34 



§ 64. Equations (122j show that in the coordinates (.ï\, a-'^, .r',, a-' j 



be 

 the sj'stem has a velocity of translation — in the direction of ,x\. 



a 



If this velocity is denoted by v, we have according- to (123j 



\y 



1 



If therefore we put 



E 



we find 



J/c" Mv 

 E'= , G'= . . . . (126) 



I/' 



1— IX 1-- 



When the system moves as a whole we may therefore ascribe 

 to it an energy and a momentum which depend on the velocity of 

 translation in the way known from the theory of relativity. The 

 quantity M, to which the energy of the gravitation field also con- 

 tributes a certain part, may be called the "mass" of the system. 

 From what has been said in § 62 it follows that within certain 

 limits it depends on the way in which the system and the gravita- 

 tion field are described. 



It must be remarked however that, if for the gravitation field we 

 had chosen the stress-energy-tensor r„ (§ 52), the total energy of the 

 system even when in motion would be zero. The same would be 

 true of the total momentum and we should have to put 3f = 0. 



At first sight it may seem strange that we may arbitrarily ascribe 

 to the moving system the momentum determined by (126) or a momen- 

 tum 0; one might be inclined to think that, when a definite system 

 of coordinates has been chosen, the momentum must have a definite 

 value, which might be determined by an experiment in which the 

 system is brought to rest by "external" forces. We must remember 

 however (comp. § 52) that in the theory of gravitation we may 

 introduce no "external" forces without considering also the material 

 system S' in which they originate. This system S' together with 

 the system S with which we were originally concerned, will form 

 an entity, in which there is a gravitation field, part of which is 

 due to aS" (and a part also to the simultaneous existence of >S and ^')> 

 There is no doubt that we may apply the above considerations to 

 the total system {S, S') without being led into contradiction with 

 any observation. 



