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

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

Lorentz transformation via quaternions and hyperbolic numbers[edit | edit source]

Quaternions[edit | edit source]

The Lorentz transformations can also be expressed in terms of w:biquaternions: A Minkowskian quaternion (or minquat) q having one real part and one purely imaginary part is multiplied by biquaternion a applied as pre- and postfactor. Using an overline to denote quaternion conjugation and * for complex conjugation, its general form (on the left) and the corresponding boost (on the right) are as follows:[1][2]






Hamilton (1844/45) and Cayley (1845) derived the quaternion transformation for 3D rotations, and Cayley (1854, 1855) gave the corresponding transformation representing 4D rotations, while the corresponding parameter were already given long before by Euler (1771). Cox (1882/83) discussed the Lorentz interval in terms of Weierstrass coordinates in the course of adapting w:William Kingdon Clifford's biquaternions a+ωb to hyperbolic geometry by setting (alternatively, 1 gives elliptic and 0 parabolic geometry). Stephanos (1883) related the imaginary part of w:William Rowan Hamilton's biquaternions to the radius of spheres, and introduced a homography leaving invariant the equations of oriented spheres or oriented planes in terms of w:Lie sphere geometry. Buchheim (1884/85) discussed the Cayley absolute and adapted Clifford's biquaternions to hyperbolic geometry similar to Cox by using all three values of . Eventually, the modern Lorentz transformation using biquaternions with as in hyperbolic geometry was given by Noether (1910) and Klein (1910) as well as Conway (1911) and Silberstein (1911).

Hyperbolic numbers[edit | edit source]

Often connected with quaternionic systems is the w:hyperbolic number , which also allows to formulate the Lorentz transformations:[3][4]






After the trigonometric expression (w:Euler's formula) was given by Euler (1748), and the hyperbolic analogue as well as hyperbolic numbers by Cockle (1848) in the framework of w:tessarines, it was shown by Cox (1882/83) that one can identify with associative quaternion multiplication. Here, is the hyperbolic w:versor with , while -1 denotes the elliptic or 0 denotes the parabolic counterpart (not to be confused with the expression in Clifford's biquaternions also used by Cox, in which -1 is hyperbolic). The hyperbolic versor was also discussed by Macfarlane (1892, 1894, 1900) in terms of w:hyperbolic quaternions. The expression for hyperbolic motions (and -1 for elliptic, 0 for parabolic motions) also appear in "biquaternions" defined by Vahlen (1901/02, 1905).

Clifford algebra[edit | edit source]

More extended forms of complex and (bi-)quaternionic systems in terms of w:Clifford algebra can also be used to express the Lorentz transformations. For instance, using a system a of Clifford numbers one can transform the following general quadratic form into itself, in which the individual values of can be set to +1 or -1 at will:[5][6]






The Lorentz interval follows if the sign of one differs from all others. The general definite form as well as the general indefinite form and their invariance under transformation (1) was discussed by Lipschitz (1885/86), while hyperbolic motions were discussed by Vahlen (1901/02, 1905) by setting in transformation (2), while elliptic motions follow with -1 and parabolic motions with 0, all of which he also related to biquaternions.

Historical notation[edit | edit source]

Euler (1771) – Quaternion rotation parameter[edit | edit source]

Euler (1771) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as w:orthogonal transformation. The transformation in three dimensions was given as[M 1]

in which the eight coefficients were related by Euler to four arbitrary parameter p,q,r,s, which where rediscovered by w:Olinde Rodrigues (1840) who related them to rotation angles[M 2]:[M 3]

They are equivalent to the parameter of an (improper) quaternion rotation in three dimensions, equivalent to (improper) Euler–Rodrigues parameters.

The orthogonal transformation in four dimensions was given by him as[M 4]

and he gave a representation of the sixteen coefficients using eight parameters a,b,c,d,p,q,r,s[M 5]

They are equivalent to the parameter of an (improper) quaternion rotation in four dimensions.

Hamilton (1844/45) – Quaternions[edit | edit source]

w:William Rowan Hamilton, in an abstract of a lecture held in November 1844 and published 1845/47, showed that spatial rotations can be formulated using his theory of w:quaternions by employing w:versors as pre- and postfactor, with α as unit vector and a as real angle:[M 6]


In a footnote added before printing, he showed that this is equivalent to Cayley's (1845) rotation formula by setting[M 7]

(2) .

Hamilton acknowledged Cayley's independent discovery and priority for first printed (February 1845) publication, but noted that he himself communicated formula (1) already in October 1844 to Charles Graves.

Formulas (1) or (2) are role models for Lorentz boost (7a ), by replacing versors and quaternions with hyperbolic versors and biquaternions.

