﻿744 Prof. Hackett on Relativity- Contraction in a Rotating 



arrival at x' = mean of departure and arrival times at the 

 origin. 



According to the reckoning of the fixed observer, however, 

 the passage of the timing aperture at x' through the plane 

 of (xz) is from (3) later than at the origin by x'vco/c 2 \Jl — v 2 /c 2 

 owing to the twist in the shaft. This is the local time effect 

 and leads to the Lorentz time-transformation adopting the 

 usual conventions. 



Counting time from the instant when the origins in each 

 system coincide and the timing aperture at the origin in S' 

 passes through the plane of (x,z), we have at every point 

 along the shaft 



. . „ . rotation of the S' shaft 



t in fe-time = — L i \ — -, — : — = r— . 



its angular velocity m o-units 



The rotation of the S' shaft is got by adding the twist 

 to the angular distance of the timing aperture from the plane 

 of (#, z). The latter is t' in the units we have adopted, and 

 from (3) and (4) we have 6 = x'vjc 2 i giving 



t in S-units = / -- 1 - Tr - g (t' + 6) = .-i-^ 2 (*' + x'v/c 2 ). 

 V 1 — « / cr v 1 - V 2 c z 



... (5) 



§ 4. The Strain in the Rotating Shaft. 



The discussion in § 2 has shown that to a fixed observer 

 a rotating shaft with a motion of translation in the direction 

 of its own axis is in a state of strain. For convenience the 

 term contraction is used here for the ratio of new length to 

 original length. We shall now proceed to find the principal 

 contractions for this state of strain. For the sake of clear- 

 ness, the assumptions involved in this discussion are set 

 forth below. 



We have as data the following deductions from the appli- 

 cation of the principle of Restricted Relativity to the systems 

 considered in § 2 : — 



(I.) The twist in the shaft is independent of its radius, 

 whether it is hollow or solid, and is given by (3). 



(II.) The FitzG-erald-Lorentz contraction of the distance 

 between two planes perpendicular to the axis is not altered 

 by the rotation, otherwise the relations deduced in (2), (3), 

 and (5) in agreement with the ordinary theory could not 

 exist. 



The contractions of the relativity theory are independent 



