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. Next, $$g^{kl}$$ being a contravariant tensor of the second order, we can deduce from (40) that $$\left(G_{im}\right)$$ is a covariant tensor of the same order. According to (41) $$G$$ is then a scalar. The same is true for $$Q dS$$.

We remark that $$g_{ba}=g_{ab}$$ 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