= Arrival-time compression for a moving source
{title2=$dt_a/dt=1-\mathbf n\cdot\boldsymbol\beta$}
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 $dt_a/dt=1-\mathbf n\cdot\mathbf u/c$, 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 $r/(c\gamma^3)$.
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