102 Dr. L. Silberstein on General Relativity 



corresponding quadratic form for the line-element let as 

 take, with iielfcrami-Killing's theorem as guide, for our 

 particular three-space just the extreme appearing in that 

 theorem, viz. a space of constant curvature equal to that of 

 the world. The iine-element dX of such a space, no matter 

 what the sign of R 2 , can be written, as is well-known, in 

 polar coordinates r, <f>, 0, for instance, 



d\ 2 = dr 2 + R 2 sin 2 ^ (dcj> 2 + sin 2 <j>d6 2 ) . 



Such being the space part of the line-element, let us use a 

 system in which g u =g 2i z=zg u = 0, which is always possible, 

 and let us tentatively take g u =l. Thus, with x^ — d, the 

 required expression for the line-element will be dx 2 — d\ 2 , 

 i. e. 



ds 2 =c 2 dt 2 -dr 2 -R 2 sm 2 ^(d<l> 2 + sm 2 (l>d0 2 ). . . (1) 



Attaching (mentally) the suffixes 1, 2, 3, 4 to the radial, the 

 meridional, the latitudinal, and the time-direction, respec- 

 tively, the equivalent fundamental tensor will conveniently 

 be written, with gu=ffi 9 



9i=-h # 2 =-^ 2 sin 2 -^> # 3 =^ 2 .sin 2 <£, ^=1, . (2) 



all other components being zero. That the form (1) or the 

 corresponding tensor (2) do actually express (in a particular, 

 convenient reference system) the said four-manifold, will be 

 seen hereafter in more than one way. 



Our original assumption is now reduced to the assumption 

 that, outside of matter, it is always possible to choose such a 

 system of coordinates in which the line-element takes the 

 form (1). We shall refer to such variables by the short 

 name of natural coordinates. 



It will be well understood, however, that we do not 

 postulate the invariance of the particular form (1) or of the 

 corresponding tensor (2) which, of course, could be preserved 

 only with respect to certain very particular transformations, 

 whereas we require all physical laws to be generally covariant. 

 Thus, in any not "natural" system of coordinates, which 

 we will generally denote by Wi, u 2 , w 3 , m 4 , the line-element (1) 

 will assume the form 



ds 2 =gijduidi(j, (3) 



where gij will be some linear homogeneous functions of the 



