Page:LorentzGravitation1916.djvu/22

In the following calculations the vector $\mathrm {N}$ has one of the directions $1^{*},\dots 4^{*}$. 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.