# Page:Grundgleichungen (Minkowski).djvu/38

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

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

 (67) $lor\ \bar{s}=\frac{\partial s_{1}}{\partial x_{1}}+\frac{\partial s_{2}}{\partial x_{2}}+\frac{\partial s_{3}}{\partial x_{3}}+\frac{\partial s_{4}}{\partial x_{4}}$

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

$lor'\ \bar{s}'=(lor\ \mathsf{A})(\mathsf{\bar{A}}\bar{s})=lor\ \bar{s}$,

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

 {A} $lor\ f = -s$

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

 {B} $lor\ F^{*} = 0$

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

 (68) $\frac{\partial s_{1}}{\partial x_{1}}+\frac{\partial s_{2}}{\partial x_{2}}+\frac{\partial s_{3}}{\partial x_{3}}+\frac{\partial s_{4}}{\partial x_{4}}=0$,