History of Lorentz transformations

The history of Lorentz transformations comprises the development of linear transformations forming the Lorentz group or Poincaré group preserving the Lorentz interval $-x_{0}^{2}+\cdots +x_{n}^{2}$ and the Minkowski inner product $-x_{0}y_{0}+\cdots +x_{n}y_{n}$ .

In mathematics, transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory of quadratic forms, hyperbolic geometry, Möbius geometry, and sphere geometry, which is connected to the fact that the group of motions in hyperbolic space, the Möbius group or projective special linear group, and the Laguerre group are isomorphic to the Lorentz group.

In physics, Lorentz transformations became known at the beginning of the 20th century, when it was discovered that they exhibit the symmetry of Maxwell's equations. Subsequently, they became fundamental to all of physics, because they formed the basis of special relativity in which they exhibit the symmetry of Minkowski spacetime, making the speed of light invariant between different inertial frames. They relate the spacetime coordinates of two arbitrary inertial frames of reference with constant relative speed v. In one frame, the position of an event is given by x,y,z and time t, while in the other frame the same event has coordinates x′,y′,z′ and t′.

Most general Lorentz transformations

The general quadratic form q(x) with coefficients of a symmetric matrix A, the associated bilinear form b(x,y), and the linear transformations of q(x) and b(x,y) into q(x′) and b(x′,y′) using the transformation matrix g, can be written as[1]

{\begin{matrix}{\begin{aligned}{\begin{aligned}q=\sum _{0}^{n}A_{ij}x_{i}x_{j}=\mathbf {x} ^{\mathrm {T} }\cdot \mathbf {A} \cdot \mathbf {x} \end{aligned}}&=q'=\mathbf {x} ^{\mathrm {\prime T} }\cdot \mathbf {A} '\cdot \mathbf {x} '\\b=\sum _{0}^{n}A_{ij}x_{i}y_{j}=\mathbf {x} ^{\mathrm {T} }\cdot \mathbf {A} \cdot \mathbf {y} &=b'=\mathbf {x} ^{\mathrm {\prime T} }\cdot \mathbf {A} '\cdot \mathbf {y} '\end{aligned}}\quad \left(A_{ij}=A_{ji}\right)\\\hline \left.{\begin{aligned}x_{i}^{\prime }&=\sum _{j=0}^{n}g_{ij}x_{j}=\mathbf {g} \cdot \mathbf {x} \\x_{i}&=\sum _{j=0}^{n}g_{ij}^{(-1)}x_{j}^{\prime }=\mathbf {g} ^{-1}\cdot \mathbf {x} '\end{aligned}}\right|\mathbf {g} ^{\rm {T}}\cdot \mathbf {A} \cdot \mathbf {g} =\mathbf {A} '\end{matrix}}

(Q1)

in which case n=1 is the binary quadratic form, n=2 is the ternary quadratic form, n=3 is the quaternary quadratic form.

Learning materials from Wikiversity: The binary quadratic form was introduced by Lagrange (1773) and Gauss (1798/1801), and the ternary quadratic form by Gauss (1798/1801).

The general Lorentz transformation follows from (Q1) by setting A=A′=diag(-1,1,...,1) and det g=±1. It forms an indefinite orthogonal group called the Lorentz group O(1,n), while the case det g=+1 forms the restricted Lorentz group SO(1,n). The quadratic form q(x) becomes the Lorentz interval in terms of an indefinite quadratic form of Minkowski space (being a special case of pseudo-Euclidean space), and the associated bilinear form b(x) becomes the Minkowski inner product:[2][3]

{\begin{matrix}{\begin{aligned}-x_{0}^{2}+\cdots +x_{n}^{2}&=-x_{0}^{\prime 2}+\dots +x_{n}^{\prime 2}\\-x_{0}y_{0}+\cdots +x_{n}y_{n}&=-x_{0}^{\prime }y_{0}^{\prime }+\cdots +x_{n}^{\prime }y_{n}^{\prime }\end{aligned}}\\\hline \left.{\begin{matrix}\mathbf {x} '=\mathbf {g} \cdot \mathbf {x} \\\downarrow \\{\begin{aligned}x_{0}^{\prime }&=x_{0}g_{00}+x_{1}g_{01}+\dots +x_{n}g_{0n}\\x_{1}^{\prime }&=x_{0}g_{10}+x_{1}g_{11}+\dots +x_{n}g_{1n}\\&\dots \\x_{n}^{\prime }&=x_{0}g_{n0}+x_{1}g_{n1}+\dots +x_{n}g_{nn}\end{aligned}}\\\\\mathbf {x} =\mathbf {g} ^{-1}\cdot \mathbf {x} '\\\downarrow \\{\begin{aligned}x_{0}&=x_{0}^{\prime }g_{00}-x_{1}^{\prime }g_{10}-\dots -x_{n}^{\prime }g_{n0}\\x_{1}&=-x_{0}^{\prime }g_{01}+x_{1}^{\prime }g_{11}+\dots +x_{n}^{\prime }g_{n1}\\&\dots \\x_{n}&=-x_{0}^{\prime }g_{0n}+x_{1}^{\prime }g_{1n}+\dots +x_{n}^{\prime }g_{nn}\end{aligned}}\end{matrix}}\right|{\begin{matrix}{\begin{aligned}\mathbf {A} \cdot \mathbf {g} ^{\mathrm {T} }\cdot \mathbf {A} &=\mathbf {g} ^{-1}\\\mathbf {g} ^{\rm {T}}\cdot \mathbf {A} \cdot \mathbf {g} &=\mathbf {A} \\\mathbf {g} \cdot \mathbf {A} \cdot \mathbf {g} ^{\mathrm {T} }&=\mathbf {A} \\\\\end{aligned}}\\{\begin{aligned}\sum _{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k}&=\left\{{\begin{aligned}-1\quad &(j=k=0)\\1\quad &(j=k>0)\\0\quad &(j\neq k)\end{aligned}}\right.\\\sum _{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0}&=\left\{{\begin{aligned}-1\quad &(i=k=0)\\1\quad &(i=k>0)\\0\quad &(i\neq k)\end{aligned}}\right.\end{aligned}}\end{matrix}}\end{matrix}}

(1a)

Learning materials from Wikiversity: Such general Lorentz transformations (1a) for various dimensions were used by Gauss (1818), Jacobi (1827, 1833), Lebesgue (1837), Bour (1856), Somov (1863), Hill (1882) in order to simplify computations of elliptic functions and integrals.[4][5] They were also used by Poincaré (1881), Cox (1881/82), Picard (1882, 1884), Killing (1885, 1893), Gérard (1892), Hausdorff (1899), Woods (1901, 1903), Liebmann (1904/05) to describe hyperbolic motions (i.e. rigid motions in the hyperbolic plane or hyperbolic space), which were expressed in terms of Weierstrass coordinates of the hyperboloid model satisfying the relation $-x_{0}^{2}+\cdots +x_{n}^{2}=-1$ or in terms of the Cayley–Klein metric of projective geometry using the "absolute" form $-x_{0}^{2}+\cdots +x_{n}^{2}=0$ .[6][7] In addition, infinitesimal transformations related to the Lie algebra of the group of hyperbolic motions were given in terms of Weierstrass coordinates $-x_{0}^{2}+\cdots +x_{n}^{2}=-1$ by Killing (1888-1897).

If $x_{i},\ x_{i}^{\prime }$ in (1a) are interpreted as homogeneous coordinates, then the corresponding inhomogenous coordinates $u_{s},\ u_{s}^{\prime }$ follow by

\left[{\frac {x_{0}}{x_{0}}},\ {\frac {x_{s}}{x_{0}}}\right]=\left[1,\ u_{s}\right],\ \left[{\frac {x_{0}^{\prime }}{x_{0}^{\prime }}},\ {\frac {x_{s}^{\prime }}{x_{0}^{\prime }}}\right]=\left[1,\ u_{s}^{\prime }\right],\ (s=1,2\dots n)

so that the Lorentz transformation becomes a homography leaving invariant the equation of the unit sphere, which John Lighton Synge called "the most general formula for the composition of velocities" in terms of special relativity (the transformation matrix g stays the same as in (1a)):[8]

{\begin{matrix}{\begin{matrix}-x_{0}^{2}+\cdots +x_{n}^{2}=-x_{0}^{\prime 2}+\dots +x_{n}^{\prime 2}&\rightarrow &{\begin{aligned}-1+u_{1}^{2}+\cdots +u_{n}^{2}&={\scriptstyle {\frac {-1+u_{1}^{\prime 2}+\cdots +u_{n}^{\prime 2}}{\left(g_{00}+g_{01}u_{1}^{\prime }+\dots +g_{0n}u_{n}^{\prime }\right)^{2}}}}\\{\scriptstyle {\frac {-1+u_{1}^{2}+\cdots +u_{n}^{2}}{\left(g_{00}-g_{10}u_{1}-\dots -g_{n0}u_{n}\right)^{2}}}}&=-1+u_{1}^{\prime 2}+\cdots +u_{n}^{\prime 2}\end{aligned}}\\\hline -x_{0}^{2}+\cdots +x_{n}^{2}=-x_{0}^{\prime 2}+\dots +x_{n}^{\prime 2}=0&\rightarrow &-1+u_{1}^{2}+\cdots +u_{n}^{2}=-1+u_{1}^{\prime 2}+\cdots +u_{n}^{\prime 2}=0\end{matrix}}\\\hline {\begin{aligned}u_{s}^{\prime }&={\frac {g_{s0}+g_{s1}u_{1}+\dots +g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots +g_{0n}u_{n}}}\\\\u_{s}&={\frac {-g_{0s}+g_{1s}u_{1}^{\prime }+\dots +g_{ns}u_{n}^{\prime }}{g_{00}-g_{10}u_{1}^{\prime }-\dots -g_{n0}u_{n}^{\prime }}}\end{aligned}}\left|{\begin{aligned}\sum _{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k}&=\left\{{\begin{aligned}-1\quad &(j=k=0)\\1\quad &(j=k>0)\\0\quad &(j\neq k)\end{aligned}}\right.\\\sum _{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0}&=\left\{{\begin{aligned}-1\quad &(i=k=0)\\1\quad &(i=k>0)\\0\quad &(i\neq k)\end{aligned}}\right.\end{aligned}}\right.\end{matrix}}

(1b)

Learning materials from Wikiversity: Such Lorentz transformations for various dimensions were used by Gauss (1818), Jacobi (1827–1833), Lebesgue (1837), Bour (1856), Somov (1863), Hill (1882), Callandreau (1885) in order to simplify computations of elliptic functions and integrals, by Picard (1882-1884) in relation to Hermitian quadratic forms, or by Woods (1901, 1903) in terms of the Beltrami–Klein model of hyperbolic geometry. In addition, infinitesimal transformations in terms of the Lie algebra of the group of hyperbolic motions leaving invariant the unit sphere $-1+u_{1}^{\prime 2}+\cdots +u_{n}^{\prime 2}=0$ were given by Lie (1885-1893) and Werner (1889) and Killing (1888-1897).

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 Lorentz transformation (1a) assumes the form of an orthogonal transformation of Euclidean space forming the orthogonal group O(n) if det g=±1 or the special orthogonal group SO(n) if det g=+1, the Lorentz interval becomes the Euclidean norm, and the Minkowski inner product becomes the dot product:[9]

{\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)

Learning materials from Wikiversity: 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 rotation in terms of trigonometric functions, which become Lorentz transformations by using an imaginary angle $\phi =i\eta$ , so that trigonometric functions become equivalent to 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 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)

Learning materials from Wikiversity: 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 (3b) while (2c) becomes (3d).

Lorentz transformation via hyperbolic functions

The case of a Lorentz transformation without spatial rotation is called a Lorentz boost. The simplest case can be given, for instance, by setting n=1 in (1a):

{\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}\\\hline \left.{\begin{aligned}\mathbf {x} '&={\begin{bmatrix}g_{00}&g_{01}\\g_{10}&g_{11}\end{bmatrix}}\cdot \mathbf {x} \\\mathbf {x} &={\begin{bmatrix}g_{00}&-g_{10}\\-g_{01}&g_{11}\end{bmatrix}}\cdot \mathbf {x} '\end{aligned}}\right|\det {\begin{bmatrix}g_{00}&g_{01}\\g_{10}&g_{11}\end{bmatrix}}=1\\\hline {\begin{aligned}x_{0}^{\prime }&=x_{0}g_{00}+x_{1}g_{01}\\x_{1}^{\prime }&=x_{0}g_{10}+x_{1}g_{11}\\\\x_{0}&=x_{0}^{\prime }g_{00}-x_{1}^{\prime }g_{10}\\x_{1}&=-x_{0}^{\prime }g_{01}+x_{1}^{\prime }g_{11}\end{aligned}}\left|{\begin{aligned}g_{01}^{2}-g_{00}^{2}&=-1\\g_{11}^{2}-g_{10}^{2}&=1\\g_{01}g_{11}-g_{00}g_{10}&=0\\g_{10}^{2}-g_{00}^{2}&=-1\\g_{11}^{2}-g_{01}^{2}&=1\\g_{10}g_{11}-g_{00}g_{01}&=0\end{aligned}}\rightarrow {\begin{aligned}g_{00}^{2}&=g_{11}^{2}\\g_{01}^{2}&=g_{10}^{2}\end{aligned}}\right.\end{matrix}}

(3a)

which resembles precisely the relations of hyperbolic functions in terms of hyperbolic angle $\eta$ . Thus by adding an unchanged $x_{2}$ -axis, a Lorentz boost or hyperbolic rotation for n=2 (being the same as a rotation around an imaginary angle $i\eta =\phi$ in (2b) or a translation in the hyperbolic plane in terms of the hyperboloid model) is given by

{\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline g_{00}=g_{11}=\cosh \eta ,\ g_{01}=g_{10}=-\sinh \eta \\\hline \left.{\begin{aligned}\mathbf {x} '&={\begin{bmatrix}\cosh \eta &-\sinh \eta \\-\sinh \eta &\cosh \eta \end{bmatrix}}\cdot \mathbf {x} \\\mathbf {x} &={\begin{bmatrix}\cosh \eta &\sinh \eta \\\sinh \eta &\cosh \eta \end{bmatrix}}\cdot \mathbf {x} '\end{aligned}}\right|\det {\begin{bmatrix}\cosh \eta &-\sinh \eta \\-\sinh \eta &\cosh \eta \end{bmatrix}}=1\\\hline \left.{\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}}\right|{\scriptstyle {\begin{aligned}\sinh ^{2}\eta -\cosh ^{2}\eta &=-1&(a)\\\cosh ^{2}\eta -\sinh ^{2}\eta &=1&(b)\\{\frac {\sinh \eta }{\cosh \eta }}&=\tanh \eta &(c)\\{\frac {1}{\sqrt {1-\tanh ^{2}\eta }}}&=\cosh \eta &(d)\\{\frac {\tanh \eta }{\sqrt {1-\tanh ^{2}\eta }}}&=\sinh \eta &(e)\\{\frac {\tanh q\pm \tanh \eta }{1\pm \tanh q\tanh \eta }}&=\tanh \left(q\pm \eta \right)&(f)\end{aligned}}}\end{matrix}}

(3b)

in which the rapidity can be composed of arbitrary many rapidities $\eta _{1},\eta _{2}\dots$ as per the 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 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 unit hyperbola:

{\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}=1\\\hline \left[\eta =\eta _{2}-\eta _{1}\right]\\{\scriptstyle {\begin{aligned}{\begin{bmatrix}x_{1}^{\prime }&x_{0}^{\prime }\\x_{0}^{\prime }&x_{1}^{\prime }\end{bmatrix}}&={\begin{bmatrix}\cosh \eta _{1}&\sinh \eta _{1}\\\sinh \eta _{1}&\cosh \eta _{1}\end{bmatrix}}={\begin{bmatrix}\cosh \left(\eta _{2}-\eta \right)&\sinh \left(\eta _{2}-\eta \right)\\\sinh \left(\eta _{2}-\eta \right)&\cosh \left(\eta _{2}-\eta \right)\end{bmatrix}}={\begin{bmatrix}\cosh \eta &-\sinh \eta \\-\sinh \eta &\cosh \eta \end{bmatrix}}\cdot {\begin{bmatrix}\cosh \eta _{2}&\sinh \eta _{2}\\\sinh \eta _{2}&\cosh \eta _{2}\end{bmatrix}}={\begin{bmatrix}\cosh \eta &-\sinh \eta \\-\sinh \eta &\cosh \eta \end{bmatrix}}\cdot {\begin{bmatrix}x_{1}&x_{0}\\x_{0}&x_{1}\end{bmatrix}}\\{\begin{bmatrix}x_{1}&x_{0}\\x_{0}&x_{1}\end{bmatrix}}&={\begin{bmatrix}\cosh \eta _{2}&\sinh \eta _{2}\\\sinh \eta _{2}&\cosh \eta _{2}\end{bmatrix}}={\begin{bmatrix}\cosh \left(\eta _{1}+\eta \right)&\sinh \left(\eta _{1}+\eta \right)\\\sinh \left(\eta _{1}+\eta \right)&\cosh \left(\eta _{1}+\eta \right)\end{bmatrix}}={\begin{bmatrix}\cosh \eta &\sinh \eta \\\sinh \eta &\cosh \eta \end{bmatrix}}\cdot {\begin{bmatrix}\cosh \eta _{1}&\sinh \eta _{1}\\\sinh \eta _{1}&\cosh \eta _{1}\end{bmatrix}}={\begin{bmatrix}\cosh \eta &\sinh \eta \\\sinh \eta &\cosh \eta \end{bmatrix}}\cdot {\begin{bmatrix}x_{1}^{\prime }&x_{0}^{\prime }\\x_{0}^{\prime }&x_{1}^{\prime }\end{bmatrix}}\end{aligned}}}\\\hline {\begin{aligned}x_{0}^{\prime }&=\sinh \eta _{1}&&=\sinh \left(\eta _{2}-\eta \right)&&=\sinh \eta _{2}\cosh \eta -\cosh \eta _{2}\sinh \eta &&=x_{0}\cosh \eta -x_{1}\sinh \eta \\x_{1}^{\prime }&=\cosh \eta _{1}&&=\cosh \left(\eta _{2}-\eta \right)&&=-\sinh \eta _{2}\sinh \eta +\cosh \eta _{2}\cosh \eta &&=-x_{0}\sinh \eta +x_{1}\cosh \eta \\\\x_{0}&=\sinh \eta _{2}&&=\sinh \left(\eta _{1}+\eta \right)&&=\sinh \eta _{1}\cosh \eta +\cosh \eta _{1}\sinh \eta &&=x_{0}^{\prime }\cosh \eta +x_{1}^{\prime }\sinh \eta \\x_{1}&=\cosh \eta _{2}&&=\cosh \left(\eta _{1}+\eta \right)&&=\sinh \eta _{1}\sinh \eta +\cosh \eta _{1}\cosh \eta &&=x_{0}^{\prime }\sinh \eta +x_{1}^{\prime }\cosh \eta \end{aligned}}\end{matrix}}

(3c)

Finally, Lorentz boost (3b) assumes a simple form by using squeeze mappings in analogy to Euler's formula in (2c):[10]

(1){\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline {\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{matrix}}\left|{\scriptstyle {\begin{aligned}X_{1}&=x_{1}+x_{0}\\X_{2}&=x_{2}\\X_{3}&=x_{1}-x_{0}\\\\a_{1}&=e^{-\eta }\\a_{2}&=1\\a_{3}&=e^{\eta }=a_{1}^{-1}\end{aligned}}}(2){\begin{matrix}X_{2}^{\prime 2}-X_{1}^{\prime }X_{3}^{\prime }=X_{2}^{2}-X_{1}X_{3}\\\hline {\begin{aligned}X_{1}^{\prime }&=a_{1}X_{1}\\X_{2}^{\prime }&=a_{2}X_{2}\\X_{3}^{\prime }&=a_{3}X_{3}\\\\X_{1}&=a_{3}X_{1}^{\prime }\\X_{2}&=a_{2}X_{2}^{\prime }\\X_{3}&=a_{1}X_{3}^{\prime }\end{aligned}}\\\left(a_{1}a_{3}-a_{2}^{2}=0\right)\end{matrix}}\right.

