# Page:Grundgleichungen (Minkowski).djvu/36

This page has been proofread, but needs to be validated.

This formula contains four equations, of which the fourth follows from the first three, since this is a space-time vector which is perpendicular to w.

Lastly, we shall transform the differential equations (A) and (B) into a typical form.

### § 12. The Differential Operator Lor.

A 4✕4 series matrix

 (62) $S=\begin{array}{cccc} S_{11}, & S_{12}, & S_{13}, & S_{1}\\ S_{21}, & S_{22}, & S_{23}, & S_{24}\\ S_{31}, & S_{32}, & S_{33}, & S_{34}\\ S_{41}, & S_{42}, & S_{43}, & S_{44}\end{array}=\left|S_{hk}\right|$

with the condition that in case of a Lorentz transformation it is to be replaced by $\mathsf{\bar{A}}S\mathsf{A}$, may be called a space-time matrix of the II. kind. We have examples of this in : —

1) the alternating matrix f, which corresponds to the space-time vector of the II. kind, —
2) the product fF of two such matrices, for by a transformation $\mathsf{A}$, it is replaced by $(\mathsf{A}^{-1}f\mathsf{A})(\mathsf{A}^{-1}F\mathsf{A})=\mathsf{A}^{-1}fF\mathsf{A}$,
3) further when $w_{1},\ w_{2},\ w_{3},\ w_{4}$ and $\Omega_{1},\ \Omega_{2},\ \Omega_{3},\ \Omega_{4}$ are two space-time vectors of the 1st kind, the 4✕4 matrix with the $S_{hk}=w_{h}\Omega_{k}$,
lastly in a multiple L of the unit matrix of 4✕4 series in which all the elements in the principal diagonal are equal to L, and the rest are zero.

We shall have to do constantly with functions of the space-time point x, y, z, it, and we may with advantage employ the 1✕4 series matrix, formed of differential symbols, —

$\left|\frac{\partial}{\partial x},\ \frac{\partial}{\partial y},\ \frac{\partial}{\partial z},\ \frac{\partial}{i\partial t}\right|$,

or

 (63) $\left|\frac{\partial}{\partial x_{1}},\ \frac{\partial}{\partial x_{2}},\ \frac{\partial}{\partial x_{3}},\ \frac{\partial}{x_{4}}\right|$