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

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

Lorentz transformation via Cayley–Klein parameters, Möbius and spin transformations[edit | edit source]

The previously mentioned Euler-Rodrigues parameter a,b,c,d (i.e. Cayley-Hermite parameter in equation (Q3) with d=1) are closely related to Cayley–Klein parameter α,β,γ,δ introduced by Helmholtz (1866/67), Cayley (1879) and Klein (1884) to connect Möbius transformations and rotations:[M 1]

thus equation (Q3) becomes:

 

 

 

 

(Q4)

Also the Lorentz transformation can be expressed with variants of the Cayley–Klein parameters: One relates these parameters to a spin-matrix D, the w:spin transformations of variables (the overline denotes w:complex conjugate), and the w:Möbius transformation of . When defined in terms of isometries of hyperblic space (hyperbolic motions), the w:Hermitian matrix u associated with these Möbius transformations produces an invariant determinant identical to the Lorentz interval. Therefore, these transformations were described by w:John Lighton Synge as being a "factory for the mass production of Lorentz transformations".[1] It also turns out that the related w:spin group Spin(3, 1) or w:special linear group SL(2, C) acts as the double cover of the Lorentz group (one Lorentz transformation corresponds to two spin transformations of different sign), while the w:Möbius group Con(0,2) or w:projective special linear group PSL(2, C) is isomorphic to both the Lorentz group and the group of isometries of hyperbolic space.

In space, the Möbius/Spin/Lorentz transformations can be written as:[2][1][3][4]

 

 

 

 

(6a)

thus:[5]

 

 

 

 

(6b)

or in line with general Lorentz transformation (1b) one can substitute so that the Möbius/Lorentz transformations become related to the unit sphere:

 

 

 

 

(6c)

The general transformation u′ in (6a ) was given by Cayley (1854), while the general relation between Möbius transformations and transformation u′ leaving invariant the w:generalized circle was pointed out by Poincaré (1883) in relation to w:Kleinian groups. The adaptation to the Lorentz interval by which (6a ) becomes a Lorentz transformation was given by Klein (1889-1893, 1896/97), Bianchi (1893), Fricke (1893, 1897). Its reformulation as Lorentz transformation (6b ) was provided by Bianchi (1893) and Fricke (1893, 1897). Lorentz transformation (6c ) was given by Klein (1884) in relation to surfaces of second degree and the invariance of the unit sphere. In relativity, (6a ) was first employed by Herglotz (1909/10).

In the plane, the transformations can be written as:[6][4]

 

 

 

 

(6d)

thus

 

 

 

 

(6e)

which includes the special case implying , reducing the transformation to a Lorentz boost in 1+1 dimensions:

 

 

 

 

(6f)

Finally, by using the Lorentz interval related to a hyperboloid, the Möbius/Lorentz transformations can be written

 

 

 

 

(6g)

The general transformation u′ and its invariant in (6d ) was already used by Lagrange (1773) and Gauss (1798/1801) in the theory of integer binary quadratic forms. The invariant was also studied by Klein (1871) in connection to hyperbolic plane geometry (see equation (3d)), while the connection between u′ and with the Möbius transformation was analyzed by Poincaré (1886) in relation to w:Fuchsian groups. The adaptation to the Lorentz interval by which (6d ) becomes a Lorentz transformation was given by Bianchi (1888) and Fricke (1891). Lorentz Transformation (6e ) was stated by Gauss around 1800 (posthumously published 1863), as well as Selling (1873), Bianchi (1888), Fricke (1891), Woods (1895) in relation to integer indefinite ternary quadratic forms. Lorentz transformation (6f ) was given by Bianchi (1886, 1894) and Eisenhart (1905). Lorentz transformation (6g ) of the hyperboloid was stated by Poincaré (1881) and Hausdorff (1899).

Historical notation[edit | edit source]

Lagrange (1773) – Binary quadratic forms[edit | edit source]

After the invariance of the sum of squares under linear substitutions was discussed by Euler (1771), the general expressions of a w:binary quadratic form and its transformation was formulated by w:Joseph-Louis Lagrange (1773/75) as follows[M 2]

The transformation of coefficients (p,q,r) is identical to transformation u′ in (6d ) and becomes the complete Lorentz transformation by setting
.

