Page:The principle of relativity (1920).djvu/151

 This hypothesis sounds rather phantastical. For the contraction is not to be thought of as a consequence of the resistance of ether, but purely as a gift from the skies, as a sort of condition always accompanying a state of motion.

I shall show in our figure that Lorentz's hypothesis is fully equivalent to the new conceptions about time and space. Thereby it may appear more intelligible. Let us now, for the sake of simplicity, neglect (y, z) and fix our attention on a two dimensional world, in which let upright strips parallel to the t-axis represent a state of rest and another parallel strip inclined to the t-axis represent a state of uniform motion for a body, which has a constant spatial extension (see fig. 1). If OA´ is parallel to the second strip, we can take t´ as the t-axis and x´ as the x-axis, then the second body will appear to be at rest, and the first body in uniform motion. We shall now assume that the first body supposed to be at rest, has the length l, i.e., the cross section PP of the first strip upon the x-axis = l^.OC, where OC is the unit measuring rod upon the x-axis—and the second body also, when supposed to be at rest, has the same length l, this means that, the cross section Q´Q´ of the second strip has a cross-section l^·OC´, when measured parallel to the x´-axis. In these two bodies, we have now images of two Lorentz-electrons, one of which is at rest and the other moves uniformly. Now if we stick to our original coordinates, then the extension of the second electron is given by the cross section QQ of the strip belonging to it measured parallel to the x-axis. Now it is clear since Q´Q´ = l^·OC´, that QQ = l^·OD´.

If (dc/dt) = v, an easy calculation gives that

OD´ = OC [sqrt](1-(v^2/c^2)), therefore (PP/QQ) = (1/[sqrt](1-(v^2/c^2))