without the Equivalence Hypothesis. 109 



the exception o£ those of the type (ikik), and these become 



(iHk) = ^gu9hk— — {hiih), . . . (15) 



for i^k. The most useful, of course, is the formula (5) 

 itself, since it enables us to write at once all the constituents 

 of the Riemann-Christoffel tensor for the assumed world in 

 any system of reference. 



4. Natural Systems of Reference. 



In Section 2 the coordinates r, cp, 6, ct in which ds 2 

 assumes the simple form (1), and in which, therefore, light 

 is propagated uniformly and isotropioally, were called natural 

 coordinates. Now, it is interesting to inquire whether, in a 

 world with any fixed curvature, there is but one or a whole 

 class of natural systems, — apart from such, of course, as can 

 be derived from the original one by mere three-space trans- 

 formations. From the older Relativity we know that for 

 R = ^o , when ds 2 becomes 



( 2 dt 2 _ dr 2 _ V 2 (d<£2 + sin 2^ ^2) = c 2 dt 2 _ ^2 _ dy 2 _, ^ 



there is an infinity of natural systems all derivable from 

 #, y, z, ct by the Lorentz transformation. It can be expected 

 that something analogous will hold for any finite R, real 

 or imaginary. Let us, therefore, try to find such natural 

 systems. More definitely, starting from the form (1), let us 

 ask for such transformations r = r(r', <£', 0', ct 1 ), etc. which 

 turn (1) into 



c 2 dt l2- dr >2 _ R 2 sin 2 T (<ty/a + s i n 20' d ffX) m 



Then, at least all these systems (if no others) will share with 

 the original one the "natural" property of simple optical 

 behaviour and other properties therewith connected. In short, 

 let us find the generalization of the Lorentz transformation 

 for a space-time of any constant curvature. 



It will be formally convenient to introduce I R I as the 



unit of length *, and similarly — ! as the unit of time, no 



matter how long these units might turn out to be when 

 compared with those usually employed by the physicist or 

 the astronomer. 



In these units, and therefore with the light velocitv c = l, 

 further with 



l = it = t x /'^l, 



* With the exception, of course, of the case of I It [ =oo . 



