Page:LorentzGravitation1916.djvu/40

where $P$ is given by (15) 1915. As the Lagrangian function defined by (11) 1915 equally falls under this form and also the sum of this function and the new term, the expression (71) may be regarded as the total function $\mathrm {L}$. The function $\varphi$ may be left indeterminate. If now with this function the calculations of §§ 5 and 6, 1915 are repeated, we find the components of the stress-energy-tensor of the matter.

The equations for the gravitation field again take the form (65). $T_{ab}$ is defined by an equation of the form (63), where on the left hand side we must differentiate while the $w_{a}$'s are kept constant. Relation (66) can again be verified without difficulty.

We shall not, however, dwell upon this, as the following considerations are more general and apply e.g. also to systems of material points that are anisotropic as regards the configuration and the molecular actions.

§ 44. At any point $P$ of the field-figure the Lagrangian function $\mathrm {L}$ will evidently be determined by the course and the mutual situation of the world-lines of the material points in the neighbourhood of $P$. This leads to the assumption that for constant $g_{ab}$'s the variation $\delta \mathrm {L}$ is a homogeneous linear function of the virtual displacements $\delta x_{a}$ of the material points and of the differential coefficients

${\frac {\partial \delta x_{a}}{\partial x_{b}}}$

these last quantities evidently determining the deformation of an infinitesimal part of the figure formed by the world-lines^[1].

The calculation becomes most simple if we put

$\mathrm {L} ={\sqrt {-g}}H$

(72)

and for constant $g_{ab}$'s

$\delta H=\sum (a)U_{a}\delta x_{a}+\sum (ab)V_{a}^{b}{\frac {\partial \delta x_{a}}{\partial x_{b}}}$

(73)

Considerations corresponding exactly to those mentioned in §§ 4 — 6, 1915, now lead to the equations of motion and to the following expressions for the components of the stress-energy-tensor

${\mathfrak {T}}_{c}^{c}=-\mathrm {L} -{\sqrt {-g}}V_{c}^{c}$

(74)

and for $b\neq c$

${\mathfrak {T}}_{c}^{b}=-{\sqrt {-g}}V_{c}^{b}$

(75)

The differential equations again take the form (65) if the quantities $T_{ab}$ are defined by

1. In the cases considered in § 43, $\delta \mathrm {L}$ can indeed be represented in this way.