$ exactly off the worldline, and $s (x, t)$ is $C^{1}$ there because $z$ is $C^{3}$ ; conversely no image point lies on $Γ$ , for $∣ w ∣ = ∣ z (t) - z (s) ∣ \leq v_{*} ∣ t - s ∣ = v_{*} ∣ w ∣ / c$ is impossible for $w \neq 0$ . The map is $C^{1}$ with derivative matrix (in the order $s, w_{1}, w_{2}, w_{3}$ ) whose last row is $(1, w / (c ∣ w ∣))$ and whose first column is $(z^{'} (s), 1)$ with the identity in the remaining block; expanding along the last row gives $\det D g (s, w) = - 1 + z^{'} (s) \cdot w / (c ∣ w ∣) = - κ (s, w)$ , so $∣ \det D g ∣ = κ$ , and $κ \geq 1 - v_{*} / c > 0$ by [F2]. If $g (s, w) = g (s^{'}, w^{'})$ then $∣ s - s^{'} ∣ = ∣ ∣ w^{'} ∣ - ∣ w ∣ ∣ / c \leq ∣ z (s^{'}) - z (s) ∣ / c \leq (v_{*} / c) ∣ s - s^{'} ∣$ , hence $s = s^{'}$ and $w = w^{'}$ . By [F9], for every nonnegative Borel $h$ on $V$ one has $\int_{V} h = \int_{U} h (g) κ d s d w$ , and for every $f \in L^{1} (V)$ , $\int_{V} f = \int_{U} f (g) κ d s d w$ .

1.4givenF2F12F13F14

Retarded derivative identities. On $V$ , with $s = s (x, t)$ the root of [F2]: $s_{t} = 1 / κ$ and $\nabla s = - n / (c κ)$ . Hence $r_{t} = n \cdot (- z^{'} (s) s_{t}) = - c N / κ$ and $\nabla r = n / κ$ ; $R_{t} = - z^{'} (s) s_{t} = - c β / κ$ and $\partial_{i} R = e_{i} + β n_{i} / κ$ ; $β_{t} = \dot{β} s_{t} = \dot{β} / κ$ and $\partial_{i} β = \dot{β} \partial_{i} s = - \dot{β} n_{i} / (c κ)$ . Since $κ = 1 - N$ and $D = κ r = r - R \cdot β$ , the product rules with the bilinearity of the inner product give $D_{t} = r_{t} - R_{t} \cdot β - R \cdot β_{t} = (c (∣ β ∣^{2} - N) - r d) / κ$ and, componentwise, $\partial_{i} D = \partial_{i} r - (\partial_{i} R) \cdot β - R \cdot \partial_{i} β = n_{i} (1 - ∣ β ∣^{2} + r d / c) / κ - β_{i}$ , that is, $\nabla D = ((1 - ∣ β ∣^{2} + r d / c) / κ) n - β$ .

1.5F14algebra

The needed triple-product identity follows directly from the coordinate definition in [F14]. For arbitrary $a, b, u \in R^{3}$ , the first component of $a \times (b \times u)$ is $a_{2} (b_{1} u_{2} - b_{2} u_{1}) - a_{3} (b_{3} u_{1} - b_{1} u_{3}) = b_{1} (a_{2} u_{2} + a_{3} u_{3}) - u_{1} (a_{2} b_{2} + a_{3} b_{3}) = b_{1} (a \cdot u) - u_{1} (a \cdot b)$ , since the terms $b_{1} a_{1} u_{1} - u_{1} a_{1} b_{1}$ cancel. Cyclic permutation gives the other two components, so $a \times (b \times u) = b (a \cdot u) - u (a \cdot b)$ . With $a = n$ , $b = n - β$ , $u = \dot{β}$ this gives $n \times ((n - β) \times \dot{β}) = d (n - β) - κ \dot{β}$ , since $n \cdot n = 1$ .

2.1step 1.2step 1.3F3F7F9F10

Scalar pairing identity. Insert the kernel identity [F3] into the definition of $Φ$ : $⟨ Φ, ψ ⟩ = \frac{q}{4 π ε_{0}} \int_{R} \int_{R^{3}} ψ (g (s, w)) ∣ w ∣^{- 1} d w d s$ . The double integral is absolutely convergent by the cylinder estimate of step 1.2, so Fubini's theorem reorders it, and the substitution identity of step 1.3 for the $L^{1}$ function $ψ / (κ r)$ on $V$ (whose absolute integral is finite by the same cylinder estimate) turns it into $⟨ Φ, ψ ⟩ = \int_{V} \frac{C ψ (x, t)}{κ (x, t) r (x, t)} d x d t$ , since $r (g (s, w)) = ∣ w ∣$ , $κ (g (s, w)) = κ (s, w)$ and the Jacobian factor $κ$ cancels $κ^{- 1}$ .