Gauss (1800)[edit | edit source]

Binary quadratic form[edit | edit source]

The theory of binary quadratic forms was considerably expanded by w:Carl Friedrich Gauss (1798, published 1801) in his w:Disquisitiones Arithmeticae. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:[M 3]

As pointed out by Gauss, F and F′ are called "proper equivalent" if αδ-βγ=1, so that F is contained in F′ as well as F′ is contained in F. In addition, if another form F″ is contained by the same procedure in F′ it is also contained in F and so forth.[M 4]

The transformation of coefficients (a,b,c) is identical to transformation u′ in (6d ) and becomes the complete Lorentz transformation by setting
.

Cayley–Klein parameter[edit | edit source]

The determination of all transformations of the Lorentz interval (as a special case of an integer ternary quadratic form) into itself was explicitly worked out by Gauss around 1800 (posthumously published in 1863), for which he provided a coefficient system α,β,γ,δ:[M 5]

Gauss' result was cited by Bachmann (1869), Selling (1873), Bianchi (1888), w:Leonard Eugene Dickson (1923).[7] The parameters α,β,γ,δ, when applied to spatial rotations, were later called Cayley–Klein parameters.

This is equivalent to Lorentz transformation (6e ), containing Lorentz boost (6f ) as a special case with and .

Cayley (1854) – Cayley–Klein parameter[edit | edit source]

Already in 1854, Cayley published an alternative method of transforming quadratic forms by using certain parameters α,β,γ,δ in relation to an improper homographic transformation of a surface of second order into itself:[M 6]

By setting and rewriting M and M' in terms of four different parameters he demonstrated the invariance of , and subsequently showed the relation to 4D quaternion transformations. Fricke & Klein (1897) credited Cayley by calling the above transformation the most general (real or complex) space collineation of first kind of an absolute surface of second kind into itself.[M 7] Parameters α,β,γ,δ are similar to what was later called Cayley–Klein parameters in relation to spatial rotations (which was done by Cayley in 1879[M 8] and before by w:Hermann von Helmholtz (1866/67)[M 9]).

Cayley's improper transformation becomes proper with some sign changes, and becomes equivalent to Lorentz transformation in (6a ) by setting M=M'=1 and:
.
Subsequently solved for it becomes Lorentz transformation (6b ).

Klein (1871–97)[edit | edit source]

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 (6d ).

Möbius transformation, spin transformation, Cayley–Klein parameter[edit | edit source]

In (1872) while devising the w:Erlangen program, Klein discussed the general relation between projective metrics, w:binary forms and conformal geometry transforming a sphere into itself in terms of linear transformations of the w:complex variable x+iy.[M 13] Following Klein, these relations were discussed by w:Ludwig Wedekind (1875) using .[M 14] Klein (1875) then showed that all finite groups of motions follow by determining all finite groups of such linear transformations of x+iy into itself.[M 15] In (1878),[M 16] Klein classified the substitutions of with αδ-βγ=1 into hyperbolic, elliptic, parabolic, and in (1882)[M 17] he added the loxodromic substitution as the combination of elliptic and hyperbolic ones. (In 1890, w:Robert Fricke in his edition of Klein's lectures of w:elliptic functions and w:Modular forms, referred to the analogy of this treatment to the theory of quadratic forms as given by Gauss and in particular Dirichlet.)[M 7]

In (1884) Klein related the linear fractional transformations (interpreted as rotations around the x+iy-sphere) to Cayley–Klein parameters [α,β,γ,δ], to Euler–Rodrigues parameters [a,b,c,d], and to the w:unit sphere by means of w:stereographic projection, and also discussed transformations preserving surfaces of second degree equivalent to the transformation given by Cayley (1854):[M 18]

The formulas on the left related to the unit sphere are equivalent to Lorentz transformation (6c ). The formulas on the right can be related to those on the left by setting

and become equivalent to Lorentz transformation (6a ) by setting
and subsequently solved for x1... it becomes Lorentz transformation (6b ).

