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Smooth scalar and vector potentials on a star-shaped space-time region

Physical scope

The fields and potentials of the classical vacuum electromagnetic framework of The electric and magnetic fields as fixed-time spatial vector fields, The microscopic vacuum Maxwell equations: smooth and weak branches and Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression on a single inertial laboratory region over a star-shaped spatial region. The only physical hypothesis used is the pair of source-free Maxwell equations (absence of magnetic charge and Faraday's law) as adopted in the postulate; no source equation, no regularity beyond the stated joint class, and no assertion that the potentials are observable are included. The star-shaped hypothesis is a genuine restriction on the spatial region, and the companion examples exhibit a divergence-free field on a non-star-shaped region with no global vector potential.

Statement

Let $c > 0$ , let $k \geq 2$ be an integer, let $Ω \subseteq R^{3}$ be an open set that is star-shaped with respect to a point $a \in Ω$ (Star-shaped open subsets of Euclidean space), and let $I \subseteq R$ be an interval with at least two elements (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length). Joint regularity on $Ω \times I$ is read in the restriction convention of Joint regularity of the radial vector and scalar potentials on a star-shaped domain: a map on $Ω \times I$ is jointly $C^{m}$ when it is the restriction of a $C^{m}$ map on some open neighbourhood of $Ω \times I$ in $R^{4}$ . For open $I$ this is the ordinary open-domain class of $C^{k}$ Euclidean maps and diffeomorphisms and agrees with the field class of The electric and magnetic fields as fixed-time spatial vector fields; for non-open $I$ it is the convention that makes the statements below meaningful, and no time jet is asserted beyond the restricted values.

Let $E, B : Ω \times I \to R^{3}$ be jointly $C^{k}$ and satisfy the two source-free Maxwell equations of The microscopic vacuum Maxwell equations: smooth and weak branches pointwise on $Ω \times I$ :

$div B = 0, curl E = - \partial_{t} B,$

with the divergence and curl of Divergence and curl of a $C^{1}$ vector field and the partial derivative $\partial_{t}$ at fixed position of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ (at included endpoints of a non-open $I$ , as relative derivatives in the restriction convention). Then there are a scalar potential $φ : Ω \times I \to R$ that is jointly $C^{k - 1}$ and a vector potential $A : Ω \times I \to R^{3}$ that is jointly $C^{k}$ , both in the restriction convention, with

$B = curl A, E = - \nabla φ - \partial_{t} A on Ω \times I,$

so that $(φ, A)$ is a potential pair whose field pair is $(E, B)$ , in the vocabulary of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression. If $E$ and $B$ are jointly $C^{\infty}$ , then $φ$ and $A$ may be chosen jointly $C^{\infty}$ . Conversely, if $φ$ and $A$ are jointly $C^{\infty}$ on $Ω \times I$ in the restriction convention and $E : = - \nabla φ - \partial_{t} A$ , $B : = curl A$ are their field pair, then $div B = 0$ and $curl E = - \partial_{t} B$ on $Ω \times I$ .

The theorem asserts existence on the stated star-shaped domain only: it claims no global potential on a general open set, no boundary, decay or growth condition on the potentials, and no uniqueness. Potentials are auxiliary objects of the framework and are not claimed to be observable.

Facts & Assumptions

Given: A constant $c > 0$ , an integer $k \geq 2$ , an open set $Ω \subseteq R^{3}$ star-shaped with respect to $a \in Ω$ , an interval $I \subseteq R$ with at least two elements, and fields $E, B : Ω \times I \to R^{3}$ that are jointly $C^{k}$ in the restriction convention and satisfy the two source-free equations of the Statement pointwise.

[F1]

Joint regularity and radial witnesses: for an open star-shaped $U \subseteq R^{3}$ with centre $a$ and a jointly $C^{m}$ field $B : U \times I \to R^{3}$ , $m \geq 1$ , with $div B (\cdot, t) = 0$ for every $t \in I$ , the radial integral $A (x, t) = \int_{0}^{1} s B (a + s (x - a), t) \times (x - a) d s$ is jointly $C^{m}$ with $curl A (\cdot, t) = B (\cdot, t)$ for every $t$ ; for a jointly $C^{m}$ field $C$ with $curl C (\cdot, t) = 0$ for every $t$ the negative radial scalar witness $ϕ (x, t) = - \int_{0}^{1} ⟨ C (a + s (x - a), t), x - a ⟩ d s$ is jointly $C^{m}$ with $\nabla ϕ (\cdot, t) = - C (\cdot, t)$ for every $t$ ; jointly $C^{\infty}$ data give jointly $C^{\infty}$ witnesses; and the restriction convention carries the same conclusions at included endpoints of a non-open interval (Joint regularity of the radial vector and scalar potentials on a star-shaped domain).

