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

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.