Page:LorentzGravitation1916.djvu/34

order. From this we can deduce that ${\displaystyle \left({\frac {1}{\sqrt {-g}}}\delta {\mathfrak {g}}^{ab}\right)}$ is also such a tensor.

Writing for it ${\displaystyle \epsilon ^{ab}}$ we find according to (46) and (47) that

${\displaystyle \sum (ab)M_{ab}\epsilon ^{ab}}$

is a scalar for every choice of ${\displaystyle \left(\epsilon ^{ab}\right)}$.

This involves that ${\displaystyle \left(M_{ab}\right)}$ is a covariant tensor of the second order and as the same is true for ${\displaystyle \left(G_{ab}\right)}$ we must prove the equation

${\displaystyle M_{ab}=G_{ab}}$

only for one special choice of coordinates.

§ 37. Now this choice can be made in such a way that at the point ${\displaystyle P}$ of the field-figure ${\displaystyle g_{11}=g_{22}=g_{33}=-1}$, ${\displaystyle g_{44}=+1}$, ${\displaystyle g_{ab}=0}$ for ${\displaystyle a\neq b}$ and that moreover all first derivatives ${\displaystyle g_{ab,e}}$ vanish. If then the values ${\displaystyle g_{ab}}$ at a point ${\displaystyle Q}$ near ${\displaystyle P}$ are developed in series of ascending powers of the differences of coordinates ${\displaystyle x_{a}(Q)-x_{a}(P)}$ the terms directly following the constant ones will be of the second order. It is with these terms that we are concerned in the calculation both of ${\displaystyle M_{ab}}$ and of ${\displaystyle G_{ab}}$ for the point ${\displaystyle P}$. As in the results the coefficients of these terms occur to the first power only, it is sufficient to show that each of the above mentioned terms separately contributes the same value to ${\displaystyle M_{ab}}$ and to ${\displaystyle G_{ab}}$.

From these considerations we may conclude that

 ${\displaystyle \delta _{1}Q=\sum (ab)G_{ab}\delta {\mathfrak {g}}^{ab}}$ (48)

Expressions containing instead of ${\displaystyle \delta {\mathfrak {g}}^{ab}}$ either the variations ${\displaystyle \delta g^{ab}}$ or ${\displaystyle \delta g_{ab}}$ might be derived from this by using the relations between the different variations. Of these we shall only mention the formula

 ${\displaystyle \delta g^{ab}={\frac {1}{\sqrt {-g}}}\delta {\mathfrak {g}}^{ab}-{\frac {g^{ab}}{2{\sqrt {-g}}}}\sum (cd)g_{cd}\delta {\mathfrak {g}}^{cd}}$ (49)

§ 38. In connexion with what precedes we here insert a consideration the purpose of which will be evident later on. Let the infinitely small quantity ${\displaystyle \xi }$ be an arbitrarily chosen continuous function of the coordinates and let the variations ${\displaystyle \delta g_{ab}}$ be defined by the condition that at some point ${\displaystyle P}$ the quantities ${\displaystyle g_{ab}}$ have after the change the values which existed before the change at the point ${\displaystyle Q}$, to which ${\displaystyle P}$ is shifted when ${\displaystyle x_{h}}$ is diminished by ${\displaystyle \xi }$, while the three other coordinates are left constant. Then we have

${\displaystyle \delta g_{ab}=-g_{ab,h}\xi }$

and similar formulae for the variations ${\displaystyle \delta {\mathfrak {g}}^{ab}}$.