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A switched point charge with no current violates the weak continuity compatibility

Physical scope

A mathematical incompatibility example for the declared distributional source model: the switched charge distribution defined by <rho,psi> = q int_0^infty psi(0,t) dt for every psi in C_c^infty(U), U = R^3 x R, with zero current has div J + partial_t rho = q delta_(0,0) nonzero, so it admits no locally integrable field pair satisfying the four formal weak Maxwell residual identities and cannot be an admitted source model under the singular-source adoption. This is a statement about the declared model, not a measurement; no apparatus, uncertainty or empirical claim is involved.

Example

Let $q \neq 0$ be a real charge parameter (coulombs) and let $U : = R^{3} \times R$ with the laboratory conventions of Distributional Maxwell equations with point and stationary-sheet sources. Define a distribution $ρ$ on $U$ by

$⟨ ρ, ψ ⟩ : = q \int_{0}^{\infty} ψ (0, t) d t, ψ \in C_{c}^{\infty} (U)$

— an explicit switched-history distribution for $z (t) = 0$ on $(0, \infty)$ , with no compensating current or source contribution at the appearance event; it does not satisfy the proper-graph hypothesis of the zero-endpoint continuity branch of Distributional Maxwell equations with point and stationary-sheet sources(b) — and let $J : = 0$ be the zero distribution. Then:

(a) $\partial_{t} ρ = q δ_{(0, 0)}$ and $div J = 0$ (Dirac delta and its derivatives), hence $\partial_{t} ρ + div J = q δ_{(0, 0)} \neq 0$ in $D^{'} (U)$ : for every test function $ψ$ , $⟨ \partial_{t} ρ, ψ ⟩ = - q \int_{0}^{\infty} \partial_{t} ψ (0, t) d t = q ψ (0, 0)$ , because $t \mapsto ψ (0, t)$ is smooth with compact support. In particular the continuity identity fails at the appearance event $(0, 0)$ .

(b) Consequently there are no locally integrable fields $E, B$ on $U$ satisfying the four weak residual identities of Distributional Maxwell equations with point and stationary-sheet sources(e) with the source pair $(ρ, 0)$ , the $ρ$ -pairing being the explicit pairing displayed above and $J = 0$ : such fields would satisfy the distributional continuity identity of Charge continuity as a compatibility condition of the vacuum Maxwell equations(b), which applies to every distributional source pair for which the four identities hold, and (a) contradicts it. The source pair $(ρ, 0)$ is therefore incompatible with the formal four residual identities themselves; it is not one of the declared point sources of Distributional Maxwell equations with point and stationary-sheet sources(b), because the properness hypothesis fails here as recorded, and no fields turn it into an admitted singular-source model under Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources.

(c) The failure is exactly the omitted endpoint term of the point-source construction: by the endpoint clause of Point-support spacetime densities: distributions, continuity equation and endpoint fluxes, the same trajectory on a bounded interval $I_{0} = (a, b)$ has $\partial_{t} ρ + div J = q δ_{γ (a)} - q δ_{γ (b)}$ with $γ (t) = (z (t), t)$ . A switched point charge with $J = 0$ and no endpoint or current term is thus inadmissible; a switched charge must be accompanied by an explicit current or endpoint source term, which the weak equations of Distributional Maxwell equations with point and stationary-sheet sources then require to satisfy continuity.

This is a mathematical incompatibility statement about a declared source model, not a report of a measurement: no apparatus, sampling model, uncertainty or empirical claim is involved.

Verification

Given: A real $q \neq 0$ , the distribution $ρ$ with $⟨ ρ, ψ ⟩ = q \int_{0}^{\infty} ψ (0, t) d t$ on $U = R^{3} \times R$ , and the zero distribution $J$ . The ambient ${A C}_{ω}$ convention is that of Distributional Maxwell equations with point and stationary-sheet sources.

[F1] Distributional derivatives act on tests by $⟨ \partial_{μ} T, ψ ⟩ = - ⟨ T, \partial_{μ} ψ ⟩$ , and a test function is smooth with compact support (Distributional derivative, Distribution, Test function space d of an open set).

