= 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> gives
$$
t_D=\frac{2D/c+\delta}{1+\beta},\qquad\tau_D=\frac{t_D}{\gamma}.
$$
The 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.
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