The T.E. Field

For the T.E. degrees of freedom the charge-current $S_\mu dx^\mu$ and the vector potential $A_\mu dx^\mu$ have the form given by

$$( S_{\tau}, S_{\xi}, S_{r}, S_{\theta})=\left(0,0, \frac{1}{... ... S}{\partial \theta}, -r\frac{\partial S}{\partial r}\right)~,$$ (29)

and

$$( A_{\tau}, A_{\xi}, A_{r}, A_{\theta})=\left(0,0, \frac{1}{... ...{\partial \theta}, -r\frac{\partial \psi}{\partial r}\right)~.$$ (30)

The electromagnetic field,

$$\begin{eqnarray*} \frac{1}{2}F_{\mu\nu}dx^\mu \wedge dx^\nu &\equiv& \hat E_{lon... ...d\theta \qquad\qquad \qquad\quad\textrm{in}~P~\textrm{and}~F~, \end{eqnarray*}$$

has the following components:

	In $I$ or in $II$	In $F$ or in $P$
$\hat E_{long.}:$	$${\frac{1}{\xi}F_{\xi\tau}=0}$$	$$\frac{1}{\xi}F_{\tau\xi}=0$$
$\hat E_r:$	$$\frac{1}{\xi}F_{r\tau}= -\frac{1}{r} \frac{\partial}{\partial\theta}\left( \frac{1}{\xi} \frac{\partial \psi}{\partial \tau} \right)$$	$$F_{r\xi}= -\frac{1}{r} \frac{\partial}{\partial\theta}\left( \frac{\partial \psi}{\partial \xi} \right)$$
$\hat E_\theta:$	$$\frac{1}{\xi r}F_{\theta\tau}= \frac{\partial}{\partial r}\left( \frac{1}{\xi}\frac{\partial \psi}{\partial \tau} \right)$$	$$\frac{1}{r}F_{\theta\xi}= \frac{\partial}{\partial r}\left( \frac{\partial \psi}{\partial \xi} \right)$$
$\hat B_{long.}:$	$$\frac{1}{r}F_{r\theta}= -\left( \frac{1}{r}\frac{\partial}{\par... ...l}{\partial r} +\frac{1}{r^2}\frac{\partial^2}{\partial\theta^2} \right) \psi$$	$$\frac{1}{r}F_{r\theta}= -\left( \frac{1}{r}\frac{\partial}{\par... ...l}{\partial r} +\frac{1}{r^2}\frac{\partial^2}{\partial\theta^2} \right) \psi$$
$\hat B_r:$	$$\frac{1}{r}F_{\theta\xi}= \frac{\partial}{\partial r}\left( \frac{\partial \psi}{\partial \xi} \right)$$	$$\frac{1}{\xi r}F_{\theta\tau}= \frac{\partial}{\partial r}\left( \frac{1}{\xi}\frac{\partial \psi}{\partial \tau} \right)$$
$\hat B_\theta:$	$$F_{\xi r}= \frac{1}{r}\frac{\partial}{\partial\theta} \left(\frac{\partial \psi}{\partial \xi} \right)$$	$$\frac{1}{\xi}F_{\tau r}= \frac{1}{r}\frac{\partial}{\partial\theta} \left( \frac{1}{\xi}\frac{\partial \psi}{\partial \tau} \right)$$

The carets in the first column serve as a reminder that these components are relative to the orthonormal basis of the metric, Eqs.(25) and (26).