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Retarded potentials, Lorenz gauge and Maxwell fields of a compactly supported conserved smooth source

Physical scope

Retarded potentials and fields of a prescribed smooth conserved source of compact spatial support in a compact time interval, on the whole laboratory spacetime in the framework's inertial chart: Lorenz gauge, the smooth vacuum Maxwell equations, zero field before the source acts, the zero-data Cauchy identification and zero total charge. Excluded: point, line, sheet or non-integrable sources; weak/distributional branches; eternal isolated charges; finite-energy, decay or boundary claims; self-force and radiation reaction; and every empirical or measurement claim.

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) as carried by the retarded-kernel and Cauchy suppliers below (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let the laboratory region, chart and units be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values, so that $c > 0$ , $μ_{0} > 0$ , $ε_{0} = (μ_{0} c^{2})^{- 1}$ and $W : = c^{- 2} \partial_{t}^{2} - Δ$ is the wave expression of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression. Let $a > 0$ , $t_{0} < T$ and let $ρ : R^{3} \times R \to R$ , $J : R^{3} \times R \to R^{3}$ be source fields of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current that are jointly of class $C^{\infty}$ in the sense of $C^{k}$ Euclidean maps and diffeomorphisms, vanish identically outside the compact set ${\overline{B}}_{a} (0) \times [t_{0}, T]$ , and satisfy the continuity equation $\partial_{t} ρ + div J = 0$ everywhere --- the compatibility condition that Charge continuity as a compatibility condition of the vacuum Maxwell equations shows to be forced by the Maxwell system, here assumed for the prescribed source. Define the retarded potentials by the explicit retarded integrals

$φ (x, t) : = \frac{1}{4 π ε_{0}} \int_{R^{3}} \frac{ρ (y, t - ∣ x - y ∣ / c)}{∣ x - y ∣} d y, A (x, t) : = \frac{μ_{0}}{4 π} \int_{R^{3}} \frac{J (y, t - ∣ x - y ∣ / c)}{∣ x - y ∣} d y,$

and put $E : = - \nabla φ - \partial_{t} A$ , $B : = \nabla \times A$ (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression, The electric and magnetic fields as fixed-time spatial vector fields). Then:

(a) [Smoothness and wave equations] $φ$ and each component of $A$ are of class $C^{\infty}$ on $R^{3} \times R$ , and $W φ = ρ / ε_{0}$ , $W A = μ_{0} J$ pointwise.

(b) [Lorenz gauge] With $G : = div A + c^{- 2} \partial_{t} φ$ one has $G = 0$ on $R^{3} \times R$ : the retarded pair is in Lorenz gauge, the continuity hypothesis supplies this identity and the total-charge conclusion (f).

(c) [Maxwell fields] $E$ and $B$ are of class $C^{\infty}$ and the tuple $(E, B, ρ, J)$ satisfies the smooth branch of the vacuum Maxwell equations of The microscopic vacuum Maxwell equations: smooth and weak branches on the whole of $R^{3} \times R$ : $div E = ρ / ε_{0}$ , $div B = 0$ , $\nabla \times E = - \partial_{t} B$ , $\nabla \times B = μ_{0} J + c^{- 2} \partial_{t} E$ .

(d) [No field before the source] $E = B = 0$ at every event with $t < t_{0}$ (indeed $φ = A = 0$ there): the selected retarded solution carries no field from before the source acted.

(e) [Zero-data Cauchy identification] For every $t_{1} < t_{0}$ the pair $(φ, A)$ restricted to $R^{3} \times [t_{1}, \infty)$ is the unique $C^{2}$ extension solution of the wave Cauchy problem of The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support at time $t_{1}$ with zero Cauchy data and sources $ρ / ε_{0}$ , $μ_{0} J$ .

(f) [Total charge] The total charge $Q (t) : = \int_{R^{3}} ρ (y, t) d y$ is zero for every $t$ : a source that is smooth, compactly supported in space and restricted to a compact time interval and satisfies continuity carries zero total charge, so this branch does not model an eternal isolated nonzero charge.

(g) [Scope] This is the compact smooth branch with an adopted causal selection; it asserts nothing about point, line or sheet sources, about weak or distributional branches, about finite-energy or decay properties, about radiation reaction or self-force, and no empirical claim. In particular the fields of a prescribed isolated point charge with an eternal worldline are not covered by this theorem and are supplied separately by the point-source items of this page.