(3d)

Learning materials from Wikiversity: Hyperbolic relations (a,b) on the right of (3b) were given by Riccati (1757), relations (a,b,c,d,e,f) by Lambert (1768–1770). Lorentz transformations (3b) were given by Laisant (1874), Cox (1882), Lindemann (1890/91), Gérard (1892), Killing (1893, 1897/98), Whitehead (1897/98), Woods (1903/05) and Liebmann (1904/05) in terms of Weierstrass coordinates of the hyperboloid model. Hyperbolic angle sum laws equivalent to Lorentz boost (3c) were given by Riccati (1757) and Lambert (1768–1770), while the matrix representation was given by Glaisher (1878) and Günther (1880/81). Lorentz transformations (3d-1) were given by Lindemann (1890/91) and Herglotz (1909), while formulas equivalent to (3d-2) by Klein (1871).

In line with equation (1b) one can use coordinates $[u_{1},\ u_{2},\ 1]=\left[{\tfrac {x_{1}}{x_{0}}},\ {\tfrac {x_{2}}{x_{0}}},\ {\tfrac {x_{0}}{x_{0}}}\right]$ inside the unit circle $u_{1}^{2}+u_{2}^{2}=1$ , thus the corresponding Lorentz transformations (3b) obtain the form:

{\begin{matrix}{\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}&\rightarrow &{\begin{aligned}-1+u_{1}^{2}+u_{2}^{2}&={\frac {-1+u_{1}^{\prime 2}+u_{2}^{\prime 2}}{\left(\cosh \eta +u_{1}^{\prime }\sinh \eta \right)^{2}}}\\{\frac {-1+u_{1}^{2}+u_{2}^{2}}{\left(\cosh \eta -u_{1}\sinh \eta \right)^{2}}}&=-1+u_{1}^{\prime 2}+u_{2}^{\prime 2}\end{aligned}}\\\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}=0&\rightarrow &-1+u_{x}^{2}+u_{y}^{2}=-1+u_{x}^{\prime 2}+u_{y}^{\prime 2}=0\end{matrix}}\\\hline {\scriptstyle {\begin{aligned}{\frac {\sinh \eta }{\cosh \eta }}&=\tanh \eta =v\\\cosh \eta &={\frac {1}{\sqrt {1-\tanh ^{2}\eta }}}\end{aligned}}}\left|{\begin{aligned}&(a)&&(b)&&(c)\\u_{1}^{\prime }&={\frac {-\sinh \eta +u_{1}\cosh \eta }{\cosh \eta -u_{1}\sinh \eta }}&&={\frac {u_{1}-\tanh \eta }{1-u_{1}\tanh \eta }}&&={\frac {u_{1}-v}{1-u_{1}v}}\\u_{2}^{\prime }&={\frac {u_{2}}{\cosh \eta -u_{1}\sinh \eta }}&&={\frac {u_{2}{\sqrt {1-\tanh ^{2}\eta }}}{1-u_{1}\tanh \eta }}&&={\frac {u_{2}{\sqrt {1-v^{2}}}}{1-u_{1}v}}\\\\u_{1}&={\frac {\sinh \eta +u_{1}^{\prime }\cosh \eta }{\cosh \eta +u_{1}^{\prime }\sinh \eta }}&&={\frac {u_{1}^{\prime }+\tanh \eta }{1+u_{1}^{\prime }\tanh \eta }}&&={\frac {u_{1}^{\prime }+v}{1+u_{1}^{\prime }v}}\\u_{2}&={\frac {u_{2}^{\prime }}{\cosh \eta +u_{1}^{\prime }\sinh \eta }}&&={\frac {u_{2}^{\prime }{\sqrt {1-\tanh ^{2}\eta }}}{1+u_{1}^{\prime }\tanh \eta }}&&={\frac {u_{2}^{\prime }{\sqrt {1-v^{2}}}}{1+u_{1}^{\prime }v}}\end{aligned}}\right.\end{matrix}}

(3e)

Learning materials from Wikiversity: These Lorentz transformations were given by Escherich (1874) and Killing (1898) (on the left), as well as Beltrami (1868) and Schur (1885/86, 1900/02) (on the right) in terms of Beltrami coordinates[11] of hyperbolic geometry.

By using the scalar product of $\left[u_{1},u_{2}\right]$ , the resulting Lorentz transformation can be seen as equivalent to the hyperbolic law of cosines:[12][R 1][13]

{\begin{matrix}&{\begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\u'^{2}=u_{1}^{\prime 2}+u_{2}^{\prime 2}\end{matrix}}\left|{\begin{matrix}u_{1}=u\cos \alpha \\u_{2}=u\sin \alpha \\\\u_{1}^{\prime }=u'\cos \alpha '\\u_{2}^{\prime }=u'\sin \alpha '\end{matrix}}\right|{\begin{aligned}u\cos \alpha &={\frac {u'\cos \alpha '+v}{1+vu'\cos \alpha '}},&u'\cos \alpha '&={\frac {u\cos \alpha -v}{1-vu\cos \alpha }}\\u\sin \alpha &={\frac {u'\sin \alpha '{\sqrt {1-v^{2}}}}{1+vu'\cos \alpha '}},&u'\sin \alpha '&={\frac {u\sin \alpha {\sqrt {1-v^{2}}}}{1-vu\cos \alpha }}\\\tan \alpha &={\frac {u'\sin \alpha '{\sqrt {1-v^{2}}}}{u'\cos \alpha '+v}},&\tan \alpha '&={\frac {u\sin \alpha {\sqrt {1-v^{2}}}}{u\cos \alpha -v}}\end{aligned}}\\\Rightarrow &u={\frac {\sqrt {v^{2}+u^{\prime 2}+2vu'\cos \alpha '-\left(vu'\sin \alpha '\right){}^{2}}}{1+vu'\cos \alpha '}},\quad u'={\frac {\sqrt {-v^{2}-u^{2}+2vu\cos \alpha +\left(vu\sin \alpha \right){}^{2}}}{1-vu\cos \alpha }}\\\Rightarrow &{\frac {1}{\sqrt {1-u^{\prime 2}}}}={\frac {1}{\sqrt {1-v^{2}}}}{\frac {1}{\sqrt {1-u^{2}}}}-{\frac {v}{\sqrt {1-v^{2}}}}{\frac {u}{\sqrt {1-u^{2}}}}\cos \alpha &(b)\\\Rightarrow &{\frac {1}{\sqrt {1-\tanh ^{2}\xi }}}={\frac {1}{\sqrt {1-\tanh ^{2}\eta }}}{\frac {1}{\sqrt {1-\tanh ^{2}\zeta }}}-{\frac {\tanh \eta }{\sqrt {1-\tanh ^{2}\eta }}}{\frac {\tanh \zeta }{\sqrt {1-\tanh ^{2}\zeta }}}\cos \alpha \\\Rightarrow &\cosh \xi =\cosh \eta \cosh \zeta -\sinh \eta \sinh \zeta \cos \alpha &(a)\end{matrix}}

(3f)

Learning materials from Wikiversity: 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).

Lorentz transformation via velocity

In the theory of relativity, Lorentz transformations exhibit the symmetry of Minkowski spacetime by using a constant c as the speed of light, and a parameter v as the relative velocity between two inertial reference frames. In particular, the hyperbolic angle $\eta$ in (3b) can be interpreted as the velocity related rapidity $\tanh \eta =\beta =v/c$ , so that $\gamma =\cosh \eta$ is the Lorentz factor, $\beta \gamma =\sinh \eta$ the proper velocity, $u'=c\tanh q$ the velocity of another object, $u=c\tanh(q+\eta )$ the velocity-addition formula, thus (3b) becomes:

{\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline {\begin{aligned}x_{0}^{\prime }&=x_{0}\gamma -x_{1}\beta \gamma \\x_{1}^{\prime }&=-x_{0}\beta \gamma +x_{1}\gamma \\x_{2}^{\prime }&=x_{2}\\\\x_{0}&=x_{0}^{\prime }\gamma +x_{1}^{\prime }\beta \gamma \\x_{1}&=x_{0}^{\prime }\beta \gamma +x_{1}^{\prime }\gamma \\x_{2}&=x_{2}^{\prime }\end{aligned}}\left|{\scriptstyle {\begin{aligned}\beta ^{2}\gamma ^{2}-\gamma ^{2}&=-1&(a)\\\gamma ^{2}-\beta ^{2}\gamma ^{2}&=1&(b)\\{\frac {\beta \gamma }{\gamma }}&=\beta &(c)\\{\frac {1}{\sqrt {1-\beta ^{2}}}}&=\gamma &(d)\\{\frac {\beta }{\sqrt {1-\beta ^{2}}}}&=\beta \gamma &(e)\\{\frac {u'+v}{1+{\frac {u'v}{c^{2}}}}}&=u&(f)\end{aligned}}}\right.\end{matrix}}

(4a)

Or in four dimensions and by setting $x_{0}=ct,\ x_{1}=x,\ x_{2}=y$ and adding an unchanged z the familiar form follows, using ${\sqrt {\tfrac {c+v}{c-v}}}$ as Doppler factor:

{\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}+z^{2}=-c^{2}t^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}\\\hline \left.{\begin{aligned}t'&=\gamma \left(t-x{\frac {v}{c^{2}}}\right)\\x'&=\gamma (x-vt)\\y'&=y\\z'&=z\end{aligned}}\right|{\begin{aligned}t&=\gamma \left(t'+x{\frac {v}{c^{2}}}\right)\\x&=\gamma (x'+vt')\\y&=y'\\z&=z'\end{aligned}}\end{matrix}}\Rightarrow {\begin{aligned}(ct'+x')&=(ct+x){\sqrt {\frac {c+v}{c-v}}}\\(ct'-x')&=(ct-x){\sqrt {\frac {c-v}{c+v}}}\end{aligned}}

(4b)

In physics, analogous transformations have been introduced by Voigt (1887) and by Lorentz (1892, 1895) who analyzed Maxwell's equations, they were completed by Larmor (1897, 1900) and Lorentz (1899, 1904), and brought into their modern form by Poincaré (1905) who gave the transformation the name of Lorentz.[14] Eventually, Einstein (1905) showed in his development of special relativity that the transformations follow from the principle of relativity and constant light speed alone by modifying the traditional concepts of space and time, without requiring a mechanical aether in contradistinction to Lorentz and Poincaré.[15] Minkowski (1907–1908) used them to argue that space and time are inseparably connected as spacetime. Minkowski (1907–1908) and Varićak (1910) showed the relation to imaginary and hyperbolic functions. Important contributions to the mathematical understanding of the Lorentz transformation were also made by other authors such as Herglotz (1909/10), Ignatowski (1910), Noether (1910) and Klein (1910), Borel (1913–14).

Learning materials from Wikiversity: In pure mathematics, similar transformations have been used by Lipschitz (1885/86).

Also Lorentz boosts for arbitrary directions in line with (1a) can be given as:[16]

\mathbf {x} '={\begin{bmatrix}\gamma &-\gamma \beta n_{x}&-\gamma \beta n_{y}&-\gamma \beta n_{z}\\-\gamma \beta n_{x}&1+(\gamma -1)n_{x}^{2}&(\gamma -1)n_{x}n_{y}&(\gamma -1)n_{x}n_{z}\\-\gamma \beta n_{y}&(\gamma -1)n_{y}n_{x}&1+(\gamma -1)n_{y}^{2}&(\gamma -1)n_{y}n_{z}\\-\gamma \beta n_{z}&(\gamma -1)n_{z}n_{x}&(\gamma -1)n_{z}n_{y}&1+(\gamma -1)n_{z}^{2}\end{bmatrix}}\cdot \mathbf {x} ,\quad \left[\mathbf {n} ={\frac {\mathbf {v} }{v}}\right]

or in vector notation

{\begin{aligned}t'&=\gamma \left(t-{\frac {v\mathbf {n} \cdot \mathbf {r} }{c^{2}}}\right)\\\mathbf {r} '&=\mathbf {r} +(\gamma -1)(\mathbf {r} \cdot \mathbf {n} )\mathbf {n} -\gamma tv\mathbf {n} \end{aligned}}

(4c)

Such transformations were formulated by Herglotz (1911) and Silberstein (1911) and others.

In line with equation (1b), one can substitute $\left[{\tfrac {u_{x}}{c}},\ {\tfrac {u_{y}}{c}},\ 1\right]=\left[{\tfrac {x}{ct}},\ {\tfrac {y}{ct}},\ {\tfrac {ct}{ct}}\right]$ in (3b) or (4a), producing the Lorentz transformation of velocities (or velocity addition formula) in analogy to Beltrami coordinates of (3e):

{\begin{matrix}{\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}=-c^{2}t^{\prime 2}+x^{\prime 2}+y^{\prime 2}&\rightarrow &{\begin{aligned}-c^{2}+u_{x}^{2}+u_{y}^{2}&={\frac {-c^{2}+u_{x}^{\prime 2}+u_{y}^{\prime 2}}{\gamma ^{2}\left(1+{\frac {v}{c^{2}}}u_{x}^{\prime }\right)^{2}}}\\{\frac {-c^{2}+u_{x}^{2}+u_{y}^{2}}{\gamma ^{2}\left(1-{\frac {v}{c^{2}}}u_{x}\right)^{2}}}&=-c^{2}+u_{x}^{\prime 2}+u_{y}^{\prime 2}\end{aligned}}\\\hline -c^{2}t^{2}+x^{2}+y^{2}=-c^{2}t^{\prime 2}+x^{\prime 2}+y^{\prime 2}=0&\rightarrow &-c^{2}+u_{x}^{2}+u_{y}^{2}=-c^{2}+u_{x}^{\prime 2}+u_{y}^{\prime 2}=0\end{matrix}}\\\hline {\scriptstyle {\begin{aligned}{\frac {\sinh \eta }{\cosh \eta }}&=\tanh \eta ={\frac {v}{c}}\\\cosh \eta &={\frac {1}{\sqrt {1-\tanh ^{2}\eta }}}\end{aligned}}}\left|{\begin{aligned}u_{x}^{\prime }&={\frac {-c^{2}\sinh \eta +u_{x}c\cosh \eta }{c\cosh \eta -u_{x}\sinh \eta }}&&={\frac {u_{x}-c\tanh \eta }{1-{\frac {u_{x}}{c}}\tanh \eta }}&&={\frac {u_{x}-v}{1-{\frac {v}{c^{2}}}u{}_{x}}}\\u_{y}^{\prime }&={\frac {cu_{y}}{c\cosh \eta -u_{x}\sinh \eta }}&&={\frac {u_{y}{\sqrt {1-\tanh ^{2}\eta }}}{1-{\frac {u_{x}}{c}}\tanh \eta }}&&={\frac {u_{y}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1-{\frac {v}{c^{2}}}u{}_{x}}}\\\\u_{x}&={\frac {c^{2}\sinh \eta +u_{x}^{\prime }c\cosh \eta }{c\cosh \eta +u_{x}^{\prime }\sinh \eta }}&&={\frac {u_{x}^{\prime }+c\tanh \eta }{1+{\frac {u_{x}^{\prime }}{c}}\tanh \eta }}&&={\frac {u_{x}^{\prime }+v}{1+{\frac {v}{c^{2}}}u_{x}^{\prime }}}\\u_{y}&={\frac {cy'}{c\cosh \eta +u_{x}^{\prime }\sinh \eta }}&&={\frac {u_{y}^{\prime }{\sqrt {1-\tanh ^{2}\eta }}}{1+{\frac {u_{x}^{\prime }}{c}}\tanh \eta }}&&={\frac {u_{y}^{\prime }{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1+{\frac {v}{c^{2}}}u_{x}^{\prime }}}\end{aligned}}\right.\end{matrix}}

