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

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History of Lorentz transformation (edit)

Lorentz transformation via hyperbolic functions[edit | edit source]

Translation in the hyperbolic plane[edit | edit source]

A ray through the unit hyperbola x2y2 = 1 at the point (cosh a, sinh a).

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 E:most general Lorentz transformation (1a):

or in matrix notation

 

 

 

 

(3a)

which resembles precisely the relations of w:hyperbolic functions in terms of w:hyperbolic angle . Thus a Lorentz boost or w:hyperbolic rotation (being the same as a rotation around an imaginary angle in E:(2b) or a translation in the hyperbolic plane in terms of the hyperboloid model) is given by

or in matrix notation

 

 

 

 

(3b)

Hyperbolic identities (a,b) on the right of (3b) were given by Riccati (1757), all identities (a,b,c,d,e,f) by Lambert (1768–1770). Lorentz transformations (3b-A) were given by Laisant (1874), Cox (1882), Lindemann (1890/91), Gérard (1892), Killing (1893, 1897/98), Whitehead (1897/98), Woods (1903/05), Elliott (1903) and Liebmann (1904/05) in terms of Weierstrass coordinates of the w:hyperboloid model, while transformations similar to (3b-C) have been used by Lipschitz (1885/86). In special relativity, hyperbolic functions were used by Frank (1909) and Varićak (1910).

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

 

 

 

 

(3c)

Lorentz transformations (3c) for arbitrary k were given by many authors (see E:Lorentz transformations via squeeze mappings), while a form similar to was given by Lipschitz (1885/86), and the exponential form was implicitly used by Mercator (1668) and explicitly by Lindemann (1890/91), Elliott (1903), Herglotz (1909).

Rapidity can be composed of arbitrary many rapidities as per the w: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 w: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

or in exponential form as squeeze mapping analogous to (3c):

 

 

 

 

(3d)

Hyperbolic angle sum laws were given by Riccati (1757) and Lambert (1768–1770) and many others, while matrix representations were given by Glaisher (1878) and Günther (1880/81).

Hyperbolic law of cosines[edit | edit source]

By adding an unchanged third coordinate in Lorentz transformation (3b) and interpreting as w:homogeneous coordinates, the Lorentz transformation can be rewritten in line with equation E:(1b) by using coordinates defined by inside the w:unit circle as follows:

 

 

 

 

(3e)

Transformations (A) were given by Escherich (1874) and Killing (1898), and transformations (C) by Beltrami (1868) and Schur (1885/86, 1900/02) 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]

 

 

 

 

(3f)

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). By further setting or it follows:

 

 

 

 

(3g)

Formulas (3g-B) are the equations of an w:ellipse of eccentricity v, w:eccentric anomaly α' and w:true anomaly α, first geometrically formulated by Kepler (1609) and explicitly written down by Euler (1735, 1748), Lagrange (1770) and many others in relation to planetary motions. They were also used by E:Darboux (1873) as a sphere transformation. In special relativity, these formulas describes the aberration of light, see E:velocity addition and aberration.

Historical notation[edit | edit source]

Mercator (1668) – hyperbolic relations[edit | edit source]

Mercator's (1668) illustration of AH·FH=AI·BI.

While deriving the w:Mercator series, w:Nicholas Mercator (1668) demonstrated the following relations on a rectangular hyperbola:[M 1]

It can be seen that Mercator's relations , with implicitly correspond to hyperbolic functions , with (which were explicitly introduced by Riccati (1757) much later). In particular, his result AH.AI::BI.FH, denoting that the ratio between AH and AI is equal to the ratio between BI and FH or in modern notation, corresponds to squeeze mapping or Lorentz boost (3c) because:

or solved for AH and FH:

and .

Furthermore, transforming Mercator's asymptotic coordinates , into Cartesian coordinates gives:


which produces the unit hyperbola as in (3d), in agreement with Mercator's result AH·FH=1/2 when the hyperbola is referred to its asymptotes.

Euler (1735) – True and eccentric anomaly[edit | edit source]

w:Johannes Kepler (1609) geometrically formulated w:Kepler's equation and the relations between the w:mean anomaly, w:true anomaly, and w:eccentric anomaly.[M 2][5] The relation between the true anomaly z and the eccentric anomaly P was algebraically expressed by w:Leonhard Euler (1735/40) as follows:[M 3]

and in 1748:[M 4]

while w:Joseph-Louis Lagrange (1770/71) expressed them as follows[M 5]

These relations resemble formulas (3g), while (3e) follows by setting in Euler's formulas or in Lagrange's formulas.

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

Riccati's (1757) illustration of hyperbolic addition laws.

w:Vincenzo Riccati (1757) introduced hyperbolic functions cosh and sinh, which he denoted as Ch. and Sh. related by with r being set to unity in modern publications, and formulated the addition laws of hyperbolic sine and cosine:[M 6][M 7]

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 (3d) with .

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 8][6]

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:[M 9]

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 E:(4a) and E:(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 (3d).

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 10][7][8]

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 E:(4e).

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 11]

(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 12]

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

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

In his French translation of w:Giusto Bellavitis' principal work on w: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]

(1)

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[9] 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]

and
This is equivalent to Lorentz transformation (3e), also equivalent to the relativistic velocity addition E:(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 E:Killing (1878–1893), E:Poincaré (1881), and E: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 expressed by matrix multiplication:[M 18]

In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix producing the hyperbolic addition is analogous to matrix producing Lorentz boost (3b) and (3d).