Cayley (1845) – Quaternions[edit | edit source]

In 1845, w:Arthur Cayley showed that the Euler-Rodrigues parameters in equation equation (Q3) representing rotations can be related to quaternion multiplication by pre- and postfactors (an equivalent rotation formula was also used by Hamilton (1844/45)):[M 8]

and in 1848 he used the abbreviated form[M 9]

In 1854 he showed how to transform the sum of four squares into itself:[M 10]

or in 1855:[M 11]

Cayley's quaternion transformation of the sum of four squares, abbreviated QqP, served as a role model for the representation of the Lorentz transformation by Noether (1910), Klein (1910), in which the scalar part is imaginary.

Cockle (1848) - Tessarines[edit | edit source]

w:James Cockle (1848) introduced the w:tessarine algebra as follows:[M 12]


While is the ordinary imaginary unit, the new unit led him to formulate the following relation:[M 13]


The real tessarine is a split-complex or hyperbolic number, a w:hyperbolic versor, and the multiplication leads to Lorentz boost (7b ).

Cox (1882) – Quaternions[edit | edit source]

Homersham Cox (1882/83) described hyperbolic geometry in terms of an analogue to w:quaternions and w:Hermann Grassmann's w:exterior algebra. To that end, he used w:hyperbolic numbers (without mentioning Cockle (1848)) as a means to transfer point P to point Q in the hyperbolic plane, which he wrote in the form:[M 14]

In (1882/83a) he showed the equivalence of PQ=-cosh(θ)+ι·sinh(θ) with "quaternion multiplication",[M 15] and in (1882/83b) he described QP−1=cosh(θ)+ι·sinh(θ) as being "associative quaternion multiplication".[M 16] He also showed that the position of point P in the hyperbolic plane may be determined by three quantities in terms of Weierstrass coordinates obeying the relation z2-x2-y2=1.[M 17]

Cox's associative quaternion multiplication using the hyperbolic versor is equivalent to the Lorentz boost (7b ) by setting and .

Cox went on to develop an algebra for hyperbolic space analogous to Clifford's w:biquaternions. While Clifford (1873) used biquaternions of the form a+ωb in which ω2=0 denotes parabolic space and ω2=1 elliptic space, Cox discussed hyperbolic space using the imaginary quantity and therefore ω2=-1.[M 18] He also obtained relations of quaternion multiplication in terms of Weierstrass coordinates:[M 19]

Stephanos (1883) – Biquaternions[edit | edit source]

w:Cyparissos Stephanos (1883)[M 20] showed that Hamilton's biquaternion a0+a1ι1+a2ι2+a3ι3 can be interpreted as an oriented sphere in terms of Lie's sphere geometry (1871), having the vector a1ι1+a2ι2+a3ι3 as its center and the scalar as its radius. Its norm is thus equal to the power of a point of the corresponding sphere. In particular, the norm of two quaternions N(Q1-Q2) (the corresponding spheres are in contact with N(Q1-Q2)=0) is equal to the tangential distance between two spheres. The general contact transformation between two spheres then can be given by a w:homography using 4 arbitrary quaternions A,B,C,D and two variable quaternions X,Y:[M 21][7][8]

(or ).

Stephanos pointed out that the special case A=0 denotes transformations of oriented planes (see Laguerre (1882)).

The Lorentz group SO(1,3) is a subgroup of the conformal group Con(1,3) in terms of Lie's transformations of orientied spheres in which the radius indicates the fourth coordinate. The Lorentz group is isomorphic to the group of Laguerre's transformation of oriented planes.

Buchheim (1884–85) – Biquaternions[edit | edit source]

w:Arthur Buchheim (1884, published 1885) applied Clifford's biquaternions and their operator ω to different forms of geometries (Buchheim mentions Cox (1882) as well). He defined the scalar ω2=e2 which in the case -1 denotes hyperbolic space, 1 elliptic space, and 0 parabolic space. He derived the following relations consistent with the Cayley–Klein absolute:[M 22]

By choosing e2=-1 for hyperbolic space, the Cayley absolute becomes the Lorentz interval.

Lipschitz (1885/86) – Clifford algebra[edit | edit source]

w:Rudolf Lipschitz used an even subalgebra of w:Clifford algebra in order to formulate the orthogonal transformation of a sum or squares into itself, for which he used real variables and constants, thus Λ becomes a real quaternion for n=3.[M 23] He went further and discussed transformations in which both variables x,y... and constants are complex, thus Λ becomes a complex quaternion (i.e. biquaternion) for n=3.[M 24] The transformation system for both real and complex quantities has the form:[M 25]

Lipschitz noticed that this corresponds to the transformations of quadratic forms given by Hermite (1854) and Cayley (1855). He then modified his equations to discuss the general indefinite quadratic form, by defining some variables and constants as real and some of them as purely imaginary:[M 26]

resulting into

By setting m=n-1 or n=m+1, the Lorentz interval and the Lorentz transformation follows

Vahlen (1901/02) – Clifford algebra and Möbius transformation[edit | edit source]

Modifying Lipschitz's (1885/86) variant of Clifford numbers, w:Theodor Vahlen (1901/02) formulated Möbius transformations (which he called vector transformations) and biquaternions in order to discuss motions in n-dimensional non-Euclidean space, leaving the following quadratic form invariant (where j2=1 represents hyperbolic motions, j2=-1 elliptic motions, j2=0 parabolic motions):[M 27]

The group of hyperbolic motions or the Möbius group are isomorphic to the Lorentz group.

