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 Lorentz transformation it is to be replaced by , 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 , it is replaced by ,
- 3) further when and are two space-time vectors of the 1st kind, the 4✕4 matrix with the ,
- 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, —