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

History of Lorentz transformation ()
Introduction Chapter 1: Most general Lorentz transformations Chapter 2: LT via imaginary orthogonal transformation Chapter 3: LT via hyperbolic functions Chapter 4: LT via velocity Chapter 5: LT via sphere transformation Chapter 6: LT via Cayley–Hermite transformation Chapter 7: LT via Cayley–Klein parameters, Möbius and spin transformations Chapter 8: LT via quaternions and hyperbolic numbers Chapter 9: LT via trigonometric functions Chapter 10: LT via squeeze mappings
History of Topics in Special Relativity () Multiple chapter topics History of Lorentz transformations History of relativistic four vectors History of relativistic tensors Single page topics History of twin paradox History of time dilation History of three-acceleration transformation Müller's history of relativity

Lorentz transformation via imaginary orthogonal transformation

By using the imaginary quantities $[{\mathfrak {x}}_{0},\ {\mathfrak {x}}'_{0}]=\left[ix_{0},\ ix_{0}^{\prime }\right]$ in x as well as $[{\mathfrak {g}}_{0s},\ {\mathfrak {g}}_{s0}]=\left[ig_{0s},\ ig_{s0}\right]$ (s=1,2...n) in g, the E: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:

${\begin{matrix}{\begin{aligned}{\mathfrak {x}}_{0}^{2}+x_{1}^{2}+\cdots +x_{n}^{2}&={\mathfrak {x}}_{0}^{\prime 2}+x_{1}^{\prime 2}+\dots +x_{n}^{\prime 2}\\{\mathfrak {x}}_{0}{\mathfrak {y}}_{0}+x_{1}y_{1}+\cdots +x_{n}y_{n}&={\mathfrak {x}}_{0}^{\prime }{\mathfrak {y}}_{0}^{\prime }+x_{1}^{\prime }y_{1}^{\prime }+\cdots +x_{n}^{\prime }y_{n}^{\prime }\end{aligned}}\\\hline {\begin{matrix}\mathbf {x} '=\mathbf {g} \cdot \mathbf {x} \\\mathbf {x} =\mathbf {\mathbf {g} ^{-1}} \cdot \mathbf {x} '\end{matrix}}\left|{\begin{aligned}\sum _{i=0}^{n}g_{ij}g_{ik}&=\left\{{\begin{aligned}1\quad &(j=k)\\0\quad &(j\neq k)\end{aligned}}\right.\\\sum _{j=0}^{n}g_{ij}g_{kj}&=\left\{{\begin{aligned}1\quad &(i=k)\\0\quad &(i\neq k)\end{aligned}}\right.\end{aligned}}\right.\end{matrix}}$

(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 $\phi =i\eta$, so that trigonometric functions become equivalent to w:hyperbolic functions:

${\begin{array}{c|c|cc}{\mathfrak {x}}_{0}^{2}+x_{1}^{2}+x_{2}^{2}={\mathfrak {x}}_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}&\left(ix_{0}\right){}^{2}+x_{1}^{2}+x_{2}^{2}=\left(ix_{0}^{\prime }\right)^{2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}&&-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline (1){\begin{aligned}{\mathfrak {x}}_{0}^{\prime }&={\mathfrak {x}}_{0}\cos \phi -x_{1}\sin \phi \\x_{1}^{\prime }&={\mathfrak {x}}_{0}\sin \phi +x_{1}\cos \phi \\x_{2}^{\prime }&=x_{2}\\\\{\mathfrak {x}}_{0}&={\mathfrak {x}}_{0}^{\prime }\cos \phi +x_{1}^{\prime }\sin \phi \\x_{1}&=-{\mathfrak {x}}_{0}^{\prime }\sin \phi +x_{1}^{\prime }\cos \phi \\x_{2}&=x_{2}^{\prime }\end{aligned}}&(2){\begin{aligned}ix_{0}^{\prime }&=ix_{0}\cos i\eta -x_{1}\sin i\eta \\x_{1}^{\prime }&=ix_{0}\sin i\eta +x_{1}\cos i\eta \\x_{2}^{\prime }&=x_{2}\\\\ix_{0}&=ix_{0}^{\prime }\cos i\eta +x_{1}^{\prime }\sin i\eta \\x_{1}&=-ix_{0}^{\prime }\sin i\eta +x_{1}^{\prime }\cos i\eta \\x_{2}&=x_{2}^{\prime }\end{aligned}}&\rightarrow &{\begin{aligned}x_{0}^{\prime }&=x_{0}\cosh \eta -x_{1}\sinh \eta \\x_{1}^{\prime }&=-x_{0}\sinh \eta +x_{1}\cosh \eta \\x_{2}^{\prime }&=x_{2}\\\\x_{0}&=x_{0}^{\prime }\cosh \eta +x_{1}^{\prime }\sinh \eta \\x_{1}&=x_{0}^{\prime }\sinh \eta +x_{1}^{\prime }\cosh \eta \\x_{2}&=x_{2}^{\prime }\end{aligned}}\end{array}}$

