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.