Page:LorentzGravitation1915.djvu/8

758

the quantities $\psi _{ab}$^[1] on the left hand side being subject to the conditions

$\psi _{aa}=0,\quad \psi _{ba}=-\psi _{ab}$

(26)

so that they represent 6 mutually independent numerical values. These are the components of the electric force $\mathrm {\textsf {E}}$ and the magnetic force $\mathrm {\textsf {H}}$. We have indeed

$\left.{\begin{array}{ccccc}\psi _{41}=\mathrm {\textsf {E}} _{x},&&\psi _{42}=\mathrm {\textsf {E}} _{y},&&\psi _{43}=\mathrm {\textsf {E}} _{z},\\\psi _{23}=\mathrm {\textsf {H}} _{x},&&\psi _{31}=\mathrm {\textsf {H}} _{y},&&\psi _{12}=\mathrm {\textsf {H}} _{z},\end{array}}\right\}$

(27)

and it is thus seen that the first three of the formulae (25) express the connection between the magnetic field and the electric current. The fourth shows how the electric field is connected with the charge.

On passing to another system of coordinates we have for $w_{a}$ the transformation formula

$w'_{a}=|p|\Sigma (b)\pi _{ba}w_{b},$

which can easily be deduced, while for $\psi _{ab}$ we shall assume the formula

$\psi '_{ab}=|p|\Sigma (cd)\pi _{ca}\pi _{db}\psi _{cd}$

(28)

In virtue of this assumption the equations (25) are covariant for any change of coordinates.

§ 8. Beside $\psi _{ab}$ we shall introduce certain other quantities ${\bar {\psi }}_{ab}$ which we define by

${\bar {\psi }}_{ab}={\frac {1}{\sqrt {-g}}}\Sigma (cd)g_{ca}g_{db}\psi _{cd}$

(29)

or with regard to (26)

${\bar {\psi }}_{ab}={\frac {1}{\sqrt {-g}}}\Sigma ({\overline {cd}})\left(g_{ca}g_{db}-g_{da}g_{cb}\right)\psi _{cd},$

(30)

in which last equation the bar over $cd$ means that in the sum each combination of two numbers occurs only once.

As a consequence of this definition we have

${\bar {\psi }}_{aa}=0,\ {\bar {\psi }}_{ba}=-{\bar {\psi }}_{ab}$

(31)

and we find by inversion^[2]

$\psi _{ab}={\sqrt {-g}}\ \Sigma (cd)\gamma _{ac}\gamma _{bd}{\bar {\psi }}_{cd}$

(32)

1. The quantities $\psi _{ab}$ are connected with the components $\varphi _{ab}$ of the tensor introduced by Einstein by the equations $\psi _{ab}={\sqrt {-g}}\cdot \varphi _{ab}$.
2. By the definition of the quantities $\gamma$ (§ 3) we have

   $\Sigma (a)g_{ab}\gamma _{ab}=1$

   (α)

   and for $b\neq c$

   $\Sigma (a)g_{ab}\gamma _{ac}=0,\ \mathrm {or} \ \Sigma (a)g_{ba}\gamma _{ca}=0$

   (β)

   Substituting for ${\bar {\psi }}_{cd}$ an expression similar to (29) with other letters as indices,