Left Null Space

The key to success is to identify the divergence

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\vec J=0$$

as an element of the to-be-diagonalized Maxwell wave operator. Relative to the o.n. spherical coordinate basis $\{ \frac{\partial}{\partial r},\frac{1}{r}\frac{\partial}{\partial \theta}, \frac{1}{r\sin\theta}\frac{\partial}{\partial \varphi} \}$ one has

$$\partial_t(1\cdot \rho)+\frac{1}{r^2}\partial_r(r^2\cdot \hat J_r... ...\cdot \hat J_\theta)+ \frac{1}{r\sin\theta}\partial_\varphi(1\cdot \hat J_\phi) =0$$ (690)
or

$$\vec {\mathcal{U}}_\ell^T \left[ \begin{array}{c} \rho\\ \hat J_r\\ \hat J_\theta\\ \hat J_\varphi \end{array} \right] =0~.$$ (691)

Here

$$\vec {\mathcal{U}}_\ell^T=[ \partial_t ~~ \frac{1}{r^2}\partial_r... ...sin\theta}\partial_\theta \sin\theta ~~ \frac{1}{r\sin\theta}\partial_\varphi ]$$ (692)

is the left nullspace element of $\mathcal A$ , the spherical representative of Eq.(6.49) on page . By inspecting the above four-dimensional divergence expression one readily identifies the following three divergenceless independent 4-vector fields

$$\left[ \begin{array}{c} \rho\\ \hat J_r\\ ~\\ \hat J_\theta\\ ~\\... ...frac{1}{r^2}\frac{\partial~ (r^2 J)}{\partial r}\\ 0\\ 0 \end{array} \right] ~.$$ (693)

Here $S^{TE}$ and $S^{TM}$ are arbitrary scalar functions, while $J$ and $I$ are required to satisfy

$$\frac{\partial^2 (r^2 J)}{\partial r^2} - \frac{\partial^2 (r^2 J... ...ta}+ \frac{1}{\sin^2\theta}\frac{\partial^2 I}{\partial \varphi^2} \right)=0~,$$

the spherical coordinate version of Eq.()