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

From Wikiversity
Jump to navigation Jump to search
History of Topics in Special Relativity: History of Lorentz transformation (edit)

Lorentz transformation via squeeze mappings[edit | edit source]

As already indicated in equation (3d) in exponential form or equation (6f) in terms of Cayley–Klein parameter, Lorentz boosts in terms of hyperbolic rotations can be expressed as w:squeeze mappings. Using asymptotic coordinates of a hyperbola (u,v), they have the general form (some authors alternatively add a factor of 2 or ):[1]

 

 

 

 

(9a)

That this equation system indeed represents a Lorentz boost can be seen by plugging (1) into (2) and solving for the individual variables:

 

 

 

 

(9b)

Lorentz transformation (9a) of asymptotic coordinates have been used by Laisant (1874) and Günther (1880/81) in relation to elliptic trigonometry, or by Lie (1879-81), Bianchi (1886, 1894), Darboux (1891/94), Eisenhart (1905) as Lie transform[1] of w:pseudospherical surfaces in terms of the w:Sine-Gordon equation, or by Lipschitz (1885/86) in transformation theory. From that, different forms of Lorentz transformation were derived: (9b) by Lipschitz (1885/86), Bianchi (1886, 1894), Eisenhart (1905), trigonometric Lorentz boost (8a) by Bianchi (1886, 1894) and Darboux (1891/94), and trigonometric Lorentz boost (8b) by Eisenhart (1905). Lorentz boost (9b) was rediscovered in the framework of special relativity by w:Hermann Bondi (1964)[2] in terms of w:Bondi k-calculus, by which k can be physically interpreted as Doppler factor. Since (9b) is equivalent to (6f) in terms of Cayley–Klein parameter by setting , it can be interpreted as the 1+1 dimensional special case of Lorentz Transformation (6e) stated by Gauss around 1800 (posthumously published 1863), Selling (1873), Bianchi (1888), Fricke (1891) and Woods (1895).

Variables u, v in (9a) can be rearranged to produce another form of squeeze mapping, resulting in Lorentz transformation (5b) in terms of Cayley-Hermite parameter:

 

 

 

 

(9c)

These Lorentz transformations were given (up to a sign change) by Laguerre (1882), Darboux (1887), Smith (1900) in relation to Laguerre geometry.

On the basis of factors k or a, all previous Lorentz boosts (3b), (4a), (8a), (8b), can be expressed as squeeze mappings as well:

 

 

 

 

(9d)

Squeeze mappings in terms of were used by Darboux (1891/94) and Bianchi (1894), in terms of by Lindemann (1891) and Herglotz (1909), in terms of by Eisenhart (1905), in terms of by Bondi (1964).

Historical notation[edit | edit source]

Laisant (1874) – Elliptic polar coordinates[edit | edit source]

w:Charles-Ange Laisant (1874) extended circular trigonometry to elliptic trigonometry. In his model, polar coordinates x, y of circular trigonometry are related to polar coordinates x', y' of elliptic trigonometry by the relation[M 1]

He noticed the geometrical implication that any elliptic polar system of coordinates obtained by this formula is located on the same equilateral hyperbola having its asymptotes as axes.

This is equivalent to Lorentz transformation (9a).

Lie (1879-84) – Transforming pseudospherical surfaces[edit | edit source]

w:Sophus Lie (1879/80) derived an operation from w:Pierre Ossian Bonnet's (1867) investigations on surfaces of constant curvatures, by which pseudospherical surfaces can be transformed into each other.[M 2] Lie gave explicit formulas for this operation in two papers published in 1881: If are asymptotic coordinates of two principal tangent curves and their respective angle, and is a solution of the Sine-Gordon equation , then the following operation (now called Lie transform) is also a solution from which infinitely many new surfaces of same curvature can be derived:[M 3]

In (1880/81) he wrote these relations as follows:[M 4]

In (1883/84) he showed that the combination of Lie transform O with Bianchi transform I produces w:Bäcklund transform B of pseudospherical surfaces:[M 5]

As shown by Bianchi (1886) and Darboux (1891/94), the Lie transform is equivalent to Lorentz transformations (9a) and (9b) in terms of null coordinates 2s=u+v and 2σ=u-v. In general, it can be shown that the Sine-Gordon equation is Lorentz invariant.

Günther (1880/81) – Elliptic polar coordinates[edit | edit source]

Following Laisant (1874), w:Siegmund Günther (1880/81) demonstrated the relation between circular polar coordinates and elliptic polar coordinates as[M 6]

showing that any elliptic polar system of coordinates obtained by this formula is located on the same equilateral hyperbola having its asymptotes as axes.

