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

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

Lorentz transformation via hyperbolic functions[edit | edit source]

Translation in the hyperbolic plane[edit | edit source]

The case of a Lorentz transformation without spatial rotation is called a w:Lorentz boost. The simplest case can be given, for instance, by setting n=1 in the most general Lorentz transformation (1a):

or in matrix notation






which resembles precisely the relations of w:hyperbolic functions in terms of w:hyperbolic angle . Thus by adding an unchanged -axis, a Lorentz boost or w:hyperbolic rotation for n=2 (being the same as a rotation around an imaginary angle in equation (2b) or a translation in the hyperbolic plane in terms of the hyperboloid model) is given by

or in matrix notation






in which the rapidity can be composed of arbitrary many rapidities as per the angle sum laws of hyperbolic sines and cosines, so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between angle sum laws of circular trigonometry and spatial rotations. Alternatively, the hyperbolic angle sum laws themselves can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the w:unit hyperbola:

or in matrix notation






Finally, Lorentz boost (3b ) assumes a simple form by using w:squeeze mappings in analogy to Euler's formula in equation (2c):[1]






Hyperbolic relations (a,b) on the right of (3b ) were given by Riccati (1757), relations (a,b,c,d,e,f) by Lambert (1768–1770). Lorentz transformations (3b ) were given by Laisant (1874), Cox (1882), Lindemann (1890/91), Gérard (1892), Killing (1893, 1897/98), Whitehead (1897/98), Woods (1903/05) and Liebmann (1904/05) in terms of Weierstrass coordinates of the w:hyperboloid model. Hyperbolic angle sum laws equivalent to Lorentz boost (3c ) were given by Riccati (1757) and Lambert (1768–1770), while the matrix representation was given by Glaisher (1878) and Günther (1880/81). Lorentz transformations (3d -1) were given by Lindemann (1890/91) and Herglotz (1909), while formulas equivalent to (3d -2) by Klein (1871).

Hyperbolic law of cosines[edit | edit source]

In line with equation (1b ) one can use coordinates inside the w:unit circle , thus the corresponding Lorentz transformations (3b ) obtain the form:






These Lorentz transformations were given by Escherich (1874) and Killing (1898) (on the left), as well as Beltrami (1868) and Schur (1885/86, 1900/02) (on the right) in terms of Beltrami coordinates[2] of hyperbolic geometry. By using the scalar product of , the resulting Lorentz transformation can be seen as equivalent to the w:hyperbolic law of cosines:[3][R 1][4]






The hyperbolic law of cosines (a) was given by Taurinus (1826) and Lobachevsky (1829/30) and others, while variant (b) was given by Schur (1900/02).

Historical notation[edit | edit source]

Riccati (1757) – hyperbolic addition[edit | edit source]

w:Vincenzo Riccati introduced hyperbolic functions in 1757,[M 1][M 2] in particular he formulated the angle sum laws for hyperbolic sine and cosine:

He furthermore showed that and follow by setting and in the above formulas.

The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost (3c ). (In modern publications, Riccati's additional factor r is set to unity.)

Lambert (1768–1770) – hyperbolic addition[edit | edit source]

While Riccati (1757) discussed the hyperbolic sine and cosine, w:Johann Heinrich Lambert (read 1767, published 1768) introduced the expression tang φ or abbreviated as the w:tangens hyperbolicus of a variable u, or in modern notation tφ=tanh(u):[M 3][5]

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:[M 4]

The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of (3b ). Relations (f,g) can also be found in (3e ). By setting tφ=v/c, formula (c) becomes the relative velocity between two frames, (d) the w:Lorentz factor, (e) the w:proper velocity, (f) or (g) becomes the Lorentz transformation of velocity (or relativistic w:velocity addition formula) for collinear velocities in (4a) and (4d).

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's "cos" and "sin" actually mean "cosh" and "sinh"):

The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost (3c ).

Taurinus (1826) – Hyperbolic law of cosines[edit | edit source]

After the addition theorem for the tangens hyperbolicus was given by Lambert (1768), w:hyperbolic geometry was used by w:Franz Taurinus (1826), and later by w:Nikolai Lobachevsky (1829/30) and others, to formulate the w:hyperbolic law of cosines:[M 5][6][7]

When solved for it corresponds to the Lorentz transformation in Beltrami coordinates (3f ), and by defining the rapidities it corresponds to the relativistic velocity addition formula (4e).