2.2step 1.2step 1.3F3F7F9F10F14

Vector pairing identity. Componentwise, $⟨ Υ_{j}, ψ ⟩ = \frac{μ_{0} q}{4 π} \int_{R} \int_{R^{3}} z_{j}^{'} (s) ψ (g (s, w)) ∣ w ∣^{- 1} d w d s$ ; the double integral is absolutely bounded by $μ_{0} ∣ q ∣ v_{*}$ times the scalar majorant of step 1.2, Fubini applies, and the substitution of step 1.3 gives $⟨ Υ_{j}, ψ ⟩ = \frac{μ_{0} q c}{4 π} \int_{V} \frac{β_{j} ψ}{κ r} d x d t$ , where $β_{j} (s) = β_{j} (g (s, w))$ and $z_{j}^{'} = c β_{j}$ . Since $\frac{μ_{0} q c}{4 π} = μ_{0} ε_{0} c \cdot \frac{q}{4 π ε_{0}} = \frac{C}{c} \cdot c^{2} μ_{0} ε_{0} = \frac{C}{c}$ , this is $⟨ Υ_{j}, ψ ⟩ = \int_{V} \frac{β_{j} (x, t)}{c} \frac{C ψ}{κ r} d x d t$ .

3.1step 1.2step 1.3step 2.1step 2.2F2F5F7F8F9F10

Potentials and regular distributions. By [F2] the root is $C^{3}$ off the worldline, so $r, n$ are $C^{3}$ , $β$ is $C^{2}$ , $κ$ is $C^{2}$ and $D = κ r$ is $C^{2}$ with $D > 0$ on $V$ ; hence $φ = C / D$ and $A = β φ / c$ are of class $C^{2}$ on $V$ , and the identities of steps 2.1 and 2.2 read $⟨ Φ, ψ ⟩ = \int_{V} φ ψ$ and $⟨ Υ_{j}, ψ ⟩ = \int_{V} A_{j} ψ$ for every test $ψ$ . The graph $Γ$ is closed, hence Borel, and null: for each time $t$ its spatial section ${z (t)}$ is a singleton, so Tonelli together with the product identification of Euclidean measure gives $λ_{4} (Γ) = \int_{R} λ_{3} ({z (t)}) d t = 0$ . Therefore $\int_{V} φ ψ = \int_{Ω} \tilde{φ} ψ$ and $\int_{V} A_{j} ψ = \int_{Ω} {\tilde{A}}_{j} ψ$ for the zero extensions. Finally $\tilde{φ}, \tilde{A}$ are locally integrable: for a compact $Q \subseteq Ω$ the substitution identity of step 1.3 and the cylinder bound of step 1.2 give $\int_{Ω} ∣ \tilde{φ} ∣ 1_{Q} \leq ∣ C ∣ \int_{U} 1_{Q} (g) ∣ w ∣^{- 1} d s d w \leq 4 π ∣ C ∣ M_{Q^{'}}$ with $M_{Q^{'}} = T^{'} σ (S^{2}) L^{' 2} / (8 π) < \infty$ for the window constants of $Q$ . Hence the regular functionals of $\tilde{φ}, \tilde{A}$ are distributions and $Φ = u_{\tilde{φ}}$ , $Υ = u_{\tilde{A}}$ .

4.1step 1.4step 1.5step 3.1F4F12F13F14

The electric field. On $V$ , $\nabla φ = - C \nabla D / D^{2}$ , $φ_{t} = - C D_{t} / D^{2}$ and $A = β φ / c$ , so $A_{t} = (\dot{β} φ / κ + β φ_{t}) / c = \dot{β} C / (c κ D) - β C D_{t} / (c D^{2})$ and $E = - \nabla φ - A_{t} = \frac{C}{D^{2}} (\nabla D + \frac{β D_{t}}{c}) - \frac{C \dot{β}}{c κ D}$ . Substituting the identities of step 1.4, the bracket equals $\frac{1}{κ} (1 - ∣ β ∣^{2} + \frac{r d}{c}) (n - β)$ , so $E = C \frac{(1 - ∣ β ∣^{2}) (n - β) + r d (n - β) / c}{κ^{3} r^{2}} - \frac{C \dot{β}}{c κ D} = \frac{C}{κ^{3} r^{2}} [(1 - ∣ β ∣^{2}) (n - β) + \frac{r}{c} (d (n - β) - κ \dot{β})]$ , using $D = κ r$ to write the second summand over the common denominator. The cross-product identity $n \times ((n - β) \times \dot{β}) = d (n - β) - κ \dot{β}$ of step 1.5 then gives the displayed electric field.

