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Source-free vacuum Maxwell fields satisfy the wave equation

Physical scope

Conditional statement about source-free regions of the smooth branch of the adopted vacuum Maxwell postulate, for jointly $C^{2}$ fields on a named inertial chart with SI constants. Excluded: the weak branch, singular sources, regions with surface sources, any converse from wave equations to Maxwell, and any decay or finite-energy hypothesis. No empirical claim.

Statement

Assume the smooth branch of the vacuum Maxwell postulate The microscopic vacuum Maxwell equations: smooth and weak branches on the laboratory region $U ≅ I \times Ω$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, with the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values and $ε_{0} = (μ_{0} c^{2})^{- 1}$ , the divergence and curl of Divergence and curl of a $C^{1}$ vector field, the partial derivatives of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ and the Laplacian $Δ$ of The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field. Let $E, B : Ω \times I \to R^{3}$ be jointly $C^{2}$ fields of The electric and magnetic fields as fixed-time spatial vector fields, let $(ρ, J)$ be the source fields of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, and let $W \subseteq Ω \times I$ be open with $ρ = 0$ and $J = 0$ on $W$ (a source-free region). Then on $W$ , componentwise,

$Δ E = c^{- 2} \partial_{t}^{2} E, Δ B = c^{- 2} \partial_{t}^{2} B;$

equivalently each of the six Cartesian components of $E$ and of $B$ satisfies the scalar wave equation $c^{- 2} \partial_{t}^{2} v = Δ v$ of speed $c$ on $W$ .

(a) [Hypotheses actually used] The derivation differentiates the Faraday and Ampère equations in time, commutes the time derivative with $curl$ (Clairaut's theorem, available for jointly $C^{2}$ fields), uses the curl-curl identity of The curl of a curl is the gradient of the divergence minus the Laplacian, and uses the two Gauss equations $div E = 0$ , $div B = 0$ on $W$ . The two evolution equations alone are not sufficient for the conclusion without the constraint equations.

(b) [One-directional] No converse is asserted. The two vector wave equations do not imply the four source-free Maxwell equations: the constraints $div E = div B = 0$ and the two curl relations are not recovered, and solutions of the six uncoupled scalar wave equations need not lie in a source-free smooth Maxwell branch. The correctly constrained initial-value problem is a separate result on this page.

(c) [Scope] Only the smooth branch is covered. No decay, boundedness, compact support, finite-energy or boundary hypothesis is imposed or needed; the conclusion is local on $W$ , asserts no value of any field or derivative at a point of singular source support, and does not cover the weak branch of The microscopic vacuum Maxwell equations: smooth and weak branches. The result is conditional on the adopted postulates and makes no empirical claim.

Facts & Assumptions

Given: The laboratory region $U ≅ I \times Ω$ and its frame, the open set $W \subseteq Ω \times I$ on which $ρ = 0$ and $J = 0$ , the jointly $C^{2}$ fields $E, B$ and continuous sources $ρ, J$ , the four smooth-branch equations of the postulate on $U$ , and the constants with $ε_{0} = (μ_{0} c^{2})^{- 1}$ .

[F1]

The fields are jointly $C^{2}$ on $Ω \times I$ in the class sense of The electric and magnetic fields as fixed-time spatial vector fields, so at every fixed $t \in I$ the maps $E (\cdot, t), B (\cdot, t) : Ω \to R^{3}$ are $C^{2}$ spatial vector fields read in the frame basis $(e_{1}, e_{2}, e_{3})$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, and all second partial derivatives including the mixed time-space ones exist and are continuous.

[F2]

On $U$ the smooth branch of the postulate holds: $div E = ρ / ε_{0}$ , $div B = 0$ , $curl E = - \partial_{t} B$ and $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ , with $ρ, J$ the fields of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current and $\partial_{t}$ taken at fixed position (The microscopic vacuum Maxwell equations: smooth and weak branches, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ); on the open subset $W$ one has $ρ = 0$ and $J = 0$ , so there $div E = 0$ , $div B = 0$ , $curl E = - \partial_{t} B$ and $curl B = c^{- 2} \partial_{t} E$ .

[F3]

The operators are those of Divergence and curl of a $C^{1}$ vector field and the Laplacian is that of The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field; for a $C^{2}$ field $F$ on an open subset of $R^{3}$ the curl-curl identity $curl curl F = \nabla div F - Δ F$ holds, with $Δ F$ componentwise (The curl of a curl is the gradient of the divergence minus the Laplacian).

