are real functions of $r$ and $t$. Notice that we turn on a magnetic field as well as an electric field. It is not possible to turn on only an electric field due to the Bianchi identity $\partial_{[+} F_{t2]}=0$, where $+$ denotes the coordinate $x_3+i x_4$. However, in the current setup, the magnetic field is determined by the electric field and is not an independent quantity.

The non-trivial linear equations of motion of the fields $a_T(r,t)$ and $a_L(r,t)$ are $$\label{E:CoupledEquationsForConductivity}
\begin{split}
-\frac{r}{H(r)} \partial_t^2 a_L + \partial_r( r H(r) \partial_r a_L) + 4 \partial_r h(r) \partial_t a_T &=0\\ 
-\frac{r}{H(r)} \partial_t^2 a_T +\partial_r(r H(r) \partial_r a_T) -\frac{k^2} r a_T + 4 k \frac{\tilde E-2\alpha k h(r)^2}{r^3} a_T -4 \partial_r h(r) \partial_t a_L& =0\;.
\end{split}$$ Here $h(r)$ is the non-linear solution we have found previously. We are interested in modes with definite frequency of the form $e^{-i \omega t}$. The fields $a_T(r)$ and $a_L(r)$ behave as $H(r)^{-\frac{i \omega} 4}$ near the horizon $r=r_+$ if we impose the in-going boundary condition. On the other hand, near the $AdS_5$ boundary $r=\infty$, they behave as $$\label{E:aa2nearAdS5}
\begin{split}
a_T&= a_{T(0)} + a_{T(2)} r^{-2} + a_{T(\log)} r^{-2} \log r/r_+ +\cdots\\
a_L&= a_{L(0)} + a_{L(2)} r^{-2} + a_{L(\log)} r^{-2} 