Page:Relativity (1931).djvu/63

 when at rest, and the more quickly it is moving, the shorter is the rod. For the velocity $$v = c$$ we should have $$\sqrt{1 - \tfrac{v^2}{c^2}} = 0$$, and for still greater velocities the square-root becomes imaginary. From this we conclude that in the theory of relativity the velocity $$c$$ plays the part of a limiting velocity, which can neither be reached nor exceeded by any real body.

Of course this feature of the velocity $$c$$ as a limiting velocity also clearly follows from the equations of the Lorentz transformation, for these become meaningless if we choose values of $$v$$ greater than $$c$$.

If, on the contrary, we had considered a metre-rod at rest in the $$x$$-axis with respect to $$K$$, then we should have found that the length of the rod as judged from $$K'$$ would have been $$\sqrt{1 - \tfrac{v^2}{c^2}}$$; this is quite in accordance with the principle of relativity which forms the basis of our considerations.

A priori it is quite clear that we must be able to learn something about the physical behaviour of measuring-rods and clocks from the equations of transformation, for the magnitudes $$x$$, $$y$$, $$z$$, $$t$$, are nothing more nor less than the results of measurements obtainable by means of measuring-rods and clocks. If we had based our considerations on the Galilei transformation we should not have obtained a contraction of the rod as a consequence of its motion.