(4d)

or using trigonometric and hyperbolic identities it becomes the hyperbolic law of cosines in terms of (3f):[12][R 1][13]

{\begin{matrix}&{\begin{matrix}u^{2}=u_{x}^{2}+u_{y}^{2}\\u'^{2}=u_{x}^{\prime 2}+u_{y}^{\prime 2}\end{matrix}}\left|{\begin{matrix}u_{x}=u\cos \alpha \\u_{y}=u\sin \alpha \\\\u_{x}^{\prime }=u'\cos \alpha '\\u_{y}^{\prime }=u'\sin \alpha '\end{matrix}}\right|{\begin{aligned}u\cos \alpha &={\frac {u'\cos \alpha '+v}{1+{\frac {v}{c^{2}}}u'\cos \alpha '}},&u'\cos \alpha '&={\frac {u\cos \alpha -v}{1-{\frac {v}{c^{2}}}u\cos \alpha }}\\u\sin \alpha &={\frac {u'\sin \alpha '{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1+{\frac {v}{c^{2}}}u'\cos \alpha '}},&u'\sin \alpha '&={\frac {u\sin \alpha {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1-{\frac {v}{c^{2}}}u\cos \alpha }}\\\tan \alpha &={\frac {u'\sin \alpha '{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{u'\cos \alpha '+v}},&\tan \alpha '&={\frac {u\sin \alpha {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{u\cos \alpha -v}}\end{aligned}}\\\Rightarrow &u={\frac {\sqrt {v^{2}+u^{\prime 2}+2vu'\cos \alpha '-\left({\frac {vu'\sin \alpha '}{c}}\right){}^{2}}}{1+{\frac {v}{c^{2}}}u'\cos \alpha '}},\quad u'={\frac {\sqrt {-v^{2}-u^{2}+2vu\cos \alpha +\left({\frac {vu\sin \alpha }{c}}\right){}^{2}}}{1-{\frac {v}{c^{2}}}u\cos \alpha }}\\\Rightarrow &{\frac {1}{\sqrt {1-{\frac {u^{\prime 2}}{c^{2}}}}}}={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{\frac {1}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}-{\frac {v/c}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{\frac {u/c}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\cos \alpha \\\Rightarrow &{\frac {1}{\sqrt {1-\tanh ^{2}\xi }}}={\frac {1}{\sqrt {1-\tanh ^{2}\eta }}}{\frac {1}{\sqrt {1-\tanh ^{2}\zeta }}}-{\frac {\tanh \eta }{\sqrt {1-\tanh ^{2}\eta }}}{\frac {\tanh \zeta }{\sqrt {1-\tanh ^{2}\zeta }}}\cos \alpha \\\Rightarrow &\cosh \xi =\cosh \eta \cosh \zeta -\sinh \eta \sinh \zeta \cos \alpha \end{matrix}}

(4e)

and by further setting u=u′=c the relativistic aberration of light follows:[17]

{\begin{matrix}\cos \alpha ={\frac {\cos \alpha '+{\frac {v}{c}}}{1+{\frac {v}{c}}\cos \alpha '}},\ \sin \alpha ={\frac {\sin \alpha '{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1+{\frac {v}{c}}\cos \alpha '}},\ \tan \alpha ={\frac {\sin \alpha '{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{\cos \alpha '+{\frac {v}{c}}}},\ \tan {\frac {\alpha }{2}}={\sqrt {\frac {c-v}{c+v}}}\tan {\frac {\alpha '}{2}}\\\cos \alpha '={\frac {\cos \alpha -{\frac {v}{c}}}{1-{\frac {v}{c}}\cos \alpha }},\ \sin \alpha '={\frac {\sin \alpha {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1-{\frac {v}{c}}\cos \alpha }},\ \tan \alpha '={\frac {\sin \alpha {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{\cos \alpha -{\frac {v}{c}}}},\ \tan {\frac {\alpha '}{2}}={\sqrt {\frac {c+v}{c-v}}}\tan {\frac {\alpha }{2}}\end{matrix}}

(4f)

The velocity addition formulas were given by Einstein (1905) and Poincaré (1905/06), the aberration formula for cos(α) by Einstein (1905), while the relations to the spherical and hyperbolic law of cosines were given by Sommerfeld (1909) and Varićak (1910).

Learning materials from Wikiversity: These formulas resemble the equations of an ellipse of eccentricity v/c, eccentric anomaly α' and 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.[18][19]

Lorentz transformation via conformal, spherical wave, and Laguerre transformation

If one only requires the invariance of the light cone represented by the differential equation $-dx_{0}^{2}+\dots +dx_{n}^{2}=0$ , which is the same as asking for the most general transformation that changes spheres into spheres, the Lorentz group can be extended by adding dilations represented by the factor λ. The result is the group Con(1,p) of spacetime conformal transformations in terms of special conformal transformations and inversions producing the relation

-dx_{0}^{2}+\dots +dx_{n}^{2}=\lambda \left(-dx_{0}^{\prime 2}+\dots +dx_{n}^{\prime 2}\right)

.

One can switch between two representations of this group by using an imaginary sphere radius coordinate x₀=iR with the interval $dx_{0}^{2}+\dots +dx_{n}^{2}$ related to conformal transformations, or by using a real radius coordinate x₀=R with the interval $-dx_{0}^{2}+\dots +dx_{n}^{2}$ related to spherical wave transformations in terms of contact transformations preserving circles and spheres. It turns out that Con(1,3) is isomorphic to the special orthogonal group SO(2,4), and contains the Lorentz group SO(1,3) as a subgroup by setting λ=1. More generally, Con(q,p) is isomorphic to SO(q+1,p+1) and contains SO(q,p) as subgroup.[20] This implies that Con(0,p) is isomorphic to the Lorentz group of arbitrary dimensions SO(1,p+1). Consequently, the conformal group in the plane Con(0,2) – known as the group of Möbius transformations – is isomorphic to the Lorentz group SO(1,3).[21][22] This can be seen using tetracyclical coordinates satisfying the form $-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0$ .

A special case of Lie's geometry of oriented spheres is the Laguerre group, transforming oriented planes and lines into each other. It's generated by the Laguerre inversion leaving invariant $x^{2}+y^{2}+z^{2}-R^{2}$ with R as radius, thus the Laguerre group is isomorphic to the Lorentz group.[23][24]

Learning materials from Wikiversity: Both representations of Lie sphere geometry and conformal transformations were studied by Lie (1871) and others. It was shown by Bateman & Cunningham (1909–1910), that the group Con(1,3) is the most general one leaving invariant the equations of Maxwell's electrodynamics. Tetracyclical coordinates were discussed by Pockels (1891), Klein (1893), Bôcher (1894). The relation between Con(1,3) and the Lorentz group was noted by Bateman & Cunningham (1909–1910) and others. The Laguerre inversion was introduced by Laguerre (1882) and discussed by Darboux (1887) and Smith (1900). A similar concept was studied by Scheffers (1899) in terms of contact transformations. Stephanos (1883) argued that Lie's geometry of oriented spheres in terms of contact transformations, as well as the special case of the transformations of oriented planes into each other (such as by Laguerre), provides a geometrical interpretation of Hamilton's biquaternions. The group isomorphism between the Laguerre group and Lorentz group was pointed out by Bateman (1910), Cartan (1912, 1915/55), Poincaré (1912/21) and others.

Lorentz transformation via Cayley–Hermite transformation

The general transformation (Q1) of any quadratic form into itself can also be given using arbitrary parameters based on the Cayley transform (I-T)⁻¹·(I+T), where I is the identity matrix, T an arbitrary antisymmetric matrix, and by adding A as symmetric matrix defining the quadratic form (there is no primed A' because the coefficients are assumed to be the same on both sides):[25][26]

{\begin{matrix}q=\mathbf {x} ^{\mathrm {T} }\cdot \mathbf {A} \cdot \mathbf {x} =q'=\mathbf {x} ^{\mathrm {\prime T} }\cdot \mathbf {A} \cdot \mathbf {x} '\\\hline \\\mathbf {x} =(\mathbf {I} -\mathbf {T} \cdot \mathbf {A} )^{-1}\cdot (\mathbf {I} +\mathbf {T} \cdot \mathbf {A} )\cdot \mathbf {x} '\\{\text{or}}\\\mathbf {x} =\mathbf {A} ^{-1}\cdot (\mathbf {A} -\mathbf {T} )\cdot (\mathbf {A} +\mathbf {T} )^{-1}\cdot \mathbf {A} \cdot \mathbf {x} '\end{matrix}}

(Q2)

For instance, the choice A=diag(1,1,1) gives an orthogonal transformation which can be used to describe spatial rotations corresponding to the Euler-Rodrigues parameters [a,b,c,d] which can be interpreted as the coefficients of quaternions. Setting d=1, the equations have the form:

{\begin{matrix}\mathbf {A} =\operatorname {diag} (1,1,1),\quad \mathbf {T} ={\scriptstyle {\begin{vmatrix}0&a&-b\\-a&0&c\\b&-c&0\end{vmatrix}}}\\\hline x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline \mathbf {x} '={\frac {1}{\kappa }}\left[{\begin{matrix}1-a^{2}-b^{2}+c^{2}&2(bc-a)&2(ac+b)\\2(bc+a)&1-a^{2}+b^{2}-c^{2}&2(ab-c)\\2(ac-b)&2(ab+c)&1+a^{2}-b^{2}-c^{2}\end{matrix}}\right]\cdot \mathbf {x} \\\left(\kappa =1+a^{2}+b^{2}+c^{2}\right)\end{matrix}}

(Q3)

Learning materials from Wikiversity: After Cayley (1846) introduced transformations related to sums of positive squares, Hermite (1853/54, 1854) derived transformations for arbitrary quadratic forms, whose result was reformulated in terms of matrices (Q2) by Cayley (1855a, 1855b). The Euler-Rodrigues parameter were discovered by Euler (1771) and Rodrigues (1840).

Also the Lorentz interval and the general Lorentz transformation in any dimension can be produced by the Cayley–Hermite formalism.[R 2][R 3][27][28] For instance, Lorentz transformation (1a) with n=1 follows from (Q2) with:

{\begin{matrix}\mathbf {A} =\operatorname {diag} (-1,1),\quad \mathbf {T} ={\scriptstyle {\begin{vmatrix}0&a\\-a&0\end{vmatrix}}}\\\hline -x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}\\\hline \mathbf {x} '={\frac {1}{1-a^{2}}}\left[{\begin{matrix}1+a^{2}&-2a\\-2a&1+a^{2}\end{matrix}}\right]\cdot \mathbf {x} \end{matrix}}\Rightarrow {\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}\\\hline \left.{\begin{aligned}x_{0}&=x_{0}^{\prime }{\frac {1+\beta _{0}^{2}}{1-\beta _{0}^{2}}}+x_{1}^{\prime }{\frac {2\beta _{0}}{1-\beta _{0}^{2}}}&=&{\frac {x_{0}^{\prime }\left(1+\beta _{0}^{2}\right)+x_{1}^{\prime }2\beta _{0}}{1-\beta _{0}^{2}}}\\x_{1}&=x_{0}^{\prime }{\frac {2\beta _{0}}{1-\beta _{0}^{2}}}+x_{1}^{\prime }{\frac {1+\beta _{0}^{2}}{1-\beta _{0}^{2}}}&=&{\frac {x_{0}^{\prime }2\beta _{0}+x_{1}^{\prime }\left(1+\beta _{0}^{2}\right)}{1-\beta _{0}^{2}}}\\\\x_{0}^{\prime }&=x_{0}{\frac {1+\beta _{0}^{2}}{1-\beta _{0}^{2}}}-x_{1}{\frac {2\beta _{0}}{1-\beta _{0}^{2}}}&=&{\frac {x_{0}\left(1+\beta _{0}^{2}\right)-x_{1}2\beta _{0}}{1-\beta _{0}^{2}}}\\x_{1}^{\prime }&=-x_{0}{\frac {2\beta _{0}}{1-\beta _{0}^{2}}}+x_{1}{\frac {1+\beta _{0}^{2}}{1-\beta _{0}^{2}}}&=&{\frac {-x_{0}2\beta _{0}+x_{1}\left(1+\beta _{0}^{2}\right)}{1-\beta _{0}^{2}}}\end{aligned}}\right|{\scriptstyle {\begin{aligned}{\frac {2\beta _{0}}{1+\beta _{0}^{2}}}&=\beta \\{\frac {1+\beta _{0}^{2}}{1-\beta _{0}^{2}}}&=\gamma \\{\frac {2\beta _{0}}{1-\beta _{0}^{2}}}&=\beta \gamma \end{aligned}}}\end{matrix}}

(5a)

This becomes Lorentz boost (4a or 4b) by setting ${\tfrac {2a}{1+a^{2}}}={\tfrac {v}{c}}$ , which is equivalent to the relation ${\tfrac {2\beta _{0}}{1+\beta _{0}^{2}}}={\tfrac {v}{c}}$ known from Loedel diagrams, thus (5a) can be interpreted as a Lorentz boost from the viewpoint of a "median frame" in which two other inertial frames are moving with equal speed $\beta _{0}$ in opposite directions.

Furthermore, Lorentz transformation (1a) with n=2 is given by:

{\begin{matrix}\mathbf {A} =\operatorname {diag} (-1,1,1),\quad \mathbf {T} ={\scriptstyle {\begin{vmatrix}0&a&-b\\-a&0&c\\b&-c&0\end{vmatrix}}}\\\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline \mathbf {x} '={\frac {1}{\kappa }}\left[{\begin{matrix}1+a^{2}+b^{2}+c^{2}&-2(bc-a)&-2(ac+b)\\2(bc+a)&1+a^{2}-b^{2}-c^{2}&2(ab-c)\\2(ac-b)&-2(ab-c)&1-a^{2}+b^{2}-c^{2}\end{matrix}}\right]\cdot \mathbf {x} \\\left(\kappa =1-a^{2}-b^{2}+c^{2}\right)\end{matrix}}

(5b)

or using n=3:

{\begin{matrix}\mathbf {A} =\operatorname {diag} (-1,1,1,1),\quad \mathbf {T} ={\scriptstyle {\begin{vmatrix}0&a&-b&c\\-a&0&d&e\\b&-d&0&f\\-c&-e&-f&0\end{vmatrix}}}\\\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}+x_{3}^{\prime 2}\\\hline \mathbf {x} '={\frac {1}{\kappa }}\left[{\scriptstyle {\begin{aligned}&1+a^{2}+b^{2}+c^{2}+&&2(-bd+a+ec+pf)&&2(-ad-b+fc-pe)&&2(pd+fb-ea+c)\\&\quad d^{2}+e^{2}+f^{2}+p^{2}&&1+a^{2}-b^{2}-c^{2}&&2(-d-ab+pc-fe)&&2(fd+pb+ca-e)\\&2(bd+a-ec+pf)&&\quad -d^{2}-e^{2}+f^{2}+p^{2}&&1-a^{2}+b^{2}-c^{2}&&2(-ed-cb+pa-f)\\&2(ad-b-fc-pe)&&2(d-ab-pc-fe)&&\quad -d^{2}+e^{2}-f^{2}+p^{2}&&1-a^{2}-b^{2}+-c^{2}\\&2(pd-fb+ea+c)&&2(fd-pb+ca+e)&&2(-ed-cb-pa+f)&&\quad +d^{2}-e^{2}-f^{2}+p^{2}\end{aligned}}}\right]\cdot \mathbf {x} \\\left({\begin{aligned}\kappa &=1-a^{2}-b^{2}-c^{2}+d^{2}+e^{2}+f^{2}-p^{2}\\p&=af+be+cd\end{aligned}}\right)\end{matrix}}

(5c)

Learning materials from Wikiversity: The transformation of a binary quadratic form of which Lorentz transformation (5a) is a special case was given by Hermite (1854), equations containing Lorentz transformations (5a, 5b, 5c) as special cases were given by Cayley (1855), Lorentz transformation (5a) was given (up to a sign change) by Laguerre (1882), Darboux (1887), Smith (1900) in relation to Laguerre geometry, and Lorentz transformation (5b) was given by Bachmann (1869). In relativity, equations similar to (5b, 5c) were first employed by Borel (1913) to represent Lorentz transformations.

As described in equation (3d), the Lorentz interval is closely connected to the alternative form $X_{2}^{2}-X_{1}X_{3}$ ,[29] which in terms of the Cayley–Hermite parameters is invariant under the transformation:

{\begin{matrix}X_{2}^{\prime 2}-X_{1}^{\prime }X_{3}^{\prime }=X_{2}^{2}-X_{1}X_{3}\\\hline \mathbf {X} '={\frac {1}{\kappa }}\left[{\begin{matrix}(b+1)^{2}&-2(b+1)c&c^{2}\\a(b+1)&1-ac-b^{2}&(b-1)c\\a^{2}&-2a(b-1)&(b-1)^{2}\end{matrix}}\right]\cdot \mathbf {X} \\\left(\kappa =1+ac-b^{2}\right)\end{matrix}}

(5d)

Learning materials from Wikiversity: This transformation was given by Cayley (1884), even though he didn't relate it to the Lorentz interval but rather to $x_{0}^{2}+x_{1}^{2}+x_{2}^{2}$ .

