do not expect any effect then. For the second case (2) the rays enter the shell before the start of the expansion and the shell is ‘ignited’ while the ray is crossing the interior. Only a final intervall in $u$ of length $u_0$ is involved. For case (1), entrance and exit of the ray occur while the expansion is taking place.

We obtain a collapsing shell by inverting the setting ($u\to-u, v\to-v, \tau \to -\tau$, etc.). To reproduce a collapse to a black hole, we need to take $R_0 \to 2M$ and $\nu \to 1$.[^1] This procedure will allow us to reproduce Hawking’s result from the setting of the expanding shell.

The transition functions, which follow from (\[ugenU\]) and (\[ugenv\]) by integration, are too complicated for an exact analytic treatment of the problem. We therefore have to find a reasonable approximation. Let us start from the inverse of Eqs. (\[ugenU\]) and (\[ugenv\]), which reads $$\begin{aligned}
\frac{{\mathrm{d}}U}{{\mathrm{d}}u} \bigg|_{U>-U_0}
             & = & \frac{[\gamma(1-\nu^2)+\nu^2]^\frac{1}{2}+\nu}{(1+\nu)}\ ,\\
\frac{{\mathrm{d}}V}{{\mathrm{d}}v} \bigg|_{V>0}
             & = & \frac{[\gamma(1-\nu^2)+\nu^2]^\frac{1}{2}-\nu}{(1-\nu)}\ .\end{aligned}$$ As a first approximation we assume that the expansion velocity $\nu$ is constant. Secondly, we expect the biggest effect when the space-time curvature is high, thus in the vicinity of the 