Page:LorentzGravitation1916.djvu/22

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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.