14.2.11∗. A light beam is formed between two lenses with a circular cross-section of radius $R$ directed along the $x$-axis. A disk of the same radius moves along the $y$-axis at a speed of $v$. The disk plane is perpendicular to the $x$-axis. In a laboratory system where the lenses are stationary, the moving disk contracts in the direction of travel and therefore cannot block the light beam. For the observer on the disk, the beam cross-section is reduced and, it would seem, the moment of complete screening of light should be observed. Explain this paradox.
Solution
The obstruction of every point on the reduced cross-section of the light beam witnessed by the observer on the disk is simultaneous. However, this obstruction is not simultaneous in the laboratory frame for any two points with different $y$ coordinates. (By Lorentz transformation, if the difference of $y$ coordinates is $\Delta y$, then the time separation of the obstruction events is $v\Delta y/c^2$.) Consequently, points on the same wavefront parallel to the disk are not obstructed at the same time in the laboratory frame, and the paradox is resolved.
