Both we and the traveller carry optical clocks, of precisely the same dimensions when placed side by side; so that for either one of us, 100,000 oscillations of our respective light rays in our respective optical clocks correspond to one second of our respective times. Let us assume that the optical clock of the travelling observer is placed crosswise to his motion with respect to us, so that his light days oscillate back and forth in a direction perpendicular to his relative motion. So far as the traveller is concerned, the light rays of his optical clock will oscillate along one same straight line 1.5 kilometres in length; and if during his entire trip back and forth he finds that his clock has oscillated
times, he will be justified in saying that according to his optical clock his trip has lasted
seconds. This, in the precise numerical example we have chosen, would turn out to be twenty years.
Now let us examine the problem from our own point of view; that is to say, let us refer all measurements to our own Galilean system attached to the earth. In this Galilean frame of ours the oscillating wave of light of the traveller will not follow a to-and-fro motion along the same straight line. It will follow a zigzag line analogous to a series of
’s placed end to end. The total length of this path followed by the light wave must therefore be longer than when referred to the frame of the traveller; in the present numerical example it would be just twice as long. The question then arises: “How long will it take for the traveller’s wave of light to describe this zigzag line back and forth when time is measured according to the standards of our Galilean frame attached to the earth?” The postulate of the invariant velocity of light gives us the answer immediately. Since the zigzag line followed by the light wave in our frame is just twice as long as the succession of superposed up-and-down lines which it follows in the traveller’s frame, and since, whether referred to one frame or the other, the velocity of the light wave along its path must be the same (invariant velocity of light in all Galilean frames), then the immediate conclusion is that the total duration of the
oscillations will be twice as long when measured by us as when measured by the traveller.