Physics. — ''Calculation of some special cases, in Einstein's theory 

 of gravitation' . By Dr. Gunnar Nordstrom. (Communicated 

 bj Prof. H. A. LoRENTz). 



(Communicated in the meeting of April 26, 1918). 



As an application of tlie theorems deduced in two preceding papers 

 for Einstein's theory ') of gravitation, we shall now calculate the 

 gravitation field and tlie stresses for some special stationary systems 

 tvith spherical symmetry. 



First the state at a surface of discontinuity will be investigated. 



^ 1 . Introductory formulae. 



In a field with spherical symmetry a surface of discontinuity 

 necessarily is a sphere. Tliis suiface will be considered as the limiting 

 case of a layer of (inite depth, and we shall oidy have to pay attention 

 to such surfaces in which in the limit some component of the material 

 stress-energy-tensor increases above every arbitrary limit so that the 

 line-integral across the layer remains finite. In general at such a 

 surface of discontinuity there evidently works a surface-tension P : 



'a 

 = Lim I i.p 



ri—ri=oJ 



^>dr (1) 



where ?'i denotes the inner radius of the layer, and r^ the outer one. 

 The radical component of the stress-tensor ^'' on the contrary we 



shall suppose never to pass every arbitrary limit; in other words 

 we assume that: 



'"2 



li7n I t'r dr=zO (2) 



'■i 



First we shall consider a general surface of discontinuity and 

 only afterwards we shall introduce special assumptions. We start 

 from the first and third formulae (38) I and from (39) I. (From these 

 three formulae the second formula (38) I may also be derived, but 



1) G. Nordstrom, On the mass of a material system according to the gravitation 

 theory of Einstein. These Proceedings XX, 1917, p. 1076 (cited further on as I) 

 and: On the energy of the gravitation field in Einstein's theory. These Proceedings 

 XX, 1918 p. 1238 (cited further on as II). 



