Page:LorentzGravitation1916.djvu/41

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

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

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

and for $$b\ne c$$

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

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

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

$\delta g^{cc}=2g^{cb}\frac{\partial\xi_{c}}{\partial x_{b}}$|undefined

and for $$a\ne c$$

$\delta g^{ac}=2g^{ca}=g^{ab}\frac{\partial\xi_{c}}{\partial x_{b}}$|undefined

The change of $$H$$ due to these variations is

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