Page:TolmanPostulate.djvu/13

on his system a centimeter apart, in the line of motion of the system, and requests B to determine the time it takes for a point on b to pass from one mark to the other. B also makes two marks on his system a centimeter apart, and A finds that it takes the same number of seconds to pass from one mark to the other as B found in his own similar experiment. Any other outcome of the trial would be contradictory to the first postulate of relativity. If, however, we again arbitrarily consider A to be at rest, A's seconds are shorter than B's in the ratio ${\sqrt {1-\beta ^{2}}}:1$, and hence the two points on B's system must have been nearer together than those on A's in this same ratio ${\sqrt {1-\beta ^{2}}}:1$. In other words, the "moving" centimeter in the longitudinal direction is shorter than a "stationary" one in the ratio ${\sqrt {1-\beta ^{2}}}:1$.

We have now derived the change in the units of length and time in a moving system, with the help of the Bucherer experiment. Before we can derive the desired principle regarding the velocity of light, we must go one step further and find out how the clocks are set in a moving system.

The observer B who is in motion with the velocity v past a system a which we consider at rest, lays off a length of one centimeter on his system in a longitudinal direction, and with the help of two clocks, one at each end of the centimeter, notes the time taken for a point on a to pass from one end of this centimeter to the other. He obtains, of course, the time 1/v. Since, however, his seconds are "longer and his centimeters shorter than stationary ones in the ratio $1:{\sqrt {1-\beta ^{2}}}$ we would have expected him to obtain the time (1-β²)·1/v, and we can account for his obtaining the longer time 1/v, only by the assumption, that, in a moving system, a clock 1 cm. to the rear of another is set ahead by the amount

${\frac {1}{v}}-{\frac {1}{v}}\left(1-\beta ^{2}\right)={\frac {v}{c^{2}}}$ seconds.

We are now ready to deduce our principle as to the velocity of light. Consider a source of light and an observer B who measures the velocity of the light coming from this source. If the observer B is at rest and marks off a length of one centimeter in the path of light, he finds, of course, that the light takes the time 1/c to pass from one mark to the other. We wish to prove, however, that