Page:LorentzGravitation1916.djvu/39

obtained by differentiating the factor ${\displaystyle {\sqrt {-g}}}$ only and the other part by keeping this factor constant.

For the calculation of the first of these parts we can use the relation

 ${\displaystyle {\frac {\partial \log \left({\sqrt {-g}}\right)}{\partial g^{ac}}}=-{\frac {1}{2}}g_{ac}}$

and for the second part we find

${\displaystyle {\frac {1}{2}}{\sqrt {-g}}\sum (pq)g^{pq}{\bar {\psi }}_{ap}{\bar {\psi }}_{cq}}$

If (32) 1915 is used (67) and (68) finally become

${\displaystyle {\begin{array}{c}\sum (q)\psi _{cq}{\bar {\psi }}_{cq}+\sum \limits _{a\neq c}(a)\psi _{ac}^{*}\psi _{a'c'}=2\mathrm {L} \\\\\sum (q){\bar {\psi }}_{cq}\psi _{bq}+\sum \limits _{a\neq c}(a)\psi _{ab}^{*}\psi _{a'c'}=0\end{array}}}$

These equations are really fulfilled. This is evident from ${\displaystyle \psi _{aa}=0}$, ${\displaystyle {\bar {\psi }}_{aa}=0}$, ${\displaystyle \psi _{ba}=-\psi _{ab}}$ and ${\displaystyle {\bar {\psi }}_{ba}=-{\bar {\psi }}_{ab}}$, besides, the meaning of ${\displaystyle \psi _{ab}^{*}}$ (§ 11, 1915) and equation (35) 1915 must be taken into consideration.

§ 43. In nearly the same way we can treat the gravitation field of a system of incoherent material points; here the quantities ${\displaystyle w_{a}}$ and ${\displaystyle u_{a}}$ (§§ 4 and 5, 1915) play a similar part as ${\displaystyle \psi _{ab}}$ and ${\displaystyle {\bar {\psi }}_{ab}}$ in what precedes. To consider a more general case we can suppose "molecular forces" to act between the material points (which we assume to be equal to each other); in such a way that in ordinary mechanics we should ascribe to the system a potential energy depending on the density only. Conforming to this we shall add to the Lagrangian function ${\displaystyle \mathrm {L} }$ (§ 4, 1915) a term which is some function of the density of the matter at the point ${\displaystyle P}$ of the field-figure, such as that density is when by a transformation the matter at that point has been brought to rest. This can also be expressed as follows. Let ${\displaystyle d\sigma }$ be an infinitely small three-dimensional extension expressed in natural units, which at the point ${\displaystyle P}$ is perpendicular to the world-line passing through that point, and ${\displaystyle {\bar {\varrho }}d\sigma }$ the number of points where ${\displaystyle d\sigma }$ intersects world-lines. The contribution of an element of the field-figure to the principal function will then be found by multiplying the magnitude of that element expressed in natural units by a function of ${\displaystyle {\bar {\varrho }}}$. Further calculation teaches us that the term to be added to ${\displaystyle \mathrm {L} }$ must have the form

 ${\displaystyle {\sqrt {-g}}\varphi \left({\frac {P}{\sqrt {-g}}}\right)}$ (71)