706 Mr. W. Wilson on Space- Time Manifolds 



where the constant c is given by 



dt _ 

 ds 

 The first o£ these equations may now be written 

 d 2 x 



showing that the manifold (2) corresponds to a uniform 

 gravitational field in the x direction. 

 In the manifold (3) we have 



<9m = -1 





(^ = -1 



#22 = — 1 





lg 22 = -l 



\ gzi = - r2 





1 



\g u = l — i 



V 



\9 U = 1 



\9u =-r 2 co 





{ff i4 = -<» 



and 



9 



The non-vanishing Ohristoffel expressions Yp y are : 



r' u =—cor, 



P3 __ 



1 13 — 



F3 -^ 



J- u — — * 



r' 41 = -» 2 r; 

 and the equations of motion (8) are : 



oV I. as J 



£ 2 </> 2 dr d((b-\-tot) 



d.s 



ds ds 



dh 

 ds 2 



dH 

 ds 2 



= 0, 



= o, 



= 0. 



The last equation gives 



ds 



