Page:Relativity (1931).djvu/161



If we call $$v$$ the velocity with which the origin of $$K'$$ is moving relative to $$K$$, we then have

The same value $$v$$ can be obtained from equation (5), if we calculate the velocity of another point of $$K'$$ relative to $$K$$, or the velocity (directed towards the negative $$x$$-axis) of a point of $$K$$ with respect to $$K'$$. In short, we can designate v as the relative velocity of the two systems.

Furthermore, the principle of relativity teaches us that, as judged from $$K$$, the length of a unit measuring-rod which is at rest with reference to $$K'$$ must be exactly the same as the length, as judged from $$K'$$, of a unit measuring-rod which is at rest relative to $$K$$. In order to see how the points of the $$x'$$-axis appear as viewed from $$K$$, we only require to take a “snapshot” of $$K'$$ from $$K$$; this means that we have to insert a particular value of $$t$$ (time of $$K$$), e.g. $$t = 0$$. For this value of $$t$$ we then obtain from the first of the equations (5)

Two points of the $$x'$$-axis which are separated by the distance $$\Delta x' = 1$$when measured in the $$K'$$ system are thus separated in our instantaneous photograph by the distance