 Solution

Radial nullness factors as $du(f,du+2dr)=0$, giving the two families. For $r<b$, $u=$ constant has $dr/dt=+f$ and is outgoing; $du/dr=-2/f$ has $dr/dt=-f$ and is ingoing. In the $r,t^{\ast }=r+u$ plane they cross the horizon according to these slopes. The cosmological horizon prevents a future-directed inward light signal sent from $r>b$ from reaching the static observer inside.