Facts & Assumptions

Given: ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )); the laboratory chart and SI constants of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values; constants $a > 0$ , $t_{0} < T$ ; a jointly $C^{\infty}$ source pair $(ρ, J)$ vanishing outside ${\overline{B}}_{a} (0) \times [t_{0}, T]$ and satisfying $\partial_{t} ρ + div J = 0$ ; the retarded integrals $φ, A$ of the Statement, with $K$ denoting the retarded kernel of [F1] so that $φ = ε_{0}^{- 1} (K * ρ)$ and $A_{k} = μ_{0} (K * J_{k})$ .

[F1]

Retarded kernel: for every $h \in C_{c}^{\infty} (R^{3} \times R)$ the function $(K * h) (x, t) = \int_{R^{3}} h (y, t - ∣ x - y ∣ / c) / (4 π ∣ x - y ∣) d y$ is well defined (absolutely convergent), of class $C^{\infty}$ on $R^{3} \times R$ , and satisfies $W (K * h) = h$ pointwise. (The retarded wave kernel is a distribution supported on the cone, solves the point-wave equation, and convolves to the retarded solution (b))

[F2]

Gauge identities and the decoupling definition: for a potential pair of class $C^{2}$ jointly on an open set, $div E = W φ - \partial_{t} G$ and $curl B - c^{- 2} \partial_{t} E = W A + \nabla G$ hold there, the source-free equations $div B = 0$ and $curl E = - \partial_{t} B$ hold automatically for any potential pair, and if in addition $G = 0$ then the sourced Maxwell system of The microscopic vacuum Maxwell equations: smooth and weak branches holds if and only if $W φ = ρ / ε_{0}$ and $W A = μ_{0} J$ ; the fields $E, B$ , the Lorenz divergence $G$ and the wave expression $W$ are those of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression, and field regularity classes are those of The electric and magnetic fields as fixed-time spatial vector fields. (Maxwell's equations in potential form: the Lorenz divergence, the decoupled wave equations and residual generators (a),(c))

[F3]

Cauchy uniqueness: if $v \in C^{2} (R^{3} \times [t_{1}, T^{'}])$ is of class $C^{2}$ in the extension sense, satisfies $W v = h$ on $R^{3} \times (t_{1}, T^{'})$ with $h \in C^{\infty} (R^{3} \times [t_{1}, T^{'}])$ in the extension sense, and has prescribed Cauchy data at $t_{1}$ , then $v$ equals the solution constructed there. (The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support (b))

[F4]

SI constants: $ε_{0} = (μ_{0} c^{2})^{- 1}$ , so $c^{- 2} ε_{0}^{- 1} = μ_{0}$ ; $c > 0$ , $μ_{0} > 0$ . (Framework-local SI units, the vacuum constants and the exact/measured status of their values)

[F5]

Source fields: smooth source fields $(ρ, J)$ with their units, the continuity equation and the compact-support convention are those of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current; the standing hypotheses of this theorem are as stated in the Given. (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current)

[F6]

$C^{1}$ change of variables for $L^{1}$ : for a $C^{1}$ diffeomorphism $T : U \to V$ of open sets and $f \in L^{1} (λ_{n})$ , $\int_{V} f = \int_{U} f \circ T ∣ \det D T ∣$ . (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions)

[F7]

Differentiation under the integral sign: if $x \mapsto f (x, s)$ is integrable for every $s$ in an open interval, $s \mapsto f (x, s)$ is differentiable for almost every $x$ , the extended partial derivative is measurable in $x$ , and $∣ \partial_{s} f (x, s) ∣ \leq g (x)$ for an integrable $g$ independent of $s$ , then $\int f (x, s) d x$ is differentiable with derivative $\int \partial_{s} f (x, s) d x$ . (Differentiation under the integral sign)

[F8]

Polar coordinates: for every nonnegative Borel $f$ on $R^{3}$ , $\int_{R^{3}} f d λ_{3} = \int_{0}^{\infty} \int_{S^{2}} f (r ω) r^{2} d σ (ω) d r$ ; in particular $\int_{B_{s} (0)} ∣ z ∣^{- 1} d z = 4 π \int_{0}^{s} r d r = 2 π s^{2} < \infty$ . (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma)

[F9]

Linearity of the integral on $L^{1}$ . (The Lebesgue integral is linear on $L^{1} (μ)$ )

[F10]

Divergence theorem for a bounded $C^{1}$ Euclidean domain $Ω$ and $F \in C^{1} (\overline{Ω}; R^{3})$ : $\int_{Ω} div F d x = \int_{\partial Ω} F \cdot ν d S$ with outward normal. (Divergence on a bounded C1 Euclidean domain)

[F11]

Parameter integrals: if $K$ is compact measurable of finite measure and $f, \partial_{t} f$ are continuous on a neighbourhood of $K \times [a, b]$ , then $t \mapsto \int_{K} f (x, t) d x$ is differentiable there with derivative $\int_{K} \partial_{t} f (x, t) d x$ . (Differentiation under the integral sign over a compact measurable set and over a compact $C^{1}$ surface)

