Page:LorentzGravitation1916.djvu/22

In the following calculations the vector $\mathrm{N}$ has one of the directions $1^{*},\dots4^{*}$. As this is also the case with the vectors $\mathrm{B}_{a^{*}}$ and $\mathrm{B}_{b^{*}}$, the vector product occurring in (22) can easily be expressed in $\xi$-units. After that we may pass to natural units and finally, as is necessary for the substitution in (10), to $x$-units.

In order to pass from $\xi$-units to natural units we have to multiply a vector in the direction $a^{*}$ by a certain coefficient $\lambda_{a}$, and a part of the extension $a^{*},b^{*},c^{*}$ by a coefficient $\lambda_{abc}$. These coefficients correspond to $l_{a}$ (§ 10) and $l_{abc}$ (§ 12). The factors $\lambda_{abc}$ e.g. can be expressed by means of the minors $\Gamma_{ab}$ of the determinant $\gamma$ of the quantities $\gamma_{ab}$. If this is worked out and if the equations

$\gamma_{ab}=\frac{G_{ab}}{g},\ g_{ab}=\frac{\Gamma_{ab}}{\gamma},\ g\gamma=1$

are taken into consideration, we obtain the following corollary, which we shall soon use:

Let $a, b, c, d$ and also $a', b', c', d'$ be the numbers 1, 2, 3, 4 in any order, $a'$ being not the same as $a$, then we have, if none of the two numbers $\alpha$ and $\alpha'$ is 4,

 $\frac{l_{bcd}\lambda b'c'd'}{l_{a'}\lambda_{a}}=-1$ (23)

and if one of the two is 4

 $\frac{l_{bcd}\lambda b'c'd'}{l_{a'}\lambda_{a}}=1$ (24)

§ 25. We shall now suppose (comp. § 24) that in $\xi$-units the vector $\mathrm{B}_{a^{*}}$ has the value +1, and we shall write $\chi_{ab}$ for the value that must then be given to $\mathrm{B}_{b^{*}}$. If the $\xi$-components of the vectors $\mathrm{A^{I}}$ etc. are denoted by $\Xi_{1}^{I},\dots\Xi_{4}^{I}$ etc., we find from (21)

 $\chi_{ab}=\left(\Xi_{a}^{I}\Xi_{b}^{II}-\Xi_{a}^{II}\Xi_{b}^{I}\right)+\left(\Xi_{a}^{III}\Xi_{b}^{IV}-\Xi_{a}^{IV}\Xi_{b}^{III}\right)$ (25)

This formula involves that

 $\chi_{ba}=-\chi_{ab}$ (26)

It may be remarked that $\chi_{ba}$ is the value that must be given to the vector $\mathrm{B}_{a^{*}}$ if $\mathrm{B}_{b^{*}}$ is taken to be 1.

The quantities $\chi_{ab}$ may be said to represent the rotations $\left[\mathrm{B}_{a^{*}}\cdot\mathrm{B}_{b^{*}}\right]$.

At the end of our calculations we shall introduce instead of $\chi_{ab}$ the quantities t$\psi_{ab}$ defined by

 $\psi_{ab}=\chi_{a'b'}(a\mp b),\ \psi_{aa}=0$ (27)

In the first of these equations $a, b, a', b'$ are supposed to be the numbers 1, 2, 3, 4, in an order obtained from 1, 2, 3, 4 by an even number of permutations.