$z \to +\infty$, we find in the small $\omega$ limit $$\varphi^{(1\rightarrow)}_{\omega l}
= 4M\omega (z+1)\left[ Q_l(z)-\frac{(z-1)}{l(l+1)}\frac{dQ_l(z)}{dz}\right]\,,
\label{cond3}$$ where this has been normalized such that Eqs. (\[cond2\]) and (\[cond3\]) are in agreement at $r\approx r_H$.

From gauge invariance of the amplitude (\[Amp\]) it is clear that we may use the modes in the spherical Coulomb gauge, $A^{(1'\rightarrow;\,\omega lm)}_\mu$, in place of those in the modified Feynman gauge $A^{(1\rightarrow;\,\omega lm)}_\mu$. Then the contribution from the $r$-components will be suppressed in the low energy limit due to extra factors of $\omega$ \[see Eq. (\[suppr\])\]. Therefore we need to consider only the $t$-component in this limit. The $t$-component can be found in the small $\omega$ limit from Eqs. (\[R1\]), (\[AtCoul\]) and (\[cond3\]) as $$A^{(1' \rightarrow;\,\omega lm)}_t
=  \frac{4i\omega(z-1)}{\sqrt{l(l+1)}}\,\frac{dQ_l(z)}{dz}
Y_{lm}(\theta,\phi) e^{-i\omega t}\,,
\label{A1'}$$ where we have used the Legendre equation satisfied by $Q_l(z)$. By using Eq. (\[A1’\]) in Eq. (\[Amp\]) and comparing the result with Eq. (\[Amp2\]), we obtain $${\cal T}^{\rightarrow}_{\omega lm}=
-i\frac{\sqrt{2\omega}\,q(z_0 -1)}{\sqrt{\pi l(l+1)}}
\frac{dQ_l(z_0)}{dz_0}\overline Y_{lm}(\theta_0,\phi_0)\,.$$ where $z_0 = r_0/M -1$. Then we find the [*proper*]{} response rate of the charge as \[see Eq. (\[res\])\] $$\frac{R_{0lm}}{\sqrt{f(r_0)}}
= \frac{q^2(z_0 -1)^2}{\pi M l(l+1)\sqrt{f(r_0)}}\left[
\frac{dQ_l(z_0)}{dz_0}\right]^2 |Y_{lm}(\theta_0,\phi_0)|^2\,,$$ where the factor $\sqrt{f(r_0)}$ appears because here we are calculating the 