5.1step 1.4step 4.1F2F4F13F14F15∎

The magnetic field and regularity. By [F15] and the chain rule, $curl A = \nabla (φ / c) \times β + (φ / c) curl β$ with $curl β = \nabla s \times \dot{β}$ , so $curl A = - \frac{C \nabla D \times β}{c D^{2}} - \frac{C n \times \dot{β}}{c^{2} κ^{2} r}$ . Substituting $\nabla D$ from step 1.4 and using $n \times (n - β) = - n \times β$ gives $curl A = - C [\frac{1 - ∣ β ∣^{2} + r d / c}{c κ^{3} r^{2}} (n \times β) + \frac{n \times \dot{β}}{c^{2} κ^{2} r}]$ . Crossing the field of step 4.1 with $n / c$ gives the same expression, because $n \times n = 0$ , $n \times (n - β) = - n \times β$ and $n \times (d (n - β) - κ \dot{β}) = - d (n \times β) - κ (n \times \dot{β})$ ; hence $B = n \times E / c$ . Since for $z \in C^{3}$ the root is $C^{3}$ and $\nabla D, D_{t}, β, \dot{β}, κ, r, n$ are as in step 1.4 with $κ r > 0$ on $V$ , both displayed fields are $C^{1}$ on $V$ ; the same computation with a $C^{k}$ trajectory uses the $C^{k}$ root branch of [F2] and gives $C^{k - 2}$ fields, and the smooth branch gives smooth fields.

Remarks

Choice accounting. ${A C}_{ω}$ is used through the polar formula and the two change-of-variables theorems (step 1.3 and the polar bound of step 1.2); the root lemma, the current continuity and the algebraic computations are choice-free as stated in their own items.
What the retarded selection does and does not do. The pairings select the retarded history $s < t$ through the kernel $K$ and the root of [F2]. No statement is made about exactly two solutions, about arbitrary nonzero Cauchy data, about finite self-energy or about a self-force law; the fields are computed from the stipulated potential pair off the worldline only.
Signs. The conventions $E = - \nabla φ - \partial_{t} A$ , $B = curl A$ and $C = q / (4 π ε_{0})$ , $μ_{0} = (c^{2} ε_{0})^{- 1}$ are those fixed in [F4] and the Statement; the formula for $B$ is the consequence $B = n \times E / c$ , not an independent convention.

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

The retarded point-history pair solves wave, Lorenz and Maxwell equations distributionally, with locally integrable field representatives off the worldline

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $c > 0$ , $ε_{0} > 0$ , $q \in R$ , $μ_{0} = (ε_{0} c^{2})^{- 1}$ , and let $z : R \to R^{3}$ be $C^{3}$ with $∣ z^{'} (s) ∣ \leq v_{*} < c$ for all $s$ . These are mathematical parameters. On $Q = R^{3} \times R$ let $ρ, J$ and $Φ, Υ$ be exactly the proper point-current and retarded-history distributions of Point-support spacetime densities: distributions, continuity equation and endpoint fluxes and Retarded potentials of a point history as the retarded convolution of the point sources, and the off-worldline Lienard-Wiechert fields, with the kernel $K$ of The retarded wave kernel is a distribution supported on the cone, solves the point-wave equation, and convolves to the retarded solution and no independent free addition. Put $W = c^{- 2} \partial_{t}^{2} - Δ$ . Then:

(a) $W Φ = ρ / ε_{0}$ , $W Υ = μ_{0} J$ and $div Υ + c^{- 2} \partial_{t} Φ = 0$ in $D^{'} (Q)$ .

