make such effects sensible. The velocities which occur in some of the newly investigated domains of physics are just as new and outside our former experience as the fifth dimension.
Returning now to the magnitude of this difference of opinion as to the distance between the clocks, it is easy to show that, from our point of view, the moving observer overestimates the distance in the ratio
So that it may be said in general that lengths in the direction of motion, which he says are equal, we say are unequal in this same ratio.
On lengths perpendicular to the direction of motion our estimates agree.
Now let us ask ourselves: What are corresponding lengths in the two systems? Corresponding lengths may with propriety be given the same name, "meter" for instance. The condition that two lengths should be "corresponding" is simply that each observer comes to the same conclusion with respect to the other length.
The lengths AB and MN are not "corresponding," for the moving observer says that MN is equal to AB, while we say AB is less than MN, in the ratio (1—β²). If, however, we mark off on our platform a length which is a mean proportion between our estimate of the length AB and the length MN, this length, say ME, will "correspond" to the length AB, for we shall then say, that AB is less than ME in the ratio √(1—β²), while the moving observer will say that ME is less than AB in the same ratio.
Thus any length, in the direction of motion, on a moving system is estimated less in the ratio √(1—β²) by a "stationary" observer.
Or, put in a better way, an observer viewing a system which is moving with respect to him, sees all lengths, in the direction of motion, shrunken in the proportion √(1—β²), where β is velocity with which the system is passing him in terms of the velocity of light.
We have now reached two results, which we may summarize thus; first, clocks which a moving observer calls in unison do not appear in unison to a "stationary" observer, the clock in advance as regards motion appearing behind the other in time, and second, distances in the moving system appear shortened in the direction of motion in the ratio √(1—β²). In the above we can, of course, interchange the words "moving" and "stationary."
Next let us turn our attention to the unit of time in each system. It is not hard to show that the unit of time in the moving system will appear to us greater than ours in the ratio √(1—β²). This is due to the fact that in the moving system forward clocks are behind in time.
In the measurement of time we assume a certain standard motion to be taking place at a constant rate and then take as a measure of time the total displacement which this motion has caused. Time measurement with an ordinary clock is obviously a special case of this general rule.
The moving observer can adopt as his unit of time the time it takes light, moving with the characteristic[1] space velocity V, to travel a certain distance d and return to him.
Suppose d is in the direction of motion, and the light after traveling a certain distance in the direction of motion is reflected back to the observer. He will then write
We, however, "know" that he is overestimating the distance d in the ratio √(1—β²)
- ↑ That the moving observer's estimate of V can not change with his velocity follows of course from the first postulate.
