Page:The principle of relativity (1920).djvu/49

 meaningless; in our theory c plays the part of infinite velocity.

It is clear that similar results hold about stationary bodies in a stationary system when considered from a uniformly moving system.

Let us now consider that a clock which is lying at rest in the stationary system gives the time t, and lying at rest relative to the moving system is capable of giving the time τ; suppose it to be placed at the origin of the moving system k, and to be so arranged that it gives the time τ. How much does the clock gain, when viewed from the stationary system K? We have,

τ = 1/([sqrt](1 - (v^2/c^2))) (t - (v/c^2)x), and x = vt,

[therefore] τ - t = [1 - ([sqrt](1 - (v^2/c^2)))]] t.

Therefore the clock loses by an amount (1/2)(v^2/c^2) per second of motion, to the second order of approximation.

From this, the following peculiar consequence follows. Suppose at two points A and B of the stationary system two clocks are given which are synchronous in the sense explained in § 3 when viewed from the stationary system. Suppose the clock at A to be set in motion in the line joining it with B, then after the arrival of the clock at B, they will no longer be found synchronous, but the clock which was set in motion from A will lag behind the clock which had been all along at B by an amount (1/2)t(v^2/c^2), where t is the time required for the journey.