15 



2:{a)Kadwa =-(fL+ 2:{ae) — (V/=^ V', (f.r«) — — {öQ - Ö,Q). (77) 



0,Ve 2x 



Let us now suppose that only the coordinate .r/, undergoes an 



infinitely small change, which has the same value at all points of 



the field-figure. Let at the same time the system of values ga/j be 



shifted" everywhere in the direction of Xh over the distance ^»/j. 



The left hand side of the equation then becomes Kuöxk and we 



have on the right hand side 



dL ^ aQ 



(f L = — T- — d'.r/,, öQ=: — - — öxh. 

 Oxh Oxh 



After dividing the equation by ^^^v, we may thus, according to 



(74) and (75), write 



— S (e) -^' = - dwnt- 



OXe 



By the same division we obtain from öQ — ö^Q the expression 

 occurring on the left hand side of (51), which we have repre- 

 sented by 



OXe 



where the complex é is defined by (52) and (53). If therefore we 

 introduce a new complex t which differs from é only by the factor 



— , so that 



fe^^e-;, (78) 



h 2x It 

 we find 



Kh == — divkt — divht ....... (79) 



The form of this equation leads us to consider t as the stress- 

 energy-complex of the gravitation field, just as 5^ is the stress-energy- 

 tensor for the matter. We need not further explain that for the 

 case K/i ^ the four equations contained in (79) express the 

 conservation of momentum and of energy for the total system, matter 

 and gravitation field taken together. 



^ 48. To learn something about the nature of the stress-energy- 

 complex r we shall consider the stationary gravitation field caused 

 by a quantity of matter without motion and distributed symmetri- 

 cally around a point 0. In this problem it is convenient to introduce 

 for the three space coordinates a'i, a\, I'j, {x^ will represent the time) 

 "polar" coordinates. By .i\ we shall therefore denote a quantity r 



