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

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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 [[../Lorentz transformation (hyperbolic)#math_3b|E:(3b)]], $\theta$ is equivalent to the w:Gudermannian function $\mathrm{g}\mathrm{d}(\eta )=2\arctan (e^{\eta })-\pi /2$, and $\vartheta$ is equivalent to the Lobachevskian w:angle of parallelism $\mathrm{\Pi }(\eta )=2\arctan (e^{-\eta })$:

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

This relation was first defined by Varićak (1910).

a) Using $\sin \theta =\frac{v}{c}$ 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 w:contact transformation in the plane (Laguerre geometry). In special relativity, it was first used by Plummer (1910), by Gruner (1921) while developing w:Loedel diagrams, and by w:Vladimir Karapetoff in the 1920s.

b) Using $\cos \vartheta =\frac{v}{c}$ 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 w:Loedel diagrams.

Historical notation

Template:Anchor Bianchi (1886) – Pseudospherical surfaces

w:Luigi Bianchi (1886) investigated [[../Lorentz_transformation_(squeeze)#Lie2|E:Lie's transformation (1880)]] of pseudospherical surfaces, obtaining the result:

$\begin{array}{cc}(1)& u+v=2\alpha ,u-v=2\beta ;\\ (2)& \mathrm{\Omega }(\alpha ,\beta )\Rightarrow \mathrm{\Omega }(k\alpha ,\frac{\beta }{k});\\ (3)& \theta (u,v)\Rightarrow \theta (\frac{u+v\sin \sigma }{\cos \sigma },\frac{u\sin \sigma +v}{\cos \sigma })={\mathrm{\Theta }}_{\sigma }(u,v);\\ & \text{Inverse:}(\frac{u-v\sin \sigma }{\cos \sigma },\frac{-u\sin \sigma +v}{\cos \sigma })\\ (4)& \frac{1}{2}(k+\frac{1}{k})=\frac{1}{\cos \sigma },\frac{1}{2}(k-\frac{1}{k})=\frac{\sin \sigma }{\cos \sigma }\end{array}$.

Template:Anchor Darboux (1891/94) – Pseudospherical surfaces

Similar to Bianchi (1886), w:Gaston Darboux (1891/94) showed that the [[../Lorentz_transformation_(squeeze)#Lie2|E:Lie's transformation (1880)]] gives rise to the following relations:

$\begin{array}{cc}(1)& u+v=2\alpha ,u-v=2\beta ;\\ (2)& \omega =\phi (\alpha ,\beta )\Rightarrow \omega =\phi (\alpha m,\frac{\beta }{m})\\ (3)& \omega =\psi (u,v)\Rightarrow \omega =\psi (\frac{u+v\sin h}{\cos h},\frac{v+u\sin h}{\cos h})\end{array}$.

Template:Anchor Scheffers (1899) – Contact transformation

w:Georg Scheffers (1899) synthetically determined all finite w:contact transformations preserving circles in the plane, consisting of dilatations, inversions, and the following one preserving circles and lines (compare with Laguerre inversion by [[../Lorentz transformation (conformal)#Laguerre|E:Laguerre (1882)]] and [[../Lorentz transformation (conformal)#Darboux2|Darboux (1887)]]):

$\begin{array}{c}{\sigma }^{\prime 2}-{\rho }^{\prime 2}={\sigma }^2-{\rho }^2\\ {\rho }^{\prime }=\frac{\rho }{\cos \omega }+\sigma \tan \omega ,{\sigma }^{\prime }=\rho \tan \omega +\frac{\sigma }{\cos \omega }\end{array}$

Template:Anchor Eisenhart (1905) – Pseudospherical surfaces

w:Luther Pfahler Eisenhart (1905) followed Bianchi (1886, 1894) and Darboux (1891/94) by writing the [[../Lorentz_transformation_(squeeze)#Lie2|E:Lie's transformation (1880)]] of pseudospherical surfaces:

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

Template:Anchor Varićak (1910) – Circular and Hyperbolic functions

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

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

Template:Anchor Plummer (1910) – Trigonometric Lorentz boosts

w:Henry Crozier Keating Plummer (1910) defined the following relations

$\begin{array}{c}\tau =t\sec \beta -x\tan \beta /U\\ \xi =x\sec \beta -Ut\tan \beta \\ \eta =y,\zeta =z,\\ \sin \beta =v/U\end{array}$

Template:Anchor Gruner (1921) – Trigonometric Lorentz boosts

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

$\begin{array}{c}v=\alpha \cdot c;\beta =\frac{1}{\sqrt{1-{\alpha }^2}}\\ \sin \phi =\alpha ;\beta =\frac{1}{\cos \phi };\alpha \beta =\tan \phi \\ x^{\prime }=\frac{x}{\cos \phi }-t\cdot \tan \phi ,t^{\prime }=\frac{t}{\cos \phi }-x\cdot \tan \phi \end{array}$

In another paper Gruner used the alternative relations:

$\begin{array}{c}\alpha =\frac{v}{c};\beta =\frac{1}{\sqrt{1-{\alpha }^2}};\\ \cos \theta =\alpha =\frac{v}{c};\sin \theta =\frac{1}{\beta };\cot \theta =\alpha \cdot \beta \\ x^{\prime }=\frac{x}{\sin \theta }-t\cdot \cot \theta ,t^{\prime }=\frac{t}{\sin \theta }-x\cdot \cot \theta \end{array}$

References

Historical mathematical sources

• {{#section:History of Topics in Special Relativity/mathsource|bia86lez}}
• {{#section:History of Topics in Special Relativity/mathsource|dar94cou}}
• {{#section:History of Topics in Special Relativity/mathsource|eis05}}
• {{#section:History of Topics in Special Relativity/mathsource|schef99}}

Historical relativity sources

Template:Reflist {{#section:History of Topics in Special Relativity/relsource|grun21a}} {{#section:History of Topics in Special Relativity/relsource|grun21b}} {{#section:History of Topics in Special Relativity/relsource|plum10}} {{#section:History of Topics in Special Relativity/relsource|var10}}

Secondary sources

Template:Reflist {{#section:History of Topics in Special Relativity/secsource|L9}}