Page:LorentzGravitation1916.djvu/12

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§ 12. In what precedes we were concerned with the volumes of parallelepipeds expressed in natural units. When we have introduced coordinates x_{1},\dots x_{4} we may also express these volumes in the "x-units" corresponding to the coordinates chosen.

Let us consider e.g. the three-dimensional extension x_{4}=const. , which cuts the conjugate indicatrix in the ellipsoid

g_{11}x_{1}^{2}+g_{22}x_{2}^{2}+g_{33}x_{3}^{2}+2g_{12}x_{1}x_{2}+2g_{23}x_{2}x_{3}+2g_{31}x_{3}x_{1}=-\epsilon^{2}

If we agree that in x-measure spaces in this extension will be represented by positive numbers and that a parallelepiped with the positive edges dx_{1},dx_{2},dx_{3} will have the volume dx_{1}\ dx_{2}\ dx_{3} , we find for that of the parallelepiped on three conjugate radius-vectors

\frac{\epsilon^{3}}{\sqrt{-G_{44}}}

where it has been taken into consideration that G_{44} is negative.

The volume of the same parallelepiped being expressed in natural measure by — -i\epsilon^{3} (§ 8), we have to multiply by

l_{123}=-i\sqrt{-G_{44}}\, (8)

if we want to pass from the expression in x-measure to that in natural measure.

For the extension \left(x_{2},x_{3},x_{4}\right) , i.e. x_{1}=0 the corresponding factor is

l_{234}=-\sqrt{G_{11}} (9)


§ 13. In the theory of electromagnetic phenomena we are concerned in the first place with the electric charge and the convection current. So far as these quantities belong to a definite element d\Omega of the field-figure they may be combined into

\mathrm{q}d\Omega

where \mathrm{q} is a vector which we may call the current vector. When it is resolved into four components having the directions of the axes, the first three components determine the convection current, while the fourth component gives the density of the electric charge.

As to the electric and the magnetic force, these two taken together can be represented at each point of the field-figure by two rotations

\mathrm{R}_{e} and \mathrm{R}_{h}

in definite, mutually conjugate two-dimensional extensions. These quantities are closely connected with the current vector, for after having introduced coordinates x_{1},\dots x_{4} we have for each closed surface \sigma the vector equation