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

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

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

This relation was first defined by Varićak (1910).

a) Using ${\displaystyle \sin \theta ={\tfrac {v}{c}}}$ one obtains the relations ${\displaystyle \sec \theta =\gamma }$ and ${\displaystyle \tan \theta =\beta \gamma }$ , and the Lorentz boost takes the form:^[1]

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

(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 ${\displaystyle \cos \vartheta ={\tfrac {v}{c}}}$ one obtains the relations ${\displaystyle \csc \vartheta =\gamma }$ and ${\displaystyle \cot \vartheta =\beta \gamma }$ , and the Lorentz boost takes the form:^[1]

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

(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.

Bianchi (1886) – Pseudospherical surfaces edit

w:Luigi Bianchi (1886) investigated E:Lie's transformation (1880) of pseudospherical surfaces, obtaining the result:^{[M 1]}

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

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

Darboux (1891/94) – Pseudospherical surfaces edit

Similar to Bianchi (1886), w:Gaston Darboux (1891/94) showed that the E:Lie's transformation (1880) gives rise to the following relations:^{[M 2]}

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

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 ${\displaystyle \sin h={\tfrac {v}{c}}}$ .

Scheffers (1899) – Contact transformation edit

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)):^{[M 3]}

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

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

Eisenhart (1905) – Pseudospherical surfaces edit

w:Luther Pfahler Eisenhart (1905) followed Bianchi (1886, 1894) and Darboux (1891/94) by writing the E:Lie's transformation (1880) of pseudospherical surfaces:^{[M 4]}

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

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 ${\displaystyle m=\alpha ^{2}}$ . Transformation (4) is equivalent to trigonometric Lorentz boost (8b), and becomes Lorentz boost E:(4b) with ${\displaystyle \cos \sigma ={\tfrac {v}{c}}}$ . Eisenhart's angle σ corresponds to ϑ of Lorentz boost E:(9d).

Varićak (1910) – Circular and Hyperbolic functions edit

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:^{[R 1]}

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

Plummer (1910) – Trigonometric Lorentz boosts edit

w:Henry Crozier Keating Plummer (1910) defined the following relations^{[R 2]}

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

Gruner (1921) – Trigonometric Lorentz boosts edit

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:^{[R 3]}

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

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

In another paper Gruner used the alternative relations:^{[R 4]}

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

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

Historical mathematical sources edit

• Bianchi, L. (1886), Lezioni di geometria differenziale, Pisa: Nistri
• Darboux, G. (1894) [1891], Leçons sur la théorie générale des surfaces. Troisième partie, Paris: Gauthier-Villars 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

Historical relativity sources edit

• Gruner, P.; Sauter, J. (1921a), "Représentation géométrique élémentaire des formules de la théorie de la relativité", Archives des sciences physiques et naturelles, 5, 3: 295–296{{citation}}: CS1 maint: multiple names: authors list (link)

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

• Gruner, P. (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 edit

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