(b) Define the vector distributions $E = - \nabla Φ - \partial_{t} Υ$ and $B = curl Υ$ using componentwise Distributional derivative and the finite-coordinate cross and curl conventions. They satisfy

$div E = ρ / ε_{0}, div B = 0, curl E = - \partial_{t} B, curl B = μ_{0} J + c^{- 2} \partial_{t} E .$

(c) On the complement of $Γ = {(z (s), s) : s \in R}$ , the distributions $E, B$ restrict to the exact $C^{1}$ fields of Retarded potentials of a point history as the retarded convolution of the point sources, and the off-worldline Lienard-Wiechert fields, with all its higher-regularity branches retained. The zero extensions of those field formulas on $Γ$ belong to $L_{l o c}^{1} (Q)$ , and their regular distributions are precisely $E, B$ (Regular distribution from a locally integrable function). The off-worldline $C^{1}$ fields also satisfy the source-free classical Maxwell equations there, because the point pair restricts to zero and each continuous regular residual is zero. Thus the equations in (b) are equations for locally integrable fields with the exact point source, not merely arbitrary field distributions. No field value on $Γ$ is physically assigned; zero extension is a representative convention on a null set. No physical adoption, nonlinear distribution product, finite self-energy, self-force, global acceleration bound, or incoming-field uniqueness theorem is asserted.

Facts & Assumptions

Given: The mathematical parameters $c, ε_{0}, q, μ_{0}$ , the all-time uniformly subluminal $C^{3}$ trajectory $z$ , the distributions $ρ, J, Φ, Υ$ and the kernel $K$ of the cited items, and a test function $ψ \in D (Q)$ in the arguments below.

[F1]

On $Ω = R^{3} \times R$ , with $C = q / (4 π ε_{0})$ and $κ, r, β, D$ as in that item: the retarded-history pairings $Φ, Υ$ are distributions given by explicit history integrals, satisfy $⟨ Φ, ψ ⟩ = \int_{Ω ∖ Γ} C ψ / (κ r) d x d t$ and $⟨ Υ, ψ ⟩ = \int_{Ω ∖ Γ} (β / c) C ψ / (κ r) d x d t$ , are finite for every test, and the potentials $φ = C / (κ r)$ , $A = β φ / c$ are $C^{2}$ off the worldline with zero extensions in $L_{l o c}^{1}$ ; the classical fields $- \nabla φ - \partial_{t} A$ and $curl A$ are $C^{1}$ off the worldline and their higher-regularity branches hold (Retarded potentials of a point history as the retarded convolution of the point sources, and the off-worldline Lienard-Wiechert fields).

[F2]

For the retarded kernel $K$ , $(K ⊛ ψ) (y, τ) = (4 π)^{- 1} \int ψ (y + w, τ + ∣ w ∣ / c) ∣ w ∣^{- 1} d w$ , and $W K = δ_{(0, 0)}$ distributionally, that is, $⟨ K, W χ ⟩ = χ (0, 0)$ for every test $χ$ ; no smoothness of $K$ on the cone is asserted (The retarded wave kernel is a distribution supported on the cone, solves the point-wave equation, and convolves to the retarded solution).

[F3]

The functionals $ρ, ψ \mapsto q \int_{R} ψ (z (s), s) d s$ and $J, ψ \mapsto q \int_{R} z^{'} (s) ψ (z (s), s) d s$ are distributions on $Ω$ with $\partial_{t} ρ + div J = 0$ (Point-support spacetime densities: distributions, continuity equation and endpoint fluxes).

[F4]

Distributional derivatives satisfy $⟨ \partial^{α} u, φ ⟩ = (- 1)^{∣ α ∣} ⟨ u, \partial^{α} φ ⟩$ , are continuous linear on $D^{'}$ for the weak and strong topologies, and satisfy $\partial^{α} u_{f} = u_{\partial^{α} f}$ for $f \in C^{k}$ with $∣ α ∣ \leq k$ under Countable Choice (Distributional derivative, Distributional differentiation is continuous and commutes).

[F5]

A locally integrable function defines the regular functional $u_{f} (φ) = \int f φ$ , the map $f \mapsto u_{f}$ is injective modulo almost-everywhere equality, and the regular functional is linear in $f$ (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions).

[F6]

For smooth fields: $div curl Z = 0$ ; $curl \nabla f = 0$ ; $curl curl Z = \nabla div Z - Δ Z$ (The divergence of the curl of a $C^{2}$ field vanishes, The curl of the gradient of a $C^{2}$ function vanishes, The curl of a curl is the gradient of the divergence minus the Laplacian).

[F7]

Dominated convergence: pointwise convergence with one integrable majorant gives convergence of integrals (Dominated convergence).

