Page:Grundgleichungen (Minkowski).djvu/38

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(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,