Maxwell Wave Equation

The first pair of Maxwell's equations, (6.39) and (6.40), imply that there exists a vector potential $\vec A$ and scalar potential $\phi$ from which one derives the electric and magnetic fields,

$$\vec B =\nabla \times \vec A$$ (643)

$$\vec E =-\nabla \phi -\frac{\partial \vec A}{\partial t}~.$$ (644)

Conversely, the existence of these potentials guarantees that the first pair of these equations is satisfied automatically. By applying these potentials to the differential expressions of the second pair of Maxwell's equations, (6.41)-(6.42), one obtains the mapping

$$\left[ \begin{array}{c} \phi\\ \vec A \end{array} \right] ~~\stac... ...leadsto} \mathcal{A}\left[ \begin{array}{c} \phi\\ \vec A \end{array} \right]~,$$ (645)

where

$$\mathcal{A}\left[ \begin{array}{c} \phi\\ \vec A \end{array} \rig... ...l\phi}{\partial t}+ \frac{\partial^2\vec A}{\partial t^2} \end{array} \right]~.$$ (646)

It follows that Maxwell's field equations reduce to Maxwell's four-component wave equation,

$$\left[ \begin{array}{c} \displaystyle -\nabla^2\phi -\nabla \cdot... ...ray} \right] = 4\pi \left[ \begin{array}{c} \rho\\ \vec J \end{array} \right]~.$$ (647)

Maxwell's wave operator is the linch pin of his theory of electromagnetism. This is because it has the following properties:

1. It is a linear map from the space of four-vector fields into itself, i.e.
   $$R^4 \stackrel{\mathcal A}{\longrightarrow} R^4$$
   at each point event $(t,\vec x)$ .
2. The map is singular. This means that there exist nonzero vectors $\vec {\mathcal U}_r$ and $\vec {\mathcal U}_\ell$ such that

   $${\mathcal A} \,\vec {\mathcal U}_r =\vec 0$$
   and

   $$\vec {\mathcal U}_\ell^T {\mathcal A} =\vec 0~.$$

   
   
   In particular, one has
   1. the fact that

      $$\mathcal{A} \left[ \begin{array}{c} -\partial_t\\ \vec \nabla \en... ...d{array} \right]\Lambda =\left[ \begin{array}{c} 0\\ \vec 0 \end{array} \right]$$ (648)

      
      
      for all three-times differentiable scalar fields $\Lambda(t,\vec x)$ . Thus
      $$\vec {\mathcal{U}}_r \equiv \left[ \begin{array}{c} -\partial_t\\ \vec \nabla \end{array}\right] ~~ \in~ \mathcal{N}(\mathcal{A})~.$$
      The null space of $\mathcal{A}$ is therefore nontrivial and 1-dimensional at each $(t,\vec x)$ .
   2. the fact that
      $$\left[ \partial_t~~\vec\nabla \cdot \right] \mathcal{A} \left[ \b... ...bla\cdot\vec A +0+\partial_t\nabla^2\phi+\vec\nabla\cdot\partial_t^2\vec A=0~,$$
      for all 4-vectors $\left[ \begin{array}{c} \phi\\ \vec A \end{array} \right]$ . Thus

      $$\vec {\mathcal{U}}_\ell^T \equiv \left[ \partial_t~~\vec\nabla \cdot \right] ~~\in~\textrm{left null space of }\mathcal{A}~,$$ (649)
      or

      $$\vec {\mathcal{U}}_\ell ~\in ~\mathcal{N}(\mathcal{A}^T)~.$$ (650)

      
      
      The left null space of $\mathcal{A}$ is therefore also 1-dimensional at each $(t,\vec x)$ .

In light of the singular nature of $\mathcal{A}$ , the four-component Maxwell wave equation

$$\mathcal{A}\left[ \begin{array}{c} \phi\\ \vec A \end{array} \right] = 4\pi \left[ \begin{array}{c} \rho\\ \vec J \end{array} \right]$$ (651)

has no solution unless the source $\left[ \begin{array}{c} \rho\\ \vec J \end{array} \right]$ also satisfies

$$\vec {\mathcal{U}}_\ell^T \left[ \begin{array}{c} \rho\\ \vec J \end{array} \right]=0~.$$

This is the linear algebra way of expressing

$$\partial_t\rho+\vec \nabla \cdot \vec J=0~,$$ (652)

the differential law of charge conservation. Thus Maxwell's equations apply if and only if the law of charge conservation holds. If charge conservation did not hold, then Maxwell's equations would be silent. They would not have a solution. Such silence is a mathematical way of expressing the fact that at its root theory is based on observation and established knowledge, and that arbitrary hypotheses must not contaminate the theoretical.