﻿750 Prof. Hackett on Relativity- Contraction in a Rotating 



The way in which the form for the simple longitudinal 

 contraction is maintained in this more complex motion sup- 

 ports the assumptions which we have made, especially 

 assumption (B), giving in (23) the direction of one of the 

 principal contractions. 



These results hold generally for any solid shaft in the 

 same state of motion, since, as stated in deductions (I.) 

 and (II.), § 4, the twist and longitudinal contraction are 

 independent of the form of the shaft. 



§ 5. The Contraction in a Rotating Disk, 



The expressions deduced in the last section hold for all 

 values of v and u. They should hold in the limiting case for 

 which v = when the shaft is rotating around an axis fixed 

 relative to the observer. In this way we derive a solution 

 of the problem of the rotating disk which enters so frequently 

 into discussions of the restricted and general principles of 

 relativity. Writing v = 0, we find 



circumferential contraction = Vl — u 2 /c 2 . 



Before discussing this result, it may be well to state the 

 solutions which have been previously given of this problem. 

 Following Ehrenfest, it has been frequently stated * that if a 

 measuring rod is applied tangentially to the edge of a disk 

 in rotation in its own plane about its centre, the rod is 

 shortened in the direction of motion, but will not experience 

 a shortening if it is applied to the disk in the direction of the 

 radius. The result was originally put forward as a speculative 

 inference from the restricted principle of relativity. It raised 

 the difficulty that the ratio of the circumference to the dia- 

 meter is no longer constant, but this has since been met by 

 the statement that the disk is no longer a Euclidean plane. 



On the other hand, Lorentz finds that both radius and 

 circumference contract in the ratio of 1 to 1 — v 2 /8c 2 from an 

 investigation based on the general principle of relativity. 



This problem is a special case of the " general question as 

 to how far the dimensions of a solid body will be changed 

 when its parts have unequal velocities, when, for example, 

 it has a rotation about a fixed axis. It is clear that in such 

 a case the different parts of the body will by their interaction 

 hinder each other in the tendency to contract to the amount 

 determined by a/1 — v 2 \c 2> "\. 



* Einstein, i Theory of Relativity/ p. 81 ; Jeans, Proc. Roy. Soc. 

 vol. 97. A, p. 68 (1920). 



t Lorentz, 'Nature,' February 17th, 1921, p. 79. 



