Page:LorentzGravitation1915.djvu/3



§ 3. In 's theory the gravitation field is determined by certain characteristic quantities $$g_{ab}$$, functions of $$x_{1},x_{2},x_{3},x_{4}$$, among which there are 10 different ones, as

A point of fundamental importance is the connection between these quantities and the corresponding coefficients $$g'_{ab}$$, with which we are concerned, when by an arbitrary substitution $$x_{1},\dots x_{4}$$ are changed for other coordinates $$x'_{1},\dots x'_{4}$$. This connection is defined by the condition that

$ds^{2}=g_{11}dx_{1}^{2}+\dots+g_{44}dx_{4}^{2}+2g_{12}dx_{1}dx_{2}+\dots$

or shorter

$ds^{2}=\Sigma(ab)g_{ab}dx_{a}dx_{b}$

be an invariant.

Putting

we find

Instead of (6) we shall also write

so that the set of quantities $$\pi_{ba}$$ may be called the inverse of the set $$p_{ab}$$. Similarly, we introduce a set of quantities $$\gamma_{ba}$$, the inverse of the set $$g_{ab}$$.

We remark here that in virtue of (5) and (7) $$g'_{ba}=g'_{ab}$$ and that likewise $$\gamma_{ba}=\gamma_{ab}$$.

Our formulae will also contain the determinant of the quantities $$g_{ab}$$, which we shall denote by $$g$$, and the determinant $$p$$ of the coefficients $$p_{ab}$$ (absolute value: $$|p|$$). The determinant $$g$$ is always negative.

We may now, as has been shown by, deduce the motion of a material point in a gravitation field from the principle expressed by (3) if we take for the Lagrangian function