1081 



integrated over a surface / enclosiji^ the material system. Therefore 

 the mass of the system is also expressed by a surface integral over 

 a surface enclosing the system, unfortunately the quasivector S, 

 the normal component ^„ of which occurs in the integral expression 

 is not covariant, even not with respect to LoRKNTZ-transformations. 



§ 3. Application to a field with spherical symmetry. 



In our discussion on a system with spherical symmetry we shall 

 principally introduce the same notation as J. Droste in his article: 

 Het zwaartekrachtsveld van een of meer lichamen volgens de theorie 

 van Einstein (further cited as: Droste, Het zwaartekrachtsveld) chapter 

 n ^1. In contradiction with Droste we shall however consider also 

 the tield within a material body. Introducing as space-coordinates 

 the polar coordinates r, 0-, rp we can at any rate represent ihe line- 

 element (Is by the same expression as Droste viz. : 



ds^ = 10^ dt" —u- dr' —v' (dd^' + sw' i> dr,:.""), . . . (22) 

 where u,v,iu are functions of r only. Here the time-coordinate 

 a\ = t has thus been chosen that everywhere 



which is always possible in a stationary field with spherical symmetry ^). 

 Instead of the polar coordinates r,i>,(f we shall now introduce 

 as space-coordinates the corresponding orthogonal coordinates 



x^ =■ r cos B- cos tp, \ 



,v^ = r cos {^ siti (p, >....... (23) 



.t*j =z r sin &, J 



while we keep the same time- coordinate as Droste. We put 



v^rp (24) 



^) Because of the spherical symmetry gu and g-^i must be zero. The system of 

 coordinates may however be chosen in such a way that gri is not zero. We 

 then have 



c/s^ z= ïo" de + 2g,.4 dt dr—ii^ dr''—v^ (d&' + sifi" ih dtp*). . (22a) 



If however the time-coordinate is transformed in the following way, while r is 

 left unchanged: 



dt^dt -f \^){r)dr, 

 we obtain 



ds*z=w*di^ \- 2{gri +\piv^)dTdr-{u' - xp'w'—2t\Hjr4)dr^~v'{d»' \ sw'»d//'). 

 If now the function 4> (*') is defined thus that 



gri 4- ^ w'" = 0, 

 g,i will be zero in the new system of coordinates. 



