Page:Grundgleichungen (Minkowski).djvu/35



i,e.

$$\Phi_{1}\Psi_{2} - \Phi_{2}\Psi_{1} = i(w_{3}\Omega_{4} - w_{4}\Omega_{3})$$, etc..

The vector $$\Omega$$ fulfills the relation

which we can write as

$$\Omega_{4}=i(\mathfrak{w}_{x}\Omega_{1}+\mathfrak{w}_{y}\Omega_{2}+\mathfrak{w}_{z}\Omega_{3})$$

and $$\Omega$$ is also normal to w. In case $$\mathfrak{w} =0$$, we have $$\Phi_{4} = 0,\ \Psi_{4} = 0,\ \Omega_{4} = 0$$, and

I shall call $$\Omega$$, which is a space-time vector 1st kind the Rest-Ray.

As for the relation E), which introduces the conductivity $$\sigma$$, we have

$$-w\bar{s}=-(w_{1}s_{1}+w_{2}s_{2}+w_{3}s_{3}+w_{4}s_{4})=\frac{-\left|\mathfrak{w}\right|s_{\mathfrak{w}}+\varrho}{\sqrt{1-\mathfrak{w}^{2}}}=\varrho'$$

This expression gives us the rest-density of electricity (see §8 and §4). Then

represents a space-time vector of the 1st kind, which since $$w\bar{w}=1$$, is normal to w, and which I may call the rest-current. Let us now conceive of the first three component of this vector as the x-, y-, z co-ordinates of the space-vector, then the component in the direction of $$\mathfrak{w}$$ is

$$\mathfrak{s_{w}}-\frac{\left|\mathfrak{w}\right|\varrho'}{\sqrt{1-\mathfrak{w}^{2}}}=\frac{\mathfrak{s_{w}}-\left|\mathfrak{w}\right|\varrho}{\sqrt{1-\mathfrak{w}^{2}}}=\frac{\mathfrak{F_{w}}}{1-\mathfrak{w}^{2}}$$

and the component in a perpendicular direction is $$\mathfrak{s_{\bar{w}}}=\mathfrak{F_{\bar{w}}}$$.

This space-vector is connected with the space-vector $$\mathfrak{F}=\mathfrak{s}-\varrho\mathfrak{w}$$, which we denoted in § 8 as the conduction-current.

Now by comparing with $$\Phi = -wF$$, the relation (E) can be brought into the form