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

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 in order to connect Möbius transformations ${\tfrac {\alpha \zeta +\beta }{\gamma \zeta +\delta }}$ and rotations:[30]

{\begin{aligned}\alpha &=1+ib,&\beta &=-a+ic,\\\gamma &=a+ic,&\delta &=1-ib.\end{aligned}}

thus (Q3) becomes:

{\begin{matrix}x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline \mathbf {x} '={\frac {1}{\kappa }}\left[{\begin{matrix}{\frac {1}{2}}\left(\alpha ^{2}-\beta ^{2}-\gamma ^{2}+\delta ^{2}\right)&\beta \delta -\alpha \gamma &{\frac {i}{2}}\left(-\alpha ^{2}+\beta ^{2}-\gamma ^{2}+\delta ^{2}\right)\\\gamma \delta +\alpha \beta &\alpha \delta +\beta \gamma &i(\alpha \beta +\gamma \delta )\\-{\frac {i}{2}}\left(-\alpha ^{2}-\beta ^{2}+\gamma ^{2}+\delta ^{2}\right)&-i(\alpha \gamma +\beta \delta )&{\frac {1}{2}}\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}+\delta ^{2}\right)\end{matrix}}\right]\cdot \mathbf {x} \\(\kappa =\alpha \delta -\beta \gamma )\end{matrix}}

(Q4)

Learning materials from Wikiversity: The Cayley-Klein parameter were introduced by Helmholtz (1866/67), Cayley (1879) and Klein (1884).

Also the Lorentz transformation can be expressed with variants of the Cayley–Klein parameters: One relates these parameters to a spin-matrix D, the spin transformations of variables $\xi ',\eta ',{\bar {\xi }}',{\bar {\eta }}'$ (the overline denotes complex conjugate), and the Möbius transformation of $\zeta ',{\bar {\zeta }}'$ . When defined in terms of isometries of hyperbolic space (hyperbolic motions), the Hermitian matrix u associated with these Möbius transformations produces an invariant determinant $\det \mathbf {u} =x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}$ identical to the Lorentz interval. Therefore, these transformations were described by John Lighton Synge as being a "factory for the mass production of Lorentz transformations".[31] It also turns out that the related spin group Spin(3, 1) or 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 Möbius group Con(0,2) or 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:[32][31][33][34]

{\begin{matrix}\zeta ={\frac {x_{1}+ix_{2}}{x_{0}-x_{3}}}={\frac {x_{0}+x_{3}}{x_{1}-ix_{2}}}\rightarrow \zeta '={\frac {\alpha \zeta +\beta }{\gamma \zeta +\delta }}\left|\zeta '={\frac {\xi '}{\eta '}}\rightarrow {\begin{aligned}\xi '&=\alpha \xi +\beta \eta \\\eta '&=\gamma \xi +\delta \eta \end{aligned}}\right.\\\hline \left.{\begin{matrix}\mathbf {u} =\left({\begin{matrix}X_{1}&X_{2}\\X_{3}&X_{4}\end{matrix}}\right)=\left({\begin{matrix}{\bar {\xi }}\xi &\xi {\bar {\eta }}\\{\bar {\xi }}\eta &{\bar {\eta }}\eta \end{matrix}}\right)=\left({\begin{matrix}x_{0}+x_{3}&x_{1}-ix_{2}\\x_{1}+ix_{2}&x_{0}-x_{3}\end{matrix}}\right)\\\det \mathbf {u} =x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}\end{matrix}}\right|{\begin{matrix}\mathbf {D} =\left({\begin{matrix}\alpha &\beta \\\gamma &\delta \end{matrix}}\right)\\{\begin{aligned}\det {\boldsymbol {\mathbf {D} }}&=1\end{aligned}}\end{matrix}}\\\hline \mathbf {u} '=\mathbf {D} \cdot \mathbf {u} \cdot {\bar {\mathbf {D} }}^{\mathrm {T} }={\begin{aligned}X_{1}^{\prime }&=X_{1}\alpha {\bar {\alpha }}+X_{2}\alpha {\bar {\beta }}+X_{3}{\bar {\alpha }}\beta +X_{4}\beta {\bar {\beta }}\\X_{2}^{\prime }&=X_{1}{\bar {\alpha }}\gamma +X_{2}{\bar {\alpha }}\delta +X_{3}{\bar {\beta }}\gamma +X_{4}{\bar {\beta }}\delta \\X_{3}^{\prime }&=X_{1}\alpha {\bar {\gamma }}+X_{2}\alpha {\bar {\delta }}+X_{3}\beta {\bar {\gamma }}+X_{4}\beta {\bar {\delta }}\\X_{4}^{\prime }&=X_{1}\gamma {\bar {\gamma }}+X_{2}\gamma {\bar {\delta }}+X_{3}{\bar {\gamma }}\delta +X_{4}\delta {\bar {\delta }}\end{aligned}}\\\hline {\begin{aligned}X_{3}^{\prime }X_{2}^{\prime }-X_{1}^{\prime }X_{4}^{\prime }&=X_{3}X_{2}-X_{1}X_{4}=0\\\det \mathbf {u} '=x_{0}^{\prime 2}-x_{1}^{\prime 2}-x_{2}^{\prime 2}-x_{3}^{\prime 2}&=\det \mathbf {u} =x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}\end{aligned}}\end{matrix}}

(6a)

thus:[35]

{\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}+x_{3}^{\prime 2}\\\hline \mathbf {x} '={\frac {1}{2}}\left[{\scriptstyle {\begin{aligned}&\alpha {\bar {\alpha }}+\beta {\bar {\beta }}+\gamma {\bar {\gamma }}+\delta {\bar {\delta }}&&\alpha {\bar {\beta }}+\beta {\bar {\alpha }}+\gamma {\bar {\delta }}+\delta {\bar {\gamma }}&&i(\alpha {\bar {\beta }}-\beta {\bar {\alpha }}+\gamma {\bar {\delta }}-\delta {\bar {\gamma }})&&\alpha {\bar {\alpha }}-\beta {\bar {\beta }}+\gamma {\bar {\gamma }}-\delta {\bar {\delta }}\\&\alpha {\bar {\gamma }}+\gamma {\bar {\alpha }}+\beta {\bar {\delta }}+\delta {\bar {\beta }}&&\alpha {\bar {\delta }}+\delta {\bar {\alpha }}+\beta {\bar {\gamma }}+\gamma {\bar {\beta }}&&i(\alpha {\bar {\delta }}-\delta {\bar {\alpha }}+\gamma {\bar {\beta }}-\beta {\bar {\gamma }})&&\alpha {\bar {\gamma }}+\gamma {\bar {\alpha }}-\beta {\bar {\delta }}-\delta {\bar {\beta }}\\&i(\gamma {\bar {\alpha }}-\alpha {\bar {\gamma }}+\delta {\bar {\beta }}-\beta {\bar {\delta }})&&i(\delta {\bar {\alpha }}-\alpha {\bar {\delta }}+\gamma {\bar {\beta }}-\beta {\bar {\gamma }})&&\alpha {\bar {\delta }}+\delta {\bar {\alpha }}-\beta {\bar {\gamma }}-\gamma {\bar {\beta }}&&i(\gamma {\bar {\alpha }}-\alpha {\bar {\gamma }}+\beta {\bar {\delta }}-\delta {\bar {\beta }})\\&\alpha {\bar {\alpha }}+\beta {\bar {\beta }}-\gamma {\bar {\gamma }}-\delta {\bar {\delta }}&&\alpha {\bar {\beta }}+\beta {\bar {\alpha }}-\gamma {\bar {\delta }}-\delta {\bar {\gamma }}&&i(\alpha {\bar {\beta }}-\beta {\bar {\alpha }}+\delta {\bar {\gamma }}-\gamma {\bar {\delta }})&&\alpha {\bar {\alpha }}-\beta {\bar {\beta }}-\gamma {\bar {\gamma }}+\delta {\bar {\delta }}\end{aligned}}}\right]\cdot \mathbf {x} \\(\alpha \delta -\beta \gamma =1)\end{matrix}}

(6b)

or in line with equation (1b) one can substitute $\left[u_{1},\ u_{2},\ u_{3},\ 1\right]=\left[{\tfrac {x_{1}}{x_{0}}},\ {\tfrac {x_{2}}{x_{0}}},\ {\tfrac {x_{3}}{x_{0}}},\ {\tfrac {x_{0}}{x_{0}}}\right]$ so that the Möbius/Lorentz transformations become related to the unit sphere:

{\begin{matrix}u_{1}^{2}+u_{2}^{2}+u_{3}^{2}=u_{1}^{\prime 2}+u_{2}^{\prime 2}+u_{3}^{\prime 2}=1\\\hline \left.{\begin{matrix}\zeta ={\frac {u_{1}+iu_{2}}{1-u_{3}}}={\frac {1+u_{3}}{u_{1}-iu_{2}}}\\\zeta '={\frac {u_{1}^{\prime }+iu_{2}^{\prime }}{1-u_{3}^{\prime }}}={\frac {1+u_{3}^{\prime }}{u_{1}^{\prime }-iu_{2}^{\prime }}}\end{matrix}}\right|\quad \zeta '={\frac {\alpha \zeta +\beta }{\gamma \zeta +\delta }}\end{matrix}}

(6c)

Learning materials from Wikiversity: The general transformation u′ in (6a) was given by Cayley (1854), while the general relation between Möbius transformations and transformation u′ leaving invariant the generalized circle was pointed out by Poincaré (1883) in relation to 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:[29][34]

{\begin{matrix}\zeta ={\frac {x_{1}}{x_{0}-x_{2}}}={\frac {x_{0}+x_{2}}{x_{1}}}\rightarrow \zeta '={\frac {\alpha \zeta +\beta }{\gamma \zeta +\delta }}\left|\zeta '={\frac {\xi '}{\eta '}}\rightarrow {\begin{aligned}\xi '&=\alpha \xi +\beta \eta \\\eta '&=\gamma \xi +\delta \eta \end{aligned}}\right.\\\hline \left.{\begin{matrix}\mathbf {u} =\left({\begin{matrix}X_{1}&X_{2}\\X_{2}&X_{3}\end{matrix}}\right)=\left({\begin{matrix}\xi ^{2}&\xi \eta \\\xi \eta &\eta ^{2}\end{matrix}}\right)=\left({\begin{matrix}x_{0}+x_{2}&x_{1}\\x_{1}&x_{0}-x_{2}\end{matrix}}\right)\\\det \mathbf {u} =x_{0}^{2}-x_{1}^{2}-x_{2}^{2}\end{matrix}}\right|{\begin{matrix}\mathbf {D} =\left({\begin{matrix}\alpha &\beta \\\gamma &\delta \end{matrix}}\right)\\{\begin{aligned}\det {\boldsymbol {\mathbf {D} }}&=1\end{aligned}}\end{matrix}}\\\hline \mathbf {u} '=\mathbf {D} \cdot \mathbf {u} \cdot \mathbf {D} ^{\mathrm {T} }={\begin{aligned}X_{1}^{\prime }&=X_{1}\alpha ^{2}+X_{2}2\alpha \beta +X_{3}\beta ^{2}\\X_{2}^{\prime }&=X_{1}\alpha \gamma +X_{2}(\alpha \delta +\beta \gamma )+X_{3}\beta \delta \\X_{3}^{\prime }&=X_{1}\gamma ^{2}+X_{2}2\gamma \delta +X_{3}\delta ^{2}\end{aligned}}\\\hline {\begin{aligned}X_{2}^{\prime 2}-X_{1}^{\prime }X_{3}^{\prime }&=X_{2}^{2}-X_{1}X_{3}=0\\\det \mathbf {u} '=x_{0}^{\prime 2}-x_{1}^{\prime 2}-x_{2}^{\prime 2}&=\det \mathbf {u} =x_{0}^{2}-x_{1}^{2}-x_{2}^{2}\end{aligned}}\end{matrix}}

(6d)

thus

{\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline \mathbf {x} '=\left[{\begin{matrix}{\frac {1}{2}}\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}+\delta ^{2}\right)&\alpha \beta +\gamma \delta &{\frac {1}{2}}\left(\alpha ^{2}-\beta ^{2}+\gamma ^{2}-\delta ^{2}\right)\\\alpha \gamma +\beta \delta &\alpha \delta +\beta \gamma &\alpha \gamma -\beta \delta \\{\frac {1}{2}}\left(\alpha ^{2}+\beta ^{2}-\gamma ^{2}-\delta ^{2}\right)&\alpha \beta -\gamma \delta &{\frac {1}{2}}\left(\alpha ^{2}-\beta ^{2}-\gamma ^{2}+\delta ^{2}\right)\end{matrix}}\right]\cdot \mathbf {x} \\(\alpha \delta -\beta \gamma =1)\end{matrix}}

(6e)

which includes the special case $\beta =\gamma =0$ implying $\delta =1/\alpha$ , reducing the transformation to a Lorentz boost in 1+1 dimensions:

{\begin{matrix}X_{1}X_{3}=X_{1}^{\prime }X_{3}^{\prime }\quad \Rightarrow \quad -x_{0}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{2}^{\prime 2}\\\hline {\begin{aligned}X_{1}&=\alpha ^{2}X_{1}^{\prime }\\X_{2}&=X_{2}^{\prime }\\X_{3}&={\frac {1}{\alpha ^{2}}}X_{3}^{\prime }\end{aligned}}\quad \Rightarrow \quad {\begin{aligned}x_{0}&={\frac {x_{0}^{\prime }\left(\alpha ^{4}+1\right)+x_{2}^{\prime }\left(\alpha ^{4}-1\right)}{2\alpha ^{2}}}\\x_{1}&=x_{1}^{\prime }\\x_{2}&={\frac {x_{0}^{\prime }\left(\alpha ^{4}-1\right)+x_{2}^{\prime }\left(\alpha ^{4}+1\right)}{2\alpha ^{2}}}\end{aligned}}\end{matrix}}

(6f)

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

{\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}=-1\\\hline \left.{\begin{matrix}\zeta ={\frac {x_{1}+ix_{2}}{x_{0}+1}}={\frac {x_{0}-1}{x_{1}-ix_{2}}}\\\zeta '={\frac {x_{1}^{\prime }+ix_{2}^{\prime }}{x_{0}^{\prime }+1}}={\frac {x_{0}^{\prime }-1}{x_{1}^{\prime }-ix_{2}^{\prime }}}\end{matrix}}\right|\quad \zeta '={\frac {\alpha \zeta +\beta }{\gamma \zeta +\delta }}\end{matrix}}

(6g)

Learning materials from Wikiversity: The general transformation u′ and its invariant $X_{2}^{2}-X_{1}X_{3}$ in (6d) was already used by Lagrange (1773) and Gauss (1798/1801) in the theory of integer binary quadratic forms. The invariant $X_{2}^{2}-X_{1}X_{3}$ was also studied by Klein (1871) in connection to hyperbolic plane geometry (see equation (3d)), while the connection between u′ and $X_{2}^{2}-X_{1}X_{3}$ with the Möbius transformation was analyzed by Poincaré (1886) in relation to 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).

Lorentz transformation via quaternions and hyperbolic numbers

The Lorentz transformations can also be expressed in terms of biquaternions: A Minkowskian quaternion (or minquat) q having one real part and one purely imaginary part is multiplied by biquaternion a applied as pre- and postfactor. Using an overline to denote quaternion conjugation and * for complex conjugation, its general form (on the left) and the corresponding boost (on the right) are as follows:[36][37]

\left.{\begin{matrix}-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}+x_{3}^{\prime 2}=-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\\\hline q'=aq{\bar {a}}^{\ast }\\\hline {\begin{aligned}q&=ix_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}\\q'&=ix_{0}^{\prime }+x_{1}^{\prime }e_{1}+x_{2}^{\prime }e_{2}+x_{3}^{\prime }e_{3}\\a&=\cos \chi +i\sin \chi =e^{i\chi }\end{aligned}}\\\left(a{\bar {a}}=1,\ \chi ={\text{imaginary}}\right)\end{matrix}}\right|{\begin{matrix}\chi ={\frac {1}{2}}i\eta \\\downarrow \\{\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},\quad x_{3}^{\prime }=x_{3}\end{aligned}}\end{matrix}}

(7a)

Learning materials from Wikiversity: Hamilton (1844/45) and Cayley (1845) derived the quaternion transformation $aqa^{-1}$ for spatial rotations, and Cayley (1854, 1855) gave the corresponding transformation $aqb$ leaving invariant the sum of four squares $x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{2}^{2}$ . Cox (1882/83) discussed the Lorentz interval in terms of Weierstrass coordinates $x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{2}^{2}=1$ in the course of adapting William Kingdon Clifford's biquaternions a+ωb to hyperbolic geometry by setting $\omega ^{2}=-1$ (alternatively, 1 gives elliptic and 0 parabolic geometry). Stephanos (1883) related the imaginary part of William Rowan Hamilton's biquaternions to the radius of spheres, and introduced a homography leaving invariant the equations of oriented spheres or oriented planes in terms of Lie sphere geometry. Buchheim (1884/85) discussed the Cayley absolute $x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{2}^{2}=0$ and adapted Clifford's biquaternions to hyperbolic geometry similar to Cox by using all three values of $\omega ^{2}$ . Eventually, the modern Lorentz transformation using biquaternions with $\omega ^{2}=-1$ as in hyperbolic geometry was given by Noether (1910) and Klein (1910) as well as Conway (1911) and Silberstein (1911).

