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

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

Lorentz transformation via trigonometric functions[edit | edit source]

The following general relation connects the speed of light and the relative velocity to hyperbolic and trigonometric functions, where is the rapidity in equation (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 used 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[edit | edit source]

Bianchi (1886–1893) – Pseudospherical surfaces[edit | edit source]

w:Luigi Bianchi (1886) investigated 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 (1881/94) – Pseudospherical surfaces[edit | edit source]

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

.

Equations (1) together with transformation (2) gives Lorentz boost (9a ) in terms of null coordinates. Transformation (3) is equivalent to trigonometric Lorentz boost (8a ), and becomes Lorentz boost (4a ) with .

Scheffers (1899) – Contact transformation[edit | edit source]

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 Laguerre (1882) and Darboux (1887)):[M 3]

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

Eisenhart (1905) – Pseudospherical surfaces[edit | edit source]

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

.

Equations (1) together with transformation (2) gives Lorentz boost (9a ) in terms of null coordinates. Transformation (3) is equivalent to Lorentz boost (9b ) in terms of Bondi's k factor, as well as Lorentz boost (6f ) with . Transformation (4) is equivalent to trigonometric Lorentz boost (8b ), and becomes Lorentz boost (4b ) with . Eisenhart's angle σ corresponds to ϑ of Lorentz boost (9d ).

Varićak (1910) – Circular and Hyperbolic functions[edit | edit source]

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]

Gruner (1921) – Trigonometric Lorentz boosts[edit | edit source]

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 2]

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

In another paper Gruner used the alternative relations:[R 3]

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

References[edit | edit source]

Historical mathematical sources[edit | edit source]

  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

Historical relativity sources[edit | edit source]

  1. Varićak (1910), p. 93
  2. Gruner (1921a)
  3. Gruner (1921b)
  • 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. 
  • Gruner, Paul (1921b). "Eine elementare geometrische Darstellung der Transformationsformeln der speziellen Relativitätstheorie". Physikalische Zeitschrift 22: 384–385. 
  • Varićak, V. (1910), "Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie"  [Application of Lobachevskian Geometry in the Theory of Relativity], Physikalische Zeitschrift, 11: 93–6

Secondary sources[edit | edit source]

  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.