[F8]

The mean value inequality bounds $∥ f (b) - f (a) ∥$ by $\sup ∣ f^{'} ∣ (b - a)$ for a continuous function differentiable on the open interval; partial derivatives are one-variable derivatives along coordinate lines; the chain rule holds for composed differentiable maps; and $C^{k}$ is the componentwise regularity class (The mean value inequality: if $f : [a, b] \to R^{m}$ is continuous and differentiable on $(a, b)$ with $∥ f^{'} ∥_{2} \leq M$ , then $∥ f (b) - f (a) ∥_{2} \leq M (b - a)$ , Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , $C^{k}$ Euclidean maps and diffeomorphisms).

[F9]

Under Countable Choice: the polar formula $\int_{R^{n}} f d λ_{n} = \int_{0}^{\infty} \int_{S^{n - 1}} f (r ω) r^{n - 1} d σ d r$ for nonnegative Borel $f$ , and change of variables $\int_{V} g = \int_{U} g (T) ∣ \det D T ∣$ for a $C^{1}$ diffeomorphism and nonnegative Borel $g$ (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, Borel change of variables from the compact-support formula and Radon uniqueness).

[F10]

Nonnegative integrals are monotone and scale; the integral is linear on $L^{1}$ and obeys the triangle inequality; a nonnegative measurable function has zero integral exactly when it vanishes almost everywhere; Euclidean balls have positive finite measure (Monotonicity and nonnegative homogeneity of the nonnegative integral, The Lebesgue integral is linear on $L^{1} (μ)$ , The modulus of an integral is bounded by the integral of the modulus, A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere, Euclidean balls have positive finite Lebesgue measure).

[F11]

Tonelli and Fubini hold on sigma-finite products; Euclidean Borel sets and Lebesgue measure agree with the product Borel structure and product measure (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product, The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}).

[F12]

Newton-Leibniz with an interior derivative, one-dimensional integration by parts, rectangle Riemann-Fubini, Riemann-to-Lebesgue comparison for compact test integrands, translations preserve Lebesgue measure, and smooth compact cutoffs exist in ZF (Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative, If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ , Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections, Riemann–Lebesgue comparison for distribution test integrands, Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, Test function cutoffs and euclidean localization).

[F13]

The Euclidean inner product and the cross product are the cited bilinear operations on $R^{3}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product in $R^{3}$ ).

[A1]

The Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1givenF2F7F8F9F10F11

Fix $ψ \in D (Q)$ and put $F : = K ⊛ ψ$ , so that $F (y, s) = (4 π)^{- 1} \int ψ (y + w, s + ∣ w ∣ / c) ∣ w ∣^{- 1} d w$ . Then $F$ is of class $C^{1}$ on $Q$ and $\partial_{y_{i}} F = K ⊛ (\partial_{x_{i}} ψ)$ , $\partial_{s} F = K ⊛ (\partial_{t} ψ)$ . Indeed, fix a bounded parameter neighbourhood $∣ y ∣ \leq R$ , $∣ s ∣ \leq R$ and choose $M$ with $supp ψ \subseteq {\overline{B}}_{M} (0) \times [- M, M]$ ; every nonzero contribution then has $w \in B_{M + R}$ , on which $∣ w ∣^{- 1}$ is integrable by the polar formula. For fixed $(y, s)$ the difference quotients $[ψ (y + h e_{i} + w, s + ∣ w ∣ / c) - ψ (y + w, s + ∣ w ∣ / c)] / h$ converge as $h \to 0$ to the partial derivative $\partial_{x_{i}} ψ (y + w, s + ∣ w ∣ / c)$ by definition of the partial derivative, and the mean value inequality bounds them by $\sup ∣ \partial_{x_{i}} ψ ∣ 1_{B_{M + R}} ∣ w ∣^{- 1}$ , an integrable majorant; dominated convergence gives $\partial_{y_{i}} F = K ⊛ (\partial_{x_{i}} ψ)$ on the parameter neighbourhood, and the same argument with the $s$ -difference quotients and the $t$ -partial derivatives gives $\partial_{s} F = K ⊛ (\partial_{t} ψ)$ ; the resulting integrals are continuous in $(y, s)$ by the same dominated convergence, so $F$ is $C^{1}$ .

