# Page:LorentzGravitation1916.djvu/36

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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

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

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

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

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

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

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

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

It is thus seen that

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

and that we have therefore

 ${\displaystyle div{\mathfrak {s}}=div{\mathfrak {s}}_{0}}$ (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 ${\displaystyle g_{ab}}$, (viz. those of the third order) are the same in ${\displaystyle div_{h}{\mathfrak {s}}}$ and ${\displaystyle div_{h}{\mathfrak {s}}_{0}}$. 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 ${\displaystyle {\mathfrak {s}}}$ and ${\displaystyle {\mathfrak {s}}_{0}}$ are determined by (52), (53) and (57) and if the function defined in § 32 is taken for ${\displaystyle G}$, 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

${\displaystyle \int \mathrm {L} dS}$

where ${\displaystyle \mathrm {L} }$ is defined by (35) (1915). From ${\displaystyle \mathrm {L} }$ 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.