[F2]

The smooth branch of the adopted vacuum Maxwell equations holds pointwise for jointly $C^{1}$ fields and contains, among its four equations, $div B = 0$ and $curl E = - \partial_{t} B$ ; the postulate contains no magnetic charge (The microscopic vacuum Maxwell equations: smooth and weak branches).

[F3]

Potential-pair vocabulary: for a potential pair $(φ, A)$ of class $C^{m}$ , $m \geq 1$ , on an open set the electric and magnetic fields are $E (φ, A) = - \nabla φ - \partial_{t} A$ and $B (φ, A) = curl A$ , and both are of class $C^{m - 1}$ ; the definition asserts no existence of potentials for a given field pair (Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression).

[F4]

Euclidean calculus: a map is of class $C^{m}$ when each component is, the class is closed under sums and compositions of class, and sums, scalar multiples and products of differentiable real functions obey the usual derivative rules ( $C^{k}$ Euclidean maps and diffeomorphisms, 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 $g (c) \neq 0$ ).

[F5]

Linearity: for $C^{1}$ fields $F, G$ on an open $U \subseteq R^{3}$ and reals $α, β$ , $curl (α F + β G) = α curl F + β curl G$ , and the divergence is likewise linear (Divergence and curl are linear and satisfy the scalar product rules).

[F6]

Clairaut--Schwarz: for a $C^{2}$ function on an open subset of $R^{m}$ mixed coordinate partial derivatives commute, so for a jointly $C^{2}$ field the time derivative of each spatial first partial derivative equals the spatial first partial derivative of the time derivative (Clairaut--Schwarz theorem for continuous second partial derivatives).

[F7]

For a $C^{2}$ field on an open subset of $R^{3}$ the divergence of its curl vanishes pointwise (The divergence of the curl of a $C^{2}$ field vanishes).

[F8]

For a $C^{2}$ scalar function on an open subset of $R^{3}$ the curl of its gradient vanishes pointwise (The curl of the gradient of a $C^{2}$ function vanishes).

[F9]

The divergence of a $C^{1}$ field is $div F = \sum_{i} \partial_{i} F_{i}$ , the curl is $curl F = (\partial_{y} F_{z} - \partial_{z} F_{y}, \partial_{z} F_{x} - \partial_{x} F_{z}, \partial_{x} F_{y} - \partial_{y} F_{x})$ , the gradient is the tuple of first coordinate partial derivatives, and partial derivatives are one-variable derivatives along coordinate lines (Divergence and curl of a $C^{1}$ vector field, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

Proof

technique · direct; apply the radial vector witness to $B$, apply the radial scalar witness to the auxiliary field $E+\partial_tA$, and verify the converse by the standard curl and divergence identities

1.1givenF1F2F9

Hypothesis check and vector step. The set $Ω$ is open and star-shaped with centre $a$ and $I$ is an interval, and by [F2] the field $B$ is jointly $C^{k}$ with $div B (\cdot, t) = 0$ on $Ω$ for every $t \in I$ , the divergence being that of [F9]. Since $k \geq 2 \geq 1$ , the vector witness of [F1] applies and produces the field $A (x, t) : = \int_{0}^{1} s B (a + s (x - a), t) \times (x - a) d s$ , which is jointly $C^{k}$ on $Ω \times I$ and satisfies $curl A (\cdot, t) = B (\cdot, t)$ for every $t \in I$ .

1.2givenF5F6F7F8F9

Converse. Let $φ$ and $A$ be jointly $C^{\infty}$ on $Ω \times I$ and put $E : = - \nabla φ - \partial_{t} A$ and $B : = curl A$ . All functions are restrictions of $C^{\infty}$ maps on an open neighbourhood of $Ω \times I$ , so the identities below may be checked there and restricted. The field $A$ is $C^{\infty}$ , hence $C^{2}$ , and [F7] gives $div B = div curl A = 0$ . For the curl of $E$ , the linearity of the curl [F5] gives $curl E = - curl \nabla φ - curl \partial_{t} A$ ; the first term vanishes by [F8] applied to the $C^{2}$ function $φ$ , and for the second, the component formula of [F9] turns each coordinate of $curl \partial_{t} A$ into a difference of mixed derivatives $\partial_{t} \partial_{x}$ and $\partial_{x} \partial_{t}$ of the $C^{2}$ components of $A$ , which [F6] identifies, so $curl \partial_{t} A = \partial_{t} curl A = \partial_{t} B$ . Hence $curl E = - \partial_{t} B$ on $Ω \times I$ .

