1243 



^ 2. The energy of the (jravitntlon field. 



In this ^ we shall calculate the disdibiition of ihe energy in a 

 gravitation field with spherical symmetry viz. we siiall calculate the 

 quantity t/ in such a field. The body whicii excites the field may 

 also be electric. Our calcuhxtion be based upon the tbrniulae (17) I 

 and (13) I, whicli give 



xf^^-ixJS-X^ + iJS--^-. . .... (16) 



We suppose 2:^^^ to be given as a function of r. When we can 



« 



calculate Ihe vector ^1 we find from this formula also f/. We shall 

 deduce an expiession for ?l, which holds at a poiirt of the A', axis 



(.r, = .i\ = 0). According to (5) I we have : 



VI, = i i/q^^ (.,.1 .,/^ - ..,^1 g:-) (^/^ + It^ - ^^ 



As in I § 3 we shall choose the time-coordinate, so that g,.^ = 0. 

 At a point of the A\ we have because of (33) 1 and (25) i : 



g^^ = -—, g^-=g'^= ^, ^^i. ^ ,,i3 ^ ,,.. ^ 0, 



Ox, 



(17) 



As to the calculation of '^Ij we first remark that the terms of the 

 summation for which n z= ^ give a contribution zero. In order that 

 a tei-m may give a coiitribntiou different from zero, eitiier a or n 

 has to be equal to 1, while the other two of the three indices 

 ct, (tt, V must be equal. 



|tt = l, a = r =1= 1 gives for 91, the contribution: 



11/ n/^.2^^" I 38^i^88 , ^,^9<,\ 



k^-9-9 [g^ VO^- +9 '^- » 



a = \ , i^i = V =1= 1 gives the contribution : 



ö.r, ö.t;,y ' '' \^ d.v, dx^J ' dx^y 

 The two contributions together give 51,. Introducing the expres- 

 sions (17) we obtain for 51, = '^1^ 



w p p' 2 / p^\ p^to' 



n r \ u y u 



Foi" a point on the A, axis, the right-hand side is e(|ual to 51,. 



If we consider a point not on tjje A^i axis, the same expression 

 holds for the radial component 51,.. 



