Page:Gruner1921.djvu/1

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

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

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.