the phase of an incoming wave is given by $$e^{- \mathrm{i}\omega v}\ .$$ Inside of the mass shell, the corresponding mode travels on a curve with constant $V$. At $r=0$ we have $U=V$ where the incoming ray $\exp(- \mathrm{i} \tilde{\omega} V)$ turns smoothly into the outgoing ray $\exp(- \mathrm{i} \tilde{\omega} U)$. At [I]{}$^+$, outgoing modes are proportional to $$e^{ -\mathrm{i}\omega' u}\ .$$ With the help of the transition functions (\[uebergang\]) we can now glue curves of constant $v$ to curves of constant $V$ and curves of constant $U$ to curves of constant $u$. Figure \[3faelle\] gives an overview of this procedure. Here we already inserted $U_0=2R_0$.

(13,4.5) (0,3.5)[$e^{-\mathrm{i}\omega v}$]{} (1,3.7)[(1,0)[1.2]{}]{} (1.2,3.9) (2.5,3.5)[$e^{-\mathrm{i}\omega \xi(V)}$]{} (4.2,3.7)[(1,0)[1]{}]{} (5.4,3.5)[$e^{-\mathrm{i}\omega \xi(U)}$]{} (7.2,3.7)[(1,0)[1.2]{}]{} (8.8,3.5)[$e^{-\mathrm{i}\omega \xi(\eta(u))}$]{} (11.2,3.5)[case (1)]{} (1,3.5)[(1,-1)[1.2]{}]{} (0.6,2.6) (2.5,2.0)[$e^{-\mathrm{i}\omega V/\sqrt{\gamma_0}}$]{} (4.2,2.2)[(1,0)[1]{}]{} (5.4,2.0)[$e^{-\mathrm{i} \omega U/\sqrt{\gamma_0}}$]{} (7.2,2.2)[(1,0)[1.2]{}]{} (7.3,2.5) (8.8,2.0)[$e^{-\mathrm{i}\omega \eta(u)/\sqrt{\gamma_0}}$]{} (11.2,2.0)[case (2)]{} (7.2,2.0)[(1,-1)[1.2]{}]{} (6.3,1.0) (7.5,0)[$e^{-\mathrm{i}\omega (u + u_0 - 2R_0/\sqrt{\gamma_0})}$]{} (11.2,0.2)[case (3)]{}

In the following we restrict our analysis to the most interesting case (1), which provides the result for the extended expansion phase of the mass shell. Case (2) is restricted to a short time after the ignition of the shell, and thus is only characteristic of the ignition phase of our problem. Its importance depends 