(2b)

or in exponential form using w:Euler's formula $e^{i\phi }=\cos \phi +i\sin \phi$:

${\begin{array}{c|c|cc}{\mathfrak {x}}_{0}^{2}+x_{1}^{2}+x_{2}^{2}={\mathfrak {x}}_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}&\left(ix_{0}\right){}^{2}+x_{1}^{2}+x_{2}^{2}=\left(ix_{0}^{\prime }\right)^{2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}&&-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline (1){\begin{aligned}x_{1}^{\prime }+i{\mathfrak {x}}_{0}^{\prime }&=e^{-i\phi }\left(x_{1}+i{\mathfrak {x}}_{0}\right)\\x_{1}^{\prime }-i{\mathfrak {x}}_{0}^{\prime }&=e^{i\phi }\left(x_{1}-i{\mathfrak {x}}_{0}\right)\\x_{2}^{\prime }&=x_{2}\\\\x_{1}+i{\mathfrak {x}}_{0}&=e^{i\phi }\left(x_{1}^{\prime }+i{\mathfrak {x}}_{0}^{\prime }\right)\\x_{1}-i{\mathfrak {x}}_{0}&=e^{-i\phi }\left(x_{1}^{\prime }-i{\mathfrak {x}}_{0}^{\prime }\right)\\x_{2}&=x_{2}^{\prime }\end{aligned}}&(2){\begin{aligned}x_{1}^{\prime }+i\left(ix_{0}^{\prime }\right)&=e^{-i(i\eta )}\left(x_{1}+i\left(ix_{0}\right)\right)\\x_{1}^{\prime }-i\left(ix_{0}^{\prime }\right)&=e^{i(i\eta )}\left(x_{1}-i\left(ix_{0}\right)\right)\\x_{2}^{\prime }&=x_{2}\\\\x_{1}+i\left(ix_{0}\right)&=e^{i(i\eta )}\left(x_{1}^{\prime }+i\left(ix_{0}^{\prime }\right)\right)\\x_{1}-i\left(ix_{0}\right)&=e^{-i(i\eta )}\left(x_{1}^{\prime }-i\left(ix_{0}^{\prime }\right)\right)\\x_{2}&=x_{2}^{\prime }\end{aligned}}&\rightarrow &{\begin{aligned}x_{1}^{\prime }-x_{0}^{\prime }&=e^{\eta }\left(x_{1}-x_{0}\right)\\x_{1}^{\prime }+x_{0}^{\prime }&=e^{-\eta }\left(x_{1}+x_{0}\right)\\x_{2}^{\prime }&=x_{2}\\\\x_{1}-x_{0}&=e^{-\eta }\left(x_{1}^{\prime }-x_{0}^{\prime }\right)\\x_{1}+x_{0}&=e^{\eta }\left(x_{1}^{\prime }+x_{0}^{\prime }\right)\\x_{2}&=x_{2}^{\prime }\end{aligned}}\end{array}}$

(2c)

Defining $[{\mathfrak {x}}_{0},\ {\mathfrak {x}}'_{0},\ \phi ]$ 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 $[{\mathfrak {x}}_{0},\ {\mathfrak {x}}'_{0},\ \phi ]$ correspond to the imaginary quantities $[ix_{0},\ ix'_{0},\ i\eta ]$ was given by Minkowski (1907) and Sommerfeld (1909). As shown in the next section using hyperbolic functions, (2b) becomes E:(3b) while (2c) becomes E:(3c).

