Page:Grundgleichungen (Minkowski).djvu/34

From Wikisource
Jump to: navigation, search
This page has been proofread, but needs to be validated.


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

The vector \Psi is perpendicular to w; we can call it the Magnetic rest-force.

Relations analogous to these hold among the quantities iwF^{*},\mathfrak{M,E,w} and Relation (D) can be replaced by the formula

{D} wF^{*} = \mu wf^{*}

We can use the relations (C) and (D) to calculate F and f from \Phi and \Psi, we have

wF = -\Phi,\ wF^{*} = -i\mu\Psi,\ wf = -\epsilon\Phi,\ wf^{*} = -i\Psi

and applying the relation (45) and (46), we have

(55) F = [w,\Phi] + i\mu[w,\Psi]^{*},
(56) f = \epsilon[w,\Phi] + i[w,\Psi]^{*},

i.e.

F_{12} = (w_{1}\Phi_{2} - w_{2}\Phi_{1}) + i\mu(w_{3}\Psi_{4} - w_{4}\Psi_{3}), etc.

f_{12} = \epsilon(w_{1}\Phi_{2} - w_{2}\Phi_{1}) + i(w_{3}\Psi_{4} - w_{4}\Psi_{3}), etc.

Let us now consider the space-time vector of the second kind [\Phi \Psi], with the components

\Phi_{2}\Psi_{3}-\Phi_{3}\Psi_{2},\ \Phi_{3}\Psi_{1}-\Phi_{1}\Psi_{3},\ \Phi_{1}\Psi_{2}-\Phi_{2}\Psi_{1},

\Phi_{1}\Psi_{4}-\Phi_{4}\Psi_{1},\ \Phi_{2}\Psi_{4}-\Phi_{4}\Psi_{2},\ \Phi_{3}\Psi_{4}-\Phi_{4}\Psi_{3},

Then the corresponding space-time vector of the first kind

w[\Phi,\Psi]=-(w\overline{\Psi})\Phi+w(\overline{\Phi})\Psi

vanishes identically owing to equations 49) and 53).

Let us now take the vector of the 1st kind

(57) |\Omega = iw[\Phi,\ \Psi]^{*}

with the components

\Omega_{1}=-i\left|\begin{array}{ccc}
w_{2}, & w_{3}, & w_{4}\\
\Phi_{2}, & \Phi_{3}, & \Phi_{4}\\
\Psi_{2}, & \Psi_{3}, & \Psi_{4}\end{array}\right|, etc.

Then by applying rule (45), we have