when at rest, and the more quickly it is moving, the shorter is the rod. For the velocity we should have , and for still greater velocities the square-root becomes imaginary. From this we conclude that in the theory of relativity the velocity 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 as a limiting velocity also clearly follows from the equations of the Lorentz transformation, for these become meaningless if we choose values of greater than .
If, on the contrary, we had considered a metre-rod at rest in the -axis with respect to , then we should have found that the length of the rod as judged from would have been ; 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 , , , , 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.
