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Perpendicular to the moving object (Einstein's light clock)

A moving object S emits a light ray perpendicular to its velocity v. According to Pythagoras, observer B then sees this ray at an angle cos(alpha) =v/c to the direction of motion. Is this always the case?
In Minkowski space, S moves in the x-direction. A plane perpendicular to x is supposed to intersect the light cone at the origin. This plane rotates around the y-axis with S by tan(beta) = s/ct = v/c. This rotated plane intersects the respective future cone of S at its apex. The conic section is always two intersecting lines. The projection onto the x-y plane yields the angle cos(alpha) = v/c to the x-axis in the direction of motion.
The perpendicular light rays are perceived obliquely, as in Einstein's light clock.
Sexl's book describes how a moving railway carriage is perceived as rotated.
The angle beta is called alpha there. The shortening of l is l*cos(beta), or l/gamma (Lorentz-factor). The rear wall of the carriage points towards an extension l*gamma. It's easy to verify; here too, cos(alpha) = v/c..
In Sexl's book, on page 107, the formula tan(e) = sin(e')*sqrt(1-vē/cē)/(cos(e')+v/c) is given. Setting e' = 90°, we get: tan(e) = sin(e)/cos(e) = 1*sqrt(1-vē/cē)/(0+v/c). With sin(e) = sqrt(1-vē/cē), we then have cos(e) = v/c.
From the observer's perspective, perpendiculars of moving objects appear oblique, and those of accelerated objects appear curved (Einstein elevator).



Ludwig Resch