Often connected with quaternionic systems is the hyperbolic number $\varepsilon ^{2}=1$ , which also allows to formulate the Lorentz transformations:[38][39]

{\begin{aligned}w'&=we^{-\varepsilon \eta }\\&=w(\cosh(-\eta )+\varepsilon \sinh(-\eta ))\\\\w&=w'e^{\varepsilon \eta }\\&=w'(\cosh \eta +\varepsilon \sinh \eta )\end{aligned}}\rightarrow {\begin{aligned}w&=x_{1}+\varepsilon x_{0}\\w'&=x_{1}^{\prime }+\varepsilon x_{0}^{\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_{0}&=x_{0}^{\prime }\cosh \eta +x_{1}^{\prime }\sinh \eta \\x_{1}&=x_{0}^{\prime }\sinh \eta +x_{1}^{\prime }\cosh \eta \end{aligned}}

(7b)

Learning materials from Wikiversity: After the trigonometric expression $e^{ix}$ (Euler's formula) was given by Euler (1748), and the hyperbolic analogue $e^{\varepsilon \eta }$ as well as hyperbolic numbers by Cockle (1848) in the framework of tessarines, it was shown by Cox (1882/83) that one can identify $ww^{\prime -1}=e^{\varepsilon \eta }$ with associative quaternion multiplication. Here, $e^{\varepsilon \eta }$ is the hyperbolic versor with $\varepsilon ^{2}=1$ , while -1 denotes the elliptic or 0 denotes the parabolic counterpart (not to be confused with the expression $\omega ^{2}$ in Clifford's biquaternions also used by Cox, in which -1 is hyperbolic). The hyperbolic versor was also discussed by Macfarlane (1892, 1894, 1900) in terms of hyperbolic quaternions. The expression $\varepsilon ^{2}=1$ for hyperbolic motions (and -1 for elliptic, 0 for parabolic motions) also appear in "biquaternions" defined by Vahlen (1901/02, 1905).

More extended forms of complex and (bi-)quaternionic systems in terms of Clifford algebra can also be used to express the Lorentz transformations. For instance, using a system a of Clifford numbers one can transform the following general quadratic form into itself, in which the individual values of $i_{1}^{2},i_{2}^{2},\dots$ can be set to +1 or -1 at will, while the Lorentz interval follows if the sign of one $i^{2}$ differs from all others.:[40][41]

{\begin{matrix}i_{1}^{2}x_{1}^{\prime 2}+\cdots +i_{n}^{2}x_{n}^{\prime 2}=i_{1}^{2}x_{1}^{2}+\cdots +i_{n}^{2}x_{n}^{2}\\\hline (1)\ x'=axa^{-1}\\(2)\ x'={\frac {ax+b}{\varepsilon ^{2}bx+a}}\end{matrix}}

(7c)

Learning materials from Wikiversity: The general definite form $x_{1}^{2}+\cdots +x_{n}^{2}$ as well as the general indefinite form $x_{1}^{2}+\cdots +x_{p}^{2}-x_{p+1}^{2}-\cdots -x_{p+q}^{2}$ and their invariance under transformation (1) was discussed by Lipschitz (1885/86), while hyperbolic motions were discussed by Vahlen (1901/02, 1905) by setting $\varepsilon ^{2}=1$ in transformation (2), while elliptic motions follow with -1 and parabolic motions with 0, all of which he also related to biquaternions.

Lorentz transformation via trigonometric functions

The following general relation connects the speed of light and the relative velocity to hyperbolic and trigonometric functions, where $\eta$ is the rapidity in (3b), $\theta$ is equivalent to the Gudermannian function ${\rm {gd}}(\eta )=2\arctan(e^{\eta })-\pi /2$ , and $\vartheta$ is equivalent to the Lobachevskian angle of parallelism $\Pi (\eta )=2\arctan(e^{-\eta })$ :

{\frac {v}{c}}=\tanh \eta =\sin \theta =\cos \vartheta

Learning materials from Wikiversity: This relation was first defined by Varićak (1910).

a) Using $\sin \theta ={\tfrac {v}{c}}$ one obtains the relations $\sec \theta =\gamma$ and $\tan \theta =\beta \gamma$ , and the Lorentz boost takes the form:[42]

{\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline \left.{\begin{aligned}x_{0}^{\prime }&=x_{0}\sec \theta -x_{1}\tan \theta &&={\frac {x_{0}-x_{1}\sin \theta }{\cos \theta }}\\x_{1}^{\prime }&=-x_{0}\tan \theta +x_{1}\sec \theta &&={\frac {x_{0}\sin \theta -x_{1}}{\cos \theta }}\\x_{2}^{\prime }&=x_{2}\\\\x_{0}&=x_{0}^{\prime }\sec \theta +x_{1}^{\prime }\tan \theta &&={\frac {x_{0}^{\prime }+x_{1}^{\prime }\sin \theta }{\cos \theta }}\\x_{1}&=x_{0}^{\prime }\tan \theta +x_{1}^{\prime }\sec \theta &&={\frac {x_{0}^{\prime }\sin \theta +x_{1}^{\prime }}{\cos \theta }}\\x_{2}&=x_{2}^{\prime }\end{aligned}}\right|{\scriptstyle {\begin{aligned}\tan ^{2}\theta -\sec ^{2}\theta &=-1\\{\frac {\tan \theta }{\sec \theta }}&=\sin \theta \\{\frac {1}{\sqrt {1-\sin ^{2}\theta }}}&=\sec \theta \\{\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}&=\tan \theta \end{aligned}}}\end{matrix}}

(8a)

Learning materials from Wikiversity: This Lorentz transformation was derived by Bianchi (1886) and Darboux (1891/94) while transforming pseudospherical surfaces, and by Scheffers (1899) as a special case of contact transformation in the plane (Laguerre geometry). In special relativity, it was used by Gruner (1921) while developing Loedel diagrams, and by Vladimir Karapetoff in the 1920s.

b) Using $\cos \vartheta ={\tfrac {v}{c}}$ one obtains the relations $\csc \vartheta =\gamma$ and $\cot \vartheta =\beta \gamma$ , and the Lorentz boost takes the form:[42]

{\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline \left.{\begin{aligned}x_{0}^{\prime }&=x_{0}\csc \vartheta -x_{1}\cot \vartheta &&={\frac {x_{0}-x_{1}\cos \vartheta }{\sin \vartheta }}\\x_{1}^{\prime }&=-x_{0}\cot \vartheta +x_{1}\csc \vartheta &&={\frac {x_{0}\cos \vartheta -x_{1}}{\sin \vartheta }}\\x_{2}^{\prime }&=x_{2}\\\\x_{0}&=x_{0}^{\prime }\csc \vartheta +x_{1}^{\prime }\cot \vartheta &&={\frac {x_{0}^{\prime }+x_{1}^{\prime }\cos \vartheta }{\sin \vartheta }}\\x_{1}&=x_{0}^{\prime }\cot \vartheta +x_{1}^{\prime }\csc \vartheta &&={\frac {x_{0}^{\prime }\cos \vartheta +x_{1}^{\prime }}{\sin \vartheta }}\\x_{2}&=x_{2}^{\prime }\end{aligned}}\right|{\scriptstyle {\begin{aligned}\cot ^{2}\vartheta -\csc ^{2}\vartheta &=-1\\{\frac {\cot \vartheta }{\csc \vartheta }}&=\cos \vartheta \\{\frac {1}{\sqrt {1-\cos ^{2}\vartheta }}}&=\csc \vartheta \\{\frac {\cos \vartheta }{\sqrt {1-\cos ^{2}\vartheta }}}&=\cot \vartheta \end{aligned}}}\end{matrix}}

(8b)

Learning materials from Wikiversity: This Lorentz transformation was derived by Eisenhart (1905) while transforming pseudospherical surfaces. In special relativity it was first used by Gruner (1921) while developing Loedel diagrams.

Lorentz transformation via squeeze mappings

As already indicated in equations (3d) in exponential form or (6f) in terms of Cayley–Klein parameter, Lorentz boosts in terms of hyperbolic rotations can be expressed as squeeze mappings. Using asymptotic coordinates of a hyperbola (u,v), they have the general form (some authors alternatively add a factor of 2 or ${\sqrt {2}}$ ):[43]

{\begin{matrix}(1)&{\begin{array}{c|c|c}u=x_{0}+x_{1}&2u=x_{0}+x_{1}&{\sqrt {2}}u=x_{0}+x_{1}\\v=x_{0}-x_{1}&2v=x_{0}-x_{1}&{\sqrt {2}}v=x_{0}-x_{1}\\u'=x_{0}^{\prime }+x_{1}^{\prime }&2u'=x_{0}^{\prime }+x_{1}^{\prime }&{\sqrt {2}}u=x_{0}^{\prime }+x_{1}^{\prime }\\v'=x_{0}^{\prime }-x_{1}^{\prime }&2v'=x_{0}^{\prime }-x_{1}^{\prime }&{\sqrt {2}}v=x_{0}^{\prime }-x_{1}^{\prime }\end{array}}\\\hline (2)&(u',v')=\left(ku,\ {\frac {1}{k}}v\right)\Rightarrow u'v'=uv\end{matrix}}

(9a)

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

{\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}\\\hline \left.{\begin{aligned}x_{0}^{\prime }&={\frac {1}{2}}\left(k+{\frac {1}{k}}\right)x_{0}-{\frac {1}{2}}\left(k-{\frac {1}{k}}\right)x_{1}&&={\frac {x_{0}\left(k^{2}+1\right)-x_{1}\left(k^{2}-1\right)}{2k}}\\x_{1}^{\prime }&=-{\frac {1}{2}}\left(k-{\frac {1}{k}}\right)x_{0}+{\frac {1}{2}}\left(k+{\frac {1}{k}}\right)x_{1}&&={\frac {-x_{0}\left(k^{2}-1\right)+x_{1}\left(k^{2}+1\right)}{2k}}\\\\x_{0}&={\frac {1}{2}}\left(k+{\frac {1}{k}}\right)x_{0}^{\prime }+{\frac {1}{2}}\left(k-{\frac {1}{k}}\right)x_{1}^{\prime }&&={\frac {x_{0}^{\prime }\left(k^{2}+1\right)+x_{1}^{\prime }\left(k^{2}-1\right)}{2k}}\\x_{1}&={\frac {1}{2}}\left(k-{\frac {1}{k}}\right)x_{0}^{\prime }+{\frac {1}{2}}\left(k+{\frac {1}{k}}\right)x_{1}^{\prime }&&={\frac {x_{0}^{\prime }\left(k^{2}-1\right)+x_{1}^{\prime }\left(k^{2}+1\right)}{2k}}\end{aligned}}\right|{\scriptstyle {\begin{aligned}{\frac {k^{2}-1}{k^{2}+1}}&=\beta \\{\frac {k^{2}+1}{2k}}&=\gamma \\{\frac {k^{2}-1}{2k}}&=\beta \gamma \end{aligned}}}\end{matrix}}

(9b)

Learning materials from Wikiversity: Lorentz transformation (9a) of asymptotic coordinates have been used Laisant (1874), Günther (1880/81) in relation to elliptic trigonometry; by Lie (1879-81), Bianchi (1886, 1894), Darboux (1891/94), Eisenhart (1905) as Lie transform)[43] of pseudospherical surfaces in terms of the Sine-Gordon equation; 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), Darboux (1891/94); trigonometric Lorentz boost (8b) by Eisenhart (1905). Lorentz boost (9b) was rediscovered in the framework of special relativity by Hermann Bondi (1964)[44] in terms of 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 $k=\alpha ^{2}$ , 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), 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:

{\begin{matrix}{\begin{matrix}u=x_{0}+x_{1}\\v=x_{0}-x_{1}\\u'=x_{0}^{\prime }+x_{1}^{\prime }\\v'=x_{0}^{\prime }-x_{1}^{\prime }\end{matrix}}\Rightarrow {\begin{matrix}u_{1}=x_{1}-x_{1}^{\prime }\\v_{1}=x_{0}+x_{0}^{\prime }\\u_{2}=x_{1}+x_{1}^{\prime }\\v_{2}=x_{0}-x_{0}^{\prime }\end{matrix}}\\\hline (u_{2},v_{2})=\left(au_{1},\ {\frac {1}{a}}v_{1}\right)\Rightarrow u_{2}v_{2}=u_{1}v_{1}\\(u',v')=\left({\frac {1+a}{1-a}}u,\ {\frac {1-a}{1+a}}v\right)\Rightarrow u'v'=uv\end{matrix}}\Rightarrow {\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}\\\hline {\begin{aligned}x_{0}^{\prime }&=x_{0}{\frac {1+a^{2}}{1-a^{2}}}-x_{1}{\frac {2a}{1-a^{2}}}&&={\frac {x_{0}\left(1+a^{2}\right)-x_{1}2a}{1-a^{2}}}\\x_{1}^{\prime }&=-x_{0}{\frac {2a}{1-a^{2}}}+x_{1}{\frac {1+a^{2}}{1-a^{2}}}&&={\frac {-x_{0}2a+x_{1}\left(1+a^{2}\right)}{1-a^{2}}}\\\\x_{0}&=x_{0}^{\prime }{\frac {1+a^{2}}{1-a^{2}}}+x_{1}^{\prime }{\frac {2a}{1-a^{2}}}&&={\frac {x_{0}^{\prime }\left(1+a^{2}\right)+x_{1}^{\prime }2a}{1-a^{2}}}\\x_{1}&=x_{0}^{\prime }{\frac {2a}{1-a^{2}}}+x_{1}^{\prime }{\frac {1+a^{2}}{1-a^{2}}}&&={\frac {x_{0}^{\prime }2a+x_{1}^{\prime }\left(1+a^{2}\right)}{1-a^{2}}}\end{aligned}}\end{matrix}}

(9c)

Learning materials from Wikiversity: 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:

{\begin{array}{c|c|c|c|c|c}&&(3b)&(4a)&(8a)&(8b)\\\hline k&{\frac {1+a}{1-a}}&e^{\eta }&{\sqrt {\tfrac {1+\beta }{1-\beta }}}&{\frac {1+\sin \theta }{\cos \theta }}&{\frac {1+\cos \vartheta }{\sin \vartheta }}=\cot {\frac {\vartheta }{2}}\\\hline {\frac {k-1}{k+1}}&a&\tanh {\frac {\eta }{2}}&{\frac {\gamma -1}{\beta \gamma }}&{\frac {1-\cos \theta }{\sin \theta }}=\tan {\frac {\theta }{2}}&{\frac {1-\sin \vartheta }{\cos \vartheta }}\\\hline {\frac {k^{2}-1}{k^{2}+1}}&{\frac {2a}{1+a^{2}}}&\tanh \eta &\beta &\sin \theta &\cos \vartheta \\\hline {\frac {k^{2}+1}{2k}}&{\frac {1+a^{2}}{1-a^{2}}}&\cosh \eta &\gamma &\sec \theta &\csc \vartheta \\\hline {\frac {k^{2}-1}{2k}}&{\frac {2a}{1-a^{2}}}&\sinh \eta &\beta \gamma &\tan \theta &\cot \vartheta \end{array}}

(9d)

Learning materials from Wikiversity: Squeeze mappings in terms of $\theta$ were used by Darboux (1891/94) and Bianchi (1894), in terms of $\eta$ by Lindemann (1891) and Herglotz (1909), in terms of $\vartheta$ by Eisenhart (1905), in terms of $\beta$ by Bondi (1964).

Electrodynamics and special relativity

Voigt (1887)

Woldemar Voigt (1887)[R 4] developed a transformation in connection with the Doppler effect and an incompressible medium, being in modern notation:[45][46]

{\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}\xi _{1}&=x_{1}-\varkappa t\\\eta _{1}&=y_{1}q\\\zeta _{1}&=z_{1}q\\\tau &=t-{\frac {\varkappa x_{1}}{\omega ^{2}}}\\q&={\sqrt {1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}}\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=x-vt\\y^{\prime }&={\frac {y}{\gamma }}\\z^{\prime }&={\frac {z}{\gamma }}\\t^{\prime }&=t-{\frac {vx}{c^{2}}}\\{\frac {1}{\gamma }}&={\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\end{aligned}}\end{matrix}}

If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation (4b). In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Optical phenomena in free space are scale, conformal (using the factor λ discussed above), and Lorentz invariant, so the combination is invariant too.[46] For instance, Lorentz transformations can be extended by using $l={\sqrt {\lambda }}$ :[R 5]

x^{\prime }=\gamma l\left(x-vt\right),\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }=\gamma l\left(t-x{\frac {v}{c^{2}}}\right)

.

l=1/γ gives the Voigt transformation, l=1 the Lorentz transformation. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a principle of relativity in general. It was demonstrated by Poincaré and Einstein that one has to set l=1 in order to make the above transformation symmetric and to form a group as required by the relativity principle, therefore the Lorentz transformation is the only viable choice.

Voigt sent his 1887 paper to Lorentz in 1908,[47] and that was acknowledged in 1909:

In a paper "Über das Doppler'sche Princip", published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (7) (§ 3 of this book) [namely $\Delta \Psi -{\tfrac {1}{c^{2}}}{\tfrac {\partial ^{2}\Psi }{\partial t^{2}}}=0$ ] a transformation equivalent to the formulae (287) and (288) [namely $x^{\prime }=\gamma l\left(x-vt\right),\ y^{\prime }=ly,\ z^{\prime }=lz,\ t^{\prime }=\gamma l\left(t-{\tfrac {v}{c^{2}}}x\right)$ ]. The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the free ether is contained in his paper.[R 6]

Also Hermann Minkowski said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in 1887. Voigt responded in the same paper by saying that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.[R 7]

Heaviside (1888), Thomson (1889), Searle (1896)

In 1888, Oliver Heaviside[R 8] investigated the properties of charges in motion according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:[48]

\mathrm {E} =\left({\frac {q\mathrm {r} }{r^{2}}}\right)\left(1-{\frac {v^{2}\sin ^{2}\theta }{c^{2}}}\right)^{-3/2}

.

