l/V β/(1-β²)
where
| l = distance between clocks A and B;
v = velocity of moving system; V = velocity of light; β = v/V. |
The way in which this difference of opinion with regard to time between the moving observer and ourselves leads to a difference of opinion with regard to length also may very easily be indicated as follows:
Suppose the moving observer desires to let us know the distance between his clocks and says he will have an assistant stationed at each clock and each of these, at a given instant, is to make a black line on our platform. He will, therefore, he says, be able to leave marked on our platform an exact measure of the length between his clocks and we can then compare it at leisure with any standard we choose to apply.
We, however, object to this measure left with us, on the ground that the two assistants did not make their marks simultaneously and hence the marks left on our platform do not, we say, represent truly the distance between his clocks. The difference is readily shown in Fig. 2, where M represents the black mark made on our platform at a certain time by the assistant at A, and N that made by the assistant at B at a later time. The latter assistant waited, we say, until his clock read the same as clock A, waited, that is, until B was at B' ; and then made the mark N. The moving observer declares, therefore, that the distance MN is equal to the distance AB, while we say that MN is greater than AB.
Again it must be emphasized that, because of the first fundamental postulate, there is no universal standard to be applied in settling such a difference of opinion. Neither the standpoint of the "moving" observer nor our standpoint is wrong. The two merely represent two different sides of reality. Any one could ask: "What is the "true" length of a metal rod? Two observers working at different temperatures come to different conclusions as to the "true length." Both are right. It depends on what is meant by "true." Again, asking a question which might have been asked centuries ago, is a man walking toward the stern of an eastbound ship really moving west! "We must answer "that depends" and we must have knowledge of the questioner's view-point before we can answer yes or no.
A similar distinction emerges from the principle of relativity. What is the distance between the two clocks? Answer: that depends. Are we to consider ourselves with the clock system when we answer, or passing the clocks with a hundredth the velocity of light or passing the clocks with a tenth the velocity of light? The answer in each case must be different, but in each case may be true.
It must be remembered that the results of the principle of relativity are as true and no truer than its postulates. If future experience bears out these postulates then the length of the body, even of a geometrical line, in fact the very meaning of "length," depends on the point of view, that is, an the relative motion of the observer and the object measured. The reason this conclusion seems at first contrary to common sense is doubtless because we, as a race, have never had occasion to observe directly velocities high enough to