1.2givenF2F4F8

For every test $ψ$ and every $(y, s) \in Q$ , $(K ⊛ W ψ) (y, s) = ψ (y, s)$ . Indeed, with the translated test $χ (z, τ) : = ψ (y + z, s + τ)$ one has $(W ψ) (y + \cdot, s + \cdot) = W χ$ by the chain rule and the constant coefficients of $W$ , so $(K ⊛ W ψ) (y, s) = ⟨ K, W χ ⟩ = ⟨ W K, χ ⟩ = ⟨ δ_{(0, 0)}, χ ⟩ = χ (0, 0) = ψ (y, s)$ , where the second equality is the definition of the distributional operator $W$ applied to $K$ and the third is $W K = δ_{(0, 0)}$ .

1.3givenF1F9F10F11F13A1

Local integrability of the classical fields. On $V : = Q ∖ Γ$ the map $g (s, w) : = (z (s) + w, s + ∣ w ∣ / c)$ is a $C^{1}$ diffeomorphism from $U : = R \times (R^{3} ∖ {0})$ onto $V$ with $∣ \det D g (s, w) ∣ = κ (s, w) \in [1 - v_{*} / c, 1 + v_{*} / c]$ : the inverse is $(x, t) \mapsto (s (x, t), x - z (s (x, t)))$ by the root of [F1], injectivity follows from $∣ s - s^{'} ∣ = ∣ ∣ w^{'} ∣ - ∣ w ∣ ∣ / c \leq (v_{*} / c) ∣ s - s^{'} ∣$ , and the determinant expansion along the last row gives $- 1 + z^{'} (s) \cdot w / (c ∣ w ∣) = - κ$ . For a compact observation window $Q_{0} \subseteq Q$ contained in ${∣ x ∣ \leq M, t_{0} \leq t \leq t_{1}}$ put $Z : = ∣ z (0) ∣$ , $S : = \max {0, (M + Z - c t_{0}) / (c - v_{*})}$ , $L : = 1 + M + Z + v_{*} \max {S, ∣ t_{1} ∣}$ and $T : = \max {0, t_{1} + S}$ ; then every $(s, w)$ with $g (s, w) \in Q_{0}$ has $s \in [- S, t_{1}]$ and $∣ w ∣ < L$ , so by [F9] $\int_{Q_{0} ∖ Γ} r^{- p} d x d t = \int_{U} 1_{Q_{0}} (g (s, w)) ∣ w ∣^{- p} κ (s, w) d s d w \leq (1 + v_{*} / c) T \int_{B_{L}} ∣ w ∣^{- p} d w$ , which is finite for $p = 1, 2$ by the polar formula ( $σ (S^{2}) L^{2} / 2$ and $σ (S^{2}) L$ respectively). On the other hand the explicit formulas of [F1] show $∣ φ ∣ + ∣ A ∣ + ∣ \nabla φ ∣ + ∣ \nabla A ∣ + ∣ \partial_{t} φ ∣ + ∣ \partial_{t} A ∣ + ∣ curl A ∣ \leq C_{0} (r^{- 2} + r^{- 1})$ on $Q_{0}$ with $C_{0}$ depending on $q, ε_{0}, c, v_{*}$ and the supremum of $∣ \dot{β} ∣$ over the compact history interval. Hence the zero extensions of $φ, A$ and of the classical fields $E_{0} : = - \nabla φ - \partial_{t} A$ , $B_{0} : = curl A$ , and of the first partial derivatives of $φ, A$ are locally integrable on $Q$ ; no local-integrability claim for derivatives of $E_{0}, B_{0}$ is needed or made. The graph $Γ$ is closed and Borel and has spatial singleton sections, so [F11] and Tonelli give spacetime measure zero; zero extension therefore does not affect these integrals.

2.1step 1.1givenF8F12

The history function $s \mapsto F (z (s), s)$ is $C^{1}$ with compact support contained in a finite interval, and its derivative is $\nabla_{y} F (z (s), s) \cdot z^{'} (s) + \partial_{s} F (z (s), s)$ . Indeed $F$ is $C^{1}$ by step 1.1, $z$ is $C^{1}$ , and the chain rule gives the derivative; the support bound ${s : g (s, w) \in supp ψ for some w} \subseteq [- S, t_{1}]$ of the same cylinder estimate shows compact support, so the total derivative integrates to zero over $R$ .