Historical notation

Euler (1771) – Orthogonal transformation

See also: History of Lorentz transformations via hyperbolic functions § Euler

See also: History of Lorentz transformations via Cayley-Hermite transformations § Euler

See also: History of Lorentz transformations via Quaternions § Euler

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

${\begin{matrix}X^{2}+Y^{2}=x^{2}+y^{2}\\\hline {\begin{aligned}X&=\alpha x+\beta y\\Y&=\gamma x+\delta y\end{aligned}}\left|{\begin{matrix}{\begin{aligned}1&=\alpha \alpha +\gamma \gamma \\1&=\beta \beta +\delta \delta \\0&=\alpha \beta +\gamma \delta \end{aligned}}\end{matrix}}\right.\\\hline {\begin{aligned}X&=x\cos \zeta +y\sin \zeta \\Y&=x\sin \zeta -y\cos \zeta \end{aligned}}\end{matrix}}$

or three dimensions

${\begin{matrix}X^{2}+Y^{2}+Z^{2}=x^{2}+y^{2}+z^{2}\\\hline {\begin{aligned}X&=Ax+By+Cz\\Y&=Dx+Ey+Fz\\Z&=Gx+Hy+Iz\end{aligned}}{\begin{matrix}\left|{\scriptstyle {\begin{aligned}1&=AA+DD+GG\\1&=BB+EE+HH\\1&=CC+FF+II\\0&=AB+DE+GH\\0&=AG+DF+GI\\0&=BC+EF+HI\end{aligned}}}\right.\end{matrix}}\\\hline {\begin{aligned}x'&=x\cos \zeta +y\sin \zeta &x''&=x'\cos \eta +z'\sin \eta \\y'&=x\sin \zeta -y\cos \zeta &y''&=y'\\z'&=z&z''&=x'\sin \eta -z'\cos \eta \\\\x'''&=x''&=X\\y'''&=y''\cos \theta +z''\sin \theta &=Y\\z'''&=y''\sin \theta -z''\cos \theta &=Z\end{aligned}}\end{matrix}}$

The orthogonal transformation in four dimensions was given by him as

${\begin{matrix}V^{2}+X^{2}+Y^{2}+Z^{2}=v^{2}+x^{2}+y^{2}+z^{2}\\\hline {\begin{aligned}V&=Av+Bx+Cy+Dz\\X&=Ev+Fx+Gy+Hz\\Y&=Iv+Kx+Ly+Mz\\Z&=Nv+Ox+Py+Qz\end{aligned}}{\begin{matrix}\left|{\scriptstyle {\begin{aligned}1&=AA+RR+II+NN&0&=AB+EF+IK+NO\\1&=BB+FF+KK+OO&0&=AC+EG+IL+NP\\1&=CC+GG+LL+PP&0&=AD+EH+IM+NQ\\1&=DD+HH+MM+QQ&0&=BC+FG+KL+OP\\0&=BD+FH+KM+OQ&0&=CD+FH+LM+PQ\end{aligned}}}\right.\end{matrix}}\\\hline {\scriptstyle {\begin{aligned}x^{I}&=x\cos \alpha +y\sin \alpha &&&x^{VI}&=x^{V}&=X\\y^{I}&=x\sin \alpha -y\cos \alpha &&&y^{VI}&=y^{V}&=Y\\z^{I}&=z&\dots &\dots &y^{VI}&=z^{V}\cos \zeta +v^{V}\sin \zeta &=Z\\v^{I}&=v&&&v^{VI}&=z^{V}\sin \zeta -v^{V}\cos \varepsilon \zeta &=V\end{aligned}}}\end{matrix}}$

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

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

$e^{+v{\sqrt {-1}}}=\cos v+{\sqrt {-1}}\sin v,\quad e^{-v{\sqrt {-1}}}=\cos v-{\sqrt {-1}}\sin v$,

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

$x''+\varepsilon z''=(x'+\varepsilon z')\cdot (\cos III+\varepsilon \sin III),\ (\varepsilon ={\sqrt {-1}})$

Replacing the real quantities by imaginary ones by setting $\left[z',z'',III\right]=\left[iz',iz'',iIII\right]$, Wessel's transformation becomes Lorentz transformation (2c).

