704 Mr, W. Wilson on Space- Time Manifolds 



So far as the writer is aware, the only gravitational field 



which has been investigated from the point of view of 



Einstein's theory is that of a single particle or of a number 



of isolated particles. On Newton's theory the intensity of 



the field in the neighbourhood of such an infinite rectilinear 



2m 

 distribution of mass is equal to — -, where m is the mass 



per unit length, using gravitational units. The following 

 investigation shows that the intensity as given by the 

 general theory of relativity is, to an exceedingly close 

 approximation, equal to the Newtonian result. 



Before proceeding to the actual investigation it will be 

 well to study the following simpler types of manifold 

 in which the square of the line element has the forms : — : 



ds 2 = -dz 2 -df-dz 2 -2atdxdt+{l-*H 2 )dt 2 , . (2) 



ds 2 = - dr 2 - dz 2 - r 2 d(f> 2 - 2cor 2 d<f>dt + (1 - rV) dt 2 , (3) 



ds 2 = -Adr 2 -dz 2 -r 2 d(l> 2 + Bdt 2 , (4) 



where a, a), A, and B are constants. 



In the relativity theory of gravitation the general form of 

 the square of the element of length is 



1, 2, 3, 4 



ds 2 = 2< gKTdx K dw T . 



KT 



The potentials g KT satisfy the equations 



<V = 0, (5) 



where 



1,2,3,4 



Gfiv = 5 (i r p — s— j 



^ \ up ve O&p pv CM ». 

 pe 



and 



= 2 (V r -^- r[ v + h — ^ log v/,9 



pv 0<-Ve J 



B.'/,,, 3.9... d.7. 



In some cases the potentials g KT may also satisfy the 

 equations 



]$ (> = 0, (6) 



* Einstein, Ann. d. Phys. xlix. p. 709 (1016). Einstein only uses 

 co-ordinates for which g = l. See also Ecldington, ' Report on Relative 

 Theory of Gravitation.' 



