Page:Grundgleichungen (Minkowski).djvu/35

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(58) |\Phi \Psi] = i[w, \Omega]^{*},

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

(59) (w\bar{\Omega})=w_{1}\Omega_{1}+w_{2}\Omega_{2}+w_{3}\Omega_{3}+w_{4}\Omega_{4}=0,

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

(60) \Omega_{1} = \Phi_{2} \Psi_{3} - \Phi_{3} \Psi_{2},\ \Omega_{2} = \Phi_{3} \Psi_{1} - \Phi_{1} \Psi_{3},\ \Omega_{3} = \Phi_{1} \Psi_{2} - \Phi_{2} \Psi_{1},

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

(61) s+(w\bar{s})w

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

(E) s+(w\bar{s})w=-\sigma wF.