Derivation of Symmetrical Gravitational Fields. 23 



are known to be 



x r + {lm, r} Xix m =Q . . . (r = l, . . . 4 ; I, m uinbral symbols), 



dots denoting differentiation with respect to the arc length 

 of the geodesic. Putting x 2 , #3, oc± all constant, # = 1, 

 we find 



{11, r] =0...(r=l,2, 3,4), 



yielding g x — constant, which constant is in fact unity 

 since ds = dx { when x 2 , #3, x± are constant (by definition 

 o£ o?i) . This argument shows, conversely, that if g 1 = constant 

 the coordinate lines X\ are geodesies. It is now easily 

 seen that it is sufficient for this that g l should be a function 

 of x\ alone, since a change of coordinates 



leaves the coordinate lines x 1 unaltered. The arc length 

 along these coordinate geodesies is in this case, of course, 

 not Xi but f ^gidx\. As an assumption of symmetry we 

 now say that g± is a function of x 1 alone and not of x 2 

 nor a? 3 , and we observe that this together with #1=1 

 makes {14, 42} = 0, so that 



G 12 = {13, 32} 



- JL B 2 «y 3 L^^3_ 1 3gs ~dgs 



"2g 3 'dx 1 'dx 2 4,g d 2 d^ "dx 2 4tg 2 g z B^i d-? 2 



_1 d /l d<73\ . 1 d#3 dff 3 I d// 2 5<y 3 



• 2 Bo?! V# 3 3^2/ fyl 3^1 d^ 2 4^2^/3 3^1 d# 2 * 



Equating this to zero and writing momentarily 

 ^2 = ^g </ 2 ; Z 3 ee # 3 , 

 it is immediately integrable with respect to x 1 and we have 



2 log -~ — -hi —1 2 = el function independent of x lt 



QX 2 



The formulae become more symmetrical from this on 

 if we write § gv = H r 2 and express our results in terms of 

 the H r . Thus the result just obtained may be written 



( s ^ ) is independent of x u 



or equivalently (on extracting the square root) 



tf -sr~ 3 i s independent of x x . . . . (A) 



±±2 0^2 



