Invariance of Transverse Lengths

Suppose that we recruit a number of observers, and provide each one with an identical meter stick. Next, let us place the different observers on trolleys that move at different velocities with respect to one another. Each time any given pair of observers make a close approach to one another, suppose that they hold up their meter sticks, orientated such that the sticks are parallel to one another, but perpendicular to their relative velocity, and then make a simultaneous measurement of the relative heights of the two top ends, and the two bottom ends, of the sticks. This is equivalent to a comparison of the lengths of the two sticks. If the two lengths are different then we have found a way of distinguishing one inertial frame from another. In fact, we can provide a unique label for each reference frame in terms of the length of its associated observer's meter stick. However, this state of affairs is forbidden by Einstein's first postulate. Hence, all meter sticks must have the same length. In other words, two observers in two inertial reference frames, moving with respect to one another, will always agree on measurements of lengths orientated perpendicular to their relative motion.