electromagnetism - What is the physical significance of the Dipole Transformation of Maxwell's Equations?

Given Maxwell's equations of the form

$$\begin{align} \bar{\nabla}\times \bar{B} = \dfrac{4\pi}{c} \bar{J} + \partial_0 \bar{E} \\ \bar{\nabla}\times \bar{E} = -\partial_0 \bar{B} \\ \bar{\nabla} \cdot \bar{B}=0 \\ \bar{\nabla} \cdot \bar{E} = 4 \pi \rho, \end{align}$$ what is the physical significance of the following transformation of Maxwell's equations: $$\begin{align} -i \bar{\nabla} \times \bar{G} + \bar{\nabla} G_0 = \dfrac{4\pi}{c} \bar{R} + \partial_{0}\bar{G} \tag{Amp-Far Dipole} \\ \bar{\nabla} \cdot \bar{G} - \partial_{0}G_0 = 4\pi R_0 , \tag{Gauss Dipole} \end{align}$$ where $\bar{G} =(i\bar{r} \times\bar{F}-x_0 \bar{F})$, $G_0 = (\bar{r} \cdot \bar{F})$, $\bar{R} = (\rho c \bar{r}-x_0\bar{J}+i \bar{r} \times\bar{J})$, $R_0 = (\bar{r}\cdot\bar{J} - x_0\rho c)/c$ and $\bar{F} = \bar{E} - i \bar{B}$.

In this post, I refer to (Amp-Far Dipole) and (Gauss Dipole) as the dipole equations, because I do not know what there actual names are or who first published these equations. I just stumbled upon them by accident on pencil and paper.

$x_0 = ct$ is the time variable multiplied by the speed of light for condensed notation purposes.

$\bar{R}$ is the complex combination of the electric dipole field density $\rho c \bar{r}-x_0\bar{J}$ and the magnetic dipole field density $\bar{r} \times\bar{J}$.

$\bar{R}$ is interpreted as a fictitious current in the dipole equations (Amp-Far Dipole). The corresponding fictitious charge density of $\bar{R}$ is $R_0$, which is equal to the Minkowski inner product of the four-position and the four-current.

Though $R_0$ and $\bar{R}$ are fictitious charge and current, they are conserved as a current when $G_0 = 0$. This implies that $G_0$ breaks charge conservation of the fictitious charge and current $R_0$ and $\bar{R}$.

An interesting consequence of the dipole equations is they are identical to Maxwell's equations when $G_0 = 0$.

I first write Ampere's law, Faraday's law and Gauss' law in complex form

$$\begin{align} -i\bar{\nabla} \times \bar{F} = \dfrac{4\pi}{c} \bar{J} + \partial_{0} \bar{F} \tag{Amp-Far} \\ \bar{\nabla} \cdot \bar{F} = 4\pi \rho \tag{Gauss} , \end{align}$$ where $\bar{F} = \bar{E} + i \bar{B}$.

I use the following differential vector calculus identity

$$\begin{align} \bar{r} \times (\bar{\nabla} \times) + \bar{r} (\bar{\nabla} \cdot) + x_0(\partial_{0}) = \bar{\nabla} \times (\bar{r} \times) + \bar{\nabla} (\bar{r} \cdot) + \partial_{0}(x_0) \end{align}$$ to transform (Amp-Far) into the following: $$\begin{align} \bar{r} \times (\bar{\nabla} \times \bar{F}) + \bar{r} (\bar{\nabla} \cdot \bar{F}) + x_0(\partial_{0} \bar{F}) =\\ \bar{r} \times \left( i\dfrac{4\pi}{c} \bar{J} + i\partial_{0} \bar{F} \right) + \bar{r} \left(4\pi \rho\right) + x_0\left(-i\bar{\nabla} \times \bar{F} - \dfrac{4\pi}{c} \bar{J}\right) =\\ \dfrac{4\pi}{c} (i\bar{r} \times\bar{J}) + \partial_{0} (i\bar{r} \times\bar{F}) + 4\pi \left( \rho \bar{r} \right) - i\bar{\nabla} \times (x_0\bar{F}) - \dfrac{4\pi}{c} (x_0\bar{J}) =\\ \bar{\nabla} \times (\bar{r} \times \bar{F}) + \bar{\nabla} (\bar{r} \cdot \bar{F}) + \partial_{0}(x_0 \bar{F}) , \end{align}$$ which reduces to the following expression $$\begin{align} -i \bar{\nabla} \times \left( i\bar{r} \times \bar{F} - x_0\bar{F} \right) + \bar{\nabla} (\bar{r} \cdot \bar{F}) =\\ \dfrac{4\pi}{c} \left( \rho c \bar{r} - x_0\bar{J} + i \bar{r} \times\bar{J} \right) + \partial_{0} \left( i\bar{r} \times\bar{F} - x_0 \bar{F} \right) . \end{align}$$ One can perform the following substitutions $\bar{G} =(i\bar{r} \times\bar{F}-x_0 \bar{F})$, $G_0 = (\bar{r} \cdot \bar{F})$, and $\bar{R} = (\rho c \bar{r}-x_0\bar{J}+i \bar{r} \times\bar{J})$ to obtain $$\begin{align} -i \bar{\nabla} \times \bar{G} + \bar{\nabla} G_0 = \dfrac{4\pi}{c} \bar{R} + \partial_{0}\bar{G} . \tag{Amp-Far Dipole} \end{align}$$

