Page:TolmanPostulate.djvu/12

Let ε' be originally so far away that the electrons do not appreciably repel each other, and let the experiment continue until they are again far apart. Owing to their mutual repulsion, while they are within each other's sphere of influence, electron ε receives a certain component velocity μ in a direction perpendicular to the general line of motion of the two systems, and ε' also receives a transverse velocity μ' . These transverse velocities are made small compared with the relative velocity of the two systems. If the mass of an electron at rest is m, then by the Bucherer experiment, the electron ε' , which we consider in motion, will have the mass $m/{\sqrt {1-\beta ^{2}}}$, where β is v/c, so that the electrons have respectively received the transverse momenta μ and $mu'\cdot /{\sqrt {1-\beta ^{2}}}$.

By the principle of the conservation of momenta

$mu=m{\frac {1}{\sqrt {1-\beta ^{2}}}}u'$

and

$u'={\sqrt {1-\beta ^{2}}}u$.

In other words, the electron ε' , since it has a larger mass than electron ε does not receive so great a transverse velocity. If, however, an observer had been traveling along with ε' , he would have been entirely unable to detect this fact that his electron had received a smaller velocity than the other one, since the first postulate of relativity states that no measurements are possible by which an observer may detect that he is in absolute motion. Since, now, to this moving observer, the velocity seems larger than it does to an observer at rest in the ratio $1:{\sqrt {1-\beta ^{2}}}$, the second which the moving observer uses must be longer than a "stationary" second in the same ratio $1:{\sqrt {1-\beta ^{2}}}$.^[1]

Having obtained the ratio between the units of time in a moving and stationary system, let us deduce the Lorentz shortening. Suppose two systems a and b moving past each other with the velocity v and two observers A and B on the systems. A makes two marks

1. It is evident that the difference of opinion of the two observers as to the transverse velocity of the electron could not be reconciled by assuming a difference in the transverse units of length, since it is perfectly possible for the observers to make a direct comparison of meter sticks held perpendicular to the line of motion of the systems. (See Lewis and Tolman, loc. cit.)