U > -U_0 \\
       1/\sqrt{\gamma_0} &\mbox{when}& U < -U_0
       \end{array}\right.\; , \\
\label{ugenv}
\left.\frac{\mathrm{d}v}{\mathrm{d}V}\right|_{r = R} &=&
\left\{\begin{array}{ccr}
       \left(\frac{\mathrm{d}t}{\mathrm{d}\tau} + \frac{\nu}{\gamma(R)}\right)
       \biggr/(1+\nu)
         &\mbox{when}& V > 0 \\
       1/\sqrt{\gamma_0} &\mbox{when}& V < 0
       \end{array}\right.\; , \end{aligned}$$ where $$\begin{aligned}
\gamma_0 := \gamma(R_0) = 1-\frac{2M}{R_0}\; .\end{aligned}$$ The matching of the inside and outside space-time demands the continuity of the line element $$\label{stetig}
(\gamma \; {\mathrm{d}}u \; {\mathrm{d}}v)|_{r = R} = 
({\mathrm{d}}U \; {\mathrm{d}}V)|_{r = R} \;,$$ which leads, with (\[ugenU\]) and (\[ugenv\]), to $$\label{z}
\frac{\mathrm{d}t}{\mathrm{d}\tau} = \frac{1}{\gamma(R)}
\sqrt{\gamma(R) (1 - \nu^2) + \nu^2}\; ,$$ which describes the relation between $t$ and $\tau$, and finally fixes the space-time geometry. Note that Eq. (\[z\]) describes nothing but the gravitational redshift $z$ for a photon that is emitted inside the shell and crosses the shell at time $\tau$, i.e. $\mathrm{d}t/\mathrm{d}\tau = 1 + z$.

To trace a radial light ray through space-time, we need to find the transition functions between the two coordinate patches on the surface of the mass shell. The setting is illustrated in Fig. \[scene\]. There are three different cases for a ray that arrives at infinity. Let us start with the trivial case (3). Everything happens before the onset of expansion. We 