Terrell rotation ($\sin \theta =v/c$)

The apparent rotation of a moving object arises when length contraction is combined with different light emission times for photons arriving at a camera together. For an edge-on plane lamina moving along x, let X,Y be its rest coordinates and put a distant camera on the negative y axis. Simultaneous reception requires $t+Y/c=\text{constant}$. Since $x=X/\gamma +vt$, its transverse image coordinate is $X/\gamma -\beta Y$. This is the projection of a stationary lamina rotated through $\theta$ with $\cos \theta =1/\gamma$, $\sin \theta =\beta$. It describes a photographic projection; simultaneous laboratory positions retain length contraction. The simple angle formula assumes the transverse distant-camera geometry.