A Mnemonic Short Cut

There is a quick way of obtaining all the physical (orthonormal) components of the electric and magnetic field. Note that the longitudinal electric and magnetic field components $\hat E_{long}$ and $\hat B_{long}$ are scalars in the Lorentz plane and in the Euclidean plane transverse to it. Consequently, for these components the transition from Minkowski to Rindler/polar coordinates could have been done without any computations. The same is true for the two-dimensional transverse electric and magnetic field vectors. As suggested by Eqs.(25) and (26), in the denominator of the partial derivatives simply make the replacements

$$\begin{eqnarray*} \partial t &\rightarrow& \xi \partial \tau\\ \partial z &\rig... ...tarrow& \partial r\\ \partial y &\rightarrow& r \partial \theta \end{eqnarray*}$$

in Rindler sectors $I$ or $II$, and

$$\begin{eqnarray*} \partial t &\rightarrow& \partial \xi\\ \partial z &\rightarr... ...tarrow& \partial r\\ \partial y &\rightarrow& r \partial \theta \end{eqnarray*}$$

in Rindler sectors $F$ or $P$. These replacements yield the computed transverse T.E. and T.M. components.

There also is a quick way of obtaining the T.M. field from the T.E. field components. Let $\psi^{TE}$ be the scalar wave function which satisfies the Klein-Gordon wave function for the T.E. field, and let $\psi^{TM}$ be that for the T.M. field. Then the corresponding field components are related as follows:

$$\begin{eqnarray*} T.E. &~& ~T.M.\\ \psi^{TE} &\rightarrow& ~~\psi^{TM} \\ \hat... ...row& ~~\hat E_r \\ \hat B_\theta &\rightarrow& ~~\hat E_\theta \end{eqnarray*}$$

This relationship holds in all four Rindler sectors. It also holds correspondingly relative to the rectilinear coordinate frame in Sections IVA2 and IVA1.