# Translation:Elementary geometric representation of the formulas of the special theory of relativity

Elementary geometric representation of the formulas of the special theory of relativity  (1921)
by Paul Gruner, translated from French by Wikisource
In French: Représentation géométrique élémentaire des formules de la théorie de la relativité, Archives des sciences physiques et naturelles (5) 3: 295–296, Scans

Gruner, P. and Sauter J. (Berne). – Elementary geometric representation of the formulas of the special theory of relativity.

The theory of special relativity, applied to two one-dimensional systems, moving relatively to each other with velocity ${\displaystyle v}$, gives the following formulas:

${\displaystyle x'=\beta (x-\alpha ct)\quad ct'=\beta (ct-\alpha x),}$

where

${\displaystyle v=\alpha \cdot c,\quad \beta ={\frac {1}{\sqrt {1-\alpha ^{2}}}}}$

The geometric representation given in a general manner by Minkowski, becomes particularly simple and elegant by choosing the axes of ${\displaystyle x}$ and ${\displaystyle t}$ for two mutually orthogonal systems.

From the attached figure, the ${\displaystyle OT}$ axis is perpendicular to axis ${\displaystyle OX'}$, and axis ${\displaystyle OT'}$ is rotated by an angle ${\displaystyle \varphi }$, such as

${\displaystyle \sin \varphi =\alpha ;\quad \beta ={\frac {1}{\cos \varphi }};\quad \alpha \beta =\tan \varphi .}$

Posing ${\displaystyle c=1}$, we immediately find that the coordinates [ 296 ] ${\displaystyle x,t,x',t'}$ of a point ${\displaystyle {\mathsf {P}}}$ satisfy the requirements of the theory of relativity:

${\displaystyle x'={\frac {x}{\cos \varphi }}-t\cdot \tan \varphi ;\quad t'={\frac {t}{\cos \varphi }}-x\cdot \tan \varphi .}$

With this mode of representation which contains no imaginary quantity, it is easy and simple to graphically demonstrate the different results of the theory of relativity (length contraction, dilatation of clocks, change in mass, energy, volume, etc. ).

Fig. 1

Furthermore, the figure immediately gives the covariant ${\displaystyle (\xi ,\tau ,\xi ',\tau ')}$ and contravariant ${\displaystyle (x,t,x',t')}$ components of a vector ${\displaystyle {\mathsf {R}}}$; it is easy to find geometrically the law of the invariance of the square of the vector:

${\displaystyle {\mathsf {R}}^{2}=x\xi +t\tau =x'\xi '+t'\tau '}$

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Original: This work is in the public domain in the United States because it was published before January 1, 1925. The author died in 1957, so this work is also in the public domain in countries and areas where the copyright term is the author's life plus 60 years or less. This work may also be in the public domain in countries and areas with longer native copyright terms that apply the rule of the shorter term to foreign works. This work is released under the Creative Commons Attribution-ShareAlike 3.0 Unported license, which allows free use, distribution, and creation of derivatives, so long as the license is unchanged and clearly noted, and the original author is attributed.