Cauchy (1829) – Orthogonal transformation

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

${\begin{matrix}x^{2}+y^{2}+z^{2}+\dots =\xi ^{2}+\eta ^{2}+\zeta ^{2}+\dots \\\hline {\begin{aligned}x&=x_{1}\xi +x_{2}\eta +x_{3}\zeta +\dots \\y&=y_{1}\xi +y_{2}\eta +y_{3}\zeta +\dots \\z&=z_{1}\xi +z_{2}\eta +z_{3}\zeta +\dots \\&\dots \\\\\xi &=x_{1}x+y_{1}y+z_{1}z+\dots \\\eta &=x_{2}x+y_{2}y+z_{2}z+\dots \\\zeta &=x_{3}x+y_{3}y+z_{3}z+\dots \\&\dots \end{aligned}}\left|{\scriptstyle {\begin{aligned}x_{1}^{2}+y_{1}^{2}+z_{1}^{2}+\dots &=1,\\x_{2}x_{1}+y_{2}y_{1}+z_{2}z_{1}+\dots &=0,\\\dots \\x_{n}x_{1}+y_{n}y_{1}+z_{n}z_{1}+\dots &=0,\\\\x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}+\dots &=0,\\x_{2}^{2}+y_{2}^{2}+z_{2}^{2}+\dots &=1,\\{\text{ }}\dots \\x_{n}x_{2}+y_{n}y_{2}+z_{n}z_{2}+\dots &=0,\\\\x_{1}x_{n}+y_{1}y_{n}+z_{1}z_{n}+\dots &=0,\\x_{2}x_{n}+y_{2}y_{n}+z_{2}z_{n}+\dots &=0,\\\dots \\x_{n}^{2}+y_{n}^{2}+z_{n}^{2}+\dots &=1\end{aligned}}}\right.\end{matrix}}$

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

See also: History of Lorentz transformations in general § Lie

See also: History of Lorentz transformations via sphere transformations § Lie

See also: History of Lorentz transformations via squeeze mappings § Lie

w:Sophus Lie (1871a) described a manifold whose elements can be represented by spheres, where the last coordinate y_n+1 can be related to an imaginary radius by iy_n+1:

${\begin{matrix}\sum _{i=1}^{i=n}(x_{i}-y_{i})^{2}+y_{n+1}^{2}=0\\\downarrow \\\sum _{i=1}^{i=n+1}(y_{i}^{\prime }-y_{i}^{\prime \prime })^{2}=0\end{matrix}}$

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

Minkowski (1907–1908) – Spacetime

See also: History of Lorentz transformations via velocity § Minkowski

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. Minkowski particularly reformulated electrodynamics in a four-dimensional way (w:Minkowski spacetime). 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):

${\begin{aligned}x'_{1}&=x_{1}\\x'_{2}&=x_{2}\\x'_{3}&=x_{3}\cos i\psi +x_{4}\sin i\psi \\x'_{4}&=-x_{3}\sin i\psi +x_{4}\cos i\psi \\\cos i\psi &={\frac {1}{\sqrt {1-q^{2}}}}\end{aligned}}$

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

$-i\tan i\psi ={\frac {e^{\psi }-e^{-\psi }}{e^{\psi }+e^{-\psi }}}=q$ with $\psi ={\frac {1}{2}}\ln {\frac {1+q}{1-q}}$.

Minkowski's expression can also by written as ψ=atanh(q) and was later called w:rapidity.

