The Lorentz contraction
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 $\sqrt{1 - u^2/c^2}$. 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 $x'\sqrt{1-u^2/c^2}$ 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 $x = x'\sqrt{1-u^2/c^2}+ut$, or \begin{equation*} x'=\frac{x-ut}{\sqrt{1-u^2/c^2}}, \end{equation*} which is the first equation of the Lorentz transformation.