Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression

Let $c>0$ and let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3\times \mathbb{R}$ be open. A scalar-vector potential pair of class $C^k$ on $\mathrm{\Omega }$, for an integer $k\ge 1$, is a pair $(\phi ,A)$ with $\phi :\mathrm{\Omega }\to \mathbb{R}$ and $A:\mathrm{\Omega }\to {\mathbb{R}}^3$ each of class $C^k$ in the sense of $C^k$ Euclidean maps and diffeomorphisms, the components of $A$ being of class $C^k$ separately; the pair is smooth when both members are $C^{\mathrm{\infty }}$, and continuous, or of class $C^0$, when both members are continuous. A pair defined on a set of the form ${\mathbb{R}}^3\times [t_0,T]$ with $t_0<T$ is of class $C^k$, $k\ge 1$, in the extension sense when there are an open interval $J\supset [t_0,T]$ and a potential pair of class $C^k$ on the open set ${\mathbb{R}}^3\times J$ whose restriction to ${\mathbb{R}}^3\times [t_0,T]$ the given pair is; the same wording is used for $C^{\mathrm{\infty }}$ and for the single functions $\phi$ and $\chi$ below.

All operators $\mathrm{\nabla }$, $\mathrm{div}$, $\mathrm{curl}$ and $\mathrm{\Delta }$ below act on the three spatial coordinates with $t$ fixed: for ${\mathrm{\Omega }}_t:=\{x\in {\mathbb{R}}^3:(x,t)\in \mathrm{\Omega }\}$, apply the cited three-dimensional operators to the restrictions $x\mapsto \phi (x,t)$, $x\mapsto A(x,t)$ or $x\mapsto u(x,t)$ on the open set ${\mathrm{\Omega }}_t$. In particular $\mathrm{\nabla }\phi =({\mathrm{\partial }}_{x_1}\phi ,{\mathrm{\partial }}_{x_2}\phi ,{\mathrm{\partial }}_{x_3}\phi )$ and $\mathrm{\Delta }u={\sum }_{i=1}^3{\mathrm{\partial }}_{x_i}^{2}u$; ${\mathrm{\partial }}_t$ differentiates only the time coordinate.

For a potential pair of class $C^k$ with $k\ge 1$ its electric field and magnetic field are

$E(\phi ,A):=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A,B(\phi ,A):=\mathrm{curl}A,$

where $\mathrm{\nabla }$ is the gradient of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, ${\mathrm{\partial }}_t$ is the partial derivative of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, and $\mathrm{div}$ and $\mathrm{curl}$ are those of Divergence and curl of a $C^1$ vector field. Both $E(\phi ,A)$ and $B(\phi ,A)$ are maps $\mathrm{\Omega }\to {\mathbb{R}}^3$ of class $C^{k-1}$.

For $k\ge 1$ the Lorenz divergence of the pair is the function

$G(\phi ,A):=\mathrm{div}A+c^{-2}{\mathrm{\partial }}_t\phi ,$

also of class $C^{k-1}$, and the pair is in Lorenz gauge on $\mathrm{\Omega }$ when $G(\phi ,A)=0$ throughout $\mathrm{\Omega }$.

For $k\ge 2$ and a function $\chi :\mathrm{\Omega }\to \mathbb{R}$ of class $C^k$, a gauge function, the gauge transform of the pair is

$T_{\chi }(\phi ,A):=(\phi -{\mathrm{\partial }}_t\chi ,A+\mathrm{\nabla }\chi ),$

again a potential pair, of class $C^{k-1}$.

Finally, for a function $u:\mathrm{\Omega }\to \mathbb{R}$ of class $C^2$ put

$Wu:=c^{-2}{\mathrm{\partial }}_t^{2}u-\mathrm{\Delta }u,$

with $\mathrm{\Delta }$ the Laplacian of The Laplacian of a $C^2$ function and of a $C^2$ vector field and ${\mathrm{\partial }}_t^2$ the second partial derivative in $t$ (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$); this is the normalization $W=c^{-2}{\mathrm{\square }}_c$ of the wave operator ${\mathrm{\square }}_c={\mathrm{\partial }}_t^2-c^2\mathrm{\Delta }$ of speed $c$ of Wave equation, Cauchy data and wave speed. For a map $U:\mathrm{\Omega }\to {\mathbb{R}}^m$ of class $C^2$ with components $U_1,\dots ,U_m$, one writes $WU:=(W{U}_1,\dots ,W{U}_m)$ componentwise.

Nothing in this definition asserts that a gauge function with prescribed auxiliary data exists, that every potential pair is gauge equivalent to a pair in Lorenz gauge, or that $Wu=g$ is solvable: those are conclusions with their own hypotheses. The signs $-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A$ and $A+\mathrm{\nabla }\chi$, the factor $c^{-2}$ in $G$, and the spelling Lorenz are part of the convention.

• Regularity is consumed exactly once. If $\phi$ and $A$ are $C^k$ with $k\ge 1$, then $\mathrm{\nabla }\phi$ and the three partial derivatives ${\mathrm{\partial }}_t{A}_i$ are $C^{k-1}$ by the cited definitions of gradient and partial derivative, so the fields $E$ and $B$ and the divergence $G$ are $C^{k-1}$; for $k=1$ this uses the convention fixed above that class $C^0$ means continuity. Likewise a $C^k$ gauge function with $k\ge 2$ has $C^{k-1}$ first partial derivatives, so $T_{\chi }(\phi ,A)$ is of class $C^{k-1}$.

• Why the pair is defined on an arbitrary open set. The definition is stated for any open $\mathrm{\Omega }\subseteq {\mathbb{R}}^3\times \mathbb{R}$ so that the full slab ${\mathbb{R}}^3\times (t_0,T)$ and the complement of a worldline are both instances; the extension sense above handles closed time slabs.

• The name. Lorenz here refers to Ludvig Lorenz, not to Hendrik Lorentz; in four-coordinate notation, set $x^0=ct$, $x^i=x_i$ for $i=1,2,3$, $A^0=\phi \mathrm{/}c$, $A^i=A_i$ and ${\mathrm{\partial }}_{\mu }=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^{\mu }$. Then ${\sum }_{\mu =0}^3{\mathrm{\partial }}_{\mu }{A}^{\mu }=c^{-2}{\mathrm{\partial }}_t\phi +\mathrm{div}A=G(\phi ,A)$, so its vanishing is precisely the defined Lorenz condition. The sign and the factor $c^{-2}$ belong to the convention and are not consequences of the definition.