Sommerfeld (1909) – Spherical trigonometry

Using an imaginary rapidity such as Minkowski, w:Arnold Sommerfeld (1909) formulated a transformation equivalent to Lorentz boost (2b), and the relativistc velocity addition E:(4d) in terms of trigonometric functions and the w:spherical law of cosines:

${\begin{matrix}\left.{\begin{array}{lrl}x'=&x\ \cos \varphi +l\ \sin \varphi ,&y'=y\\l'=&-x\ \sin \varphi +l\ \cos \varphi ,&z'=z\end{array}}\right\}\\\left(\operatorname {tg} \varphi =i\beta ,\ \cos \varphi ={\frac {1}{\sqrt {1-\beta ^{2}}}},\ \sin \varphi ={\frac {i\beta }{\sqrt {1-\beta ^{2}}}}\right)\\\hline \beta ={\frac {1}{i}}\operatorname {tg} \left(\varphi _{1}+\varphi _{2}\right)={\frac {1}{i}}{\frac {\operatorname {tg} \varphi _{1}+\operatorname {tg} \varphi _{2}}{1-\operatorname {tg} \varphi _{1}\operatorname {tg} \varphi _{2}}}={\frac {\beta _{1}+\beta _{2}}{1+\beta _{1}\beta _{2}}}\\\cos \varphi =\cos \varphi _{1}\cos \varphi _{2}-\sin \varphi _{1}\sin \varphi _{2}\cos \alpha \\v^{2}={\frac {v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos \alpha -{\frac {1}{c^{2}}}v_{1}^{2}v_{2}^{2}\sin ^{2}\alpha }{\left(1+{\frac {1}{c^{2}}}v_{1}v_{2}\cos \alpha \right)^{2}}}\end{matrix}}$

References

Historical mathematical sources

• Cauchy, A.L. (1829), "Sur l'équation à l'aide de laquelle on détermine les inégalités séculaires des mouvements des planètes", Exercises de Mathématiques, IV: 140–160. Reprinted with corrections Œuvres complètes, série 2, tome 9.
• Euler, L. (1748b), Introductio in analysin infinitorum, volume 1, Lausanne
• Euler, L. (1771) [1770], "Problema algebraicum ob affectiones prorsus singulares memorabile", Novi Commentarii Academiae Scientiarum Petropolitanae, 15: 75–106
• Lie, S. (1871a), "Ueber diejenige Theorie eines Raumes mit beliebig vielen Dimensionen, die der Krümmungs-Theorie des gewöhnlichen Raumes entspricht", Göttinger Nachrichten: 191–209
• Wessel, C. (1799) [1797], "Om directionens analytiske betegning", Royal Danish Academy of Sciences and Letters, 7 (4): 469–518; French translation (1897): Essai sur la représentation analytique de la direction

Historical relativity sources

• Minkowski, H. (1915) [1907], "Das Relativitätsprinzip", Annalen der Physik, 352 (15): 927–938

• Das Relativitätsprinzip on German Wikisource

• Minkowski, H. (1908) [1907], "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern", Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111

• Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern on German Wikisource
• The Fundamental Equations for Electromagnetic Processes in Moving Bodies on English Wikisource

• Minkowski, H. (1909) [1908], "Raum und Zeit", Physikalische Zeitschrift, 10: 75–88

• Raum und Zeit on German Wikisource
• Space and Time on English Wikisource

• Sommerfeld, A. (1909), "Über die Zusammensetzung der Geschwindigkeiten in der Relativtheorie", Verh. Der DPG, 21: 577–582

• On the Composition of Velocities in the Theory of Relativity on English Wikisource

Secondary sources

• von Laue, M. (1921), Die Relativitätstheorie, Band 1 (fourth edition of "Das Relativitätsprinzip" ed.), Vieweg; First edition 1911, second expanded edition 1913, third expanded edition 1919.
• Walter, S. A. (1999a), "Minkowski, mathematicians, and the mathematical theory of relativity", in H. Goenner; J. Renn; J. Ritter; T. Sauer (eds.), The Expanding Worlds of General Relativity – Einstein Studies, vol. 7, Boston: Birkhäuser, pp. 45–86, ISBN 978-0-8176-4060-6