I use the following differential vector calculus identity

$$\begin{align} -x_0 (\nabla\cdot) + \bar{r}\cdot (-i\bar{\nabla}\times) = \bar{\nabla} \cdot (i(\bar{r}\times)- x_0) \end{align}$$ to transform (Gauss) into the following: $$\begin{align} -x_0 (\nabla\cdot\bar{F}) + \bar{r}\cdot (-i\bar{\nabla}\times\bar{F}) =\\ -x_0 (4\pi \rho) + \bar{r}\cdot \left(\dfrac{4\pi}{c} \bar{J} + \partial_{0} \bar{F}\right) =\\ \dfrac{4\pi}{c} (\bar{r}\cdot\bar{J} - x_0\rho c) + \partial_{0} (\bar{r}\cdot\bar{F}) =\\ \bar{\nabla} \cdot (i\bar{r}\times\bar{F} - x_0\bar{F}) , \end{align}$$ which reduces to the following expression $$\begin{align} \bar{\nabla} \cdot (i\bar{r}\times\bar{F} - x_0\bar{F}) - \partial_{0} (\bar{r}\cdot\bar{F}) = \dfrac{4\pi}{c} (\bar{r}\cdot\bar{J} - x_0\rho c) . \end{align}$$ One can perform the following substitutions $R_0 = (\bar{r}\cdot\bar{J} - x_0\rho c)/c$ to obtain $$\begin{align} \bar{\nabla} \cdot \bar{G} - \partial_{0}G_0 = 4\pi R_0 . \tag{Gauss Dipole} \end{align}$$

Answer

No explicit complexification is needed to derive this breakdown of Maxwell's equations. This can be understood wholly through the real vector space of special relativity.

Let's start with Maxwell's equations for the EM field, in the clifford algebra language called STA: the spacetime algebra. Maxwell's equations take the form

$$\nabla F = -J$$

where $\nabla F = \nabla \cdot F + \nabla \wedge F$, $F = e_0 E + B \epsilon_3$, in the $(-, +, +, +)$ sign convention.

Let $x$ be the spacetime position vector. It's generally true that, for a vector $v$ and a constant bivector $C$,

$$\nabla (C \cdot x) = -2 C, \quad \nabla (C \wedge x) = 2C \implies \nabla (Cx) = 0$$

One can then evaluate the expression

$$\nabla (Fx) = (\nabla F)x + \dot \nabla (F \dot x)$$

where the overdot means that only $x$ is differentiated in the second term; using the product rule, $F$ is "held constant" and so the above formulas apply. We just argued that the second term is zero, so we get $\nabla (Fx) = (\nabla F) x$. Thus, we arrive at the following transformation of Maxwell's equations:

$$\nabla (Fx) = -Jx$$

Now, we could always write $F$ as a "complex bivector" in the sense that, using $\epsilon = e_0 \epsilon_3$, and $\epsilon \epsilon = -1$, we have

$$F = e_0 E - B \epsilon_3 e_0 \epsilon_3 \epsilon = e_0 (E + \epsilon B)$$

It's crucial to note that $\epsilon$ does not commute with any vector.

What are the components of $Fx$? Write $x = t e_0 + r$ and we can write them as

$$Fx = e_0 (Ex + \epsilon Bx) = e_0 (E \cdot r + E \wedge r - e_0 Et + \epsilon B \cdot r - e_0 B \times r + \epsilon B t e_0)$$

This too can be written in a "complex" form:

$$Fx = (e_0 E \cdot r + Et + B \times r) + \epsilon (E \times r + e_0 B\cdot r + Bt)$$

We seem to differ on some signs, but this is recognizably the same quantity you have called $G$.

Now, to talk about how these equations break down, let's write $G = G_1 + G_3$, where $G_1 = (e_0 E \cdot r + \ldots)$ and $G_3 = \epsilon (E \times r + \ldots)$. Let's also write for $R = Jx = R_0 + R_2$.

Maxwell's equations then become

$$\nabla \cdot G_1= R_0, \quad \nabla \wedge G_1 + \nabla \cdot G_3 = R_2, \quad \nabla \wedge G_3 = 0$$

The first and third equations are the components of the Gauss dipole; the second equation is the Ampere-Faraday dipole equation.

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Now, what does it all mean? The expression for $G = Fx$ includes both rotational moments of the EM field as well as some dot products, so it measures both how much the spacetime position is in the same plane as the EM field as well as how much the spacetime position is out of the plane.

It's probably more instructive to look at the source term $-Jx$. This tells us both about the moments of the four-current as well as how it goes toward or away from the coordinate origin. The description for the moments is wholly in the Ampere-Faraday dipole equation. What kinds of moments would this describe? A pair of two opposite point charges at rest, separated by a spatial vector $2 \hat v$ and centered on the origin, each with current at rest $j_0$, would create a $R = Jx = + j_0 e_t \hat v - j_0 e_t (-\hat v) = 2 j_0 e_t \hat v$, so this would be described wholly by the A-F dipole equation.

That's at time zero, however. At later times, $R$ will pick up these weird time terms. Say we're at time $\tau$. Then $R = 2 j_0 e_t \hat v + j_0 e_t (\tau e_t) - j_0 e_t (\tau e_t)$. So for this case, there's no problem: the extra stuff will just cancel. A single charge, however, would start picking up this term.

In a few words, these equations are weird.