77 



We consider again a stationary system with spherical symmetry. 

 In the same way as above we may define the distance r from the 

 centre of .symmetry by putting p = 1 viz. by demanding that tiie 

 periphery of a circle with its centre at the centre of symmetry is 

 2 jr r, when expressed in natural imits. If we do so in the case in. 

 question, the state in the field is not a single-valued but within a 

 certain interval at least a more-valued function of r. It is therefore 

 useful to introduce a new radial space-coordinate of which the 

 quantities in the field are single-valued functions. As such a 

 coordinate the distance 5 from the centre of symmetry expressed in 

 natural units suggests itself. In order to specialize our discussion we 

 can prescribe a relation between tiie radius defined by the condition 

 p =^ 1 and s and investigate afterwards whether this is in agreement 

 with a possible disti'ibution of the components T', of the stress- 

 energy- tensor. 



As a trial Ave put 



•■=^"(f?-') ••••■•• (3') 



where a is a constant, and we choose the sign thus that a positive 

 value of r corresponds to a positive value of s. For small values of 

 s r and s are proportional and the three-dimensional space is dilated 

 when we come farther away from the centre (viz. from the point 

 s = 0). For s = a r reaches however a maximum and when .v 

 increases still further the space is contracted and crosses itself at a 

 point in the neighbourhood of .s' = y'S a. For still higher values of 

 s the space is again dilated. 



Before proceeding we still remark that in fact the sign of 

 r does not play a role. Inversing the sign of r in our fundamental 

 formulae (5), (6) and (7) and interchanging also the signs of dr and 

 tv' we find from the formulae the same values as above for all 

 remaining quantities. For this reason we take in (37) everywhere 

 the -f- sign, so that r is taken negati\'e in the interval 0<^.v<^ 1^3 a. 



While the following discussions will be based on the fundamental 

 equations (5), (6), (7), we suppose u, lo, r, T',., 7'y!, 1\ to be functions 

 of .y. As .s' is the distance from the centre of symmetry expressed 

 in natural units we obtain, attending to the meaning of the quantity 

 u (see I § 3) 



ds = ndr, (38) 



As (37) gives by differentiation 



dr=('---l]ds (89) 



