Arrival-time compression for a moving source ($d{t}_a/dt=1-\mathbf{n}\cdot \mathbf{\beta }$)

For a stationary distant detector, the arrival time of a light signal emitted at $(t,\mathbf{x}(t))$ is $t_a=t+|{\mathbf{x}}_D-\mathbf{x}(t)|/c$. Its derivative is $d{t}_a/dt=1-\mathbf{n}\cdot \mathbf{u}/c$, where $\mathbf{n}$ points from the source to the detector. This exact relation follows by differentiating the distance; a constant viewing direction is a far-field approximation. For a source approaching near the speed of light, a long emission interval can correspond to a short arrival interval. Combining it with the $1/\gamma$ cone of relativistic beaming yields synchrotron pulse compression of order $r/(c{\gamma }^3)$.