Now let us return to the Lorentz transformation (15.3) and try to get a better understanding of the relationship between the (x,y,z,t) and the (x′,y′,z′,t′) coordinate systems, which we shall call the S and S′ systems, or Joe and Moe systems, respectively. We have already noted that the first equation is based on the Lorentz suggestion of contraction along the x–direction; how can we prove that a contraction takes place? In the Michelson–Morley experiment, we now appreciate that the transverse arm BC cannot change length, by the principle of relativity; yet the null result of the experiment demands that the times must be equal. So, in order for the experiment to give a null result, the longitudinal arm BE must appear shorter, by the square root  What does this contraction mean, in terms of measurements made by Joe and Moe? Suppose that Moe, moving with the S′ system in the x–direction, is measuring the x′–coordinate of some point with a meter stick. He lays the stick down x′ times, so he thinks the distance is x′ meters. From the viewpoint of Joe in the S system, however, Moe is using a foreshortened ruler, so the “real” distance measured is x′meters. Then if the S′ system has travelled a distance ut away from the S system, the S observer would say that the same point, measured in his coordinates, is at a distance , or

which is the first equation of the Lorentz transformation.