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

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

with the condition that in case of a Lorentz transformation it is to be replaced by ${\displaystyle {\mathsf {\bar {A}}}S{\mathsf {A}}}$, 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 ${\displaystyle {\mathsf {A}}}$, it is replaced by ${\displaystyle ({\mathsf {A}}^{-1}f{\mathsf {A}})({\mathsf {A}}^{-1}F{\mathsf {A}})={\mathsf {A}}^{-1}fF{\mathsf {A}}}$,
3) further when ${\displaystyle w_{1},\ w_{2},\ w_{3},\ w_{4}}$ and ${\displaystyle \Omega _{1},\ \Omega _{2},\ \Omega _{3},\ \Omega _{4}}$ are two space-time vectors of the 1st kind, the 4✕4 matrix with the ${\displaystyle S_{hk}=w_{h}\Omega _{k}}$,
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, —

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

or

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