﻿Shaft moving with Uniform Speed along its Axis. 743 

 Using (1) we get 



T = TWl-v*/c 2 , (2) 



q _ lv 2tt _ lv(o .„v 



" c 2 \/ l-v 2 /c 2 * T' ~ c 2 Vi-v'/c 2 ' 



For the moving observer, however, there is no twist in the 

 shaft. His units of time and length have altered, so that 

 equation (1) holds giving the speed of light as c. 



§ 3. Relativity Clocks. 



This combination of rotating shaft and disks may be 

 regarded as a set of relativity clocks regulated by the 

 property that the shaft must rotate with the slowest speed co 

 for which it can transmit light through the apertures in 

 either direction. The fixed observer considers that such 

 clocks in the moving system run slow according to (2). 

 It will be shown below that there is an automatic synchroni- 

 zation of the clocks. This is produced by the twist in 

 che mechanical coupling, and satisfies Einstein's test for 

 synchronism. 



In the simplest form, each disk can serve as a local clock. 

 To give the same value to <?, the same method of fixing the 

 unit of time must be adopted in all systems. It is con- 

 venient, here, to take as the unit the time of describing one 

 radian. Using this unit, the angular position in radians 

 of a special aperture with respect to (x, z) gives the time 

 directly, and from (2) 



co = sjl — v 2 /c 2 = angular velocity of S' shaft in S-units. (4) 



To the moving observer there is no twist in the shaft. 

 If he arranges that the timing aperture shall lie for every 

 disk in the plane of (#, z) when the shaft is not rotating, he 

 will conclude that in rotation the timing apertures pass 

 through this plane " simultaneously ". 



It is easily seen that Einstein's test for synchronism is 

 satisfied. The first disk may be taken as the origin, and the 

 second at x' — ZS'-units since S' ignores the contraction. 

 Let all the apertures be numbered in the opposite sense to 

 the rotation 0, 1, 2, 3, etc., beginning from the timing aper- 

 ture, and let mirrors be fitted into the apertures in the second 

 disk. For the speed co, a ray leaving the first disk by No. 

 aperture is reflected at the second disk by No. 1 mirror at x' 

 and returns through No. 2 aperture. Thus the time of 



