component in the form of spherical harmonics $Y_{lm}(\theta,\phi)$: $$\begin{aligned}
\varphi(r,t,\theta,\phi)=
\sum_{l,m} \frac{1}{r}Y_{lm}(\theta,\phi)\Psi_l(t,r) \;\;.\end{aligned}$$ With this ansatz, the wave equation in null coordinates becomes $$\begin{aligned}
 \label{wglSS}
\frac{\partial^2 \Psi_l}{\partial u \partial v}
=\gamma(r)\left( \frac{2M}{r^3} + \frac{l(l+1)}{r^2} \right)\Psi_l \;\;.\end{aligned}$$ The right-hand side has a minimum outside the Schwarzschild radius and acts like a potential well. Because of this, incoming rays may be reflected at the gravitational potential. To a good approximation one may neglect this effect of backscattering by simply dropping the right-hand side of (\[wglSS\]). In the same way we neglect potential terms in the inside. The index $l$ is left out in the following, since $\Psi$ no longer depends on $l$ if the centrifugal potential is neglected.

We are searching for solutions of the equations $$\begin{aligned}
\frac{\partial^2 \Psi}{\partial u \partial v} =0
\; \; \;
\mbox{outside,}
\; \; \;
\frac{\partial^2 \Psi}{\partial U \partial V} =0
\; \; \;
\mbox{inside.}\end{aligned}$$ Any function that does not mix the two variables $u,v$ or $U,V$ respectively is a solution. We are looking for an incoming mode which turns into an outgoing one in a smooth manner and passes through $r=0$. These solutions must behave like spherically symmetric Minkowski modes at [I]{}$^-$. At $r \to \infty$ 