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

History of Lorentz transformation (edit)
History of Topics in Special Relativity (edit)

Lorentz transformation via trigonometric functions

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The following general relation connects the speed of light and the relative velocity to hyperbolic and trigonometric functions, where   is the rapidity in E:(3b),   is equivalent to the w:Gudermannian function  , and   is equivalent to the Lobachevskian w:angle of parallelism  :

 

This relation was first defined by Varićak (1910).

a) Using   one obtains the relations   and  , and the Lorentz boost takes the form:[1]

 

 

 

 

 

(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   one obtains the relations   and  , and the Lorentz boost takes the form:[1]

 

 

 

 

 

(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

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Bianchi (1886) – Pseudospherical surfaces

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

 .
Transformation (3) and its inverse are equivalent to trigonometric Lorentz boost (8a), and becomes Lorentz boost of velocity with  .

Darboux (1891/94) – Pseudospherical surfaces

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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]

 .
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  .

Scheffers (1899) – Contact transformation

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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]

 
This is equivalent to Lorentz transformation (8a) by the identity  .

Eisenhart (1905) – Pseudospherical surfaces

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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]

 .
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  . Transformation (4) is equivalent to trigonometric Lorentz boost (8b), and becomes Lorentz boost E:(4b) with  . Eisenhart's angle σ corresponds to ϑ of Lorentz boost E:(9d).

Varićak (1910) – Circular and Hyperbolic functions

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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]

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

Plummer (1910) – Trigonometric Lorentz boosts

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

 
This is equivalent to Lorentz transformation (8a).

Gruner (1921) – Trigonometric Lorentz boosts

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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]

 
This is equivalent to Lorentz transformation (8a) by the identity  

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

 
This is equivalent to Lorentz Lorentz boost (8b) by the identity  .

References

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Historical mathematical sources

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  1. Bianchi (1886), eq. 1 can be found on p. 226, eq. (2) on p. 240, eq. (3) on pp. 240–241, and for eq. (4) see the footnote on p. 240.
  2. Darboux (1891/94), pp. 381–382
  3. Scheffers (1899), p. 158
  4. Eisenhart (1905), p. 126
  • 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

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  1. Varićak (1910), p. 93
  2. Plummer (1910), p. 256
  3. Gruner (1921a)
  4. Gruner (1921b)
  • Gruner, P. (1921b), "Eine elementare geometrische Darstellung der Transformationsformeln der speziellen Relativitätstheorie", Physikalische Zeitschrift, 22: 384–385
  • 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
  • Varićak, V. (1910), "Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie", Physikalische Zeitschrift, 11: 93–6

Secondary sources

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  1. 1.0 1.1 Majerník (1986), 536–538
  • Majerník, V. (1986), "Representation of relativistic quantities by trigonometric functions", American Journal of Physics, 54 (6): 536–538, doi:10.1119/1.14557