Consequently, Joseph John Thomson (1889)[R 9] found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the Galilean transformation z-vt in his equation[49]):

{\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}z&=\left\{1-{\frac {\omega ^{2}}{v^{2}}}\right\}^{\frac {1}{2}}z'\end{aligned}}\right|&{\begin{aligned}z^{\ast }=z-vt&={\frac {z'}{\gamma }}\end{aligned}}\end{matrix}}

Thereby, inhomogeneous electromagnetic wave equations are transformed into a Poisson equation.[49] Eventually, George Frederick Charles Searle[R 10] noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called "Heaviside-Ellipsoid" of axial ratio

{\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}&{\sqrt {\alpha }}:1:1\\\alpha =&1-{\frac {u^{2}}{v^{2}}}\end{aligned}}\right|&{\begin{aligned}&{\frac {1}{\gamma }}:1:1\\{\frac {1}{\gamma ^{2}}}&=1-{\frac {v^{2}}{c^{2}}}\end{aligned}}\end{matrix}}

[49]

Lorentz (1892, 1895)

In order to explain the aberration of light and the result of the Fizeau experiment in accordance with Maxwell's equations, Lorentz in 1892 developed a model ("Lorentz ether theory") in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson)[R 11][50]

{\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}{\mathfrak {x}}&={\frac {V}{\sqrt {V^{2}-p^{2}}}}x\\t'&=t-{\frac {\varepsilon }{V}}{\mathfrak {x}}\\\varepsilon &={\frac {p}{\sqrt {V^{2}-p^{2}}}}\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=\gamma x^{\ast }=\gamma (x-vt)\\t^{\prime }&=t-{\frac {\gamma ^{2}vx^{\ast }}{c^{2}}}=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)\\\gamma {\frac {v}{c}}&={\frac {v}{\sqrt {c^{2}-v^{2}}}}\end{aligned}}\end{matrix}}

where x^* is the Galilean transformation x-vt. Except the additional γ in the time transformation, this is the complete Lorentz transformation (4b).[50] While t is the "true" time for observers resting in the aether, t′ is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in two steps. At first an implicit Galilean transformation, and later the expansion into the "fictitious" electromagnetic system with the aid of the Lorentz transformation. In order to explain the negative result of the Michelson–Morley experiment, he (1892b)[R 12] introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced length contraction in his theory (without proof as he admitted). The same hypothesis was already made by George FitzGerald in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.

In 1895, Lorentz further elaborated on his theory and introduced the "theorem of corresponding states". This theorem states that a moving observer (relative to the ether) in his "fictitious" field makes the same observations as a resting observers in his "real" field for velocities to first order in v/c. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:[R 13]

{\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x&=x^{\prime }{\sqrt {1-{\frac {{\mathfrak {p}}^{2}}{V^{2}}}}}\\y&=y^{\prime }\\z&=z^{\prime }\\t&=t^{\prime }\end{aligned}}\right|&{\begin{aligned}x^{\ast }=x-vt&={\frac {x^{\prime }}{\gamma }}\\y&=y^{\prime }\\z&=z^{\prime }\\t&=t^{\prime }\end{aligned}}\end{matrix}}

For solving optical problems Lorentz used the following transformation, in which the modified time variable was called "local time" (German: Ortszeit) by him:[R 14]

{\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x&=\mathrm {x} -{\mathfrak {p}}_{x}t\\y&=\mathrm {y} -{\mathfrak {p}}_{y}t\\z&=\mathrm {z} -{\mathfrak {p}}_{z}t\\t^{\prime }&=t-{\frac {{\mathfrak {p}}_{x}}{V^{2}}}x-{\frac {{\mathfrak {p}}_{y}}{V^{2}}}y-{\frac {{\mathfrak {p}}_{z}}{V^{2}}}z\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=x-v_{x}t\\y^{\prime }&=y-v_{y}t\\z^{\prime }&=z-v_{z}t\\t^{\prime }&=t-{\frac {v_{x}}{c^{2}}}x'-{\frac {v_{y}}{c^{2}}}y'-{\frac {v_{z}}{c^{2}}}z'\end{aligned}}\end{matrix}}

With this concept Lorentz could explain the Doppler effect, the aberration of light, and the Fizeau experiment.[51]

Larmor (1897, 1900)

In 1897, Larmor extended the work of Lorentz and derived the following transformation[R 15]

{\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x_{1}&=x\varepsilon ^{\frac {1}{2}}\\y_{1}&=y\\z_{1}&=z\\t^{\prime }&=t-vx/c^{2}\\dt_{1}&=dt^{\prime }\varepsilon ^{-{\frac {1}{2}}}\\\varepsilon &=\left(1-v^{2}/c^{2}\right)^{-1}\end{aligned}}\right|&{\begin{aligned}x_{1}&=\gamma x^{\ast }=\gamma (x-vt)\\y_{1}&=y\\z_{1}&=z\\t^{\prime }&=t-{\frac {vx^{\ast }}{c^{2}}}=t-{\frac {v(x-vt)}{c^{2}}}\\dt_{1}&={\frac {dt^{\prime }}{\gamma }}\\\gamma ^{2}&={\frac {1}{1-{\frac {v^{2}}{c^{2}}}}}\end{aligned}}\end{matrix}}

Larmor noted that if it is assumed that the constitution of molecules is electrical then the FitzGerald–Lorentz contraction is a consequence of this transformation, explaining the Michelson–Morley experiment. It's notable that Larmor was the first who recognized that some sort of time dilation is a consequence of this transformation as well, because "individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ".[52][53] Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than (v/c)² – when his 1897 paper was reprinted in 1929, Larmor added the following comment in which he described how they can be made valid to all orders of v/c:[R 16]

Nothing need be neglected: the transformation is exact if v/c² is replaced by εv/c² in the equations and also in the change following from t to t′, as is worked out in Aether and Matter (1900), p. 168, and as Lorentz found it to be in 1904, thereby stimulating the modern schemes of intrinsic relational relativity.

In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time t″=t′-εvx′/c² instead of the 1897 expression t′=t-vx/c² by replacing v/c² with εv/c², so that t″ is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the x′, y′, z′, t′ coordinates:[R 17]

{\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{\prime }&=x-vt\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t\\t^{\prime \prime }&=t^{\prime }-\varepsilon vx^{\prime }/c^{2}\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=x-vt\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t\\t^{\prime \prime }=t^{\prime }-{\frac {\gamma ^{2}vx^{\prime }}{c^{2}}}&=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}

Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor (v/c)², and so he sought the transformations which were "accurate to second order" (as he put it). Thus he wrote the final transformations (where x′=x-vt and t″ as given above) as:[R 18]

{\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x_{1}&=\varepsilon ^{\frac {1}{2}}x^{\prime }\\y_{1}&=y^{\prime }\\z_{1}&=z^{\prime }\\dt_{1}&=\varepsilon ^{-{\frac {1}{2}}}dt^{\prime \prime }=\varepsilon ^{-{\frac {1}{2}}}\left(dt^{\prime }-{\frac {v}{c^{2}}}\varepsilon dx^{\prime }\right)\\t_{1}&=\varepsilon ^{-{\frac {1}{2}}}t^{\prime }-{\frac {v}{c^{2}}}\varepsilon ^{\frac {1}{2}}x^{\prime }\end{aligned}}\right|&{\begin{aligned}x_{1}&=\gamma x^{\prime }=\gamma (x-vt)\\y_{1}&=y'=y\\z_{1}&=z'=z\\dt_{1}&={\frac {dt^{\prime \prime }}{\gamma }}={\frac {1}{\gamma }}\left(dt^{\prime }-{\frac {\gamma ^{2}vdx^{\prime }}{c^{2}}}\right)=\gamma \left(dt-{\frac {vdx}{c^{2}}}\right)\\t_{1}&={\frac {t^{\prime }}{\gamma }}-{\frac {\gamma vx^{\prime }}{c^{2}}}=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}

by which he arrived at the complete Lorentz transformation (4b). Larmor showed that Maxwell's equations were invariant under this two-step transformation, "to second order in v/c" – it was later shown by Lorentz (1904) and Poincaré (1905) that they are indeed invariant under this transformation to all orders in v/c.

Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term "Lorentz transformation" for Lorentz's first order transformations of coordinates and field configurations:

p. 583: [..] Lorentz's transformation for passing from the field of activity of a stationary electrodynamic material system to that of one moving with uniform velocity of translation through the aether.
p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..][R 19]
p. 622: [..] the transformation first developed by Lorentz: namely, each point in space is to have its own origin from which time is measured, its "local time" in Lorentz's phraseology, and then the values of the electric and magnetic vectors [..] at all points in the aether between the molecules in the system at rest, are the same as those of the vectors [..] at the corresponding points in the convected system at the same local times.[R 20]

Lorentz (1899, 1904)

Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 (again, x* must be replaced by x-vt):[R 21]

{\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{\prime }&={\frac {V}{\sqrt {V^{2}-{\mathfrak {p}}_{x}^{2}}}}x\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t-{\frac {{\mathfrak {p}}_{x}}{V^{2}-{\mathfrak {p}}_{x}^{2}}}x\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=\gamma x^{\ast }=\gamma (x-vt)\\y^{\prime }&=y\\z^{\prime }&=z\\t^{\prime }&=t-{\frac {\gamma ^{2}vx^{\ast }}{c^{2}}}=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}

Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows (where the above value of t′ has to be inserted):[R 22]

{\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x&={\frac {\varepsilon }{k}}x^{\prime \prime }\\y&=\varepsilon y^{\prime \prime }\\z&=\varepsilon x^{\prime \prime }\\t^{\prime }&=k\varepsilon t^{\prime \prime }\\k&={\frac {V}{\sqrt {V^{2}-{\mathfrak {p}}_{x}^{2}}}}\end{aligned}}\right|&{\begin{aligned}x^{\ast }=x-vt&={\frac {\varepsilon }{\gamma }}x^{\prime \prime }\\y&=\varepsilon y^{\prime \prime }\\z&=\varepsilon z^{\prime \prime }\\t^{\prime }=\gamma ^{2}\left(t-{\frac {vx}{c^{2}}}\right)&=\gamma \varepsilon t^{\prime \prime }\\\gamma &={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\end{aligned}}\end{matrix}}

This is equivalent to the complete Lorentz transformation (4b) when solved for x″ and t″ and with ε=1. Like Larmor, Lorentz noticed in 1899[R 23] also some sort of time dilation effect in relation to the frequency of oscillating electrons "that in S the time of vibrations be kε times as great as in S₀", where S₀ is the aether frame.[54]

In 1904 he rewrote the equations in the following form by setting l=1/ε (again, x* must be replaced by x-vt):[R 24]

{\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{\prime }&=klx\\y^{\prime }&=ly\\z^{\prime }&=lz\\t'&={\frac {l}{k}}t-kl{\frac {w}{c^{2}}}x\end{aligned}}\right|&{\begin{aligned}x^{\prime }&=\gamma lx^{\ast }=\gamma l(x-vt)\\y^{\prime }&=ly\\z^{\prime }&=lz\\t^{\prime }&={\frac {lt}{\gamma }}-{\frac {\gamma lvx^{\ast }}{c^{2}}}=\gamma l\left(t-{\frac {vx}{c^{2}}}\right)\end{aligned}}\end{matrix}}

Under the assumption that l=1 when v=0, he demonstrated that l=1 must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor l to unity, Lorentz's transformations now assumed the same form as Larmor's and are now completed. Unlike Larmor, who restricted himself to show the covariance of Maxwell's equations to second order, Lorentz tried to widen its covariance to all orders in v/c. He also derived the correct formulas for the velocity dependence of electromagnetic mass, and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones.[R 25] However, he didn't achieve full covariance of the transformation equations for charge density and velocity.[55] When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:[56]

One will notice that in this work the transformation equations of Einstein’s Relativity Theory have not quite been attained. [..] On this circumstance depends the clumsiness of many of the further considerations in this work.

Lorentz's 1904 transformation was cited and used by Alfred Bucherer in July 1904:[R 26]

x^{\prime }={\sqrt {s}}x,\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t'={\frac {t}{\sqrt {s}}}-{\sqrt {s}}{\frac {u}{v^{2}}}x,\quad s=1-{\frac {u^{2}}{v^{2}}}

or by Wilhelm Wien in July 1904:[R 27]

x=kx',\quad y=y',\quad z=z',\quad t'=kt-{\frac {v}{kc^{2}}}x

or by Emil Cohn in November 1904 (setting the speed of light to unity):[R 28]

x={\frac {x_{0}}{k}},\quad y=y_{0},\quad z=z_{0},\quad t=kt_{0},\quad t_{1}=t_{0}-w\cdot r_{0},\quad k^{2}={\frac {1}{1-w^{2}}}

or by Richard Gans in February 1905:[R 29]

x^{\prime }=kx,\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t'={\frac {t}{k}}-{\frac {kwx}{c^{2}}},\quad k^{2}={\frac {c^{2}}{c^{2}-w^{2}}}

Local time

Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, Henri Poincaré in 1900 commented on the origin of Lorentz's "wonderful invention" of local time.[57] He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed $c$ in both directions, which lead to what is nowadays called relativity of simultaneity, although Poincaré's calculation does not involve length contraction or time dilation.[R 30] In order to synchronise the clocks here on Earth (the x*, t* frame) a light signal from one clock (at the origin) is sent to another (at x*), and is sent back. It's supposed that the Earth is moving with speed v in the x-direction (= x*-direction) in some rest system (x, t) (i.e. the luminiferous aether system for Lorentz and Larmor). The time of flight outwards is

\delta t_{a}={\frac {x^{\ast }}{\left(c-v\right)}}

and the time of flight back is

\delta t_{b}={\frac {x^{\ast }}{\left(c+v\right)}}

.

The elapsed time on the clock when the signal is returned is δt_a+δt_b and the time t*=(δt_a+δt_b)/2 is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time t=δt_a is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus

t^{\ast }=t-{\frac {\gamma ^{2}vx^{*}}{c^{2}}}

identical to Lorentz (1892). By dropping the factor γ² under the assumption that ${\tfrac {v^{2}}{c^{2}}}\ll 1$ , Poincaré gave the result t*=t-vx*/c², which is the form used by Lorentz in 1895.

Similar physical interpretations of local time were later given by Emil Cohn (1904)[R 31] and Max Abraham (1905).[R 32]

Lorentz transformation

On June 5, 1905 (published June 9) Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form (4b):[R 33]

{\begin{aligned}x^{\prime }&=kl(x+\varepsilon t)\\y^{\prime }&=ly\\z^{\prime }&=lz\\t'&=kl(t+\varepsilon x)\\k&={\frac {1}{\sqrt {1-\varepsilon ^{2}}}}\end{aligned}}

.

Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name "Lorentz transformation".[58][59] Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by setting l=1, and modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity, i.e. making them fully Lorentz covariant.[60]

In July 1905 (published in January 1906)[R 34] Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the principle of least action; he demonstrated in more detail the group characteristics of the transformation, which he called Lorentz group, and he showed that the combination x²+y²+z²-t² is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing $ct{\sqrt {-1}}$ as a fourth imaginary coordinate, and he used an early form of four-vectors. He also formulated the velocity addition formula (4d), which he had already derived in unpublished letters to Lorentz from May 1905:[R 35]

\xi '={\frac {\xi +\varepsilon }{1+\xi \varepsilon }},\ \eta '={\frac {\eta }{k(1+\xi \varepsilon )}}

.

Einstein (1905) – Special relativity

On June 30, 1905 (published September 1905) Einstein published what is now called special relativity and gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. While Lorentz considered "local time" to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in v/c this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations concern the nature of space and time.[61][62][63]

The notation for this transformation is equivalent to Poincaré's of 1905 and (4b), except that Einstein didn't set the speed of light to unity:[R 36]

{\begin{aligned}\tau &=\beta \left(t-{\frac {v}{V^{2}}}x\right)\\\xi &=\beta (x-vt)\\\eta &=y\\\zeta &=z\\\beta &={\frac {1}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}\end{aligned}}

Einstein also defined the velocity addition formula (4d, 4e):[R 37]

{\begin{matrix}x={\frac {w_{\xi }+v}{1+{\frac {vw_{\xi }}{V^{2}}}}}t,\ y={\frac {\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}{1+{\frac {vw_{\xi }}{V^{2}}}}}w_{\eta }t\\U^{2}=\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2},\ w^{2}=w_{\xi }^{2}+w_{\eta }^{2},\ \alpha =\operatorname {arctg} {\frac {w_{y}}{w_{x}}}\\U={\frac {\sqrt {\left(v^{2}+w^{2}+2vw\cos \alpha \right)-\left({\frac {vw\sin \alpha }{V}}\right)^{2}}}{1+{\frac {vw\cos \alpha }{V^{2}}}}}\end{matrix}}\left|{\begin{matrix}{\frac {u_{x}-v}{1-{\frac {u_{x}v}{V^{2}}}}}=u_{\xi }\\{\frac {u_{y}}{\beta \left(1-{\frac {u_{x}v}{V^{2}}}\right)}}=u_{\eta }\\{\frac {u_{z}}{\beta \left(1-{\frac {u_{x}v}{V^{2}}}\right)}}=u_{\zeta }\end{matrix}}\right.

and the light aberration formula (4f):[R 38]

\cos \varphi '={\frac {\cos \varphi -{\frac {v}{V}}}{1-{\frac {v}{V}}\cos \varphi }}

Minkowski (1907–1908) – Spacetime

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 hyperboloid model by Hermann Minkowski in 1907 and 1908.[R 39][R 40] Minkowski particularly reformulated electrodynamics in a four-dimensional way (Minkowski spacetime).[64] 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 41]

{\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[R 41] directly used the 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 rapidity. He also wrote the Lorentz transformation in matrix form equivalent to (2a) (n=3):[R 42]

{\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}^{\prime 2}+x_{2}^{\prime 2}+x_{3}^{\prime 2}+x_{4}^{\prime 2}\\\left(x_{1}^{\prime }=x',\ x_{2}^{\prime }=y',\ x_{3}^{\prime }=z',\ x_{4}^{\prime }=it'\right)\\-x^{2}-y^{2}-z^{2}+t^{2}=-x^{\prime 2}-y^{\prime 2}-z^{\prime 2}+t^{\prime 2}\\\hline x_{h}=\alpha _{h1}x_{1}^{\prime }+\alpha _{h2}x_{2}^{\prime }+\alpha _{h3}x_{3}^{\prime }+\alpha _{h4}x_{4}^{\prime }\\\mathrm {A} =\mathrm {\left|{\begin{matrix}\alpha _{11},&\alpha _{12},&\alpha _{13},&\alpha _{14}\\\alpha _{21},&\alpha _{22},&\alpha _{23},&\alpha _{24}\\\alpha _{31},&\alpha _{32},&\alpha _{33},&\alpha _{34}\\\alpha _{41},&\alpha _{42},&\alpha _{43},&\alpha _{44}\end{matrix}}\right|,\ {\begin{aligned}{\bar {\mathrm {A} }}\mathrm {A} &=1\\\left(\det \mathrm {A} \right)^{2}&=1\\\det \mathrm {A} &=1\\\alpha _{44}&>0\end{aligned}}} \end{matrix}}

As a graphical representation of the Lorentz transformation he introduced the Minkowski diagram, which became a standard tool in textbooks and research articles on relativity:[R 43]

Original spacetime diagram by Minkowski in 1908.

