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

. and. (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 $$v$$, gives the following formulas:

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

where

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

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

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

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

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

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

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. ).



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

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