Page:LorentzGravitation1916.djvu/31

${\frac {1}{2\varkappa }}\int QdS$

where

$Q={\sqrt {-g}}G$

In the integral $dS$, the element of the field-figure, is expressed in $x$-units. The integration has to be extended over the domain within a certain closed surface $\sigma$; $\varkappa$ is a positive constant.

§ 33. When we pass from the system of coordinates $x_{1},\dots x_{4}$ to another, the value of $G$ proves to remain unaltered; it is a scalar quantity. This may be verified by first proving that the quantities $ik,lm$ form a covariant tensor of the fourth order^[1]. Next, $g^{kl}$ being a contravariant tensor of the second order^[2], we can deduce from (40) that $\left(G_{im}\right)$ is a covariant tensor of the same order^[3]. According to (41) $G$ is then a scalar. The same is true^[4] for $QdS$.

We remark that $g_{ba}=g_{ab}$^[5] and $g_{ab,fe}=g_{ab,ef}$. We shall suppose $Q$ to be written in such a way that its form is not altered by interchanging $g_{ba}$ and $g_{ab}$ or $g_{ab,fe}$ and $g_{ab,ef}$. If originally this condition is not fulfilled it is easy to pass to a "symmetrical" form of this kind.

It is clear that $Q$ may also be expressed in the quantities $g_{ab}$ and their first and second derivatives and in the same way in the ${\mathfrak {g}}_{ab}$ and first and second derivatives of these quantities.

If the necessary substitutions are executed with due care, these new forms of $Q$ will also be symmetrical.

§ 34. We shall first express the quantity $Q$ in the $g_{ab}$'s and their

1. Namely:

   $g'^{lk}=\sum (ab)\pi _{ak}\pi _{bl}g^{ab}$

   The symbol $\left(g^{kl}\right)$ denotes the complex of all the quantities $g^{kl}$.

2. Namely:

   $G'_{im}=\sum (ab)p_{ai}p_{bm}G_{ab}$

3. On account of the relation

   ${\sqrt {-g'}}dS'={\sqrt {-g}}dS$

4. Similarly:

   $g^{ba}=g^{ab},\ {\mathfrak {g}}^{ba}={\mathfrak {g}}^{ab}$

5. This means that the transformation formulae for these quantities have the form

   $(ik,lm)'=\sum (abce)p_{ai}p_{bk}p_{cl}p_{em}(ab,ce)$

   See for the notations used here and for some others to be used later on my communication in Zittingsverslag Akad Amsterdam 23 (1915), p. 1073 (translated in Proceedings Amsterdam 19 (1916), p. 751). In referring to the equations and the articles of this paper I shall add the indication 1915.