1079 



factor depending on the system of units (comp. the next §, equation (15)). 



§ 2. E?ier(jy of a statmiary system. 



We shall now consider a material system of finite dimensions and 

 especially one for which there exists (at least) one system of coordi- 

 nates in uihich the gravitation field is stationary. Let us first consider 

 what must be understood by the mass of the system. The material 

 system having finite dimensions it is evident that its gravitation 

 field may be considered as being caused by a material point, the 

 mass of which. has a definite meaning, and all that holds with greater 

 accuracy according as the distance to the system is greater. The best 

 way of defining the mass of the system is based on the properties of 

 the created gravitation field at points at a great distance. According to 

 the theory of relativity however the mass of the system is equal to 

 its total energy when at rest divided by the square of the universal 

 constant c which represents the velocity of light in natural units. If 

 according to our assumption we use a system of coordinates in 

 which the gravitation field is stationary we find for the energy at 

 rest the expression 



Jjji 



(I/ + t,')dx,dx,dx. 



where the integration has to be extended over the whole three- 

 dimensional space. Possibly a universal constant factor has to be 

 added in order that we may obtain the energy expressed in the 

 desired units (comp. § 1 end). It is easy to see whether this is 

 necessary. First of all we choose the time-coordinate in such a way 

 that at an infinite distance g^^ gets the value c^ Of course the value 

 of the universal constant c depends again on the system of units, 

 which can be chosen thus that c = l. Further we remark, that 

 together with a change of the unit of the time-coordinate the 

 numerical values of ^^ — g and of all ^/'s changes proportionally 

 to the numerical value of c. The energy having the dimensions 

 l/L'7'-2, it is now evident that the factor c must be added to our 

 integral expression in order that it may express the energy inde- 

 pendently of the choice of the unit of time in the corresponding 

 unit. We thus have for the energy at rest E: 



E = ^JjT(^\' + f/) ^-^1 ^'^\ ^'^. (1Ö) 



integrated over the whole three-dimensional space. 



This expression gives the total energy at rest for a definite material 

 system when this is the only one within the domain of integration. 



