Page:LewisTolmanMechanics.djvu/6

passes over the path mnm. If, however, the entire system is considered to be in absolute motion with a velocity v, the light must pass over a different path mn'm' where nn' is the distance through which the mirror moves before the light reaches it, and mm' is the distance traversed by the source before the light returns to it.

Obviously then,

${\frac {nn'}{mn'}}={\frac {v}{c}}$,

and

${\frac {mm'}{mn'm'}}={\frac {v}{c}}$.

Also from the figure,

$mn'=mn+nn'$, $mn'm'=mnm+2nn'-mm'$.

Combining, we have

${\frac {mn'm'}{mnm}}={\frac {1}{1-{\frac {v^{2}}{c^{2}}}}}={\frac {1}{1-\beta ^{2}}}$.

Hence if we call the system in motion, instead of at rest, the calculated path of the light is greater in the ratio ${\frac {1}{1-\beta ^{2}}}$.

Now the velocity of light must seem the same to the observer, whether he is at rest or in motion. His measurements of velocity depend upon his units of length and time. We have already seen that a second on a moving clock is lengthened in the ratio ${\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$, and therefore if the path of the beam of light were also greater in this same ratio, we should expect that the moving observer would find no discrepancy in his determination of the velocity of light. From the point of view of a person considered at rest, however, we have just seen that the path is increased by the larger ratio ${\frac {1}{1-\beta ^{2}}}$. In order to account for this larger difference, we must assume that the unit of length in the moving system has been shortened in the ratio ${\frac {\sqrt {1-\beta ^{2}}}{1}}$.