106 Dr. L. Silberstein on General Relativity 



These are the constituents of a covariant tensor of rank two, 

 derived by Einstein from the Riemann-Ohristoffel tensor 

 by " contraction " (equating two indices to ono another and 

 summing over them). 



In (7 a), the general definition of the three-index symbols 

 ofOhristoffel is 





H~H=^[!] 



(8) 



where 



and, therefore^ for the form (6), and with ui = %i, say, 

 #i> 25 3? 4 — r > <£> #> c^ ; as in the preceding section, — 



and I ^ =0 when *,/, /c are all different. In our case (8) 



becomes 



so that finally, 



Ivl-i^ fit 1 - _ ! d£ r;^n 



(10) 



and < ^ j- =0 when all indices are different. Thus far ^ x 



etc. were any functions of x x , etc. or r, </>, 0, ct. Hence- 

 forth it will be enough to develop the sub-case in which 



gi=9i{r)> ff2=02(r), ^3=^sin 2 (f», g^=g^(r), 



(11) 



where, however, g u g 2 , 9± continue to be any functions of 

 r = x 1 . Then, by (7a) and (10), with dashes used for deri- 

 vatives, and introducing the abbreviations 



h l = \og(-g l ), h 2 = log(-g 2 ), K = logg i , A=log^^, 



B n = W+jV'+KV-V)/«i'+iV(V-V) ] 



B-iS^B^^W+W*')-! 