Noether (1910), Klein (1910) – Biquaternions[edit | edit source]

w:Felix Klein (1908) described Cayley's (1854) 4D quaternion multiplications as "Drehstreckungen" (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor), and pointed out that the modern principle of relativity as provided by Minkowski is essentially only the consequent application of such Drehstreckungen, even though he didn't provide details.[R 1]

In an appendix to Klein's and Sommerfeld's "Theory of the top" (1910), w:Fritz Noether showed how to formulate hyperbolic rotations using biquaternions with , which he also related to the speed of light by setting ω2=-c2. He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations equivalent to (7a ):[R 2]

Besides citing quaternion related standard works such as Cayley (1854), Noether referred to the entries in Klein's encyclopedia by w:Eduard Study (1899) and the French version by w:Élie Cartan (1908).[9] Cartan's version contains a description of Study's w:dual numbers, Clifford's biquaternions (including the choice for hyperbolic geometry), and Clifford algebra, with references to Stephanos (1883), Buchheim (1884/85), Vahlen (1901/02) and others.

Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:[R 3]

or in March 1911[R 4]

Conway (1911), Silberstein (1911) – Biquaternions[edit | edit source]

w:Arthur W. Conway in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:[R 5]

Also w:Ludwik Silberstein in November 1911[R 6] as well as in 1914,[10] formulated the Lorentz transformation in terms of velocity v:

Silberstein cites Cayley (1854, 1855) and Study's encyclopedia entry (in the extended French version of Cartan in 1908), as well as the appendix of Klein's and Sommerfeld's book.

References[edit | edit source]

Historical mathematical sources[edit | edit source]

  1. Euler (1771), pp. 75f
  2. Rodrigues (1840), p. 405
  3. Euler (1771), p. 101
  4. Euler (1771), pp. 89–91
  5. Euler (1771), p. 102
  6. Hamilton (1844/45), p. 13
  7. Hamilton (1844/45), p. 14
  8. Cayley (1845), p. 142
  9. Cayley (1848), p. 196
  10. Cayley (1854), p. 211
  11. Cayley (1855b), p. 312
  12. Cockle (1848), p. 437
  13. Cockle (1848), p. 438
  14. Cox (1882/83a), pp. 85–86
  15. Cox (1882/83a), p. 88
  16. Cox (1882/83b), p. 195
  17. Cox (1882/83a), p. 97
  18. On pp. 104-105 he started using the term v2=-1, on p. 106 he noted that one can simply use instead of v, and on p. 112 he adopted Clifford's notation by setting ω2=-1.
  19. Cox (1882/83a), pp. 108-109
  20. Stephanos (1883), p. 590ff
  21. Stephanos (1883), p. 592
  22. Buchheim (1885), p. 309
  23. Lipschitz (1886), pp. 75–79
  24. Lipschitz (1886), pp. 134–138
  25. Lipschitz (1886), p. 76; p. 137
  26. Lipschitz (1886), pp. 145–147
  27. Vahlen (1902), pp. 586–587, 590; (1905), p. 282

Historical relativity sources[edit | edit source]

  1. Klein (1908), p. 165
  2. Noether (1910), pp. 939–943
  3. Klein (1910), p. 300
  4. Klein (1911), pp. 602ff.
  5. Conway (1911), p. 8
  6. Silberstein (1911/12), p. 793
  • Klein, Felix (1921) [1910]. Über die geometrischen Grundlagen der Lorentzgruppe. 1. 533–552. doi:10.1007/978-3-642-51960-4_31. ISBN 978-3-642-51898-0. 
  • Klein, F.; Sommerfeld A. (1910). Noether, Fr.. ed. Über die Theorie des Kreisels. Heft IV. Leipzig: Teuber. https://archive.org/details/fkleinundasommer019696mbp. 
  • Klein, F. (1911). Hellinger, E.. ed. Elementarmethematik vom höheren Standpunkte aus. Teil I (Second Edition). Vorlesung gehalten während des Wintersemesters 1907-08. Leipzig: Teubner. 
  • Silberstein, L. (1912) [1911], "Quaternionic form of relativity", The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 23 (137): 790–809, doi:10.1080/14786440508637276

Secondary sources[edit | edit source]

  1. Synge (1972), pp. 13, 19, 24
  2. Girard (1984), pp. 28–29
  3. Sobczyk (1995)
  4. Fjelstad (1986)
  5. Cartan & Study (1908), section 36
  6. Rothe (1916), section 16
  7. Cartan & Study (1908), p. 460
  8. Rothe (1916), p. 1399
  9. Cartan & Study (1908), sections 35–36
  10. Silberstein (1914), p. 156