120 Dr. L. Silberstein on General Relativity 



any orthogonal curvilinear coordinates, such coordinates 

 being indispensable for the treatment o£ a non-homaloidal 

 world. 



Let F tK be a six-vector or a covariant antisymmetric tensor 

 of rank two, so that F =0, F, — — F . Then, with any co- 

 ordinates u u ...u 4 , the four equations 



^ + ^ + ^i =0 (la) 



0% ou t du K 



are generally covariant, because their left-hand members are 

 the (only four different) components of a tensor, to wit of 

 an antisymmetric oue of rank three. Using ordinary Car- 

 tesians, for instance, it will be seen at once that (la), i. e. 



'dF 2 zl'du i -\-'dF z J'bu 2 —'d¥ 2 J'du 3 = 0, etc., 



bF 2 3/Bw l +9F3 1 /BM 2 -f-BF 12 /f3w 3 =0, 



are exactly of the form of the group (1) of electromagnetic 

 equations. It requires, however, some care to find the 

 correct translation of F 23 etc. into the electro-magnetic com- 

 ponents along coordinates of: a more general kind. Such 

 translation will be given presently. Meanwhile, to cover 

 the group (II) of equations, consider the contravariant of 

 F lK , that is, with the usual prescription as to summations, 



F « =r ^ F ^ =%iVF ^ (35) 



where <y lK is the contravariant fundamental tensor corre- 

 sponding to the chosen system u.. Further, let cr K be a 

 contravariant four-vector, embodying in its space-part the 

 convection current, and g the determinant of the g { -. Then 

 the four equations 



^r g iry-i^)— ■ ■ • ( IIa > 



whose left hand members are the components of a contra- 

 variant four-vector, will again be generally contravariant *. 

 That these equations are exactly of the form of the group 

 (II) of electromagnetic equations is seen at a glance, at 

 least for Cartesians, and with some attention, also for more 

 general coordinates. 



Thus, the eight equations contained in (I a), (II a), together 

 with the six relations (35) express a set of electromagnetic 



* Einstein usually employs a system with g— —1, so that (II a) are 

 simplified to ~b^ lK l'bu K = <T K . But to fix thus tbe value of the funda- 

 mental determinant would hamper us unnecessarily. 



