176 Significance of Einstein's Gravitational Equations. 



involve B^^ linearly ; so we need consider only one z at a 

 time, and shall accordingly shorten the notation by dropping 

 the r. Thus 



Hence 



• Gv = g^B^p = — <vOn + S2 + a 33 + a u) + a ^a va - 



by substituting the Euclidean values of the g's at the origin 

 given above. 

 In particular 



G"ll = _ a il( Cl n + a 22 +■ S3 + a A4) + ^ll 2 + «12 2 + a J + «14 2 



= (V - «li a 2 2 ) +" ( a i3 2 ~ a ii a 33) + («U - tt l AJ ' ' ( 1) 



Also 



G=^"G- MV = (a n + a 22 + a 33 + a 44 ) 2 - (a n 2 + « 22 2 + . . . -f 2a 12 2 4- . . . 

 = -2{(a l2 2 -a n a 22 ) + ... (six terms)}. . . (2) 

 Hence, since g n = — 1, 



G ll - i9lS* = " i ( a 23 2 _ a 22 a 3s) + (S/ ~ ^22%) 



-f-(a 3 /-a 33 a 44 )}. . . . (3) 



Consider the three-dimensional continuum, which is the 

 section of the world by the plane ^ = 0. This is described 

 by three coordinates <r 2 , x 3 , x 4 and its Gaussian curvature, 

 which we shall denote by G (1) , is formed by dropping the 

 terms in G which contain the suffix 1. We see by (2) and 

 (3) that ' 



iG^Gu-fe^ (4) 



Since the 2 r 's contribute linearly to each term in this 

 equation, it holds for six g?s as well as for one z. 



The radius of (spherical) curvature of a manifold is denned 

 as the radius of a sphere which has the same Gaussian 

 curvature as the manifold. It is easily shown that for the 

 Gaussian curvature G (1) of a three-dimensional manifold, the 

 corresponding radius p ± is given by 



r — — 



^(1)— 2* 



ri 

 Thus the result (4) may be written 



Gii-W*-A (5) 



ri 



Consider the quadric 



(Gr^ — ig f L V G)da: li .dx v = 2 (6) 



