Translation:An elementary geometrical representation of the transformation formulas of the special theory of relativity

by Paul Gruner

An elementary geometrical representation of the transformation formulas of the special theory of relativity.

By P. Gruner.

The different geometrical representations of the Lorentz transformation either use rectangular coordinate systems with imaginary rotation angles (or imaginary time coordinates) or oblique coordinate systems with real magnitudes, though in which the units are different on every axis and have to be determined by Minkowski's unit hyperboloid. However, a geometric representation in which no imaginary quantities and no different units arise, is desirable for the introduction of beginners. Such a representation is easy to find, if one confines oneself to the comparison of two one-dimensional space-coordinate-systems only.

Two linear and straight reference systems $OX$ and $O'X'$ shall mutually move with constant velocity $v$ (one can think of two infinitely long, parallel trains, which travel past each other in absolutely dark night), let the clock indications $t$ and $t'$ in every system be regulated, so that points $O$ and $O'$ indicate the times $t=0$ and $t'=0$ at the moment of encounter, then for every encounter (coincidence) of any two points with abscissas $x$ and $x'$ and clock indications $t$ and $t'$, the following system of Lorentz transformation equations is given:

${\begin{aligned}x'&=\beta (x-\alpha \cdot c\cdot t)&&&x&=\beta (x'+\alpha \cdot c\cdot t')\\c\cdot t'&=\beta (c\cdot t-\alpha \cdot x)&&&c\cdot t&=\beta (c\cdot t'+\alpha \cdot x)\end{aligned}}$

in which

$\alpha ={\frac {v}{c}},\ \beta ={\frac {1}{\sqrt {1-\alpha ^{2}}}}$ (steadily $\beta >1$).

If one sets $c=1$ for simplification's sake^[1]‚ thus $\alpha =v$, then one can relate the events to two oblique spacetime coordinate systems $OXT$ and $O'X'T'$ with common origin; according to Minkowski they have to represent a pair of conjugate diameters of invariant unit hyperbola

$x^{2}-t^{2}=x^{\prime 2}-t^{\prime 2}=1$

It can be seen without further ado, that there are two such pairs, at which the diameters are all of same length, and which are symmetrically located with respect to the axes of the hyperbola, thus they are mutually orthogonal. They form particularly convenient spacetime coordinate systems for the geometric representation.

These systems can be (as I was informed in a friendly way by Dr. Sauter, engineer in Bern) directly derived.

Axis $OT$ is put $\perp$ to axis $OX'$, and axis $OT'$ with inclination $\theta$ to $OX'$, so that

$\cos \theta =\alpha ={\frac {v}{c}}$

thus

$\sin \theta ={\frac {1}{\beta }},\ \cot \theta =\alpha \cdot \beta$

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.)

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1. The relativity formulas are presented most conveniently, if one choses 1 kilometer as unit of length and 1/300000 second as unit of time. We propose the name light second for this unit of time.