Cayley (1859-84) – Cayley absolute and hyperbolic geometry[edit | edit source]

In 1859, w:Arthur Cayley found out that a quadratic form or projective w:quadric can be used as an "absolute", serving as the basis of a projective metric (the w:Cayley–Klein metric).[M 6] For instance, using the absolute x2+y2+z2=0, he defined the distance of two points as follows

and he also alluded to the case of the unit sphere x2+y2+z2=1. In the hands of Klein (1871), all of this became essential for the discussion of non-Euclidean geometry (in particular the Cayley–Klein or Beltrami–Klein model of hyperbolic geometry) and associated quadratic forms and transformations, including the Lorentz interval and Lorentz transformation.

Cayley (1884) himself also discussed some properties of the Beltrami–Klein model and the pseudosphere, and formulated coordinate transformations using the Cayley-Hermite formalism:[M 7]

The form PQ-Z2 and its transformation is equivalent to and its transformation in (3d ), and becomes related to the Lorentz interval by setting P=x0+x2, Q=x0-x2, Z=x1.

Beltrami (1868) – Beltrami coordinates[edit | edit source]

w:Eugenio Beltrami (1868a) introduced coordinates of the w:Beltrami–Klein model of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:[M 8]

(where the disk radius a and the w:radius of curvature R are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)[M 9]

Setting a=a0 Beltrami's (1868a) formulas become formulas (3e ), or in his (1868b) formulas one sets a=b for arbitrary dimensions.

Klein (1871) –Cayley absolute and non-Euclidean geometry[edit | edit source]

Elaborating on Cayley's (1859) definition of an "absolute" (w:Cayley–Klein metric), w:Felix Klein (1871) defined a "fundamental w:conic section" in order to discuss motions such as rotation and translation in the non-Euclidean plane,[M 10] and another fundamental form by using w:homogeneous coordinates x,y related to a circle with radius 2c with measure of curvature . When c is positive, the measure of curvature is negative and the fundamental conic section is real, thus the geometry becomes hyperbolic (w:Beltrami–Klein model):[M 11]

In (1873) he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to an ellipsoid or two-sheet w:hyperboloid.[M 12]

Using positive c in in line with hyperbolic geometry or directly by setting , Klein's two quadratic forms can be related to expressions and for the Lorentz interval in (3d ).

Laisant (1874) – Equipollences[edit | edit source]

In his French translation of w:Giusto Bellavitis' principal work on equipollences, w:Charles-Ange Laisant (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as[M 13]


Here, OA and OB are conjugate semi-diameters of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

producing the invariant relation


Substituting into (1), he showed that OM retains its form

He also defined velocity and acceleration by differentiation of (1).

These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with (3b ).

Escherich (1874) – Beltrami coordinates[edit | edit source]

w:Gustav von Escherich (1874) discussed the plane of constant negative curvature[8] based on the w:Beltrami–Klein model of hyperbolic geometry by Beltrami (1868). Similar to w:Christoph Gudermann (1830)[M 14] who introduced axial coordinates x=tan(a) and y=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions x=tanh(a/k) and y=tanh(b/k)[M 15] in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:[M 16]


This is equivalent to Lorentz transformation (3e ), also equivalent to the relativistic velocity addition (4d) by setting and multiplying [x,y,x′,y′] by 1/c, and equivalent to Lorentz boost (3b ) by setting . This is the relation between the Beltrami coordinates in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the w:hyperboloid model introduced by Killing (1878–1893), Poincaré (1881), and Cox (1881). Both coordinate systems were compared by Cox (1881).[M 17]

Glaisher (1878) – hyperbolic addition[edit | edit source]

It was shown by w:James Whitbread Lee Glaisher (1878) that the hyperbolic addition laws can be written as matrix multiplication[M 18]

This is equivalent to Lorentz boost (3c ).

Günther (1880/81) – hyperbolic addition[edit | edit source]

Following Glaisher (1878), w:Siegmund Günther (1880/81) formulated the hyperbolic addition laws in matrix form as[M 19]

This is equivalent to Lorentz boost (3c ).

Schur (1885/86, 1900/02) – Beltrami coordinates[edit | edit source]

w:Friedrich Schur (1885/86) discussed spaces of constant Riemann curvature, and by following Beltrami (1868) he used the transformation[M 20]

This is equivalent to Lorentz transformation (3e ) and therefore also equivalent to the relativistic velocity addition (4d) in arbitrary dimensions by setting R=c as the speed of light and a1=v as relative velocity.

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:[M 21]

where can have values >0, <0 or ∞.

This is equivalent to Lorentz transformation (3e ) and therefore also equivalent to the relativistic velocity addition (4d) by setting a=v and .

He also defined the triangle[M 22]

This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition (3f , b) or (4e) by setting .

Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute[edit | edit source]

w:Ferdinand von Lindemann discussed hyperbolic geometry in his (1890/91) edition of the lectures on geometry of w:Alfred Clebsch. Citing Killing (1885) and Poincaré (1887) in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set[M 23]

In addition, following Klein (1871) he employed the Cayley absolute related to surfaces of second degree, by using the following quadratic form and its transformation[M 24]

into which he put[M 25]

This is equivalent to Lorentz boost (3d ) with and 2k=1 .

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (a=0) or translations (α=0):[M 26]

This is equivalent to Lorentz boost (3b ) with α=0 and 2k=1.

Gérard (1892) – Weierstrass coordinates[edit | edit source]

w:Louis Gérard (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane and gave the case of translation as follows:[M 27]

This is equivalent to Lorentz boost (3b ).

Killing (1893,97) – Weierstrass coordinates[edit | edit source]

w:Wilhelm Killing (1878–1880) gave case of translation in the form[M 28]

This is equivalent to Lorentz boost (3b ).

In 1898, Killing wrote that relation in a form similar to Escherich (1874), and derived the corresponding Lorentz transformation for the two cases were v is unchanged or u is unchanged:[M 29]

The upper transformation system is equivalent to Lorentz transformation (3e ) and the velocity addition (4d) with l=c and , the system below is equivalent to Lorentz boost (3b ).

Woods (1903) – Weierstrass coordinates[edit | edit source]

w:Frederick S. Woods (1903, published 1905) gave the case of translation in hyperbolic space:[M 30]

This is equivalent to Lorentz boost (3b ) with k2=-1.

and the loxodromic substitution for hyperbolic space:[M 31]

This is equivalent to Lorentz boost (3b ) with β=0.

Whitehead (1897/98) – Universal algebra[edit | edit source]

w:Alfred North Whitehead (1898) discussed the kinematics of hyperbolic space as part of his study of w:universal algebra, and obtained the following transformation:[M 32]

This is equivalent to Lorentz boost (3b ) with α=0.

Liebmann (1904–05) – Weierstrass coordinates[edit | edit source]

w:Heinrich Liebmann (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:[M 33]

This is equivalent to Lorentz boost (3b ).

Herglotz (1909/10) – Möbius transformation[edit | edit source]

w:Gustav Herglotz (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:[R 2]

This is equivalent to Lorentz boost (3d ).

References[edit | edit source]

Historical mathematical sources[edit | edit source]

  1. Riccati (1757), p. 71
  2. Günther (1880/81), pp. 7–13
  3. Lambert (1761/68), pp. 309–318
  4. Lambert (1770), p. 335
  5. Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)
  6. Cayley (1859), sections 209–229
  7. Cayley (1884), section 16.
  8. Beltrami (1868a), pp. 287-288; Note I; Note II
  9. Beltrami (1868b), pp. 232, 240–241, 253–254
  10. Klein (1871), pp. 601–602
  11. Klein (1871), p. 618
  12. Klein (1873), pp. 127-128
  13. Laisant (1874b), pp. 134–135
  14. Gudermann (1830), §1–3, §18–19
  15. Escherich (1874), p. 508
  16. Escherich (1874), p. 510
  17. Cox (1881), p. 186
  18. Glaisher (1878), p. 30
  19. Günther (1880/81), p. 405
  20. Schur (1885/86), p. 167
  21. Schur (1900/02), p. 290; (1909), p. 83
  22. Schur (1900/02), p. 291; (1909), p. 83
  23. Lindemann & Clebsch (1890/91), pp. 477–478, 524
  24. Lindemann & Clebsch (1890/91), pp. 361–362
  25. Lindemann & Clebsch (1890/91), p. 496
  26. Lindemann & Clebsch (1890/91), pp. 477–478
  27. Gérard (1892), pp. 40–41
  28. Killing (1893), p. 331
  29. Killing (1898), p. 133
  30. Woods (1903/05), p. 55
  31. Woods (1903/05), p. 72
  32. Whitehead (1898), pp. 459–460
  33. Liebmann (1904/05), p. 174

Historical relativity sources[edit | edit source]

  1. Varićak (1912), p. 108
  2. Herglotz (1909/10), pp. 404-408
  • Herglotz, Gustav (1910) [1909], "Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper]", Annalen der Physik, 336 (2): 393–415, Bibcode:1910AnP...336..393H, doi:10.1002/andp.19103360208

Secondary sources[edit | edit source]

  1. Rindler (1969), p. 45
  2. Rosenfeld (1988), p. 231
  3. Pauli (1921), p. 561
  4. Barrett (2006), chapter 4, section 2
  5. Barnett (2004), pp. 22–23
  6. Bonola (1912), p. 79
  7. Gray (1979), p. 242
  8. Sommerville (1911), p. 297