Page:Grundgleichungen (Minkowski).djvu/38



If $$s=\left|s_{1},\ s_{2},\ s_{3},\ s_{4}\right|$$ is a space-time vector of the 1st kind, then

In case of a transformation $$\mathsf{A}$$, we have

i.e., lor s is an invariant in a {sc|Lorentz}}-transformation.

In all these operations the operator lor plays the part of a space-time vector of the first kind.

If f represents a space-time vector of the second kind, -lor f denotes a space-time vector of the first kind with the components

$$\begin{array}{ccccccc} & & \frac{\partial f_{12}}{\partial x_{2}} & + & \frac{\partial f_{13}}{\partial x_{3}} & + & \frac{\partial f_{14}}{\partial x_{4}},\\ \\\frac{\partial f_{21}}{\partial x_{1}} & &  & + & \frac{\partial_{23}}{\partial x_{3}} & + & \frac{\partial_{24}}{\partial x_{4}},\\ \\\frac{\partial f_{31}}{\partial x_{1}} & + & \frac{\partial_{32}}{\partial x_{2}} & &  & + & \frac{\partial_{34}}{\partial x_{4}},\\ \\\frac{\partial f_{41}}{\partial x_{1}} & + & \frac{\partial_{42}}{\partial x_{2}} & + & \frac{\partial_{43}}{\partial x_{3}},\end{array}$$

So the system o£ differential equations (A) can be expressed in the concise form

and the system (B) can be expressed in the form

Referring back to the definition (67) for $$lor\ \bar{s}$$, we find that the combinations $$lor (\overline{lor\ f})$$ and $$lor (\overline{lor\ F^{*}})$$ vanish identically, when f and F* are alternating matrices. Accordingly it follows out of (A), that