2.1step 1.1F1F4F9

The auxiliary field $C : = E + \partial_{t} A$ . The fields $E$ and $A$ are restrictions to $Ω \times I$ of $C^{k}$ maps on a common open neighbourhood; the time partial derivative of each component of the $C^{k}$ map extending $A$ exists and is $C^{k - 1}$ there by [F4] and [F9], and the class is closed under sums by [F4], so $C$ is jointly $C^{k - 1}$ on $Ω \times I$ in the restriction convention.

2.2step 1.1F2F5F6F9

The auxiliary field is curl-free at every time. Fix $t \in I$ . By linearity of the curl [F5] and the componentwise commutation of $\partial_{t}$ with the spatial partial derivatives of the jointly $C^{2}$ field $A$ , which is [F6] applied to the component functions, $curl C (\cdot, t) = curl E (\cdot, t) + \partial_{t} (curl A) (\cdot, t)$ , the operators and the derivative being those of [F9]. By step 1.1, $curl A (\cdot, t) = B (\cdot, t)$ as functions of the position variable, so $\partial_{t} (curl A) (\cdot, t) = \partial_{t} B (\cdot, t)$ ; by [F2], $curl E (\cdot, t) = - \partial_{t} B (\cdot, t)$ . Substituting gives $curl C (\cdot, t) = - \partial_{t} B (\cdot, t) + \partial_{t} B (\cdot, t) = 0$ .

3.1step 1.1step 2.1step 2.2F1F3

Scalar step and the pair identities. By step 2.1 the field $C$ is jointly $C^{k - 1}$ with $k - 1 \geq 1$ , and by step 2.2 it satisfies $curl C (\cdot, t) = 0$ on $Ω$ for every $t \in I$ ; the scalar witness of [F1] therefore gives the jointly $C^{k - 1}$ function $φ (x, t) : = - \int_{0}^{1} ⟨ C (a + s (x - a), t), x - a ⟩ d s$ with $\nabla φ (\cdot, t) = - C (\cdot, t)$ for every $t \in I$ . Since $C = E + \partial_{t} A$ by step 2.1, this reads $\nabla φ = - E - \partial_{t} A$ , that is $E = - \nabla φ - \partial_{t} A$ on $Ω \times I$ ; together with $curl A = B$ from step 1.1 this exhibits $(φ, A)$ as a potential pair with field pair $(E, B)$ in the sense of [F3], with $φ$ jointly $C^{k - 1}$ and $A$ jointly $C^{k}$ .

4.1step 3.1F1F4

Smooth clause. If $E$ and $B$ are jointly $C^{\infty}$ , the vector witness of [F1] applied to $B$ gives a jointly $C^{\infty}$ field $A$ with $curl A = B$ , the field $C = E + \partial_{t} A$ is jointly $C^{\infty}$ by the closure and differentiation rules of [F4], and the scalar witness of [F1] applied to $C$ gives a jointly $C^{\infty}$ scalar $φ$ with $\nabla φ = - C$ ; the identities of step 3.1 then hold with both potentials jointly $C^{\infty}$ .

5.1step 1.1step 1.2step 3.1step 4.1F1F2F3∎

Conclusion and bookkeeping. Steps 1.1 through 3.1 produce, for every integer $k \geq 2$ , a jointly $C^{k - 1}$ scalar $φ$ and a jointly $C^{k}$ vector $A$ with $B = curl A$ and $E = - \nabla φ - \partial_{t} A$ on $Ω \times I$ ; step 4.1 supplies the jointly $C^{\infty}$ choice when the data are jointly $C^{\infty}$ ; and step 1.2 proves the converse implication. The construction used star-shapedness and the joint class only through the witnesses of [F1], used the source-free equations through [F2], and used the potential vocabulary of [F3]; no global potential on a general open set, no boundary or decay condition, no uniqueness and no observability claim is made, and at included endpoints of a non-open interval all conclusions are those of the restriction convention recorded in [F1].

Remarks

The star-shaped hypothesis is substantive. On a general open $Ω$ the two source-free equations need not admit a global potential pair; the divergence-free radial witness of [F1] and its scalar companion both use the radial segments from the centre, and the companion examples page records the obstruction on a punctured region.
No uniqueness is claimed. The witnesses of [F1] are single specimens. Gauge transformations preserving their fields and the domain hypotheses for a converse are analyzed later on this page; this existence theorem does not classify all representations of one field pair.

Smooth scalar and vector potentials on a star-shaped space-time region

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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