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

History of Topics in Special Relativity: History of Lorentz transformation ()
Introduction Most general Lorentz transformations LT via imaginary orthogonal transformation LT via hyperbolic functions LT via velocity LT via sphere transformation LT via Cayley–Hermite transformation LT via Cayley–Klein parameters, Möbius and spin transformations LT via quaternions and hyperbolic numbers LT via trigonometric functions LT via squeeze mappings

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 w:Gudermannian function ${\rm {gd}}(\eta )=2\arctan(e^{\eta })-\pi /2$, and $\vartheta$ is equivalent to the Lobachevskian w: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:

${\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline \left.{\begin{aligned}x_{0}^{\prime }&=x_{0}\sec \theta -x_{1}\tan \theta &&={\frac {x_{0}-x_{1}\sin \theta }{\cos \theta }}\\x_{1}^{\prime }&=-x_{0}\tan \theta +x_{1}\sec \theta &&={\frac {x_{0}\sin \theta -x_{1}}{\cos \theta }}\\x_{2}^{\prime }&=x_{2}\\\\x_{0}&=x_{0}^{\prime }\sec \theta +x_{1}^{\prime }\tan \theta &&={\frac {x_{0}^{\prime }+x_{1}^{\prime }\sin \theta }{\cos \theta }}\\x_{1}&=x_{0}^{\prime }\tan \theta +x_{1}^{\prime }\sec \theta &&={\frac {x_{0}^{\prime }\sin \theta +x_{1}^{\prime }}{\cos \theta }}\\x_{2}&=x_{2}^{\prime }\end{aligned}}\right|{\scriptstyle {\begin{aligned}\tan ^{2}\theta -\sec ^{2}\theta &=-1\\{\frac {\tan \theta }{\sec \theta }}&=\sin \theta \\{\frac {1}{\sqrt {1-\sin ^{2}\theta }}}&=\sec \theta \\{\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}&=\tan \theta \end{aligned}}}\end{matrix}}$

(8a)

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

${\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline \left.{\begin{aligned}x_{0}^{\prime }&=x_{0}\csc \vartheta -x_{1}\cot \vartheta &&={\frac {x_{0}-x_{1}\cos \vartheta }{\sin \vartheta }}\\x_{1}^{\prime }&=-x_{0}\cot \vartheta +x_{1}\csc \vartheta &&={\frac {x_{0}\cos \vartheta -x_{1}}{\sin \vartheta }}\\x_{2}^{\prime }&=x_{2}\\\\x_{0}&=x_{0}^{\prime }\csc \vartheta +x_{1}^{\prime }\cot \vartheta &&={\frac {x_{0}^{\prime }+x_{1}^{\prime }\cos \vartheta }{\sin \vartheta }}\\x_{1}&=x_{0}^{\prime }\cot \vartheta +x_{1}^{\prime }\csc \vartheta &&={\frac {x_{0}^{\prime }\cos \vartheta +x_{1}^{\prime }}{\sin \vartheta }}\\x_{2}&=x_{2}^{\prime }\end{aligned}}\right|{\scriptstyle {\begin{aligned}\cot ^{2}\vartheta -\csc ^{2}\vartheta &=-1\\{\frac {\cot \vartheta }{\csc \vartheta }}&=\cos \vartheta \\{\frac {1}{\sqrt {1-\cos ^{2}\vartheta }}}&=\csc \vartheta \\{\frac {\cos \vartheta }{\sqrt {1-\cos ^{2}\vartheta }}}&=\cot \vartheta \end{aligned}}}\end{matrix}}$

(8b)

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

Bianchi (1886) – Pseudospherical surfaces

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

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

Transformation (3) and its inverse are equivalent to trigonometric Lorentz boost (8a), and becomes Lorentz boost of velocity with $\sin \sigma ={\tfrac {v}{c}}$.

Darboux (1891/94) – Pseudospherical surfaces

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

${\begin{aligned}(1)\quad &u+v=2\alpha ,\ u-v=2\beta ;\\(2)\quad &\omega =\varphi \left(\alpha ,\beta \right)\Rightarrow \omega =\varphi \left(\alpha m,\ {\frac {\beta }{m}}\right)\\(3)\quad &\omega =\psi (u,v)\Rightarrow \omega =\psi \left({\frac {u+v\sin h}{\cos h}},\ {\frac {v+u\sin h}{\cos h}}\right)\end{aligned}}$.

Equations (1) together with transformation (2) gives Lorentz boost E:(9a) in terms of null coordinates. Transformation (3) is equivalent to trigonometric Lorentz boost (8a), and becomes Lorentz boost E:(4a) with $\sin h={\tfrac {v}{c}}$.