2.2step 1.3givenF1F4F5F10F12

Zero-extension representatives of the first derivatives. Let $f$ be one of the components of $φ$ and $A$ on $V$ , with zero extension $\tilde{f}$ , and let $D \in {\partial_{t}, \partial_{1}, \partial_{2}, \partial_{3}}$ . Since $\tilde{f}$ and $D f$ (zero extended) are locally integrable by step 1.3, both $u_{\tilde{f}}$ and $u_{(D f)^{\sim}}$ are distributions. Then $D u_{\tilde{f}} = u_{(D f)^{\sim}}$ : choose the spatial cutoff $χ_{ε} (x, t) : = χ ((x - z (t)) / ε)$ built from a smooth cutoff $χ$ that is $0$ on ${\overline{B}}_{1} (0)$ and $1$ outside $B_{2} (0)$ ; it is $C^{3}$ and its first derivatives satisfy $∣ D χ_{ε} ∣ \leq C^{'} / ε$ with support in the shell $ε \leq d \leq 2 ε$ , where $d : = ∣ x - z (t) ∣$ and $(1 - v_{*} / c) r \leq d \leq (1 + v_{*} / c) r$ ; the product $χ_{ε} \tilde{f}$ is $C^{1}$ on $Q$ because it vanishes on a neighbourhood of the singular set. Classical integration by parts in each coordinate over a large box containing $supp ψ$ (boundary terms vanish, Riemann-Fubini iterates the one-dimensional identities, and the Riemann-Lebesgue comparison converts to Lebesgue integrals) gives $\int_{Q} χ_{ε} \tilde{f} D ψ = - \int_{Q} χ_{ε} (D f)^{\sim} ψ - \int_{Q} \tilde{f} (D χ_{ε}) ψ$ , and dominated convergence with the locally integrable bounds of step 1.3 lets $ε \to 0$ in the first two integrals; the third is $O (ε)$ because $∣ \tilde{f} ∣ \leq C_{0}^{'} / ε$ on the shell of volume $O (ε^{3})$ in space and $O (1)$ in time. Hence $- \int_{Q} \tilde{f} D ψ = \int_{Q} (D f)^{\sim} ψ$ , that is, $⟨ D u_{\tilde{f}}, ψ ⟩ = ⟨ u_{(D f)^{\sim}}, ψ ⟩$ for every test, proving the claim. Consequently $E = - \nabla Φ - \partial_{t} Υ = u_{E_{0}^{\sim}}$ and $B = curl Υ = u_{B_{0}^{\sim}}$ by linearity of the regular functional, with $E_{0}, B_{0}$ the classical fields of step 1.3.

3.1step 1.2step 2.1givenF1F3F4F7

The wave equations. For every test $ψ$ , $⟨ Φ, W ψ ⟩ = \frac{q}{ε_{0}} \int_{R} (K ⊛ W ψ) (z (s), s) d s = \frac{q}{ε_{0}} \int_{R} ψ (z (s), s) d s = \frac{1}{ε_{0}} ⟨ ρ, ψ ⟩$ , the first equality by the definition of $Φ$ in [F1], the second by the pointwise identity of step 1.2 applied under the history integral (whose integrand is bounded by the cylinder estimate of step 2.1, so the interchange is legitimate), and the last by [F3]. Since $⟨ W Φ, ψ ⟩ = ⟨ Φ, W ψ ⟩$ for the even-order constant-coefficient operator $W$ , this gives $W Φ = ρ / ε_{0}$ . The same computation componentwise with the prefactor $μ_{0} q$ and the weight $z_{j}^{'}$ gives $⟨ Υ_{j}, W ψ ⟩ = μ_{0} q \int_{R} z_{j}^{'} (s) ψ (z (s), s) d s = μ_{0} ⟨ J_{j}, ψ ⟩$ , hence $W Υ = μ_{0} J$ .

3.2step 1.1step 2.1givenF1F4F12

The Lorenz identity. For every test $ψ$ , using the definitions of the distributional derivatives and the identities of step 1.1, $⟨ div Υ + c^{- 2} \partial_{t} Φ, ψ ⟩ = - μ_{0} q \int_{R} [z^{'} (s) \cdot (K ⊛ \nabla ψ) (z (s), s) + (K ⊛ \partial_{t} ψ) (z (s), s)] d s = - μ_{0} q \int_{R} [z^{'} (s) \cdot \nabla_{y} F (z (s), s) + \partial_{s} F (z (s), s)] d s$ , that is, $- μ_{0} q$ times the integral of the total derivative of $s \mapsto F (z (s), s)$ , where $c^{- 2} ε_{0}^{- 1} = μ_{0}$ ; by step 2.1 that total derivative is continuous with compact support, so Newton-Leibniz gives $\int_{R} \frac{d}{d s} [F (z (s), s)] d s = 0$ and the pairing vanishes for every test. Hence $div Υ + c^{- 2} \partial_{t} Φ = 0$ .

