 Retarded radio reply to an approaching inertial observer

Let a receiver be initially at distance $D$ from a stationary responder and approach at speed $\beta c$. A light signal takes $D/c$ to reach the responder, which waits time $\delta$ and replies. Intersecting the reply's null worldline with the receiver's worldline givesThe receiver's proper time is ${\tau }_D$. These expressions describe reception before arrival when the reply is sent before the receiver reaches the responder; otherwise the trajectory after arrival must be specified.