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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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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