4.1step 3.1step 3.2F4F6

The four Maxwell equations. Using the distributional algebra for finite combinations of coordinate derivatives (which commutes in the sense $\partial^{α} \partial^{β} = \partial^{α + β}$ and is linear) together with the classical identities [F6] applied to smooth tests, one computes, with $G : = div Υ + c^{- 2} \partial_{t} Φ = 0$ from step 3.2: $div E = - Δ Φ - \partial_{t} div Υ = W Φ - c^{- 2} \partial_{t}^{2} Φ - \partial_{t} div Υ = ρ / ε_{0} - \partial_{t} G = ρ / ε_{0}$ by step 3.1; $div B = div curl Υ = 0$ ; $curl E = - curl \nabla Φ - \partial_{t} curl Υ = - \partial_{t} B$ ; and $curl B = curl curl Υ = \nabla div Υ - Δ Υ = μ_{0} J - c^{- 2} \partial_{t}^{2} Υ + \nabla div Υ = μ_{0} J + c^{- 2} \partial_{t} E + \nabla G = μ_{0} J + c^{- 2} \partial_{t} E$ , where the fourth step uses $- Δ Υ = W Υ - c^{- 2} \partial_{t}^{2} Υ$ and $\partial_{t} E = - \nabla \partial_{t} Φ - \partial_{t}^{2} Υ$ .

5.1step 2.2step 4.1givenF1F4F5F10∎

Off the worldline. For a test $ψ$ supported in $V$ one has $⟨ ρ, ψ ⟩ = q \int ψ (z (s), s) d s = 0$ and likewise $⟨ J, ψ ⟩ = 0$ , since $ψ$ vanishes on $Γ$ . By step 2.2 the restrictions of $E, B$ to $V$ are the regular distributions of the $C^{1}$ fields $E_{0}, B_{0}$ of [F1]. The classical-compatibility clause of [F4], applied on the open set $V$ , identifies their first distributional derivatives there with the regular distributions of their continuous classical first derivatives; hence the four equations of step 4.1, restricted to $V$ , read $div E_{0} = 0$ , $div B_{0} = 0$ , $curl E_{0} = - \partial_{t} B_{0}$ , $curl B_{0} = c^{- 2} \partial_{t} E_{0}$ as identities of continuous regular distributions. If a continuous function on $V$ has zero regular distribution, then it vanishes almost everywhere and hence, by continuity and the positivity of ball measure, at every point; applying this to each component of the classical residuals gives the source-free Maxwell equations pointwise on $V$ .

Remarks

Choice accounting. ${A C}_{ω}$ is used through the polar formula and change of variables in step 1.3, the Riemann-to-Lebesgue comparison in step 2.2, and the regular-distribution injectivity in step 5.1; the root, the point-current pair and the algebraic identities are choice-free as stated in their own items.
What "the equations hold" means here. The identities of (a) and (b) are distributional identities tested against $D (Q)$ ; the point source is the functional of [F3], and no product of singular distributions, no field value on $Γ$ and no finite self-energy is asserted. The identification in (c) is with the zero extensions of locally integrable classical fields; the value on the null graph is a representative convention only.
Not asserted. No physical adoption of these equations, no radiation-reaction law, no global bound on the acceleration and no uniqueness theorem for incoming fields beyond the retarded construction are claimed.

5 · Examples, experiments and counterexamples

None yet.

Causal kernels, gauge and point fields after wave existence

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression

Definition

Remarks

Gauge reachability of Lorenz gauge by the free-space wave solution, with field invariance

Statement

Facts & Assumptions

Proof

Remarks

The retarded wave kernel is a distribution supported on the cone, solves the point-wave equation, and convolves to the retarded solution

Statement

Facts & Assumptions

Proof

Remarks

The smooth forced Maxwell Cauchy system on a slab: existence, uniqueness, Gauss constraints and the cone of dependence

Statement

Facts & Assumptions

Proof

Remarks

Retarded potentials of a point history as the retarded convolution of the point sources, and the off-worldline Lienard-Wiechert fields

Statement

Facts & Assumptions

Proof

Remarks

The retarded point-history pair solves wave, Lorenz and Maxwell equations distributionally, with locally integrable field representatives off the worldline

Statement

Facts & Assumptions

Proof

Remarks

5 · Examples, experiments and counterexamples

Sources