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Charge continuity as a compatibility condition of the vacuum Maxwell equations

Physical scope

The conditional compatibility statement that any field and source tuple satisfying the framework's vacuum Maxwell equations, in the smooth C2 branch or in the distributional branch including the separately adopted singular-source models, satisfies div J + partial_t rho = 0 pointwise or in the sense of distributions. It is a compatibility condition on sources, not a conservation law of matter, not an existence statement and not empirical evidence; it inherits the regularity and source hypotheses of the branch used.

Statement

Let $U ≅ I \times Ω$ be the laboratory region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields with the constants and units of Framework-local SI units, the vacuum constants and the exact/measured status of their values, let $E, B$ be the fields of The electric and magnetic fields as fixed-time spatial vector fields and $ρ, J$ the source fields of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, with $ε_{0} = (μ_{0} c^{2})^{- 1}$ .

(a) [Smooth branch] Let $E, B$ be jointly of class $C^{2}$ on $Ω \times I$ and $ρ, J$ of class $C^{1}$ , and suppose the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches holds with these fields and sources, in particular $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ . Then the continuity equation

$\partial_{t} ρ + div J = 0$

holds pointwise on $Ω \times I$ , where $div$ is the divergence of Divergence and curl of a $C^{1}$ vector field and $\partial_{t}$ the partial time derivative of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ .

(b) [Distributional branch] Let $E, B, ρ, J$ be distributions on $Ω \times I$ (Distribution, Distributional derivative); in particular they may be locally integrable fields and sources identified with their regular distributions (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions). Suppose the four equations

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

hold in $D^{'} (Ω \times I)$ with distributional derivatives. This includes the original locally integrable weak branch of The microscopic vacuum Maxwell equations: smooth and weak branches and the separately adopted declared singular-source models of Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources, when their source and field hypotheses hold. More generally the formal compatibility implication applies to every distributional source pair for which the four identities hold, without asserting physical adoption of arbitrary distributions. Then $\partial_{t} ρ + div J = 0$ in $D^{'} (Ω \times I)$ .

The conclusion is a compatibility condition: it does not assert that every source pair is compatible with the Maxwell equations, and by itself it proves no conservation of physical charge. The derivative compatibility argument is choice-free. An application to the declared singular-source model retains the Countable Choice assumption of its defining source and surface-measure formulation (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Facts & Assumptions

Given: The laboratory region, fields, sources and constants of the Statement; in clause (a) the joint $C^{2}$ and $C^{1}$ regularity and the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches; in clause (b) the four distributional identities.

[F1]

The smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches holds pointwise, in particular $div E = ρ / ε_{0}$ and $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ , with $ε_{0} = (μ_{0} c^{2})^{- 1}$ and $μ_{0} > 0$ (The microscopic vacuum Maxwell equations: smooth and weak branches, Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F2]

The separately adopted distributional postulate requires the four identities in $D^{'} (U)$ for the declared singular-source models (Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources).

[F3]

For a $C^{2}$ vector field on an open set the divergence of its curl vanishes pointwise (The divergence of the curl of a $C^{2}$ field vanishes), and mixed partial derivatives commute (Clairaut--Schwarz theorem for continuous second partial derivatives); derivatives of sums and scalar multiples are computed termwise (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 $g (c) \neq 0$ , Divergence and curl of a $C^{1}$ vector field, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

[F4]

Distributional derivatives are defined by $⟨ \partial_{μ} T, ψ ⟩ = - ⟨ T, \partial_{μ} ψ ⟩$ on tests $ψ \in C_{c}^{\infty} (U)$ , are linear in $T$ and commute with one another (Distribution, Distributional derivative, Test function space d of an open set, Distributional differentiation is continuous and commutes).

