Page:LorentzGravitation1916.djvu/36

 If we take other coordinates the right hand side of this equation is transformed according to a formula which can be found easily. Hence we can also write down the transformation formula for the left hand side. It is as follows

§ 39. We shall now consider a second complex $$\mathfrak{s}_{0}$$, the components of which are defined by

Taking also the divergency of this complex we find that the difference

$div'_{h}\mathfrak{s}'_{0}-p\sum(m)p_{mh}div_{m}\mathfrak{s}_{0}$

has just the value which we can deduce from (56) for the corresponding difference

$div'_{h}\mathfrak{s}'-p\sum(m)p_{mh}div_{m}\mathfrak{s}$

It is thus seen that

$div'_{h}\mathfrak{s}'-div'_{h}\mathfrak{s}'_{0}=p\sum(m)p_{mh}\left(div_{m}\mathfrak{s}-div_{m}\mathfrak{s}_{0}\right)$

and that we have therefore

for all systems of coordinates as soon as this is the case for one system.

Now a direct calculation starting from (52), (53) and (57) teaches us that the terms with the highest derivatives of the quantities $$g_{ab}$$, (viz. those of the third order) are the same in $$div_{h}\mathfrak{s}$$ and $$div_{h}\mathfrak{s}_{0}$$. Further it is evident that in the system of coordinates introduced in § 37 these terms with the third derivatives are the only ones. This proves the general validity of equation (58). It is especially to be noticed that if $$\mathfrak{s}$$ and $$\mathfrak{s}_{0}$$ are determined by (52), (53) and (57) and if the function defined in § 32 is taken for $$G$$, the relation is an identity.

§ 40. We shall now derive the differential equations for the gravitation field, first for the case of an electromagnetic system. For the part of the principal function belonging to it we write

$\int\mathrm{L}dS$

where $$\mathrm{L}$$ is defined by (35) (1915). From $$\mathrm{L}$$ we can derive the stresses, the momenta, the energy-current and the energy of the