[F2] The point evaluation $δ_{(0, 0)} (ψ) = ψ (0, 0)$ defines the Dirac distribution at the origin, and it is nonzero (Dirac delta and its derivatives); some compactly supported smooth test equals one at the origin (Compactly supported scaled Euclidean bumps).

[F3] The charge-continuity theorem states that any distributional source pair satisfying the four distributional Maxwell equations satisfies $\partial_{t} ρ + div J = 0$ in $D^{'} (U)$ (Charge continuity as a compatibility condition of the vacuum Maxwell equations(b)); the weak solution predicate and the source pairings are those of Distributional Maxwell equations with point and stationary-sheet sources(b)-(e), and the singular-source adoption is Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources; the excluded original locally integrable branch is The microscopic vacuum Maxwell equations: smooth and weak branches.

[F4] The endpoint clause of the point-current supplier gives, for a finite-endpoint $C^{1}$ history with finite absolute derivative integral, $\partial_{t} ρ_{p t} + div J_{p t} = q δ_{γ (a)} - q δ_{γ (b)}$ (Point-support spacetime densities: distributions, continuity equation and endpoint fluxes).

1.1F1F2F4givenalgebra

For each compact test support $K$ , choose $T > 0$ larger than its positive time coordinates. Then $∣ ⟨ ρ, ψ ⟩ ∣ \leq ∣ q ∣ T \sup_{K} ∣ ψ ∣$ for tests supported in $K$ , so the displayed linear functional is a distribution. For every test $ψ$ , choose such a $T$ and apply the finite-endpoint branch [F4] to the constant trajectory $z = 0$ on $(0, T)$ : its derivative integral is zero and both endpoint events lie in $U$ . Its density agrees with the switched density on $\partial_{t} ψ$ because that test vanishes for $t \geq T$ , and its zero current agrees with $J$ . Thus $⟨ \partial_{t} ρ, ψ ⟩ = q ψ (0, 0) - q ψ (0, T) = q ψ (0, 0)$ , the last term vanishing by the choice of $T$ ; hence $\partial_{t} ρ = q δ_{(0, 0)}$ as distributions, and this is nonzero because $q \neq 0$ and a compactly supported smooth test equals one at the origin. The divergence of the zero distribution is zero, so $\partial_{t} ρ + div J = q δ_{(0, 0)} \neq 0$ in $D^{'} (U)$ .

2.1F3step 1.1given

Suppose locally integrable fields $E, B$ on $U$ satisfied the four weak residual identities with the source pair $(ρ, 0)$ in the formal residual reading of the solution predicate. The fields' regular distributions and the pair $(ρ, 0)$ are then a distributional source pair satisfying the four identities, so the distributional branch of the charge-continuity theorem, which applies to every distributional source pair for which the four identities hold, would give $\partial_{t} ρ + div J = 0$ in $D^{'} (U)$ . This contradicts step 1.1, so no such fields exist; moreover the pair is not among the declared point sources of the framework, so it cannot be admitted through the singular-source adoption either.

2.2F4step 1.1given

The failure is exactly the omitted endpoint term: the endpoint clause of the point-current supplier evaluates the same trajectory over a bounded history $I_{0} = (a, b)$ as $\partial_{t} ρ_{p t} + div J_{p t} = q δ_{γ (a)} - q δ_{γ (b)}$ , so a switched charge with $J = 0$ and no endpoint term is precisely the case in which that endpoint contribution has been dropped; retaining continuity therefore requires an explicit current or endpoint source term, which the weak identities then force to be the continuity-compatible one.

3.1step 1.1step 2.1step 2.2given∎

Steps 1.1-2.2 establish: the switched-history pair has $\partial_{t} ρ + div J = q δ_{(0, 0)} \neq 0$ ; no locally integrable field pair satisfies the four weak identities with it; and the failing term is exactly the endpoint contribution of the declared point-source construction. The argument is a conditional computation inside the declared distributional source model and makes no empirical claim.

A switched point charge with no current violates the weak continuity compatibility

Example

Verification

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