In his lecture in the winter semester of 1889/90 (published 1892–93), he discussed the hyperbolic plane by using (as in 1871) the Lorentz interval in terms of a circle with radius 2k as the basis of hyperbolic geometry, and another quadratic form to discuss the "kinematics of hyperbolic geometry" consisting of motions and congruent displacements of the hyperbolic plane into itself:[M 19]

Klein's Lorentz interval can be connected with the other interval by setting
,
by which the transformation system on the right becomes equivalent to Lorentz transformation (6d ) with 2k=1, and subsequently solved for x1... it becomes equivalent to Lorentz transformation (6e ).

In his lecture in the summer semester of 1890 (published 1892–93), he discussed general surfaces of second degree, including an "oval" surface corresponding to hyperbolic space and its motions:[M 20]

The transformation of the unit sphere on the right is equivalent to Lorentz transformation (6c ). Plugging the values for λ,μ,λ′,μ′,... from the right into the transformations on the left, and relating them to Klein's homogeneous coordinates by leads to Lorentz transformation (6a ). Subsequently solved for x1... it becomes Lorentz transformation (6b ).

In (1896/97), Klein again defined hyperbolic motions and explicitly used t as time coordinate:[M 21]

This is equivalent to Lorentz transformation (6a ).

Klein's work was summarized and extended by Bianchi (1888-1893) and Fricke (1893-1897), obtaining equivalent Lorentz transformations.

Selling (1873–74) – Quadratic forms[edit | edit source]

Continuing the work of Gauss (1801) on definite ternary quadratic forms and Hermite (1853) on indefinite ternary quadratic forms, w:Eduard Selling (1873) used the auxiliary coefficients ξ,η,ζ by which a definite form and an indefinite form f can be rewritten in terms of three squares:[M 22][8]

In addition, Selling showed that auxiliary coefficients ξ,η,ζ can be geometrically interpreted as point coordinates which are in motion upon one sheet of a two-sheet hyperboloid, which is related to Selling's formalism for the reduction of indefinite forms by using definite forms.[M 23]

Selling also reproduced the Lorentz transformation given by Gauss (1800/63), to whom he gave full credit, and called it the only example of a particular indefinite ternary form known to him that has ever been discussed:[M 24]

This is equivalent to Lorentz transformation (6e ), containing Lorentz boost (6f ) or (9b ) as a special case with and .

Poincaré (1881-86) – Möbius transformation[edit | edit source]

w:Henri Poincaré (1881a) demonstrated the connection of his formulas of the hyperboloid model to Möbius transformations:[M 25]

This is equivalent to Lorentz transformation (6g ).

Poincaré (1881b) also used the Möbius transformation in relation to w:Fuchsian functions and the discontinuous w:Fuchsian group, being a special case of the hyperbolic group leaving invariant the "fundamental circle" (w:Poincaré disk model and w:Poincaré half-plane model of hyperbolic geometry).[M 26] He then extended Klein's (1878-1882) study on the relation between Möbius transformations and hyperbolic, elliptic, parabolic, and loxodromic substitutions, and while formulating w:Kleinian groups (1883) he used the following transformation leaving invariant the w:generalized circle:[M 27]

Setting this becomes transformation u′ in (6a ) and becomes the complete Lorentz transformation by setting .

In 1886, Poincaré investigated the relation between indefinite ternary quadratic forms and Fuchsian functions and groups:[M 28]

This is equivalent to transformation u′ in (6d ) and becomes the complete Lorentz transformation by suitibly choosing the coefficients a,b,c,... so that [X,Y,Z]=[x+z, y, -x+z].

Bianchi (1888-93) – Möbius and spin transformations[edit | edit source]

Related to Klein's (1871) and Poincaré's (1881-1887) work on non-Euclidean geometry and indefinite quadratic forms, w:Luigi Bianchi (1888) analyzed the differential Lorentz interval in term of conic sections and hyperboloids, alluded to the linear fractional transformation of and its conjugate with parameters α,β,γ,δ in order to preserve the Lorentz interval, and gave credit to Gauss (1800/63) who obtained the same coefficient system:[M 29]

The is equivalent to Lorentz transformations (6d ) and (6e ), containing Lorentz boost (6f ) or (9b ) as a special case with and .

In 1893, Bianchi gave the coefficients in the case of four dimensions:[M 30]

This is equivalent to Lorentz transformation (6a ).

Solving for Bianchi obtained:[M 30]

This is equivalent to Lorentz transformation (6b ).

