Page:LorentzGravitation1915.djvu/3

753

$\Sigma (a)v_{a}K_{a}=0$

(4)

§ 3. In Einstein'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

$g_{ba}=g_{ab}$

(5)

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

$dx_{a}=\Sigma (b)p_{ab}dx'_{b}$

(6)

we find

$g'_{ab}=\Sigma (cd)p_{ca}p_{db}g_{cd}$

(7)

Instead of (6) we shall also write

$dx'_{a}=\Sigma (b)\pi _{ba}dx_{b}$

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 $g_{ab}$^[1].

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 Einstein, deduce the motion of a material point in a gravitation field from the principle expressed by (3) if we take for the Lagrangian function

$L=-m{\frac {ds}{dt}}=-m{\sqrt {\Sigma (ab)g_{ab}v_{a}v_{b}}}$

(8)

—————

1. Suppose

   $x_{a}=\Sigma (b)\nu _{ba}\xi _{b}$

   to follow from the equations

   $\xi _{a}=\Sigma (b)n_{ab}x_{b}$

   then the set $\nu _{ab}$ is the inverse of the set $n_{ab}$.