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A homogeneous-wave gauge generator changes Lorenz potentials without changing their fields

Physical scope

An example on the free-space branch of the framework's gauge structure: a cosinusoidal gauge generator satisfying the homogeneous wave equation changes a Lorenz potential pair while preserving the Lorenz condition and the field pair. It asserts no bounded-domain or prescribed-data gauge statement and no empirical claim.

Example

Let $c > 0$ , let $k \in R^{3} ∖ {0}$ , put $ω : = c ∥ k ∥_{2} > 0$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ), and let

$χ (x, t) : = \cos (k \cdot x - ω t),$

the real part, by $\exp (i θ) = \cos θ + i \sin θ$ of Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ , of the complex plane wave $e^{i (k \cdot x - ω t)}$ of Complex plane-wave fields: real parts of complex exponentials with wave vector $k$ and frequency $ω = c ∥ k ∥_{2}$ . For the SI interpretation, use numerical components $x$ in m, $t$ in s, $k$ in m $^{- 1}$ and $ω$ in s $^{- 1}$ , so the phase is dimensionless, and give this gauge generator amplitude one V s as in Potential representations, gauge transformations and the Lorenz and Coulomb conditions; the displayed $χ$ is its numerical value in V s. Then $χ$ is of class $C^{\infty}$ on $R^{3} \times R$ and $W χ = 0$ , where $W = c^{- 2} \partial_{t}^{2} - Δ$ is the wave expression of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression (the normalization $W = c^{- 2} □_{c}$ of the wave operator of Wave equation, Cauchy data and wave speed, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field); indeed $\partial_{t}^{2} χ = - ω^{2} χ$ , $Δ χ = - ∥ k ∥_{2}^{2} χ$ and $ω^{2} = c^{2} ∥ k ∥_{2}^{2}$ , so $W χ = c^{- 2} (- c^{2} ∥ k ∥_{2}^{2} χ) + ∥ k ∥_{2}^{2} χ = 0$ (The derivatives of sine and cosine are cosine and minus sine, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , 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 $g (c) \neq 0$ ).

Consequently, for every potential pair $(φ, A)$ of class $C^{2}$ jointly on $R^{3} \times R$ (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression), the gauge-transformed pair satisfies

$G (T_{χ} (φ, A)) = G (φ, A) and E (T_{χ} (φ, A)) = E (φ, A), B (T_{χ} (φ, A)) = B (φ, A)$

on $R^{3} \times R$ (Maxwell's equations in potential form: the Lorenz divergence, the decoupled wave equations and residual generators, Gauge invariance of the electric and magnetic fields, Potential representations, gauge transformations and the Lorenz and Coulomb conditions); in particular, if $(φ, A)$ is in Lorenz gauge then $T_{χ} (φ, A)$ is again in Lorenz gauge with the same field pair. Explicitly $\partial_{t} χ = ω \sin (k \cdot x - ω t)$ and $\nabla χ = - k \sin (k \cdot x - ω t)$ , so $φ^{'} = φ - ω \sin (k \cdot x - ω t)$ and $A^{'} = A - k \sin (k \cdot x - ω t)$ . The zero pair is in Lorenz gauge, and its transform $(- ω \sin (k \cdot x - ω t), - k \sin (k \cdot x - ω t))$ is a nonzero pair that remains in Lorenz gauge and still represents the zero field pair: it is nonzero because otherwise $\partial_{t} χ$ would vanish identically, forcing $\partial_{t}^{2} χ = 0$ , whereas $\partial_{t}^{2} χ (0, 0) = - ω^{2} \neq 0$ . Thus the Lorenz condition does not determine the potential pair. The example is the residual generator $W χ = 0$ of Maxwell's equations in potential form: the Lorenz divergence, the decoupled wave equations and residual generators on the free-space branch; on a bounded domain with prescribed gauge boundary data the admissible generators are restricted, and no such data-dependent claim is made here.

Facts & Assumptions

Given: A constant $c > 0$ , a vector $k \in R^{3} ∖ {0}$ , the frequency $ω : = c ∥ k ∥_{2} > 0$ , the function $χ (x, t) = \cos (k \cdot x - ω t)$ on $R^{3} \times R$ , an arbitrary potential pair $(φ, A)$ of class $C^{2}$ jointly on $R^{3} \times R$ , and the zero potential pair.

