this limit, the equation of motion for $a_T$ reduces to the linearized version of the equation for the phase rotation of the inhomogeneous background solution . Especially, $a_T$ goes to $h(r)$ as we take a static limit. Since $h(r)$ is a solution that does not have a non-normalizable mode, if we start with a source such that $a_{T(0)}=0$, implies that $\Omega_{TT}$ should diverge in the static limit. Therefore, we expect to see a pole at $\omega=0$ in $G^R_{TT}$.

Figure \[fig:sigma12\] shows the off-diagonal components of the conductivity $\mathrm{Im} \, \sigma_{TL}$ and $\mathrm{Im}\, \sigma_{LT}$. Note that they behave as $1/\omega$ with opposite coefficients. The antisymmetry of $\sigma_{TL}$ and $\sigma_{LT}$ is due to the fact that remains invariant under the change $t\rightarrow -t$ if we simultaneously change the sign of $a_T(r)$. The poles at $\omega=0$ in the imaginary part of the conductivity indicate that there are delta function contributions at $\omega=0$ to $\mathrm{Re}\, \sigma_{TL}$ and $\mathrm{Re} \,\sigma_{LT}$, which implies that there is an off-diagonal infinite DC conductivity.

![The real parts of the diagonal components of the diagonalized conductivity matrix.[]{data-label="fig:DiagonalizedConductivities"}](diagonalconductivities.jpg){width="7cm"}

It is also instructive to diagonalize the conductivity matrix. Figure \[fig:DiagonalizedConductivities\] shows the real parts of the two diagonal components of the diagonalized 