Page:LorentzGravitation1916.djvu/25

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§ 28. Between the differentials of the original coordinates x_{a} and the new coordinates x'_{a} which we are going to introduce we have the relations

dx'_{a}=\sum(b)\pi_{ba}dx_{b} (30)

and formulae of the same form (comp. § 10) may be written down for the components of a vector expressed in x-measure. As the quantities \mathrm{q}_{a} constitute a vector and as

\sqrt{-g'}=p\sqrt{-g}

we have according to (28)[1]

\frac{1}{\sqrt{-g'}}w'_{a}=\frac{1}{\sqrt{-g}}\sum(b)\pi_{ba}w_{b}

or

w'_{a}=p\sum(b)\pi_{ba}w_{b}

Further we have for the infinitely small quantities \xi_{a}[2] defined by (19)

\xi'_{a}=\sum(b)p_{ba}\xi_{b}

and in agreement with this for the components of a vector expressed in \xi-units

\Xi'_{a}=\sum(b)p_{ba}\Xi_{b}

so that we find from (25)[3]

\chi'_{ab}=\sum(cd)p_{ca}p_{db}\chi_{cd}

Interchanging here c and d, we obtain

\chi'_{ab}=\sum(cd)p_{da}p_{cb}\chi_{dc}=-\sum(cd)p_{da}p_{cb}\chi_{cd}

and

\chi'_{ab}=\frac{1}{2}\sum(cd)\left(p_{ca}p_{db}-p_{da}p_{cb}\right)\chi_{cd} (31)

The quantity between brackets on the right hand side is a second order minor of the determinant p and as is well known this minor

  1. Comp. § 7, l. c.
  2. For the infinitesimal quantities x_a occurring in (19) we have namely (comp. (30))

    x'_{a}=\sum(b)\pi_{ba}x_{b}

    and taking into consideration (19) and (20), i e.

    \xi_{a}=\sum(b)g_{ab}x_{b},\ x_{a}=\sum(b)\gamma_{ba}\xi_{b}

    and formula (7) l. c, we may write (comp. note 2, p. 758, l. c.)

    \begin{array}{l}
\xi'_{a}=\sum(b)g'_{ab}x'_{b}=\sum(bcde)p_{ca}p_{db}\pi_{eb}g_{cd}x_{e}=\\
\\
\qquad=\sum(cd)p_{ca}g_{cd}x_{d}=\sum(cdf)p_{ca}g_{cd}\gamma_{fd}\xi_{f}=\sum(c)p_{ca}\xi_{c}
\end{array}

  3. Put \Xi_{a}^{I}\Xi_{b}^{II}=\vartheta_{ab}. Then we have

    \vartheta'_{ab}=\Xi_{a}^{'I}\Xi_{b}^{'II}=\sum(cd)p_{ca}p_{db}\Xi_{c}^{I}\Xi_{d}^{II}=\sum(cd)p_{ca}p_{db}\vartheta_{cd}

    and similar formulae for the other three parts of (25).