[F1]

Wave vocabulary: for a $C^{2}$ function $u$ one has $W u = c^{- 2} \partial_{t}^{2} u - Δ u$ , the normalization $W = c^{- 2} □_{c}$ of the wave operator $□_{c} = \partial_{t}^{2} - c^{2} Δ$ of speed $c$ ; for a potential pair $G = div A + c^{- 2} \partial_{t} φ$ , the pair is in Lorenz gauge when $G = 0$ , and a class $C^{m}$ generator acts by $T_{χ} (φ, A) = (φ - \partial_{t} χ, A + \nabla χ)$ on a class $C^{m}$ pair with $m \geq 2$ , giving a class $C^{m - 1}$ pair (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression, Wave equation, Cauchy data and wave speed, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field).

[F2]

Plane-wave vocabulary: for $k \in R^{3}$ and a real parameter $ω$ the phase is $θ (x, t) = ⟨ k, x ⟩ - ω t$ and the complex plane wave is $\exp (i θ (x, t))$ , with $⟨ \cdot, \cdot ⟩$ the Euclidean inner product and the norm $∥ x ∥_{2} = \sqrt{⟨ x, x ⟩}$ (Complex plane-wave fields: real parts of complex exponentials, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ); Euler's formula $\exp (i θ) = \cos θ + i \sin θ$ identifies $χ$ with the real part (Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ ).

[F3]

Sine and cosine: $(\sin u)^{'} = \cos u$ and $(\cos u)^{'} = - \sin u$ on $R$ , with $\sin 0 = 0$ and $\cos 0 = 1$ ; differentiating the two formulas repeatedly exhibits every higher derivative as $\pm \sin$ or $\pm \cos$ , which are continuous, so both functions are $C^{\infty}$ (The derivatives of sine and cosine are cosine and minus sine).

[F4]

Chain rule and algebra of derivatives: composition of $C^{m}$ maps is $C^{m}$ , the chain rule computes $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , and partial derivatives are the one-variable derivatives along coordinate lines and obey the sum and scalar rules (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , 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 $g (c) \neq 0$ , Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , $C^{k}$ Euclidean maps and diffeomorphisms).

[F5]

Gauge shift of the Lorenz divergence: if $(φ, A)$ is a potential pair of class $C^{2}$ jointly and $χ$ a gauge function of class $C^{2}$ jointly, then $G (T_{χ} (φ, A)) = G (φ, A) - W χ$ on the region of definition (Maxwell's equations in potential form: the Lorenz divergence, the decoupled wave equations and residual generators).

[F6]

Gauge invariance: for a potential pair and a gauge function of class $C^{k}$ jointly with $k \geq 2$ , the transform $T_{χ} (φ, A)$ is a potential pair of class $C^{k - 1}$ with the same field pair as $(φ, A)$ (Gauge invariance of the electric and magnetic fields, Potential representations, gauge transformations and the Lorenz and Coulomb conditions).

Verification

technique · direct; compute the first and second derivatives of the cosinusoidal generator, conclude $W\chi=0$ from the dispersion relation $\omega=c\lVert k\rVert_2$, apply the gauge-shift and gauge-invariance theorems, and instantiate at the zero pair with an explicit nonzero witness

1.1givenF2F3F4

The generator is smooth with the displayed first derivatives. The phase $θ (x, t) = ⟨ k, x ⟩ - ω t$ is an affine function of $(x, t)$ , hence $C^{\infty}$ ; the cosine is $C^{\infty}$ by [F3], and the composition $χ = \cos \circ θ$ is $C^{\infty}$ on $R^{3} \times R$ by [F4]. The chain rule of [F4] gives $\partial_{t} χ = - \sin θ \cdot \partial_{t} θ = ω \sin θ$ and $\partial_{i} χ = - \sin θ \cdot \partial_{i} θ = - k_{i} \sin θ$ for $i = 0, 1, 2$ in the cited Euclidean coordinate convention, because $\partial_{t} θ = - ω$ and $\partial_{i} θ = k_{i}$ ; in vector form $\partial_{t} χ = ω \sin (k \cdot x - ω t)$ and $\nabla χ = - k \sin (k \cdot x - ω t)$ , the inner product and coordinates being those of [F2].

