= f(r)dt^2 - h(r)dr^2 - r^2 ds^2_p\,,
\label{ssle}$$ where $ds^2_p$ is the line element of a unit $p$-sphere. We assume that $f(r)$ and $h(r)$ are positive for $r >r_H$ and that both $f(r)$ and $h(r)^{-1}$ have simple zeroes or both have double zeroes at $r=r_H$, where $r_H$ is the horizon radius. We also assume that $f(r), h(r)^{-1} \to 1$ as $r\to \infty$. (Most results in this section, however, are independent of these assumptions.) Let us introduce the Wheeler tortoise coordinate $r^*$ by $$\frac{dr^*}{dr} = \sqrt{\frac{h}{f}}\,. \label{tortoise}$$ Then $r^*(r)$ is a monotonic function with domain $(r_H, +\infty)$ and range in $(-\infty,+\infty)$. Let us define for any two functions $q_1(r^*)$ and $q_2(r^*)$ the inner product $$\langle q_1,q_2\rangle = 
\int_{-\infty}^{+\infty}dr^* \overline{q_1(r^*)}q_2(r^*)\,, \label{2dinner}$$ where the overline denotes complex conjugation.

The Lagrangian density of the electromagnetic field in a modified Feynman gauge is $${\cal L}_{\rm F} = \sqrt{-g}\left[ -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
- \frac{1}{2}G^2\right]\,$$ with $$G = \nabla^\mu A_\mu + K^\mu A_\mu\,,  \label{eqforG}$$ where the vector $K^\mu$ is independent of $A_\mu$. Hence the equations of motion are $$\nabla_\nu F^{\nu\mu} +\nabla^\mu G - K^\mu G = 0\,. \label{geneq}$$ Here we choose $$K^\mu = ( 0, f'/(fh),0,0)\,, \label{Kchoice}$$ in which case Eq. (\[eqforG\]) is written as $$G = \frac{1}{f}\partial_t A_t 