Page:Grundgleichungen (Minkowski).djvu/36

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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|