2.1step 1.1F1F2F3F4

The generator satisfies the homogeneous wave equation. Differentiating the first derivatives of step 1.1 by the chain rule [F4] gives $\partial_{t}^{2} χ = ω \cos θ \cdot (- ω) = - ω^{2} \cos θ = - ω^{2} χ$ and, for each $i$ , $\partial_{i} \partial_{i} χ = - k_{i} \cos θ \cdot k_{i} = - k_{i}^{2} \cos θ$ , so summing over $i$ and using $⟨ k, k ⟩ = ∥ k ∥_{2}^{2} = \sum_{i} k_{i}^{2}$ [F2] gives $Δ χ = - ∥ k ∥_{2}^{2} \cos θ = - ∥ k ∥_{2}^{2} χ$ by the definition of the Laplacian [F1]. Substituting into $W$ [F1] and using $ω^{2} = c^{2} ∥ k ∥_{2}^{2}$ , $W χ = c^{- 2} (- ω^{2} χ) - (- ∥ k ∥_{2}^{2} χ) = (- ω^{2} c^{- 2} + ∥ k ∥_{2}^{2}) χ = 0$ on $R^{3} \times R$ .

3.1step 2.1F5F6

The gauge shift preserves the Lorenz divergence and the field pair. The pair $(φ, A)$ and the generator $χ$ are of class $C^{2}$ (indeed $χ$ is $C^{\infty}$ by step 1.1), so by [F5] $G (T_{χ} (φ, A)) = G (φ, A) - W χ = G (φ, A)$ by step 2.1, and by [F6] with $k = 2$ the transformed pair is a class $C^{1}$ potential pair with the same field pair as $(φ, A)$ . In particular, if $G (φ, A) = 0$ then $G (T_{χ} (φ, A)) = 0$ , so a pair in Lorenz gauge is transformed into a pair in Lorenz gauge with unchanged fields.

4.1step 1.1step 2.1step 3.1F1F3

The zero-pair instance and its nonvanishing. The zero pair $(0, 0)$ is a $C^{2}$ potential pair with $G (0, 0) = div 0 + c^{- 2} \partial_{t} 0 = 0$ by [F1], so it is in Lorenz gauge; by step 3.1 its transform is in Lorenz gauge with the zero field pair, and by step 1.1 it is $(φ^{'}, A^{'}) = (- \partial_{t} χ, \nabla χ) = (- ω \sin θ, - k \sin θ)$ with $θ (x, t) = k \cdot x - ω t$ . This pair is not the zero pair: if it were, then $φ^{'} = - \partial_{t} χ$ would vanish identically, hence $\partial_{t}^{2} χ$ would vanish identically, whereas the computation of step 2.1 gives $\partial_{t}^{2} χ (0, 0) = - ω^{2} \cos 0 = - ω^{2} \neq 0$ because $ω > 0$ and $\cos 0 = 1$ [F3]. Hence the zero field pair has two distinct Lorenz-gauge representatives, the zero pair and its transform, and the Lorenz condition does not determine the potential pair.

5.1step 1.1step 2.1step 3.1step 4.1F1∎

Conclusion and scope. Steps 1.1 and 2.1 verify that the cosinusoidal generator solves $W χ = 0$ on all of $R^{3} \times R$ ; step 3.1 records that such a generator shifts the Lorenz divergence by $- W χ = 0$ and preserves the field pair; and step 4.1 exhibits the explicit nonzero transform of the zero pair, a pair with the zero field pair that is still in Lorenz gauge. The construction uses no boundary condition, no choice principle, no Maxwell equation and no empirical premise; it is the free-space branch, and bounded domains with prescribed gauge data are outside its scope.

Remarks

Why this is the residual freedom. The gauge-reachability theorem solves $W χ = G$ to reach Lorenz gauge; a generator with $W χ = 0$ changes a Lorenz pair without leaving Lorenz gauge, so the gauge condition alone cannot pin down the representation. The example realises the homogeneous equation by the elementary plane-wave dispersion relation $ω = c ∥ k ∥_{2}$ .

A homogeneous-wave gauge generator changes Lorenz potentials without changing their fields

Example

Facts & Assumptions

Verification

Remarks

Depends on

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