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

Following Glaisher (1878), w:Siegmund Günther (1880/81) expressed the hyperbolic addition laws by matrix multiplication:[M 19]

In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix producing the hyperbolic addition is analogous to matrix producing Lorentz boost (3b) and (3d).

Cox (1881/82) – Weierstrass coordinates[edit | edit source]

w:Homersham Cox (1881/82) defined the case of translation in the hyperbolic plane with the y-axis remaining unchanged:[M 20]

This is equivalent to Lorentz boost (3b).

Lipschitz (1885/86) – Quadratic forms[edit | edit source]

w:Rudolf Lipschitz (1885/86) discussed transformations leaving invariant the sum of squares

which he rewrote as

.

This led to the problem of finding transformations leaving invariant the pairs (where a=1...n) for which he gave the following solution:[M 21]

Lipschitz's transformations (c) and (a) are equivalent to Lorentz boosts (3b-C) and (3c) by the identity . That is, by substituting in (3b-C) or (3c) we obtain Lipschitz's transformations.

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 22]

This is equivalent to Lorentz transformation (3e) and therefore also equivalent to the relativistic velocity addition E:(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 23]

where can have values >0, <0 or ∞.

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

He also defined the triangle[M 24]

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

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

w:Ferdinand von Lindemann discussed hyperbolic geometry in terms of the w:Cayley–Klein metric in his (1890/91) edition of the lectures on geometry of w:Alfred Clebsch. Citing E: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 25]

and used the following transformation[M 26]

into which he put[M 27]

This is equivalent to Lorentz boost (3c) 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 27]

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 28]

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 29]

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 30]

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

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 31]

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

Elliott (1903) – Invariant theory[edit | edit source]

w:Edwin Bailey Elliott (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:[M 32]

This is equivalent to Lorentz boost (3b) and (3c). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.

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

w:Frederick S. Woods (1903, published 1905) gave the case of translation in hyperbolic space:[M 33]

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

and the loxodromic substitution for hyperbolic space:[M 34]

This is equivalent to Lorentz boost (3b) with β=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 35]

This is equivalent to Lorentz boost (3b).

Frank (1909) – Special relativity[edit | edit source]

In special relativity, hyperbolic functions were used by w:Philipp Frank (1909), who derived the Lorentz transformation using ψ as rapidity:[R 2]

This is equivalent to Lorentz boost (3b).

Herglotz (1909/10) – Special relativity[edit | edit source]

In special relativity, w:Gustav Herglotz (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:[R 3]

This is equivalent to Lorentz boost (3c).

Varićak (1910) – Special relativity[edit | edit source]

In special relativity, hyperbolic functions were used 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, by setting l=ct and v/c=tanh(u) with u as rapidity he wrote the Lorentz transformation in agreement with (4b):[R 4]

This is equivalent to Lorentz boost (3b).

He showed the relation of rapidity to the w:Gudermannian function and the w:angle of parallelism:[R 4]

He also related the velocity addition to the w:hyperbolic law of cosines:[R 5]

This is equivalent to Lorentz boost (3f).

References[edit | edit source]

Historical mathematical sources[edit | edit source]

  1. Mercator (1667), prop. XIV, pp. 28-29. (He used this result to derive the Mercator series in prop. XV).
  2. Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to and and
  3. Euler (1735/40), § 19
  4. Euler (1748a), section VIII
  5. Lagrange (1770/71), section I
  6. Riccati (1757), p. 71
  7. Günther (1880/81), pp. 7–13
  8. Lambert (1761/68), pp. 309–318
  9. Lambert (1770), p. 335
  10. Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)
  11. Beltrami (1868a), pp. 287-288; Note I; Note II
  12. Beltrami (1868b), pp. 232, 240–241, 253–254
  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. Cox (1881/82), p. 194
  21. Lipschitz (1886), pp. 90–92
  22. Schur (1885/86), p. 167
  23. Schur (1900/02), p. 290; (1909), p. 83
  24. Schur (1900/02), p. 291; (1909), p. 83
  25. Lindemann & Clebsch (1890/91), pp. 477–478, 524
  26. Lindemann & Clebsch (1890/91), pp. 361–362
  27. 27.0 27.1 Lindemann & Clebsch (1890/91), p. 496
  28. Gérard (1892), pp. 40–41
  29. Killing (1893), p. 331
  30. Killing (1898), p. 133
  31. Whitehead (1898), pp. 459–460
  32. Elliott (1903), p. 109
  33. Woods (1903/05), p. 55
  34. Woods (1903/05), p. 72
  35. Liebmann (1904/05), p. 174

Historical relativity sources[edit | edit source]

  1. Varićak (1912), p. 108
  2. Frank (1909), pp. 423-425
  3. Herglotz (1909/10), pp. 404-408
  4. 4.0 4.1 Varićak (1910), p. 93
  5. Varićak (1910), p. 94
  • Varićak, V. (1910), "Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie", Physikalische Zeitschrift, 11: 93–6
  • Varičak, V. (1912), "Über die nichteuklidische Interpretation der Relativtheorie", Jahresbericht der Deutschen Mathematiker-Vereinigung, 21: 103–127

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. Volk (1976), p. 366
  6. Barnett (2004), pp. 22–23
  7. Bonola (1912), p. 79
  8. Gray (1979), p. 242
  9. Sommerville (1911), p. 297