# Page:LorentzGravitation1916.djvu/41

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

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

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

and for ${\displaystyle b\neq c}$

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

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

 ${\displaystyle 2\sum (a)g^{ab}\left({\frac {\partial H}{\partial g^{ac}}}\right)_{x}+V_{c}^{b}=0}$ (76)

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

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

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

the term ${\displaystyle \xi _{c}}$ depends on the coordinate ${\displaystyle x_{b}}$ only and is zero at the point in question of the field-figure. The quantities ${\displaystyle g^{ab}}$ 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 ${\displaystyle H}$ would change, but all together must leave it unaltered. As to the first change we remark that, according to the transformation formula for ${\displaystyle g^{ab}}$, the variation ${\displaystyle \delta g^{ab}}$ vanishes when the two indices are different from ${\displaystyle c}$, while

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

and for ${\displaystyle a\neq c}$

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

The change of ${\displaystyle H}$ due to these variations is

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