# Page:LorentzGravitation1916.djvu/25

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

 ${\displaystyle dx'_{a}=\sum (b)\pi _{ba}dx_{b}}$ (30)

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

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

we have according to (28)[1]

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

or

${\displaystyle w'_{a}=p\sum (b)\pi _{ba}w_{b}}$

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

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

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

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

so that we find from (25)[3]

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

Interchanging here ${\displaystyle c}$ and ${\displaystyle d}$, we obtain

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

and

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

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

1. Comp. § 7, l. c.
2. For the infinitesimal quantities ${\displaystyle x_{a}}$ occurring in (19) we have namely (comp. (30))

${\displaystyle x'_{a}=\sum (b)\pi _{ba}x_{b}}$

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

${\displaystyle \xi _{a}=\sum (b)g_{ab}x_{b},\ x_{a}=\sum (b)\gamma _{ba}\xi _{b}}$

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

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

3. Put ${\displaystyle \Xi _{a}^{I}\Xi _{b}^{II}=\vartheta _{ab}}$. Then we have

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

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