Page:Grundgleichungen (Minkowski).djvu/46

components as the first three components of the space-time vector

(98)	$K+(w{\overline {K}})w$

This vector is perpendicular to w; the law of Energy finds its expression in the fourth relation.

The establishment of this opinion is reserved for a separate tract.

In the limitting case $\epsilon =1,\ \mu =1,\ \sigma =0$, the vector $N=0,\ {\mathfrak {s}}=\varrho {\mathfrak {w}}$, and we obtain the ordinary equations in the theory of electrons.

APPENDIX. Mechanics and the Relativity-Postulate.

It would be very unsatisfactory if the new way of looking at the time-concept, which permits a Lorentz transformation, were to be confined to a single part of Physics.

Now many authors say that classical mechanics stand in opposition to the relativity postulate, which is taken to be the basic of the new Electro-dynamics.

In order to decide this let us fix our attention upon a special Lorentz transformation represented by (10), (11), (12), with a vector ${\mathfrak {r}}$ in any direction and of any magnitude q < 1, but different from zero. For a moment we shall not suppose any special relation to hold between the unit of length and the unit of time, so that instead of t, t',q we shall write ct, ct', and ${\frac {q}{c}}$, where c represents a certain positive constant, and $q<c$. The above mentioned equations are transformed into

${\mathfrak {r'_{\bar {v}}}}={\mathfrak {r_{\bar {v}}}},\ {\mathfrak {r'_{v}}}={\frac {c({\mathfrak {r_{v}}}-qt)}{\sqrt {c^{2}-q^{2}}}},\ t'={\frac {-q{\mathfrak {r_{v}}}+c^{2}t}{c{\sqrt {c^{2}-q^{2}}}}}$;

They denote, as we remember, that ${\mathfrak {r}}$ is the space-vector x, y, z and ${\mathfrak {r}}'$ the space-vector x', y', z'.

If in these equations, keeping ${\mathfrak {v}}$ constant, we approach the limit $c=\infty$, then we obtain from these

${\mathfrak {r'_{\bar {v}}}}={\mathfrak {r_{\bar {v}}}},\ {\mathfrak {r'_{v}}}={\mathfrak {r_{v}}}-qt,\ t'=t$.