V}{d_0[R_0(1+\nu)+\nu V] - d_0' 2M(1+\nu)} \;\;.\end{aligned}$$

We denote the approximate transition functions around $\gamma_0$ at the entrance with $\xi_0(V)$, around $\gamma_0$ at the exit with $\eta_0(u)$, and around $\gamma_1$ at the exit with $\eta_1(u)$. The inverse functions of the last two are written as $\eta_0^{-1}(U)$ and $\eta_1^{-1}(U)$, respectively. Integration of (\[dudU\]), (\[c1\]) and (\[dvdV\]) provides the functions that will be relevant to our calculation $$\begin{aligned}
\label{uebergang}
\eta_0^{-1}(U) &=& \frac{U}{c_0} + \frac{c_0'2M (1-\nu)}{c_0^2\nu}
    \ln\left[1+\frac{c_0\nu U}{(c_0 R_0 - c_0'2M)(1-\nu) 
                               + c_0\nu U_0}\right],\\
\eta_1^{-1}(U) &=& \frac{U}{c_1},\\
\xi_0(V) &=& \frac{V}{d_0} +\frac{d_0'2M (1+\nu)}{d_0^2\nu}
    \ln\left[1+\frac{d_0\nu V}{(d_0 R_0 - d_0'2M)(1+\nu)}\right].\end{aligned}$$

Particle production
===================

To compute the particle production caused by the moving mass shell, we consider an observer who defines vacuum before the onset of expansion (at [I]{}$^-$) and ask how this vacuum is seen by an observer at [I]{}$^+$, i.e. we have to calculate the Bogoliubov transformations between the vacua at [I]{}$^-$ and [I]{}$^+$. We can do this by following the track of the light-rays.

We restrict our attention to a massless, minimally coupled scalar field, which obeys the equation of motion $\Box \varphi = 0$. We expect a damping of the amplitude with $r^{-1}$ because of the spherical symmetry. Furthermore we split off an angular 