Page:LorentzGravitation1916.djvu/39

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

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

and for the second part we find

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

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

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

These equations are really fulfilled. This is evident from $$\psi_{aa}=0$$, $$\bar{\psi}_{aa}=0$$, $$\psi_{ba}=-\psi_{ab}$$ and $$\bar{\psi}_{ba}=-\bar{\psi}_{ab}$$, besides, the meaning of $$\psi_{ab}^{*}$$ (§ 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 $$w_{a}$$ and $$u_{a}$$ (§§ 4 and 5, 1915) play a similar part as $$\psi_{ab}$$ and $$\bar{\psi}_{ab}$$ 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 $$\mathrm{L}$$ (§ 4, 1915) a term which is some function of the density of the matter at the point $$P$$ 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 $$d\sigma$$ be an infinitely small three-dimensional extension expressed in natural units, which at the point $$P$$ is perpendicular to the world-line passing through that point, and $$\bar{\varrho}d\sigma$$ the number of points where $$d\sigma$$ 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 $$\bar{\varrho}$$. Further calculation teaches us that the term to be added to $$\mathrm{L}$$ must have the form