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

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History of Topics in Special Relativity: History of Lorentz transformation

1 - Most general Lorentz transformations

2 - Lorentz transformation via imaginary orthogonal transformation

3 - Lorentz transformation via hyperbolic functions

4 - Lorentz transformation via velocity

5 - Lorentz transformation via conformal, spherical wave, and Laguerre transformation

6 - Lorentz transformation via Cayley–Hermite transformation

7 - Lorentz transformation via Cayley–Klein parameters, Möbius and spin transformations

8 - Lorentz transformation via quaternions and hyperbolic numbers

9 - Lorentz transformation via trigonometric functions

10 - Lorentz transformation via squeeze mappings

Lorentz transformation via imaginary orthogonal transformation[edit | edit source]

By using the imaginary quantities in x as well as (s=1,2...n) in g, the most general Lorentz transformation (1a) assumes the form of an w:orthogonal transformation of w:Euclidean space forming the w:orthogonal group O(n) if det g=±1 or the special orthogonal group SO(n) if det g=+1, the Lorentz interval becomes the w:Euclidean norm, and the Minkowski inner product becomes the w:dot product:[1]

 

 

 

 

(2a)

The cases n=1,2,3,4 of orthogonal transformations in terms of real coordinates were discussed by Euler (1771) and in n dimensions by Cauchy (1829). The case in which one of these coordinates is imaginary and the other ones remain real was alluded to by Lie (1871) in terms of spheres with imaginary radius, while the interpretation of the imaginary coordinate as being related to the dimension of time as well as the explicit formulation of Lorentz transformations with n=3 was given by Minkowski (1907) and Sommerfeld (1909).

A well known example of this orthogonal transformation is spatial w:rotation in terms of w:trigonometric functions, which become Lorentz transformations by using an imaginary angle , so that trigonometric functions become equivalent to w:hyperbolic functions:

 

 

 

 

(2b)

or in exponential form using w:Euler's formula :

 

 

 

 

(2c)

Defining as real, spatial rotation in the form (2b -1) was introduced by Euler (1771) and in the form (2c -1) by Wessel (1799). The interpretation of (2b ) as Lorentz boost (i.e. Lorentz transformation without spatial rotation) in which correspond to the imaginary quantities was given by Minkowski (1907) and Sommerfeld (1909). As shown in the next section using hyperbolic functions, (2b ) becomes (3b ) while (2c ) becomes (3d ).

Historical notation[edit | edit source]

Euler (1771) – Orthogonal transformation[edit | edit source]

w:Leonhard 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, as well as under rotations using w:Euler angles. The case of two dimensions is given by[M 1]

or three dimensions[M 2]

The orthogonal transformation in four dimensions was given by him as[M 3]

As shown by Minkowski (1907), the orthogonal transformation can be directly used as Lorentz transformation (2a ) or (2b ) by making one variable as well as six of the sixteen coefficients imaginary.

Wessel (1799) – Euler's formula and rotation[edit | edit source]

The above orthogonal transformations representing Euclidean rotations can also be expressed by using w:Euler's formula. After this formula was derived by Euler in 1748[M 4]

,

it was used by w:Caspar Wessel (1799) to describe Euclidean rotations in the complex plane:[M 5]

Replacing the real quantities by imaginary ones by setting , Wessel's transformation becomes Lorentz transformation (2c ).

Cauchy (1829) – Orthogonal transformation[edit | edit source]

w:Augustin-Louis Cauchy (1829) extended the orthogonal transformation of Euler (1771) to arbitrary dimensions[M 6]

The orthogonal transformation can be directly used as Lorentz transformation (2a ) by making one of the variables as well as certain coefficients imaginary.

Lie (1871) – Imaginary orthogonal transformations[edit | edit source]

w:Sophus Lie (1871a) described a manifold whose elements can be represented by spheres, where the last coordinate yn+1 can be related to an imaginary radius by iyn+1:[M 7]

If the second equation is satisfied, two spheres y′ and y″ are in contact. Lie then defined the correspondence between contact transformations in Rn and conformal point transformations in Rn+1: The sphere of space Rn consists of n+1 parameter (coordinates plus imaginary radius), so if this sphere is taken as the element of space Rn, it follows that Rn now corresponds to Rn+1. Therefore, any transformation (to which he counted orthogonal transformations and inversions) leaving invariant the condition of contact between spheres in Rn, corresponds to the conformal transformation of points in Rn+1.

Minkowski (1907–1908) – Spacetime[edit | edit source]

The work on the principle of relativity by Lorentz, Einstein, Planck, together with Poincaré's four-dimensional approach, were further elaborated and combined with the w:hyperboloid model by w:Hermann Minkowski in 1907 and 1908.[R 1][R 2] Minkowski particularly reformulated electrodynamics in a four-dimensional way (w:Minkowski spacetime).[2] For instance, he wrote x, y, z, it in the form x1, x2, x3, x4. By defining ψ as the angle of rotation around the z-axis, the Lorentz transformation assumes a form (with c=1) in agreement with (2b ):[R 3]

Even though Minkowski used the imaginary number iψ, he for once[R 3] directly used the w:tangens hyperbolicus in the equation for velocity

with .

Minkowski's expression can also by written as ψ=atanh(q) and was later called w:rapidity. He also wrote the Lorentz transformation in matrix form equivalent to (2a ) (n=3):[R 4]

As a graphical representation of the Lorentz transformation he introduced the w:Minkowski diagram, which became a standard tool in textbooks and research articles on relativity:[R 5]

Original spacetime diagram by Minkowski in 1908.

Sommerfeld (1909) – Spherical trigonometry[edit | edit source]

Using an imaginary rapidity such as Minkowski, w:Arnold Sommerfeld (1909) formulated a transformation equivalent to Lorentz boost (3b ), and the relativistc velocity addition (4d ) in terms of trigonometric functions and the w:spherical law of cosines:[R 6]

References[edit | edit source]

Historical mathematical sources[edit | edit source]

  1. Euler (1771), pp. 84-85
  2. Euler (1771), pp. 77, 85-89
  3. Euler (1771), pp. 89–91
  4. Euler (1748b), section 138.
  5. Wessel (1799), § 28.
  6. Cauchy (1829), eq. 22, 98, 99, 101; Some misprints were corrected in Œuvres complètes, série 2, tome 9, pp. 174–195.
  7. Lie (1871a), pp. 199–209

Historical relativity sources[edit | edit source]

  1. Minkowski (1907/15), pp. 927ff
  2. Minkowski (1907/08), pp. 53ff
  3. 3.0 3.1 Minkowski (1907/08), p. 59
  4. Minkowski (1907/08), pp. 65–66, 81–82
  5. Minkowski (1908/09), p. 77
  6. Sommerfeld (1909), p. 826ff.

Secondary sources[edit | edit source]

  1. Laue (1921), pp. 79–80 for n=3
  2. Walter (1999a)