from the inside or from the outside. We can thus choose the same radial coordinate $r$ on both sides of the shell and we obviously can choose the same angular coordinates $\theta$ and $\phi$. Outside the shell we denote the time coordinate by $t$, which is the time measured by an asymptotic observer, who is at rest with respect to the centre of mass of the shell. The relation between the time coordinates $t$ and $\tau$ remains to be fixed by the boundary conditions on the mass shell. We take null coordinates and ignore the $r^2 d\Omega^2$ part of the line element for the moment. Inside the shell these coordinates are denoted by $(U,V)$, and by $(u,v)$ outside, where the line element is given by $$\label{Meau}
\mathrm{d}s^2 = \gamma(r)\;\mathrm{d}u\;\mathrm{d}v\; , \quad 
\gamma(r) = 1-\frac{2M}{r}\; ,$$ with $$\label{trafouv}
u + u_0 := t - \int^r_{R_0}\frac{\mathrm{d}r'}{\gamma(r')}\; , \quad 
v + v_0 := t + \int^r_{R_0}\frac{\mathrm{d}r'}{\gamma(r')}\; .$$ Inside we simply have $$\mathrm{d}s^2 = {\mathrm{d}}U\;{\mathrm{d}}V\; ,$$ with $$\label{trafoUV}
V + V_0 := \tau + r - R_0\; , \quad U + U_0 := \tau - r + R_0\; .$$ The outgoing light rays run on world lines with constant $u$ and $U$. We fix 