Page:LorentzGravitation1915.djvu/4

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Motion of a system of incoherent material points.

§ 4. Let us now, following Einstein, consider a very large number of material points wholly free from each other, which are moving in a gravitation field in such a way that at a definite moment the velocity components of these points are continuous functions of the coordinates. By taking the number very large we may pass to the limiting case of a continuously distributed matter without internal forces.

Evidently the laws of motion for a system of this kind follow immediately from those for a single material point. If $\varrho$ is the density and $dy\ dy\ dz$ an element of volume we may write instead of (8)

$-\varrho {\sqrt {\Sigma (ab)g_{ab}v_{a}v_{b}}}\cdot dx\ dy\ dz$

(9)

If now we wish to extend equation (3) to the whole system we must multiply (9) by $dt$ and integrate with respect to $x,y,z$ and $t$.

In the last term of (3) we shall do so likewise after having replaced the components $K_{a}$ by $K_{a}dx\ dy\ dz$, so that in what follows $K$ will represent the external force per unit of volume.

If further we replace $dx\ dy\ dz\ dt$ by $dS$, an element of the four-dimensional extension $x_{1},\dots x_{4}$, and put

$\varrho v_{a}=w_{a}$

(10)

$\mathrm {L} =-{\sqrt {\Sigma (ab)g_{ab}w_{a}w_{b}}}$

(11)

we find the following form of the fundamental theorem.

Let a variation of the motion of the system of material points be defined by the infinitely small quantities $\delta x_{a}$, which are arbitrary continuous functions of the coordinates within an arbitrarily chosen finite space $S$, at the limits of which they vanish. Then we have, if the integrals are taken over the space $S$, and the quantities $g_{ab}$ are left unchanged,

$\delta \int \mathrm {L} dS+\int \Sigma (a)K_{a}\delta x_{a}\cdot dS=0$

(12)

For the first term we may write

$\int \delta \mathrm {L} \cdot dS,$

if $\delta \mathrm {L}$ denotes the change of $\mathrm {L}$ at a fixed point of the space $S$.

The quantity $\mathrm {L} dS$ and therefore also the integral $\int \mathrm {L} dS$ is invariant when we pass to another system of coordinates.^[1]

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1. This follows from the invariancy of $ds^{2}$, combined with the relations

   ${\frac {\varrho '}{dx'_{4}}}=|p|{\frac {\varrho }{dx_{4}}},\quad dS'={\frac {1}{|p|}}dS$