[F5]

For $f \in L_{l o c}^{1} (Ω \times I)$ , its regular functional is $⟨ u_{f}, ψ ⟩ = \int_{Ω \times I} f ψ$ , well-defined modulo almost-everywhere equality. For tests supported in a compact $K \subseteq Ω \times I$ , $∣ ⟨ u_{f}, ψ ⟩ ∣ \leq (\int_{K} ∣ f ∣) \sup_{K} ∣ ψ ∣$ ; this choice-free order-zero estimate makes $u_{f}$ a distribution, componentwise for vector fields (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions, proof step 1.1). The cited theorem's injectivity conclusion assumes Countable Choice and is not used here.

Proof

1.1F1F3givenalgebra

In the smooth branch, the identity $div (curl B) = 0$ holds pointwise. Taking the divergence of the Ampere equation and using linearity of the coordinate derivatives gives $0 = div (curl B) = μ_{0} div J + c^{- 2} div (\partial_{t} E)$ , and the mixed partials commute, so $div (\partial_{t} E) = \partial_{t} (div E)$ ; the Gauss law gives $div E = ρ / ε_{0}$ and, $ρ$ being $C^{1}$ , $\partial_{t} (div E) = ε_{0}^{- 1} \partial_{t} ρ$ . Substituting gives $0 = μ_{0} div J + c^{- 2} ε_{0}^{- 1} \partial_{t} ρ = μ_{0} (div J + \partial_{t} ρ)$ , because $c^{2} ε_{0} μ_{0} = 1$ by the definition of $ε_{0}$ , and division by $μ_{0} > 0$ gives $\partial_{t} ρ + div J = 0$ pointwise on $Ω \times I$ .

1.2F4givenalgebra

In the distributional branch, fix $ψ \in C_{c}^{\infty} (Ω \times I)$ . The distributional curl is built from the components $\partial_{j} B_{k} - \partial_{k} B_{j}$ , so $⟨ div (curl B), ψ ⟩ = \sum_{i, j, k} ε_{i j k} ⟨ \partial_{i} \partial_{j} B_{k}, ψ ⟩$ , where $i, j, k$ range over $1, 2, 3$ and $ε_{i j k}$ is $+ 1$ for cyclic permutations of $(1, 2, 3)$ , $- 1$ for the other permutations and $0$ for repeated indices; the distributional derivatives commute and $ε_{i j k}$ is antisymmetric, so exchanging the summation indices $i$ and $j$ presents this number as its own negative, and hence it is zero. The Ampere identity in $D^{'}$ , linearity of the pairing and commutativity with $\partial_{t}$ give $0 = μ_{0} ⟨ div J, ψ ⟩ + c^{- 2} ⟨ \partial_{t} div E, ψ ⟩$ ; the Gauss identity in $D^{'}$ reads $div E = ρ / ε_{0}$ , so $\partial_{t} div E = ε_{0}^{- 1} \partial_{t} ρ$ as distributions, and substitution with $c^{2} ε_{0} μ_{0} = 1$ and $μ_{0} > 0$ gives $⟨ \partial_{t} ρ + div J, ψ ⟩ = 0$ . As $ψ$ was arbitrary, $\partial_{t} ρ + div J = 0$ in $D^{'} (Ω \times I)$ ; every ingredient is either a distributional identity or a constant multiple, so no choice principle is used.

2.1F2F5step 1.2given∎

For locally integrable fields and sources, apply the regular-distribution construction of [F5] componentwise and then step 1.2; injectivity of that construction is unnecessary. The physical adoption of the singular-source models remains the separate postulate [F2], and the Countable Choice assumption is carried only by their defining source and surface-measure formulation.

Source notes

Continuity as a compatibility condition of the Maxwell equations, obtained by the divergence-of-curl cancellation, is the standard textbook derivation cited in the frontmatter. The distributional form uses the library's own distribution interface; the singularity of point and sheet sources is handled only through the separate adopted postulate, never through an undefined product of distributions.

Charge continuity as a compatibility condition of the vacuum Maxwell equations

Statement

Facts & Assumptions

Proof

Source notes

Depends on

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Sources