Page:Grundgleichungen (Minkowski).djvu/55

summing the four, we obtain $\varkappa =K{\overline {w}}$, and therefore clearly $K+(K{\overline {w}})w$ will be a space-time vector of the 1st kind which is normal to w. Let us write out the components of this vector as

$X,\ Y,\ Z,\ iT,$

Then we arrive at the following equations for the motion of matter,

(21)	${\begin{array}{c}\nu {\frac {d}{d\tau }}{\frac {dx}{d\tau }}=X,\\\\\nu {\frac {d}{d\tau }}{\frac {dy}{d\tau }}=Y,\\\\\nu {\frac {d}{d\tau }}{\frac {dz}{d\tau }}=Z,\\\\\nu {\frac {d}{d\tau }}{\frac {dt}{d\tau }}=T.\end{array}}$

and we have also

$\left({\frac {dx}{d\tau }}\right)^{2}+\left({\frac {dy}{d\tau }}\right)^{2}+\left({\frac {dz}{d\tau }}\right)^{2}=\left({\frac {dt}{d\tau }}\right)^{2}-1$

and

$X{\frac {dx}{d\tau }}+Y{\frac {dy}{d\tau }}+Z{\frac {dz}{d\tau }}=T{\frac {dt}{d\tau }}$,

On the basis of this condition, the fourth of equations (21) is to be regarded as a direct consequence of the first three.

From (21), we can deduce the law for the motion of a material point, i.e, the law for the career of an infinitely thin space-time filament.

Let x, y, z, t denote a point on a principal line chosen in any manner within the filament. We shall form the equations (21) for the points of the normal cross section of the filament through x, y, z, t, and integrate them, multiplying by the elementary contents of the cross section over the whole space of the normal section. If the integrals of the right side be $R_{x},\ R_{y},\ R_{z},\ R_{t}$, and if m be the constant mass of the filament, we obtain

(22)	${\begin{array}{c}m{\frac {d}{d\tau }}{\frac {dx}{d\tau }}=R_{x},\\\\m{\frac {d}{d\tau }}{\frac {dy}{d\tau }}=R_{y},\\\\m{\frac {d}{d\tau }}{\frac {dz}{d\tau }}=R_{z},\\\\m{\frac {d}{d\tau }}{\frac {dt}{d\tau }}=R_{t},\end{array}}$