	\begin{align}
		\label{eq:Gmodesum}
		&G(x,x')=\frac{1}{8 \pi^2}\sum_{l=0}^{\infty}(2l+1)P_l(\cos\gamma)\sum_{n=-\infty}^{\infty}e^{in\kappa\Delta\tau}g_{nl}(r),
	\end{align}
	where $\Delta x\equiv x'-x\sim\mathcal{O}(\epsilon)$ is the coordinate separation, $\gamma$ is the geodesic distance on the 2-sphere and $\kappa$ is the surface gravity of the black hole horizon. We have also taken $P_{l}(z)$ to be the Legendre polynomial of the first kind and $g_{nl}=\kappa\,N_{nl}p_{nl}(r)\,q_{nl}(r)$ is the one-dimensional radial Green function evaluated at the same spacetime point $r$. The radial modes $p_{nl}(r)$, $q_{nl}(r)$ are solutions of the homogeneous radial equation:
