# Page:Grundgleichungen (Minkowski).djvu/38

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

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

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

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

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

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

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

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

 {A} ${\displaystyle lor\ f=-s}$

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

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

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

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