Page:Grundgleichungen (Minkowski).djvu/36

 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

with the condition that in case of a 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