Sommerfeld (1909) – Spherical trigonometry

Using an imaginary rapidity such as Minkowski, Arnold Sommerfeld (1909) formulated a transformation equivalent to Lorentz boost (3b), and the relativistic velocity addition (4d) in terms of trigonometric functions and the spherical law of cosines:[R 44]

{\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}}

Bateman and Cunningham (1909–1910) – Spherical wave transformation

In line with Lie's (1871) research on the relation between sphere transformations with an imaginary radius coordinate and 4D conformal transformations, it was pointed out by Bateman and Cunningham (1909–1910), that by setting u=ict as the imaginary fourth coordinates one can produce spacetime conformal transformations. Not only the quadratic form $\lambda \left(dx^{2}+dy^{2}+dz^{2}+du^{2}\right)$ , but also Maxwells equations are covariant with respect to these transformations, irrespective of the choice of λ. These variants of conformal or Lie sphere transformations were called spherical wave transformations by Bateman.[R 45][R 46] However, this covariance is restricted to certain areas such as electrodynamics, whereas the totality of natural laws in inertial frames is covariant under the Lorentz group.[R 47] In particular, by setting λ=1 the Lorentz group SO(1,3) can be seen as a 10-parameter subgroup of the 15-parameter spacetime conformal group Con(1,3).

Bateman (1910/12)[65] also alluded to the identity between the Laguerre inversion and the Lorentz transformations. In general, the isomorphism between the Laguerre group and the Lorentz group was pointed out by Élie Cartan (1912, 1915/55),[24][R 48] Henri Poincaré (1912/21)[R 49] and others.

Herglotz (1909/10) – Möbius transformation

Following Klein (1889–1897) and Fricke & Klein (1897) concerning the Cayley absolute, hyperbolic motion and its transformation, Gustav Herglotz (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic. The general case (on the left) equivalent to Lorentz transformation (6a) and the hyperbolic case (on the right) equivalent to Lorentz transformation (3d) or squeeze mapping (9d) are as follows:[R 50]

\left.{\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\z_{1}=x,\ z_{2}=y,\ z_{3}=z,\ z_{4}=t\\Z={\frac {z_{1}+iz_{2}}{z_{4}-z_{3}}}={\frac {x+iy}{t-z}},\ Z'={\frac {x'+iy'}{t'-z'}}\\Z={\frac {\alpha Z'+\beta }{\gamma Z'+\delta }}\end{matrix}}\right|{\begin{matrix}Z=Z'e^{\vartheta }\\{\begin{aligned}x&=x',&t-z&=(t'-z')e^{\vartheta }\\y&=y',&t+z&=(t'+z')e^{-\vartheta }\end{aligned}}\end{matrix}}

Varićak (1910) – Hyperbolic functions

Following Sommerfeld (1909), hyperbolic functions were used by Vladimir Varićak in several papers starting from 1910, who represented the equations of special relativity on the basis of 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 (3b):[R 51]

{\begin{aligned}l'&=-x\operatorname {sh} u+l\operatorname {ch} u,\\x'&=x\operatorname {ch} u-l\operatorname {sh} u,\\y'&=y,\quad z'=z,\\\operatorname {ch} u&={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}\end{aligned}}

and showed the relation of rapidity to the Gudermannian function and the angle of parallelism:[R 51]

{\frac {v}{c}}=\operatorname {th} u=\operatorname {tg} \psi =\sin \operatorname {gd} (u)=\cos \Pi (u)

He also related the velocity addition to the hyperbolic law of cosines:[R 52]

{\begin{matrix}\operatorname {ch} {u}=\operatorname {ch} {u_{1}}\operatorname {c} h{u_{2}}+\operatorname {sh} {u_{1}}\operatorname {sh} {u_{2}}\cos \alpha \\\operatorname {ch} {u_{i}}={\frac {1}{\sqrt {1-\left({\frac {v_{i}}{c}}\right)^{2}}}},\ \operatorname {sh} {u_{i}}={\frac {v_{i}}{\sqrt {1-\left({\frac {v_{i}}{c}}\right)^{2}}}}\\v={\sqrt {v_{1}^{2}+v_{2}^{2}-\left({\frac {v_{1}v_{2}}{c}}\right)^{2}}}\ \left(a={\frac {\pi }{2}}\right)\end{matrix}}

Subsequently, other authors such as E. T. Whittaker (1910) or Alfred Robb (1911, who coined the name rapidity) used similar expressions, which are still used in modern textbooks.[10]

Ignatowski (1910)

While earlier derivations and formulations of the Lorentz transformation relied from the outset on optics, electrodynamics, or the invariance of the speed of light, Vladimir Ignatowski (1910) showed that it is possible to use the principle of relativity (and related group theoretical principles) alone, in order to derive the following transformation between two inertial frames:[R 53][R 54]

{\begin{aligned}dx'&=p\ dx-pq\ dt\\dt'&=-pqn\ dx+p\ dt\\p&={\frac {1}{\sqrt {1-q^{2}n}}}\end{aligned}}

The variable n can be seen as a space-time constant whose value has to be determined by experiment or taken from a known physical law such as electrodynamics. For that purpose, Ignatowski used the above-mentioned Heaviside ellipsoid representing a contraction of electrostatic fields by x/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when n=1/c², resulting in p=γ and the Lorentz transformation (4b). With n=0, no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved by Philipp Frank and Hermann Rothe (1911, 1912),[R 55] with various authors developing similar methods in subsequent years.[66]

Noether (1910), Klein (1910) – Quaternions

Felix Klein (1908) described Cayley's (1854) 4D quaternion multiplications as "Drehstreckungen" (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor), and pointed out that the modern principle of relativity as provided by Minkowski is essentially only the consequent application of such Drehstreckungen, even though he didn't provide details.[R 56]

In an appendix to Klein's and Sommerfeld's "Theory of the top" (1910), Fritz Noether showed how to formulate hyperbolic rotations using biquaternions with $\omega ={\sqrt {-1}}$ , which he also related to the speed of light by setting ω²=-c². He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations equivalent to (7a):[R 57]

{\begin{matrix}V={\frac {Q_{1}vQ_{2}}{T_{1}T_{2}}}\\\hline X^{2}+Y^{2}+Z^{2}+\omega ^{2}S^{2}=x^{2}+y^{2}+z^{2}+\omega ^{2}s^{2}\\\hline {\begin{aligned}V&=Xi+Yj+Zk+\omega S\\v&=xi+yj+zk+\omega s\\Q_{1}&=(+Ai+Bj+Ck+D)+\omega (A'i+B'j+C'k+D')\\Q_{2}&=(-Ai-Bj-Ck+D)+\omega (A'i+B'j+C'k-D')\\T_{1}T_{2}&=T_{1}^{2}=T_{2}^{2}=A^{2}+B^{2}+C^{2}+D^{2}+\omega ^{2}\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)\end{aligned}}\end{matrix}}

Besides citing quaternion related standard works such as Cayley (1854), Noether referred to the entries in Klein's encyclopedia by Eduard Study (1899) and the French version by Élie Cartan (1908).[67] Cartan's version contains a description of Study's dual numbers, Clifford's biquaternions (including the choice $\omega ={\sqrt {-1}}$ for hyperbolic geometry), and Clifford algebra, with references to Stephanos (1883), Buchheim (1884/85), Vahlen (1901/02) and others.

Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:[R 58]

{\begin{matrix}{\begin{aligned}&\left(i_{1}x'+i_{2}y'+i_{3}z'+ict'\right)\\&\quad -\left(i_{1}x_{0}+i_{2}y_{0}+i_{3}z_{0}+ict_{0}\right)\end{aligned}}={\frac {\left[{\begin{aligned}&\left(i_{1}(A+iA')+i_{2}(B+iB')+i_{3}(C+iC')+i_{4}(D+iD')\right)\\&\quad \cdot \left(i_{1}x+i_{2}y+i_{3}z+ict\right)\\&\quad \quad \cdot \left(i_{1}(A-iA')+i_{2}(B-iB')+i_{3}(C-iC')-(D-iD')\right)\end{aligned}}\right]}{\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)-\left(A^{2}+B^{2}+C^{2}+D^{2}\right)}}\\\hline {\text{where}}\\AA'+BB'+CC'+DD'=0\\A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\end{matrix}}

or in March 1911[R 59]

{\begin{matrix}g'={\frac {pg\pi }{M}}\\\hline {\begin{aligned}g&={\sqrt {-1}}ct+ix+jy+kz\\g'&={\sqrt {-1}}ct'+ix'+jy'+kz'\\p&=(D+{\sqrt {-1}}D')+i(A+{\sqrt {-1}}A')+j(B+{\sqrt {-1}}B')+k(C+{\sqrt {-1}}C')\\\pi &=(D-{\sqrt {-1}}D')-i(A-{\sqrt {-1}}A')-j(B-{\sqrt {-1}}B')-k(C-{\sqrt {-1}}C')\\M&=\left(A^{2}+B^{2}+C^{2}+D^{2}\right)-\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)\\&AA'+BB'+CC'+DD'=0\\&A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\end{aligned}}\end{matrix}}

Conway (1911), Silberstein (1911) – Quaternions

Arthur W. Conway in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:[R 60]

{\begin{matrix}{\begin{aligned}{\mathtt {D}}&=\mathbf {a} ^{-1}{\mathtt {D}}'\mathbf {a} ^{-1}\\{\mathtt {\sigma }}&=\mathbf {a} {\mathtt {\sigma }}'\mathbf {a} ^{-1}\end{aligned}}\\e=\mathbf {a} ^{-1}e'\mathbf {a} ^{-1}\\\hline a=\left(1-hc^{-1}\lambda \right)^{\frac {1}{2}}\left(1+c^{-2}\lambda ^{2}\right)^{-{\frac {1}{4}}}\end{matrix}}

Also Ludwik Silberstein in November 1911[R 61] as well as in 1914,[68] formulated the Lorentz transformation in terms of velocity v:

{\begin{matrix}q'=QqQ\\\hline {\begin{aligned}q&=\mathbf {r} +l=xi+yj+zk+\iota ct\\q&'=\mathbf {r} '+l'=x'i+y'j+z'k+\iota ct'\\Q&={\frac {1}{\sqrt {2}}}\left({\sqrt {1+\gamma }}+\mathrm {u} {\sqrt {1-\gamma }}\right)\\&=\cos \alpha +\mathrm {u} \sin \alpha =e^{\alpha \mathrm {u} }\\&\left\{\gamma =\left(1-v^{2}/c^{2}\right)^{-1/2},\ 2\alpha =\operatorname {arctg} \ \left(\iota {\frac {v}{c}}\right)\right\}\end{aligned}}\end{matrix}}

Silberstein cites Cayley (1854, 1855) and Study's encyclopedia entry (in the extended French version of Cartan in 1908), as well as the appendix of Klein's and Sommerfeld's book.

Herglotz (1911), Silberstein (1911) – Vector transformation

Gustav Herglotz (1911)[R 62] showed how to formulate the transformation equivalent to (4c) in order to allow for arbitrary velocities and coordinates v=(v_x, v_y, v_z) and r=(x, y, z):

{\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}x^{0}&=x+\alpha u(ux+vy+wz)-\beta ut\\y^{0}&=y+\alpha v(ux+vy+wz)-\beta vt\\z^{0}&=z+\alpha w(ux+vy+wz)-\beta wt\\t^{0}&=-\beta (ux+vy+wz)+\beta t\\&\alpha ={\frac {1}{{\sqrt {1-s^{2}}}\left(1+{\sqrt {1-s^{2}}}\right)}},\ \beta ={\frac {1}{\sqrt {1-s^{2}}}}\end{aligned}}\right|&{\begin{aligned}x'&=x+\alpha v_{x}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{x}t\\y'&=y+\alpha v_{y}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{y}t\\z'&=z+\alpha v_{z}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{z}t\\t'&=-\gamma \left(v_{x}x+v_{y}y+v_{z}z\right)+\gamma t\\&\alpha ={\frac {\gamma ^{2}}{\gamma +1}},\ \gamma ={\frac {1}{\sqrt {1-v^{2}}}}\end{aligned}}\end{matrix}}

This was simplified using vector notation by Ludwik Silberstein (1911 on the left, 1914 on the right):[R 63]

{\begin{array}{c|c}{\begin{aligned}\mathbf {r} '&=\mathbf {r} +(\gamma -1)(\mathbf {ru} )\mathbf {u} +i\beta \gamma lu\\l'&=\gamma \left[l-i\beta (\mathbf {ru} )\right]\end{aligned}}&{\begin{aligned}\mathbf {r} '&=\mathbf {r} +\left[{\frac {\gamma -1}{v^{2}}}(\mathbf {vr} )-\gamma t\right]\mathbf {v} \\t'&=\gamma \left[t-{\frac {1}{c^{2}}}(\mathbf {vr} )\right]\end{aligned}}\end{array}}

Equivalent formulas were also given by Wolfgang Pauli (1921),[69] with Erwin Madelung (1922) providing the matrix form[70]

{\begin{array}{c|c|c|c|c}&x&y&z&t\\\hline x'&1-{\frac {v_{x}^{2}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{x}v_{y}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{x}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&{\frac {-v_{x}}{\sqrt {1-\beta ^{2}}}}\\y'&-{\frac {v_{x}v_{y}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&1-{\frac {v_{y}^{2}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{y}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&{\frac {-v_{y}}{\sqrt {1-\beta ^{2}}}}\\z'&-{\frac {v_{x}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&-{\frac {v_{y}v_{z}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&1-{\frac {v_{z}^{2}}{v^{2}}}\left(1-{\frac {1}{\sqrt {1-\beta ^{2}}}}\right)&{\frac {-v_{z}}{\sqrt {1-\beta ^{2}}}}\\t'&{\frac {-v_{x}}{c^{2}{\sqrt {1-\beta ^{2}}}}}&{\frac {-v_{y}}{c^{2}{\sqrt {1-\beta ^{2}}}}}&{\frac {-v_{z}}{c^{2}{\sqrt {1-\beta ^{2}}}}}&{\frac {1}{\sqrt {1-\beta ^{2}}}}\end{array}}

These formulas were called "general Lorentz transformation without rotation" by Christian Møller (1952),[71] who in addition gave an even more general Lorentz transformation in which the Cartesian axes have different orientations, using a rotation operator ${\mathfrak {D}}$ . In this case, v′=(v′_x, v′_y, v′_z) is not equal to -v=(-v_x, -v_y, -v_z), but the relation $\mathbf {v} '=-{\mathfrak {D}}\mathbf {v}$ holds instead, with the result

{\begin{array}{c}{\begin{aligned}\mathbf {x} '&={\mathfrak {D}}^{-1}\mathbf {x} -\mathbf {v} '\left\{\left(\gamma -1\right)(\mathbf {x\cdot v} )/v^{2}-\gamma t\right\}\\t'&=\gamma \left(t-(\mathbf {v} \cdot \mathbf {x} )/c^{2}\right)\end{aligned}}\end{array}}

Borel (1913–14) – Cayley–Hermite parameter

Borel (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and Cayley's (1846) parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions equivalent to (5b):[R 64]

{\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\\hline {\scriptstyle {\begin{aligned}\delta a&=\lambda ^{2}+\mu ^{2}+\nu ^{2}-\rho ^{2},&\delta b&=2(\lambda \mu +\nu \rho ),&\delta c&=-2(\lambda \nu +\mu \rho ),\\\delta a'&=2(\lambda \mu -\nu \rho ),&\delta b'&=-\lambda ^{2}+\mu ^{2}+\nu ^{2}-\rho ^{2},&\delta c'&=2(\lambda \rho -\mu \nu ),\\\delta a''&=2(\lambda \nu -\mu \rho ),&\delta b''&=2(\lambda \rho +\mu \nu ),&\delta c''&=-\left(\lambda ^{2}+\mu ^{2}+\nu ^{2}+\rho ^{2}\right),\end{aligned}}}\\\left(\delta =\lambda ^{2}+\mu ^{2}-\rho ^{2}-\nu ^{2}\right)\\\lambda =\nu =0\rightarrow {\text{Hyperbolic rotation}}\end{matrix}}

In four dimensions equivalent to (5c):[R 65]