Fricke (1891–97) – Möbius and spin transformations[edit | edit source]

w:Robert Fricke (1891) – following the work of his teacher Klein (1878–1882) as well as Poincaré (1881–1887) on automorphic functions and group theory – obtained the following transformation for an integer ternary quadratic form[M 31][9]

By setting q=1, the first part is equivalent to Lorentz transformation (6d ) and the second part is equivalent to (6e ), containing Lorentz boost (6f ) or (9b ) as a special case with and .

And the general case of four dimensions in 1893:[M 32]

By setting p=q=r=s=1, the first part is equivalent to Lorentz transformation (6a ) and the second part to (6b ).

Supported by Felix Klein, Fricke summarized his and Klein's work in a treatise concerning w:automorphic functions (1897). Using a sphere as the absolute, in which the interior of the sphere is denoted as hyperbolic space, they defined hyperbolic motions, and stressed that any hyperbolic motion corresponds to "circle relations" (now called Möbius transformations):[M 7]

This is equivalent to Lorentz transformation (6a ).

Woods (1895) – Spin transformation[edit | edit source]

In a thesis supervised by Felix Klein, w:Frederick S. Woods (1895) further developed Bianchi's (1888) treatment of surfaces satisfying the Lorentz interval (pseudominimal surface), and used the transformation of Gauss (1800/63) and Bianchi (1888) while discussing automorphisms of that surface:[M 33]

The expressions within the brackets are equivalent to Lorentz transformations (6e ), containing Lorentz boost (6f ) as a special case with and .

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

Following Klein (1889–1897) and Fricke & Klein (1897) concerning the Cayley absolute, hyperbolic motion and its transformation, w:Gustav Herglotz (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic:[R 1]

TThe general case is equivalent to Lorentz transformation (6a ).

References[edit | edit source]

Historical mathematical sources[edit | edit source]

  1. Klein (1896/97), p. 12
  2. Lagrange (1773/75), section 22
  3. Gauss (1798/1801), articles 157–158;
  4. Gauss (1798/1801), section 159
  5. Gauss (1800/1863), p. 311
  6. Cayley (1854), p. 135
  7. 7.0 7.1 7.2 Fricke & Klein (1897), §12–13
  8. Cayley (1879), p. 238f.
  9. Helmholtz (1866/67), p. 513
  10. Klein (1871), pp. 601–602
  11. Klein (1871), p. 618
  12. Klein (1873), pp. 127-128
  13. Klein (1872), 6
  14. Wedekind (1875), 1
  15. Klein (1875), §1–2
  16. Klein (1878), 8.
  17. Klein (1882), p. 173.
  18. Klein (1884), Part I, Ch. I, §1–2; Part II, Ch. II, 10
  19. Klein (1893a), p. 109ff; pp. 138–140; pp. 249–250
  20. Klein (1893b); general surface: pp. 61–66, 116–119, hyperbolic space: pp. 82, 86, 143–144
  21. Klein (1896/97), pp. 13–14
  22. Selling (1873), p. 174 and p. 179
  23. Selling (1873), pp. 182–183
  24. Selling (1873/74), p. 227 (see also p. 225 for citation).
  25. Poincaré (1881a), pp. 133–134
  26. Poincaré (1881b), p. 333
  27. Poincaré (1883), pp. 49–50; 53–54
  28. Poincaré (1886), p. 735ff.
  29. Bianchi (1888), pp. 547; 562–563 (especially footnote on p. 563); 571–572
  30. 30.0 30.1 Bianchi (1893), § 3
  31. Fricke (1891), §§ 1, 6
  32. Fricke (1893), pp. 706, 710–711
  33. Woods (1895), pp. 2–3; 10–11; 34–35

Historical relativity sources[edit | edit source]

  1. 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. 1.0 1.1 Synge (1956), ch. IV, 11
  2. Klein (1928), § 3A
  3. Penrose & Rindler (1984), section 2.1
  4. 4.0 4.1 Lorente (2003), section 4
  5. Penrose & Rindler (1984), p. 17
  6. Klein (1928), § 2A
  7. Dickson (1923), p. 210
  8. Bachmann (1923), chapter 16
  9. Dickson (1923), pp. 221, 232