History of Topics in Special Relativity/Lorentz transformation (trigonometric)

Lorentz transformation via trigonometric functions
The following general relation connects the speed of light and the relative velocity to hyperbolic and trigonometric functions, where $$\eta$$ is the rapidity in E:(3b), $$\theta$$ is equivalent to the Gudermannian function $${\rm gd}(\eta)=2\arctan(e^{\eta})-\pi/2$$, and $$\vartheta$$ is equivalent to the Lobachevskian angle of parallelism $$\Pi(\eta)=2\arctan(e^{-\eta})$$:


 * $$\frac{v}{c}=\tanh\eta=\sin\theta=\cos\vartheta$$

This relation was first defined by Varićak (1910).

a) Using $$\sin\theta=\tfrac{v}{c}$$ one obtains the relations $$\sec\theta=\gamma$$ and $$\tan\theta=\beta\gamma$$, and the Lorentz boost takes the form:

This Lorentz transformation was derived by Bianchi (1886) and Darboux (1891/94) while transforming pseudospherical surfaces, and by Scheffers (1899) as a special case of contact transformation in the plane (Laguerre geometry). In special relativity, it was first used by Plummer (1910), by Gruner (1921) while developing Loedel diagrams, and by Vladimir Karapetoff in the 1920s.

b) Using $$\cos\vartheta=\tfrac{v}{c}$$ one obtains the relations $$\csc\vartheta=\gamma$$ and $$\cot\vartheta=\beta\gamma$$, and the Lorentz boost takes the form:

This Lorentz transformation was derived by Eisenhart (1905) while transforming pseudospherical surfaces. In special relativity it was first used by Gruner (1921) while developing Loedel diagrams.

Bianchi (1886) – Pseudospherical surfaces
Luigi Bianchi (1886) investigated E:Lie's transformation (1880) of pseudospherical surfaces, obtaining the result:


 * $$\begin{align}(1)\quad & u+v=2\alpha,\ u-v=2\beta;\\

(2)\quad & \Omega\left(\alpha,\beta\right)\Rightarrow\Omega\left(k\alpha,\ \frac{\beta}{k}\right);\\ (3)\quad & \theta(u,v)\Rightarrow\theta\left(\frac{u+v\sin\sigma}{\cos\sigma},\ \frac{u\sin\sigma+v}{\cos\sigma}\right)=\Theta_{\sigma}(u,v);\\ & \text{Inverse:}\left(\frac{u-v\sin\sigma}{\cos\sigma},\ \frac{-u\sin\sigma+v}{\cos\sigma}\right)\\ (4)\quad & \frac{1}{2}\left(k+\frac{1}{k}\right)=\frac{1}{\cos\sigma},\ \frac{1}{2}\left(k-\frac{1}{k}\right)=\frac{\sin\sigma}{\cos\sigma} \end{align} $$.

Darboux (1891/94) – Pseudospherical surfaces
Similar to Bianchi (1886), Gaston Darboux (1891/94) showed that the E:Lie's transformation (1880) gives rise to the following relations:


 * $$\begin{align}(1)\quad & u+v=2\alpha,\ u-v=2\beta;\\

(2)\quad & \omega=\varphi\left(\alpha,\beta\right)\Rightarrow\omega=\varphi\left(\alpha m,\ \frac{\beta}{m}\right)\\ (3)\quad & \omega=\psi(u,v)\Rightarrow\omega=\psi\left(\frac{u+v\sin h}{\cos h},\ \frac{v+u\sin h}{\cos h}\right) \end{align} $$.

Scheffers (1899) – Contact transformation
Georg Scheffers (1899) synthetically determined all finite contact transformations preserving circles in the plane, consisting of dilatations, inversions, and the following one preserving circles and lines (compare with Laguerre inversion by E:Laguerre (1882) and Darboux (1887)):


 * $$\begin{matrix}\sigma^{\prime2}-\rho^{\prime2}=\sigma^{2}-\rho^{2}\\

\hline \rho'=\frac{\rho}{\cos\omega}+\sigma\tan\omega,\quad\sigma'=\rho\tan\omega+\frac{\sigma}{\cos\omega} \end{matrix}$$

Eisenhart (1905) – Pseudospherical surfaces
Luther Pfahler Eisenhart (1905) followed Bianchi (1886, 1894) and Darboux (1891/94) by writing the E:Lie's transformation (1880) of pseudospherical surfaces:


 * $$\begin{align}(1)\quad & \alpha=\frac{u+v}{2},\ \beta=\frac{u-v}{2}\\

(2)\quad & \omega\left(\alpha,\beta\right)\Rightarrow\omega\left(m\alpha,\ \frac{\beta}{m}\right)\\ (3)\quad & \omega(u,v)\Rightarrow\omega(\alpha+\beta,\ \alpha-\beta)\Rightarrow\omega\left(\alpha m+\frac{\beta}{m},\ \alpha m-\frac{\beta}{m}\right)\\ & \Rightarrow\omega\left[\frac{\left(m^{2}+1\right)u+\left(m^{2}-1\right)v}{2m},\ \frac{\left(m^{2}-1\right)u+\left(m^{2}+1\right)v}{2m}\right]\\ (4)\quad & m=\frac{1-\cos\sigma}{\sin\sigma}\Rightarrow\omega\left(\frac{u-v\cos\sigma}{\sin\sigma},\ \frac{v-u\cos\sigma}{\sin\sigma}\right) \end{align}$$.

Varićak (1910) – Circular and Hyperbolic functions
Relativistic velocity in terms of trigonometric functions and its relation to hyperbolic functions was demonstrated by Vladimir Varićak in several papers starting from 1910, who represented the equations of special relativity on the basis of hyperbolic geometry in terms of Weierstrass coordinates. For instance, he showed the relation of rapidity to the Gudermannian function and the angle of parallelism:


 * $$\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)$$

Plummer (1910) – Trigonometric Lorentz boosts
Henry Crozier Keating Plummer (1910) defined the following relations


 * $$\begin{matrix}\tau=t\sec\beta-x\tan\beta/U\\

\xi=x\sec\beta-Ut\tan\beta\\ \eta=y,\ \zeta=z,\\ \hline \sin\beta=v/U \end{matrix}$$

Gruner (1921) – Trigonometric Lorentz boosts
In order to simplify the graphical representation of Minkowski space, Paul Gruner (1921) (with the aid of Josef Sauter) developed what is now called Loedel diagrams, using the following relations:


 * $$\begin{matrix}v=\alpha\cdot c;\quad\beta=\frac{1}{\sqrt{1-\alpha^{2}}}\\

\sin\varphi=\alpha;\quad\beta=\frac{1}{\cos\varphi};\quad\alpha\beta=\tan\varphi\\ \hline x'=\frac{x}{\cos\varphi}-t\cdot\tan\varphi,\quad t'=\frac{t}{\cos\varphi}-x\cdot\tan\varphi \end{matrix}$$

In another paper Gruner used the alternative relations:


 * $$\begin{matrix}\alpha=\frac{v}{c};\ \beta=\frac{1}{\sqrt{1-\alpha^{2}}};\\

\cos\theta=\alpha=\frac{v}{c};\ \sin\theta=\frac{1}{\beta};\ \cot\theta=\alpha\cdot\beta\\ \hline x'=\frac{x}{\sin\theta}-t\cdot\cot\theta,\quad t'=\frac{t}{\sin\theta}-x\cdot\cot\theta \end{matrix}$$