[F12]

$C^{k}$ means continuous partial derivatives up to order $k$ ; finite sums and products of $C^{k}$ maps are $C^{k}$ . ( $C^{k}$ Euclidean maps and diffeomorphisms)

[F13]

The framework adopts the retarded branch of Adopted retarded solution selection: the physical branch carries no incoming free field: the selected physical potential pair is the retarded one with no free addition, and the theorem below computes exactly that branch; it asserts no existence of its own. (Adopted retarded solution selection: the physical branch carries no incoming free field (b),(c))

[F14]

For $C^{2}$ fields and $C^{1}$ sources satisfying the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches, the continuity equation is forced; here continuity is assumed for the prescribed source. (Charge continuity as a compatibility condition of the vacuum Maxwell equations (a))

Proof

technique · direct; identify the potentials with the retarded kernel of [F1], differentiate the kernel pairing to obtain the Lorenz gauge from continuity, read off the Maxwell system from [F2], localize the retarded integrals for the vanishing and Cauchy claims, and integrate the continuity equation for the total charge

1.1F1F4F5F12

Smoothness and wave equations. The source pair $(ρ, J)$ is jointly $C^{\infty}$ and vanishes outside the compact set ${\overline{B}}_{a} (0) \times [t_{0}, T]$ , so $ρ / ε_{0}$ and each $h = μ_{0} J_{k}$ belongs to $C_{c}^{\infty} (R^{3} \times R)$ [F5, F12]. Applying [F1] to these functions and using the identification $φ = ε_{0}^{- 1} (K * ρ)$ , $A_{k} = μ_{0} (K * J_{k})$ of the Given gives that $φ$ and every component of $A$ are of class $C^{\infty}$ on $R^{3} \times R$ , with $W φ = ρ / ε_{0}$ and $W A = μ_{0} J$ pointwise. This is (a).

1.2F6F7F8

Derivative identities for the retarded kernel. Fix $h \in C_{c}^{\infty} (R^{3} \times R)$ . For fixed $(x, t)$ the change of variable $y \mapsto z = x - y$ , a $C^{1}$ diffeomorphism of $R^{3}$ with $∣ \det D (x - \cdot) ∣ = 1$ , is legitimate for the integrable function $y \mapsto h (y, t - ∣ x - y ∣ / c) / (4 π ∣ x - y ∣)$ [F6], so $(K * h) (x, t) = \int h (x - z, t - ∣ z ∣ / c) / (4 π ∣ z ∣) d z$ . Choose $b_{h} > 0$ such that the spatial projection of $supp h$ lies in ${\overline{B}}_{b_{h}} (0)$ , and let $M_{h}$ be a finite global bound on the first derivatives of $h$ (they are continuous and compactly supported). On a parameter neighbourhood $∣ x ∣ < R$ , $∣ t ∣ < R$ , with $R > 0$ , each differentiated integrand vanishes unless $∣ x - z ∣ \leq b_{h}$ , hence unless $∣ z ∣ < R + b_{h}$ . Its absolute value is bounded by $M_{h} 1_{B_{R + b_{h} + 1}} (z) / (4 π ∣ z ∣)$ , which is integrable by [F8]. The zero extension at $z = 0$ affects a null set only; this majorant is independent of the parameter being differentiated. The derivative is measurable as a limit of measurable difference quotients, so [F7] applies once per parameter $x_{1}, x_{2}, x_{3}, t$ with the other three parameters held fixed, and gives $\partial_{x_{i}} (K * h) = K * (\partial_{y_{i}} h)$ and $\partial_{t} (K * h) = K * (\partial_{t} h)$ on $R^{3} \times R$ , where $\partial_{y_{i}} h$ is the partial derivative of $h$ in its $i$ -th spatial argument.

1.3F4F5F9algebra

No field before the source. Suppose $t < t_{0}$ . For every $y$ the retarded argument satisfies $t - ∣ x - y ∣ / c \leq t < t_{0}$ , where $ρ$ and $J$ vanish identically by the support hypothesis; hence both integrands in the definitions of $φ$ and $A$ vanish for every $y$ , and $φ (x, t) = 0$ , $A (x, t) = 0$ [F9]. Therefore $E = - \nabla φ - \partial_{t} A = 0$ and $B = \nabla \times A = 0$ at every event with $t < t_{0}$ , which is (d).

