Page:LorentzGravitation1916.djvu/23

§ 26. We have now to calculate the left hand side of equation (10) for the case that $\sigma$ is the surface of an element $\left(dx_{1},\dots dx_{4}\right)$. For this purpose we shall each time take together two opposite sides, calculating for each pair the contributions due to the different terms on the right hand side of (22), or as we may say to the different rotations $\chi _{ab}$. It is convenient now to denote by $a,b,c$ the numbers 1, 2, 3 either in this order or in any other derived from it by a cyclic permutation, while the $x$-components of the vector we are calculating and which stands on the left hand side of (10) will be represented by $X_{1},\dots X_{4}$.

a. Let us first consider that one of the sides $\left(dx_{a},dx_{b},dx_{c}\right)$ which faces towards the side of the positive $x_{4}$. The vector $\mathrm {N}$ drawn outward has the direction $4^{*}$ and in $\xi$-units the magnitude ${\frac {1}{\lambda _{4}}}$. As the direction $c$ corresponds to $a^{*},b^{*},4^{*}$, the rotation $\chi _{ab}$ gives with $\mathrm {N}$ a vector product represented by a vector in the direction $c$. The magnitude of this vector is in $\xi$-units

${\frac {1}{\lambda _{4}}}\chi _{ab}$

and in natural units

${\frac {\chi _{ab4}}{\lambda _{4}}}\chi _{ab}$

This must be multiplied by $l_{abc}dx_{a}dx_{b}dx_{c}$, the magnitude of the side under consideration in natural units, and finally by ${\tfrac {1}{l_{c}}}$ to express the vector product in $x$-units. Because of (24) we may write for the result

$\chi _{abc}dx_{a}dx_{b}dx_{c}=\psi _{c4}dx_{a}dx_{b}dx_{c}$

The opposite side gives a similar result with the opposite sign ($\mathrm {N}$ having for that side the direction $-4^{*}$), so that together the sides contribute the term

${\frac {\partial \psi _{c4}}{\partial x_{4}}}dW$

to the component $X_{c}$. For shortness sake we have put here

$dx_{1}dx_{2}dx_{3}dx_{4}=dW$

Finally we may take, $c=1,2,3$.

b. Secondly we consider a side $\left(dx_{a},dx_{b},dx_{4}\right)$ facing towards the positive $x_{c}$. The vector $\mathrm {N}$ has now the direction $-c^{*}$. We consider the vector products of this vector with the rotations $\chi _{b4}$, $\chi _{4a}$ and $\chi _{ba}$, which vector products have the directions $a,b$ and 4. A calculation exactly similar to the one we performed just now gives the contributions to $X_{a},X_{b},X_{4}$. For these we thus find the products of $dx_{a}dx_{b}dx_{4}$ by