Page:LorentzGravitation1916.djvu/41

$\left({\frac {\partial \mathrm {L} }{\partial g^{ab}}}\right)_{x}={\frac {1}{2}}{\sqrt {-g}}T_{ab}$

in the differentiation on the left hand side the coordinates of the material points are kept constant. To show that $T_{ab}$ and ${\mathfrak {T}}_{c}^{b}$ satisfy equation (66) we must now show that

$-\mathrm {L} -{\sqrt {-g}}V_{c}^{c}=2\sum (a)g^{ac}\left({\frac {\partial L}{\partial g^{ac}}}\right)_{x}$

and for $b\neq c$

$-{\sqrt {-g}}V_{c}^{b}=2\sum (a)g^{ab}\left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right)_{x}$

If here the value (72) is substituted for $\mathrm {L}$ and if (70) is taken into account, these equations say that for all values of $b$ and $c$ we must have

$2\sum (a)g^{ab}\left({\frac {\partial H}{\partial g^{ac}}}\right)_{x}+V_{c}^{b}=0$

(76)

Now this relation immediately follows from a condition, to which $\mathrm {L}$ must be subjected at any rate, viz. that $\mathrm {L} dS$ is a scalar quantity. This involves that in a definite case we must find for $H$ always the same value whatever be the choice of coordinates.

§ 45. Let us suppose that instead of only one coordinate $x_{c}$ a new one $x'_{c}$ has been introduced, which differs infinitely little from $x_{c}$, with the restriction that if

$x'_{c}=x_{c}+\xi _{c}$

the term $\xi _{c}$ depends on the coordinate $x_{b}$ only and is zero at the point in question of the field-figure. The quantities $g^{ab}$ then take other values and in the new system of coordinates the world-lines of the material points will have a slightly changed course.

By each of these circumstances separately $H$ would change, but all together must leave it unaltered. As to the first change we remark that, according to the transformation formula for $g^{ab}$, the variation $\delta g^{ab}$ vanishes when the two indices are different from $c$, while

$\delta g^{cc}=2g^{cb}{\frac {\partial \xi _{c}}{\partial x_{b}}}$

and for $a\neq c$

$\delta g^{ac}=2g^{ca}=g^{ab}{\frac {\partial \xi _{c}}{\partial x_{b}}}$

The change of $H$ due to these variations is

$2{\frac {\partial \xi _{c}}{\partial x_{b}}}\sum (a)g^{ab}\left({\frac {\partial H}{\partial g^{ac}}}\right)_{x}$