{\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\\hline {\scriptstyle {\begin{aligned}&\left(\mu ^{2}+\nu ^{2}-\alpha ^{2}\right)\cos \varphi +\left(\lambda ^{2}-\beta ^{2}-\gamma ^{2}\right)\operatorname {ch} {\theta }&&-(\alpha \beta +\lambda \mu )(\cos \varphi -\operatorname {ch} {\theta })-\nu \sin \varphi -\gamma \operatorname {sh} {\theta }\\&-(\alpha \beta +\lambda \mu )(\cos \varphi -\operatorname {ch} {\theta })-\nu \sin \varphi +\gamma \operatorname {sh} {\theta }&&\left(\mu ^{2}+\nu ^{2}-\beta ^{2}\right)\cos \varphi +\left(\mu ^{2}-\alpha ^{2}-\gamma ^{2}\right)\operatorname {ch} {\theta }\\&-(\alpha \gamma +\lambda \nu )(\cos \varphi -\operatorname {ch} {\theta })+\mu \sin \varphi -\beta \operatorname {sh} {\theta }&&-(\beta \mu +\mu \nu )(\cos \varphi -\operatorname {ch} {\theta })+\lambda \sin \varphi +\alpha \operatorname {sh} {\theta }\\&(\gamma \mu -\beta \nu )(\cos \varphi -\operatorname {ch} {\theta })+\alpha \sin \varphi -\lambda \operatorname {sh} {\theta }&&-(\alpha \nu -\lambda \gamma )(\cos \varphi -\operatorname {ch} {\theta })+\beta \sin \varphi -\mu \operatorname {sh} {\theta }\\\\&\quad -(\alpha \gamma +\lambda \nu )(\cos \varphi -\operatorname {ch} {\theta })+\mu \sin \varphi +\beta \operatorname {sh} {\theta }&&\quad (\beta \nu -\mu \nu )(\cos \varphi -\operatorname {ch} {\theta })+\alpha \sin \varphi -\lambda \operatorname {sh} {\theta }\\&\quad -(\beta \mu +\mu \nu )(\cos \varphi -\operatorname {ch} {\theta })-\lambda \sin \varphi -\alpha \operatorname {sh} {\theta }&&\quad (\lambda \gamma -\alpha \nu )(\cos \varphi -\operatorname {ch} {\theta })+\beta \sin \varphi -\mu \operatorname {sh} {\theta }\\&\quad \left(\lambda ^{2}+\mu ^{2}-\gamma ^{2}\right)\cos \varphi +\left(\nu ^{2}-\alpha ^{2}-\beta ^{2}\right)\operatorname {ch} {\theta }&&\quad (\alpha \mu -\beta \lambda )(\cos \varphi -\operatorname {ch} {\theta })+\gamma \sin \varphi -\nu \operatorname {sh} {\theta }\\&\quad (\beta \gamma -\alpha \mu )(\cos \varphi -\operatorname {ch} {\theta })+\gamma \sin \varphi -\nu \operatorname {sh} {\theta }&&\quad -\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)\cos \varphi +\left(\lambda ^{2}+\mu ^{2}+\nu ^{2}\right)\operatorname {ch} {\theta }\end{aligned}}}\\\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}-\lambda ^{2}-\mu ^{2}-\nu ^{2}=-1\right)\end{matrix}}

Gruner (1921) – Trigonometric Lorentz boosts

In order to simplify the graphical representation of Minkowski space, Paul Gruner (1921) (with the aid of Josef Sauter) developed what is now called Loedel diagrams, using the following relations:[R 66]

{\displaystyle {\begin{matrix}v=\alpha \cdot c;\quad \beta ={\frac {1}{\sqrt {1-\alpha ^{2}}}}\\\sin \varphi =\alpha

This is equivalent to Lorentz transformation (8a) by the identity $\sec \varphi ={\tfrac {1}{\cos \varphi }}$

In another paper Gruner used the alternative relations:[R 67]

{\begin{matrix}\alpha ={\frac {v}{c}};\ \beta ={\frac {1}{\sqrt {1-\alpha ^{2}}}};\\\cos \theta =\alpha ={\frac {v}{c}};\ \sin \theta ={\frac {1}{\beta }};\ \cot \theta =\alpha \cdot \beta \\\hline x'={\frac {x}{\sin \theta }}-t\cdot \cot \theta ,\quad t'={\frac {t}{\sin \theta }}-x\cdot \cot \theta \end{matrix}}

This is equivalent to Lorentz Lorentz boost (8b) by the identity $\csc \theta ={\tfrac {1}{\sin \theta }}$ .

Euler's gap

In pursuing the history in years before Lorentz enunciated his expressions, one looks to the essence of the concept. In mathematical terms, Lorentz transformations are squeeze mappings, the linear transformations that turn a square into a rectangles of the same area. Before Euler, the squeezing was studied as quadrature of the hyperbola and led to the hyperbolic logarithm. In 1748 Euler issued his precalculus textbook where the number e is exploited for trigonometry in the unit circle. The first volume of Introduction to the Analysis of the Infinite had no diagrams, allowing teachers and students to draw their own illustrations.

There is a gap in Euler's text where Lorentz transformations arise. A feature of natural logarithm is its interpretation as area in hyperbolic sectors. In relativity the classical concept of velocity is replaced with rapidity, a hyperbolic angle concept built on hyperbolic sectors. A Lorentz transformation is a hyperbolic rotation which preserves differences of rapidity, just as the circular sector area is preserved with a circular rotation. Euler's gap is the lack of hyperbolic angle and hyperbolic functions, later developed by Johann H. Lambert. Euler described some transcendental functions: exponentiation and circular functions. He used the exponential series $\sum _{0}^{\infty }x^{n}/n!.$ With the imaginary unit i² = – 1, and splitting the series into even and odd terms, he obtained

e^{ix}=\sum _{0}^{\infty }(ix)^{2n}/(2n)!\ +\ \sum _{0}^{\infty }(ix)^{2n+1}/(2n+1)!=

=\sum _{0}^{\infty }(-1)^{n}x^{2n}/2n!+i\sum _{0}^{\infty }(-1)^{n}x^{2n+1}/(2n+1)!\ =\ \cos x+i\sin x.

This development misses the alternative:

e^{x}=\cosh x+\sinh x

(even and odd terms), and

e^{jx}=\cosh x+j\sinh x\quad (j^{2}=+1)

which parametrizes the unit hyperbola.

Here Euler could have noted split-complex numbers along with complex numbers.

For physics, one space dimension is insufficient. But to extend split-complex arithmetic to four dimensions leads to hyperbolic quaternions, and opens the door to abstract algebra's hypercomplex numbers. Reviewing the expressions of Lorentz and Einstein, one observes that the Lorentz factor is an algebraic function of velocity. For readers uncomfortable with transcendental functions cosh and sinh, algebraic functions may be more to their liking.

References

Historical mathematical sources

Learning materials related to History of Topics in Special Relativity/mathsource at Wikiversity

Historical relativity sources

Varićak (1912), p. 108
Borel (1914), pp. 39–41
Brill (1925)
Voigt (1887), p. 45
Lorentz (1915/16), p. 197
Lorentz (1915/16), p. 198
Bucherer (1908), p. 762
Heaviside (1888), p. 324
Thomson (1889), p. 12
Searle (1886), p. 333
Lorentz (1892a), p. 141
Lorentz (1892b), p. 141
Lorentz (1895), p. 37
Lorentz (1895), p. 49 for local time and p. 56 for spatial coordinates.
Larmor (1897), p. 229
Larmor (1897/1929), p. 39
Larmor (1900), p. 168
Larmor (1900), p. 174
Larmor (1904a), p. 583, 585
Larmor (1904b), p. 622
Lorentz (1899), p. 429
Lorentz (1899), p. 439
Lorentz (1899), p. 442
Lorentz (1904), p. 812
Lorentz (1904), p. 826
Bucherer, p. 129; Definition of s on p. 32
Wien (1904), p. 394
Cohn (1904a), pp. 1296-1297
Gans (1905), p. 169
Poincaré (1900), pp. 272–273
Cohn (1904b), p. 1408
Abraham (1905), § 42
Poincaré (1905), p. 1505
Poincaré (1905/06), pp. 129ff
Poincaré (1905/06), p. 144
Einstein (1905), p. 902
Einstein (1905), § 5 and § 9
Einstein (1905), § 7
Minkowski (1907/15), pp. 927ff
Minkowski (1907/08), pp. 53ff
Minkowski (1907/08), p. 59
Minkowski (1907/08), pp. 65–66, 81–82
Minkowski (1908/09), p. 77
Sommerfeld (1909), p. 826ff.
Bateman (1909/10), pp. 223ff
Cunningham (1909/10), pp. 77ff
Klein (1910)
Cartan (1912), p. 23
Poincaré (1912/21), p. 145
Herglotz (1909/10), pp. 404-408
Varićak (1910), p. 93
Varićak (1910), p. 94
Ignatowski (1910), pp. 973–974
Ignatowski (1910/11), p. 13
Frank & Rothe (1911), pp. 825ff; (1912), p. 750ff.
Klein (1908), p. 165
Noether (1910), pp. 939–943
Klein (1910), p. 300
Klein (1911), pp. 602ff.
Conway (1911), p. 8
Silberstein (1911/12), p. 793
Herglotz (1911), p. 497
Silberstein (1911/12), p. 792; (1914), p. 123
Borel (1913/14), p. 39
Borel (1913/14), p. 41
Gruner (1921a),
Gruner (1921b)

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Borel, Émile (1914). Introduction Geometrique à quelques Théories Physiques. Paris.
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Secondary sources

Bôcher (1907), chapter X
Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1
Naimark (1964), 2 in four dimensions
Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.
Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.
Müller (1910), p. 661, in particular footnote 247.
Sommerville (1911), p. 286, section K6.
Synge (1955), p. 129 for n=3
Laue (1921), pp. 79–80 for n=3
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Pauli (1921), p. 561
Barrett (2006), chapter 4, section 2
Miller (1981), chapter 1
Miller (1981), chapter 4–7
Møller (1952/55), Chapter II, § 18
Pauli (1921), pp. 562; 565–566
Plummer (1910), pp. 258-259: After deriving the relativistic expressions for the aberration angles φ' and φ, Plummer remarked on p. 259: Another geometrical representation is obtained by assimilating φ' to the eccentric and φ to the true anomaly in an ellipse whose eccentricity is v/U = sin β.
Robinson (1990), chapter 3-4, analyzed the relation between "Kepler's formula" and the "physical velocity addition formula" in special relativity.
Schottenloher (2008), section 2.2
Kastrup (2008), section 2.4.1
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Klein (1896/97), p. 12
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Jannsen (1995), Chap. 3.5.6
Darrigol (2005), Kap. 4
Pais (1982), Chap. 6c
Katzir (2005), 280–288
Miller (1981), Chap. 1.14
Miller (1981), Chap. 6
Pais (1982), Kap. 7
Darrigol (2005), Chap. 6
Walter (1999a)
Bateman (1910/12), pp. 358–359
Baccetti (2011), see references 1–25 therein.
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Touma, J. R.; Tremaine, S. & Kazandjian, M. V. (2009). "Gauss's method for secular dynamics, softened". Monthly Notices of the Royal Astronomical Society. 394 (2): 1085–1108. arXiv:0811.2812. doi:10.1111/j.1365-2966.2009.14409.x.
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External links

Mathpages: 1.4 The Relativity of Light

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[var-13] Varićak (1912), p. 108

[28] Borel (1914), pp. 39–41

[29] Brill (1925)

[48] Voigt (1887), p. 45

[51] Lorentz (1915/16), p. 197

[53] Lorentz (1915/16), p. 198

[54] Bucherer (1908), p. 762

[55] Heaviside (1888), p. 324

[57] Thomson (1889), p. 12

[59] Searle (1886), p. 333

[60] Lorentz (1892a), p. 141

[62] Lorentz (1892b), p. 141

[63] Lorentz (1895), p. 37

[64] Lorentz (1895), p. 49 for local time and p. 56 for spatial coordinates.

[66] Larmor (1897), p. 229

[69] Larmor (1897/1929), p. 39

[70] Larmor (1900), p. 168

[71] Larmor (1900), p. 174

[72] Larmor (1904a), p. 583, 585

[73] Larmor (1904b), p. 622

[74] Lorentz (1899), p. 429

[75] Lorentz (1899), p. 439

[76] Lorentz (1899), p. 442

[78] Lorentz (1904), p. 812

[79] Lorentz (1904), p. 826

[82] Bucherer, p. 129; Definition of s on p. 32

[83] Wien (1904), p. 394

[84] Cohn (1904a), pp. 1296-1297

[85] Gans (1905), p. 169

[87] Poincaré (1900), pp. 272–273

[88] Cohn (1904b), p. 1408

[89] Abraham (1905), § 42

[90] Poincaré (1905), p. 1505

[94] Poincaré (1905/06), pp. 129ff

[95] Poincaré (1905/06), p. 144

[99] Einstein (1905), p. 902

[100] Einstein (1905), § 5 and § 9

[101] Einstein (1905), § 7

[102] Minkowski (1907/15), pp. 927ff

[103] Minkowski (1907/08), pp. 53ff

[mink1-105] Minkowski (1907/08), p. 59

[106] Minkowski (1907/08), pp. 65–66, 81–82

[107] Minkowski (1908/09), p. 77

[108] Sommerfeld (1909), p. 826ff.

[109] Bateman (1909/10), pp. 223ff

[110] Cunningham (1909/10), pp. 77ff

[111] Klein (1910)

[113] Cartan (1912), p. 23

[114] Poincaré (1912/21), p. 145

[115] Herglotz (1909/10), pp. 404-408

[var1-116] Varićak (1910), p. 93

[117] Varićak (1910), p. 94

[118] Ignatowski (1910), pp. 973–974

[119] Ignatowski (1910/11), p. 13

[120] Frank & Rothe (1911), pp. 825ff; (1912), p. 750ff.

[122] Klein (1908), p. 165

[123] Noether (1910), pp. 939–943

[125] Klein (1910), p. 300

[126] Klein (1911), pp. 602ff.

[127] Conway (1911), p. 8

[128] Silberstein (1911/12), p. 793

[130] Herglotz (1911), p. 497

[131] Silberstein (1911/12), p. 792; (1914), p. 123

[135] Borel (1913/14), p. 39

[136] Borel (1913/14), p. 41

[137] Gruner (1921a),

[138] Gruner (1921b)

[1] Bôcher (1907), chapter X

[ratcliffe-2] Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1

[3] Naimark (1964), 2 in four dimensions

[4] Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.

[5] Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.

[6] Müller (1910), p. 661, in particular footnote 247.

[7] Sommerville (1911), p. 286, section K6.

[8] Synge (1955), p. 129 for n=3

[9] Laue (1921), pp. 79–80 for n=3

[rind-10] Rindler (1969), p. 45

[11] Rosenfeld (1988), p. 231

[pau-12] Pauli (1921), p. 561

[barr-14] Barrett (2006), chapter 4, section 2

[15] Miller (1981), chapter 1

[16] Miller (1981), chapter 4–7

[17] Møller (1952/55), Chapter II, § 18

[18] Pauli (1921), pp. 562; 565–566

[plum-19] Plummer (1910), pp. 258-259: After deriving the relativistic expressions for the aberration angles φ' and φ, Plummer remarked on p. 259: Another geometrical representation is obtained by assimilating φ' to the eccentric and φ to the true anomaly in an ellipse whose eccentricity is v/U = sin β.

[robin-20] Robinson (1990), chapter 3-4, analyzed the relation between "Kepler's formula" and the "physical velocity addition formula" in special relativity.

[21] Schottenloher (2008), section 2.2

[22] Kastrup (2008), section 2.4.1

[23] Schottenloher (2008), section 2.3

[24] Coolidge (1916), p. 370

[ReferenceA-25] Cartan & Fano (1915/55), sections 14–15

[26] Hawkins (2013), pp. 210–214

[27] Meyer (1899), p. 329

[30] Klein (1928), § 2B

[31] Lorente (2003), section 3.3

[k28-32] Klein (1928), § 2A

[33] Klein (1896/97), p. 12

[synge-34] Synge (1956), ch. IV, 11

[35] Klein (1928), § 3A

[36] Penrose & Rindler (1984), section 2.1

[Lorente_2003,_section_4-37] Lorente (2003), section 4

[38] Penrose & Rindler (1984), p. 17

[39] Synge (1972), pp. 13, 19, 24

[40] Girard (1984), pp. 28–29

[41] Sobczyk (1995)

[42] Fjelstad (1986)

[43] Cartan & Study (1908), section 36

[44] Rothe (1916), section 16

[maj-45] Majerník (1986), 536–538

[terng-46] Terng & Uhlenbeck (2000), p. 21

[47] Bondi (1964), p. 118

[49] Miller (1981), 114–115

[pais-50] Pais (1982), Kap. 6b

[52] Voigt's transformations and the beginning of the relativistic revolution, Ricardo Heras, arXiv:1411.2559

[56] Brown (2003)

[mil-58] Miller (1981), 98–99

[milf-61] Miller (1982), 1.4 & 1.5

[65] Janssen (1995), 3.1

[67] Darrigol (2000), Chap. 8.5

[68] Macrossan (1986)

[77] Jannsen (1995), Kap. 3.3

[80] Miller (1981), Chap. 1.12.2

[81] Jannsen (1995), Chap. 3.5.6

[86] Darrigol (2005), Kap. 4

[91] Pais (1982), Chap. 6c

[92] Katzir (2005), 280–288

[93] Miller (1981), Chap. 1.14

[96] Miller (1981), Chap. 6

[97] Pais (1982), Kap. 7

[98] Darrigol (2005), Chap. 6

[104] Walter (1999a)

[112] Bateman (1910/12), pp. 358–359

[baccetti-121] Baccetti (2011), see references 1–25 therein.

[124] Cartan & Study (1908), sections 35–36

[129] Silberstein (1914), p. 156

[132] Pauli (1921), p. 555

[133] Madelung (1921), p. 207

[134] Møller (1952/55), pp. 41–43