Page:Gruner1921.djvu/2

Axis $OX$ is also put $\perp$ to $OT'$. Then indeed it is given with respect to the polar coordinates, in accordance with Fig. 1,

${\begin{aligned}x'&={\frac {x}{\sin \theta }}-t\cdot \cot \theta ,\\t'&={\frac {t}{\sin \theta }}-x\cdot \cot \theta ,\end{aligned}}$

i.e, the transformation formulas for $c=1$ given above.

With this easily constructed coordinate systems, length contraction and clock retardation can be seen without further ado.

Fig. 1

In the "primed" system $OT'X'$ (Fig. 2) the world-lines parallel to the time axis $OT'$ provide the "world history" of the point resting in this system. $A'B'$ always represents the length $l'$ of a rod resting in it.

Fig. 2

The observers resting in the "unprimed" system $OTX$ can only measure this length $l'$, by finding out the location $A$ and $B$ of the endpoints of $l'$ at equal clock-indication $t$ (thus upon a line parallel to $OX$); they find

$l=l'\cdot \sin \theta ={\frac {1}{\beta }}l'$

i.e. the known Lorentz contraction.

In the same way, the rate of one clock $C'$ (Fig. 2) of the primed system can only be evaluated from the other system, when two observers $C_{1}$ and $C_{2}$ of the latter system compare their clock indications with the readings of the clock $C'$ that travels past them. If the latter indicates the time interval

$\Delta t'=C'_{1}C'_{2}$

then the unprimed observer find the time interval

$\Delta t=DC_{2}$

which is determined by the two lines which are parallel to $OX$ and which go through $C_{1}$ and $C_{2}$. The image gives:

$\Delta t={\frac {\Delta t'}{\sin \theta }}=\beta \cdot \Delta t'$

i.e. Einstein's known retardation of the rate of a moving clock.

These example may suffice to show the clearness of this simple geometrical method. Though it should (also according to Dr. Sauter) be alluded to the circumstance, that by this choice of coordinate system, the somewhat abstract concept of covariant and contravariant components of a vector can be illustrated.

Namely (Fig. 1):

the parallel projections of vector $R:x,t;\ x',t'$ denote its contravariant components

the orthogonal projections: $\xi ,\tau ;\ \xi ',\tau '$ denote its covariant components

Because it can easily be seen, that

$x\cdot \xi =x'\cdot \xi ',\ t\cdot \tau =t'\cdot \tau '$,

thus the necessary invariance condition

$x\cdot \xi +t\cdot \tau =x'\cdot \xi '+t'\cdot \tau '$

persists. It is obvious that various illustrative consequences, also with respect to the fundamental tensor $g_{\iota \varkappa }$, can be drawn from that.

Bern, 19 May 1921.

(Received 21 May 1921.)