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

 $div'_{h}\mathfrak{s}'=p\sum(m)p_{mh}div_{m}\mathfrak{s}-Q\sum(a)p_{ah}\frac{\partial p}{\partial x_{a}}+2p\sum(abc)p_{ah,c}\mathfrak{g}^{bc}G_{ab}$ (56)

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

 $\mathfrak{s}_{0h}^{e}=-G\sum(a)\mathfrak{g}^{ae}g_{ah}+2\sum(a)\mathfrak{g}^{ae}G_{ah}$ (57)

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

 $div\mathfrak{s}=div\mathfrak{s}_{0}$ (58)

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.[1] 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

1. This has also been done by de Donder, Zittingsverslag Akad. Amsterdam, 35 (1916), p. 153.