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

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 , gives the following formulas:

where

The geometric representation given in a general manner by Minkowski, becomes particularly simple and elegant by choosing the axes of and for two mutually orthogonal systems.

From the attached figure, the axis is perpendicular to axis , and axis is rotated by an angle , such as

Posing , we immediately find that the coordinates of a point satisfy the requirements of the theory of relativity:

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. ).

Furthermore, the figure immediately gives the covariant and contravariant components of a vector ; it is easy to find geometrically the law of the invariance of the square of the vector:

**Original:**

This work is in the **public domain** in the **United States** because it was published before January 1, 1923.

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 50 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.

**Translation:**

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