114 Dr. L. Silberstein on General Relativity 



Four of the eqs. (21) will then become a 51 = a 52 = a 5i = a 54 = 0, 

 so that the group ultimately becomes 



x/=a lK a; K (L,K = l,...4:);a! 5 , = x 5 : . . (19 6) 



There being now 10 conditions for 16 coefficients, the group 

 with fixed origin is six-parametric. Moreover, apart from 

 a? 5 = a? 5 ', the eqs. (19 b) are now exactly of the form of those 

 expressing the Lorentz transformation. 



In short, the natural systems, for any constant world- 

 curvature, are obtained by subjecting the Weierstrass co- 

 ordinates of any one of them (with x-J — x 5 ) to a Lorentz 

 transformation. 



Of the six-parametric group the pure space-rotations are 

 of no interest. It will thus be enough to take the case of 



x 2 ' = x 2 , -V = ^s; V = #5- 

 The conditions (21) then become 



a ll 2 + a i i 2 = a u 2 + a u 2 = l ; ai 1 a u -\-a^a u = i 



and are satisfied by a n = a u = cos S, a H = — a 41 = sin 5, with a> 

 as the only parameter, so that the transformation ultimately 

 becomes 



; , ,_ [ ( 22 ) 



X 2 , X3 , 05% — X 2 , #3, X 5 . J 



The first line is familiar from the older relativity, the only 

 difference being that now it holds for the Weierstrass co- 

 ordinates of the world-point. To translate this result into 

 our original coordinates, put 



r ~\ 



%i, x 2 , x 3 = R sin yfr. sin ^ [cos <£, sin <f> cos 6, sin (/> sin 0], I 



r ? • ( 23 ) 



X i ^=it = R COS i}r ; x 5 = R sin yfr. cos ^, 1 



SO that a?i 2 + ^2 + a , 3 2 + ^2 = i2 2 sin2 ^ and ^2 = ^ 2? 



1 * 



identically, as required. Then we get, as a translation of 

 (22), keeping for the moment R, to avoid confusion % the 

 following equations between r, </>, 0, ^ and their dashed 



* Since | R | was taken as unit length, R will here stand for 1 or 

 s/ — 1 according as the curvature is positive or negative. 



