Page:LewisTolmanMechanics.djvu/7

We thus see that a meter-stick, which, when held perpendicular to its line of motion, has the same length as a meter-stick at rest, will be shortened when turned parallel to the line of motion in the ratio ${\frac {\sqrt {1-\beta ^{2}}}{1}}$, and indeed any moving body must be shortened in the direction of its motion in the same ratio.^[1]

Let us emphasize once more, that these changes in the units of time and length, as well as the changes in the units of mass, force, and energy which we are about to discuss, possess in a certain sense a purely factitious significance; although, as we shall show, this is equally true of other universally accepted physical conceptions. We are only justified in speaking of a body in motion when we have in mind some definite though arbitrarily chosen point as a point of rest. The distortion of a moving body is not a physical change in the body itself, but is a scientific fiction.

When Lorentz first advanced the idea that an electron, or in fact any moving body, is shortened in the line of its motion, he pictured a real

1. Certain of Einstein's other deductions from the principle of relativity will not be needed in the development of this paper, but may be directly obtained by the methods here employed. For example, the principle of relativity leads to certain curious conclusions as to the comparative readings of clocks in a system assumed to be in motion.

   Consider two systems in relative motion. An observer on system a places two carefully compared clocks, unit distance apart, in the line of motion, and has the time on each clock read when a given point on the other system passes it. An observer on system b performs a similar experiment. The difference between the readings of the two clocks in one system must be the same as the difference in the other system, for by the principle of relativity the relative velocity v of the systems must appear the same to an observer in either. However, the observer A, considering himself at rest, and familiar with the change in the units of length and time in the moving system which we have already deduced, expects that the velocity determined by B will be greater than that which he himself observes in the ratio ${\frac {1}{1-\beta ^{2}}}$, since he has concluded that B's unit of time is longer, and his unit of length in this direction is shorter, each by a factor involving ${\sqrt {1-\beta ^{2}}}$. The only possible way in which A can explain this discrepancy is to assume that the clocks which B claims to have set together are not so in reality. In other words he has to conclude that clocks, which in a moving system appear to be set together, really read differently at any instant (in stationary time), and that a given clock is "slower" than one immediately to the rear of it by an amount proportional to the distance. From what has preceded it can be readily shown that if in a moving system two clocks are situated, one in front of the other by a distance l, in units of this system, the difference in setting will be ${\frac {lv}{c^{2}}}$. From this point Einstein's equations concerning the addition of velocities also follow directly.