1.4F5F9F10F11algebra

Total charge. By the support hypothesis, $Q (t) = \int_{R^{3}} ρ (y, t) d y = \int_{B_{a} (0)} ρ (y, t) d y$ for every $t$ , and $Q (t) = 0$ for $t < t_{0}$ ; the integral depends smoothly on $t$ with $Q^{'} (t) = \int_{B_{a} (0)} \partial_{t} ρ (y, t) d y$ by [F11] applied to the fixed compact ${\overline{B}}_{a} (0)$ (the integrals over the open and closed balls agree since the smooth source vanishes on the boundary) (the integrands $ρ, \partial_{t} ρ$ are continuous on a neighbourhood of ${\overline{B}}_{a} (0) \times [t - 1, t + 1]$ ) [F5, F12]. Fix $t$ and $R > a$ ; the continuity equation and the divergence theorem [F10] give $\int_{B_{a} (0)} div J (y, t) d y = \int_{B_{R} (0)} div J (y, t) d y = \int_{\partial B_{R} (0)} J (y, t) \cdot ν d S = 0$ , because $J$ vanishes outside ${\overline{B}}_{a} (0)$ and $R > a$ ; hence $Q^{'} (t) = - \int_{B_{a} (0)} div J (y, t) d y = 0$ . Thus $Q$ is constant on $R$ and, being zero at times before $t_{0}$ , is identically zero, which is (f).

2.1step 1.2F4F5F9algebra

Lorenz gauge. By step 1.2 applied to each component $J_{i}$ and to $ρ$ , together with linearity of the integral [F9], $div A = μ_{0} \sum_{i = 1}^{3} \partial_{x_{i}} (K * J_{i}) = μ_{0} \sum_{i = 1}^{3} K * (\partial_{y_{i}} J_{i}) = μ_{0} K * (div J), c^{- 2} \partial_{t} φ = c^{- 2} ε_{0}^{- 1} K * (\partial_{t} ρ) = μ_{0} K * (\partial_{t} ρ),$ the second equality using $c^{- 2} ε_{0}^{- 1} = μ_{0}$ [F4]. Adding, $G = div A + c^{- 2} \partial_{t} φ = μ_{0} K * (\partial_{t} ρ + div J) = 0$ pointwise on $R^{3} \times R$ by the assumed continuity equation and linearity of the kernel integral [F9]. This is (b); continuity is also used in step 1.4 to establish the total-charge conclusion.

2.2step 1.1step 1.3F3F12algebra

Zero-data Cauchy identification. Fix $t_{1} < t_{0}$ and $T^{'} > t_{0}$ . By step 1.3, $φ (\cdot, t_{1}) = 0$ , $A (\cdot, t_{1}) = 0$ and also $\partial_{t} φ (\cdot, t_{1}) = 0$ , $\partial_{t} A (\cdot, t_{1}) = 0$ (the same vanishing argument applies to the time derivatives, whose integrands contain $ρ_{t}, J_{t}$ , vanishing for retarded arguments $< t_{0}$ as well). By step 1.1 the pair is $C^{\infty}$ , hence $C^{2}$ , on $R^{3} \times [t_{1}, T^{'}]$ in the extension sense and satisfies $W φ = ρ / ε_{0}$ , $W A = μ_{0} J$ on $R^{3} \times (t_{1}, T^{'})$ ; [F3] (with its final time renamed $T^{'}$ and its Cauchy data $a = b = 0$ ) therefore identifies each component with the zero-data solution constructed there. As $T^{'} > t_{0}$ is arbitrary, the identification extends to $R^{3} \times [t_{1}, \infty)$ , which is (e).

3.1step 1.1step 2.1step 2.2F2F13F14algebra∎

Maxwell fields. $E$ and $B$ are $C^{\infty}$ because $φ$ and $A$ are [step 1.1, F12]. By [F2] with the Lorenz divergence $G$ of step 2.1, $div E = W φ - \partial_{t} G = ρ / ε_{0} - 0$ and $curl B - c^{- 2} \partial_{t} E = W A + \nabla G = μ_{0} J + 0$ ; the two source-free equations $div B = 0$ and $curl E = - \partial_{t} B$ are automatic for the potential pair [F2]. Hence the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches holds for $(E, B, ρ, J)$ on $R^{3} \times R$ with $E, B$ of class $C^{\infty}$ , which is (c). The remaining clause (g) is scope bookkeeping: every hypothesis used above is one of the displayed compact smooth source hypotheses, no point, line, sheet or distributional source is involved, no product of distributions or self-field is formed, no finite-energy, decay, self-force or measurement claim is made, and the only choice assumption is ${A C}_{ω}$ carried by the kernel, polar and Cauchy suppliers.

Retarded potentials, Lorenz gauge and Maxwell fields of a compactly supported conserved smooth source

Statement

Facts & Assumptions

Proof

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