1088 



the system of coordinates is fixed by the condition that everywhere 

 V — g=:i, then we have /> =|= 1 . 



§ 4. (jreneralization of the obtained result. 



In the preceding \ we have chosen the time-coordinate so that every- 

 where y,^ =: (/,^ = (7,^ := 0. Now we shall show how the formulae 

 (41), — (45) can be generalized, so thai they also hold when this condition 

 is not fulfilled. Because of the spherical symmetry we can write 



X ,j. 

 9yA = ~'-ffri, /t— 1,2, 3, (47) 



r 



where (/,„ has the same meaning as in formula (22öf) of the note 

 on p. J081. To generalize one formulae to the case (/,^ =|= we must 

 evidently transform the time-coordiiuite in the opposite way as in 

 the note on p. 1081. The quantities referring to the original four- 

 dimensional system of coordinates, in which ^,^ = 0, will now be 

 denoted by letters with a dash oxerfhem. The expiession of the line- 

 element in polar coordinates from which we start becomes then : 



We transform the lime-coordinate by putting 



dt = dt — If' (/) dr, 

 while ?' := r, ih =z d^, (f =z ff> are left unchanged. 

 (/.>•'•' being an invariant, we obtain by substitution 



ds" = wde—2i}'~^c' dt dr — (u^ ip'w') dr'-J>' r" {dik' -- siii' i)^ d(p'). 



The components of the fundamental tensor are then transformed 

 according to the formulae 



Ml* =: w", g,4 = -" M' w'% u- =z u" — ^}^ iv^, p^ ^P*- 



These formulae firstly give 



U^ 111'' p^ z= (ir w" + flri^)p*- 



This equation shows that the determinant g of the components 

 ^^v is not changed by our transformation of the time-coordinate. 

 We have namely 



u- iv' p"^ z= — p , (?r tv"^ -1- gri^)p* = — .9' • • • (^S) 

 where g and g denote the above mentioned determinant for the 

 ortlitigonnl system of coordinates (which through the formulae (23) 

 is connected with the polar system of coordinates) before and after 

 the transformation of the time-coordinate. That both members have 