This is equivalent to Lorentz transformation (9a).

Laguerre (1882) – Laguerre inversion[edit | edit source]

After previous work by w:Albert Ribaucour (1870),[M 7] a transformation which transforms oriented spheres into oriented spheres, oriented planes into oriented planes, and oriented lines into oriented lines, was explicitly formulated by w:Edmond Laguerre (1882) as "transformation by reciprocal directions" which was later called "Laguerre inversion/transformation". It can be seen as a special case of the conformal group in terms of Lie's transformations of oriented spheres. In two dimensions the transformation or oriented lines has the form (R being the radius):[M 8]

This is equivalent (up to a sign change for R) to a squeeze mapping in terms of Lorentz boost (9c).

Darboux (1883–1891)[edit | edit source]

Transforming pseudospherical surfaces[edit | edit source]

w:Gaston Darboux (1883) represented Lie's transformation (1879/81) of pseudospheres into each other as follows:[M 9]

This becomes Lorentz boost (9a) by interpreting x, y as null coordinates.

Similar to Bianchi (1886), Darboux (1891/94) showed that the Lie transform gives rise to the following relations:[M 10]

.

Equations (1) together with transformation (2) gives Lorentz boost (9a) in terms of null coordinates.

Laguerre inversion[edit | edit source]

Following Laguerre (1882), w:Gaston Darboux (1887) formulated the Laguerre inversions in four dimensions using coordinates x,y,z,R:[M 11]

This is equivalent (up to a sign change for R) to a squeeze mapping in terms of Lorentz boost (9c) and (9d) where Darboux's k corresponds to a.

Lipschitz (1885/86)[edit | edit source]

w:Rudolf Lipschitz (1885/86) formulated transformations leaving invariant the sum of squares , which he rewrote as . This led to the problem of finding transformations leaving invariant the pairs (a=1...n) for which he gave the following solution:[M 12]

Equation system (1) represents Lorentz boost or squeeze mapping (9a), and (2) represents Lorentz boost (9b).

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

w:Luigi Bianchi (1886) investigated Lie's transformation (1880) of pseudospheres into each other, obtaining the result:[M 13]

.

Equations (1) together with transformation (2) gives Lorentz boost (9a) in terms of null coordinates. Plugging equations (4) into (3) gives Lorentz boost (9b) in terms of Bondi's k factor.

In 1894, Bianchi redefined the variables u,v as asymptotic coordinates, by which the transformation obtains the form:[M 14]

.

This is consisten with one of the choices in (9d) where Bianchi's angle σ corresponds to θ.

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

w:Ferdinand von Lindemann (1890/91) employed the Cayley absolute related to surfaces of second degree and its transformation[M 15]

into which he put[M 16]

This is equivalent to squeeze mapping (9a) and equation (9d) with and 2k=1 .

Smith (1900) – Laguerre inversion[edit | edit source]

w:Percey F. Smith (1900) followed Laguerre (1882) and Darboux (1887) and defined the Laguerre inversion as follows:[M 17]

This is equivalent (up to a sign change) to Lorentz transformation (9c).

Eisenhart (1905) – Transforming pseudospherical surfaces[edit | edit source]

w:Luther Pfahler Eisenhart (1905) followed Bianchi (1886, 1894) and Darboux (1891/94) by writing Lie's transformation (1879/81) of pseudospherical surfaces:[M 18]

.

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. Eisenhart's angle σ corresponds to ϑ in (9d).

References[edit | edit source]

Historical mathematical sources[edit | edit source]

  1. Laisant (1874a), pp. 73–76
  2. Lie (1879/80), Collected papers, vol. 3, p. 389
  3. Lie (1879/81), Collected papers, vol. 3, p. 393
  4. Lie (1880/81), Collected papers, vol. 3, pp. 477–478
  5. Lie (1883/84), Collected papers, vol. 3, p. 556
  6. Günther (1880/81), pp. 383–385
  7. Ribaucour (1870)
  8. Laguerre (1882), pp. 550–551.
  9. Darboux (1883), p. 849
  10. Darboux (1891/94), pp. 381–382
  11. Darboux (1887)
  12. Lipschitz (1886), pp. 90–92
  13. 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.
  14. Bianchi (1894), pp. 433–434
  15. Lindemann & Clebsch (1890/91), pp. 361–362
  16. Lindemann & Clebsch (1890/91), p. 496
  17. Smith (1900), p. 159
  18. Eisenhart (1905), p. 126

Secondary sources[edit | edit source]

  1. 1.0 1.1 Terng & Uhlenbeck (2000), p. 21
  2. Bondi (1964), p. 118