transition rate per proper time of the charge. We can sum over $l$ and $m$ by using $$\sum_{m=-l}^{l} |Y_{lm}(\theta_0,\phi_0)|^2
= {(2l+1)}/{(4\pi)}$$ and $$\sum_{l=1}^\infty \frac{2l+1}{l(l+1)}\left[ \frac{dQ_l(z)}{dz}\right]^2
= \frac{2Q_1(z)}{(z^2-1)^2}\,.  \label{QFORM}$$ A derivation of this formula is given in Appendix B. Thus, the total transition probability per proper time of the charge is $$R^{\rm tot} = \sum_{l=1}^{+\infty} \sum_{m=-l}^{l}
\frac{R_{0lm}}{\sqrt{f(r_0)}}
= \frac{q^2 a(r_0) }{2\pi^2}Q_1(r_0/M-1)\,,$$ where $a(r_0) = Mr_0^{-2}/\sqrt{f(r_0)}$ is the proper acceleration of the charge.

It is of course possible to work with the modes in the modified Feynman gauge directly and obtain the same result. If we do so, the contribution will come from the $r$-component of the current because the $t$-component of physical modes I vanish.

Low energy absorption cross section of the Schwarzschild black hole {#sec:Schwarz}
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Here we consider the Schwarzschild black hole in $p+2$ dimensions given by the line element (\[ssle\]) with $f(r) = 1- (r_H/r)^{p-1}$. If the absorption probability of the black hole associated with physical modes I and II are ${\cal P}_l^{(1)}$ and ${\cal P}_l^{(2)}$, respectively, then the absorption cross section is given by $$\sigma = \frac{(2\pi)^p}{p\Omega_p \omega^p}\sum_{l=1}^\infty
\left[ M^{(1)}_l {\cal P}_l^{(1)} + M^{(2)}_l{\cal P}_l^{(2)}\right]\,,
\label{ABSCR}$$ where $l$ and $\omega$ are the angular momentum and frequency, respectively, 