[F4]

For a jointly $C^{2}$ field the mixed partial derivatives commute: $\partial_{t} \partial_{i} F_{j} = \partial_{i} \partial_{t} F_{j}$ for every spatial index $i$ and component $j$ (Clairaut--Schwarz theorem for continuous second partial derivatives).

[F5]

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

Proof

1.1F1F2F5given

On $W$ the source terms vanish, so the Faraday and Ampère equations become $\partial_{t} B = - curl E$ and $\partial_{t} E = c^{2} curl B$ , the second obtained from $curl B = c^{- 2} \partial_{t} E$ by multiplying by $c^{2}$ ; both displays hold pointwise on $W$ , and each side is a continuous combination of first partial derivatives of $E, B$ , and is therefore $C^{1}$ because $E, B$ are jointly $C^{2}$ there.

1.2F2given

The conclusion is local on the open set $W$ and uses only the smooth branch: no decay, boundedness, compact-support, finite-energy or boundary hypothesis was imposed or needed, the weak branch and singular source support are excluded, and no empirical or existence claim is made.

2.1step 1.1F1F4algebra

Differentiating the two displays of step 1.1 in time gives $\partial_{t}^{2} E = c^{2} \partial_{t} (curl B)$ and $\partial_{t}^{2} B = - \partial_{t} (curl E)$ on $W$ ; by step 1.1's differentiability and [F4] the time derivative commutes with every spatial partial derivative, so $\partial_{t} (curl B) = curl (\partial_{t} B)$ and $\partial_{t} (curl E) = curl (\partial_{t} E)$ ; substituting the displays of step 1.1 yields $\partial_{t}^{2} E = c^{2} curl (\partial_{t} B) = - c^{2} curl curl E$ and $\partial_{t}^{2} B = - curl (\partial_{t} E) = - c^{2} curl curl B$ on $W$ .

3.1step 2.1F1F2F3algebra

Fix $(x, t) \in W$ ; by [F1] the spatial field $E (\cdot, t)$ is $C^{2}$ on the open set $Ω$ , so [F3] gives $curl curl E = \nabla div E - Δ E$ at $(x, t)$ , and the Gauss equation $div E = 0$ of [F2] on $W$ turns this into $curl curl E = - Δ E$ ; the identical computation with $B$ gives $curl curl B = - Δ B$ . Substituting these into step 2.1 gives $\partial_{t}^{2} E = c^{2} Δ E$ and $\partial_{t}^{2} B = c^{2} Δ B$ on $W$ , that is $Δ E = c^{- 2} \partial_{t}^{2} E$ and $Δ B = c^{- 2} \partial_{t}^{2} B$ componentwise: each component $v$ of $E$ or of $B$ satisfies $c^{- 2} \partial_{t}^{2} v = Δ v$ on $W$ .

4.1step 3.1F1F2F3givenalgebra

The derivation used both Gauss constraints through step 3.1; without $div E = div B = 0$ one obtains only the curl-curl forms of step 2.1. No converse holds: on $W = R^{3} \times R$ the time-independent field $E (x) = (x_{1}, 0, 0)$ , $B = 0$ is $C^{\infty}$ , is affine in $x_{1}$ so that $Δ E = 0$ and $\partial_{t}^{2} E = 0$ , and therefore satisfies both vector wave equations, while $div E = 1$ violates the constraint $div E = ρ / ε_{0}$ with $ρ = 0$ ; hence it lies in no source-free smooth Maxwell branch and the wave equations do not imply the four Maxwell equations.

5.1step 3.1step 4.1step 1.2given∎

Steps 1.1, 2.1, 3.1 and 4.1 derive the two displayed vector wave equations on $W$ from the four smooth-branch equations with vanishing sources, record the hypotheses actually used (time differentiation, Clairaut commutation, the curl-curl identity and the two Gauss constraints), and exhibit a witness showing that the implication is one-directional; step 1.2 records the scope.

Source notes

The cited treatments derive the source-free vacuum wave equation by exactly this route: differentiate the two evolution equations, commute derivatives, apply the curl-curl identity and use $div E = div B = 0$ in the source-free region. The textbook statements are the motivation for the claim; the deduction above uses only the imported divergence, curl, Laplacian, Clairaut and curl-curl interfaces together with the adopted smooth-branch postulate. The one-directional warning is the standard observation that solutions of the wave equation need not satisfy the constraints.

Source-free vacuum Maxwell fields satisfy the wave equation

Statement

Facts & Assumptions

Proof

Source notes

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