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 E:Laguerre (1882) and Darboux (1887)):

${\begin{matrix}\sigma ^{\prime 2}-\rho ^{\prime 2}=\sigma ^{2}-\rho ^{2}\\\hline \rho '={\frac {\rho }{\cos \omega }}+\sigma \tan \omega ,\quad \sigma '=\rho \tan \omega +{\frac {\sigma }{\cos \omega }}\end{matrix}}$

This is equivalent to Lorentz transformation (8a) by the identity $\sec \omega ={\tfrac {1}{\cos \omega }}$.

Eisenhart (1905) – Pseudospherical surfaces

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

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

Equations (1) together with transformation (2) gives Lorentz boost E:(9a) in terms of null coordinates. Transformation (3) is equivalent to Lorentz boost E:(9b) in terms of Bondi's k factor, as well as Lorentz boost E:(6f) with $m=\alpha ^{2}$. Transformation (4) is equivalent to trigonometric Lorentz boost (8b), and becomes Lorentz boost E:(4b) with $\cos \sigma ={\tfrac {v}{c}}$. Eisenhart's angle σ corresponds to ϑ of Lorentz boost E:(9d).

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:

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

This is the foundation of Lorentz transformation (8a) and (8b).

Plummer (1910) – Trigonometric Lorentz boosts

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

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

This is equivalent to Lorentz transformation (8a).

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:

${\begin{matrix}v=\alpha \cdot c;\quad \beta ={\frac {1}{\sqrt {1-\alpha ^{2}}}}\\\sin \varphi =\alpha ;\quad \beta ={\frac {1}{\cos \varphi }};\quad \alpha \beta =\tan \varphi \\\hline x'={\frac {x}{\cos \varphi }}-t\cdot \tan \varphi ,\quad t'={\frac {t}{\cos \varphi }}-x\cdot \tan \varphi \end{matrix}}$

This is equivalent to Lorentz transformation (8a) by the identity $\sec \varphi ={\tfrac {1}{\cos \varphi }}$

In another paper Gruner used the alternative relations:

${\begin{matrix}\alpha ={\frac {v}{c}};\ \beta ={\frac {1}{\sqrt {1-\alpha ^{2}}}};\\\cos \theta =\alpha ={\frac {v}{c}};\ \sin \theta ={\frac {1}{\beta }};\ \cot \theta =\alpha \cdot \beta \\\hline x'={\frac {x}{\sin \theta }}-t\cdot \cot \theta ,\quad t'={\frac {t}{\sin \theta }}-x\cdot \cot \theta \end{matrix}}$

This is equivalent to Lorentz Lorentz boost (8b) by the identity $\csc \theta ={\tfrac {1}{\sin \theta }}$.

References

Historical mathematical sources

• Bianchi, L. (1886). Lezioni di geometria differenziale. Pisa: Nistri. http://resolver.sub.uni-goettingen.de/purl?PPN577800191. 
• Darboux, G. (1894) [1891]. Leçons sur la théorie générale des surfaces. Troisième partie. Paris: Gauthier-Villars. https://archive.org/details/leonssurlathorie03darb.  This third part of his lectures was initially published in three steps: première fascicule (1890), deuxième fascicule (1891), and troisième fascicule (1895). The discussion of the Lie transform appears in the deuxième fascicule published in 1891.
• Eisenhart, L. P. (1905). "Surfaces with the same Spherical Representation of their Lines of Curvature as Pseudospherical Surfaces". American Journal of Mathematics 27 (2): 113–172. doi:10.2307/2369977. 
• Scheffers, G. (1899). "Synthetische Bestimmung aller Berührungstransformationen der Kreise in der Ebene". Leipziger Math.-Phys. Berichte 51: 145–160. https://hdl.handle.net/2027/hvd.32044092889328. 

Historical relativity sources

• Gruner, Paul & Sauter, Josef (1921a). "Représentation géométrique élémentaire des formules de la théorie de la relativité". Archives des sciences physiques et naturelles. 5 3: 295–296. http://gallica.bnf.fr/ark:/12148/bpt6k2991536/f295.image. 

• Elementary geometric representation of the formulas of the special theory of relativity on English Wikisource

• Gruner, Paul (1921b). "Eine elementare geometrische Darstellung der Transformationsformeln der speziellen Relativitätstheorie". Physikalische Zeitschrift 22: 384–385. 

• An elementary geometrical representation of the transformation formulas of the special theory of relativity on English Wikisource

• Plummer, H.C.K. (1910), "On the Theory of Aberration and the Principle of Relativity", Monthly Notices of the Royal Astronomical Society, 40: 252–266, Bibcode:1910MNRAS..70..252P

• On the Theory of Aberration and the Principle of Relativity on English Wikisource

• Varićak, V. (1910), "Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie", Physikalische Zeitschrift, 11: 93–6

• Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie on German Wikisource
• Application of Lobachevskian Geometry in the Theory of Relativity on English Wikisource

Secondary sources

• Majerník, V. (1986). "Representation of relativistic quantities by trigonometric functions". American Journal of Physics 54 (6): 536–538. doi:10.1119/1.14557.