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 velocity of light invariant between different inertial frames. They relate the spacetime coordinates, which specify the position $x,y,z$ and time $t$ of an event, relative to a particular inertial frame of reference (the "rest system"), and the coordinates $x',y',z'$ and $t'$ of the same event relative to another coordinate system moving in the positive x-direction at a constant speed $v$, relative to the rest system.

Overview

Most general Lorentz transformations

The general quadratic form $q(x)$ with coefficients of a symmetric matrix $\mathbf {A}$, 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 $\mathbf {g}$, can be written as

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

(Q1)

The case $n=1$ is the binary quadratic form introduced by Lagrange (1773) and Gauss (1798/1801), $n=2$ is the ternary quadratic form introduced by Gauss (1798/1801), $n=3$ is the quaternary quadratic form etc.

The general Lorentz transformation follows from (Q1) by setting $\mathbf {A} =\mathbf {A} '=\operatorname {diag} (-1,1,\dots ,1)$ and $\det \mathbf {g} =1$. It forms an indefinite orthogonal group called the Lorentz group SO(n, 1), the quadratic form $q(x)$ becomes the Lorentz interval in terms of an indefinite quadratic form in terms of pseudo-Euclidean space, and the associated bilinear form $b(x)$ becomes the Minkowski inner product:

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

(1a)

Such general Lorentz transformations (1a) for various dimensions were used by Gauss (1818), Jacobi (1833/34), Lebesgue (1837), Bour (1856), Somov (1863), Hill (1882) in order to simplify computations of elliptic functions and integrals. They were also used by Poincaré (1881), Cox (1881/82), 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 $-x_{0}^{2}+\cdots +x_{n}^{2}=0$. 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 dividing $x_{i},\ x_{i}^{\prime }$ by $x_{0},\ x_{0}^{\prime }$, 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 $\mathbf {g}$ stays the same as in (1a)):

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

(1b)

Such Lorentz transformations were used by Gauss (1818), Jacobi (1833/34), Lebesgue (1837), Bour (1856), Somov (1863), Hill (1882), Callandreau (1885) in order to simplify computations of elliptic functions and integrals. Particular forms of homographies were also used by Beltrami (1868) in terms of the Beltrami–Klein model of hyperbolic geometry, while Woods (1901, 1903) directly used (1b) with $n=3$. In addition, infinitesimal transformations related to the Lie algebra of the group of hyperbolic motions leaving invariant the unit sphere $x_{1}^{2}+\cdots +x_{n}^{2}-1=0$ were given by Lie (1885-1893) and Werner (1889) and Killing (1888-1897).

Particular forms of Lorentz transformations or relativistic velocity additions, mostly restricted to 2, 3 or 4 dimensions, have been formulated by many authors using:

• § orthogonal transformations – Lie (1871), Minkowski (1907), Sommerfeld (1909)

• § hyperbolic functions – Lambert (1768–1770), Taurinus (1826), Beltrami (1868), Escherich (1874), Cox (1882), Lindemann (1890/91), Gérard (1892), Killing (1893, 1897/98), Whitehead (1897/98), Schur (1885,1900), Woods (1903/05), Liebmann (1904/05)

• § velocities – Voigt (1887), Lorentz (1892, 1895), Larmor (1897, 1900), Lorentz (1899, 1904), Poincaré (1905), Einstein (1905), Minkowski (1907–1908), Varićak (1910), Herglotz (1909/10), Ignatowski (1910), Herglotz (1911) and Silberstein (1911)

• § spherical wave transformations – Lie (1871), Laguerre (1882), Stephanos (1883), Darboux (1887), Bateman & Cunningham (1909–1910)

• § Cayley–Hermite parameter – Cayley (1855), Borel (1913)

• § Cayley–Klein parameter – Gauss (1800), Cayley (1854), Klein (1884, 1889/90, 1896/97), Selling (1873), Bianchi (1888), Fricke (1891), Hausdorff (1899), Herglotz (1909/10)

• § quaternions and hyperbolic numbers – Cox (1882/83), Stephanos (1883), Buchheim (1884/85), Lipschitz (1885/86), Macfarlane (1892, 1894, 1900), Vahlen (1901/02, 1905), Noether (1910), Klein (1910), Conway (1911), Silberstein (1911)

Lorentz transformation via orthogonal transformation

By using the imaginary quantities $[{\mathfrak {x}}_{0},\ {\mathfrak {x}}'_{0}]=\left[ix_{0},\ ix_{0}^{\prime }\right]$, Lorentz transformation (1a) assumes the form of an orthogonal transformation, the Lorentz interval becomes the Euclidean norm, and the Minkowski inner product becomes the dot product:

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

(2a)

The cases $n=1,2,3,4$ of this quadratic form with real numbers and its transformation was discussed by Euler (1771) and in $n$ dimensions by Cauchy (1829). Its interpretation as leaving invariant the equation of the sphere with imaginary radius was given by Lie (1871), its interpretation as a Lorentz transformation with $n=3$ using one imaginary coordinate was given by Minkowski (1907) and Sommerfeld (1909).

A well known example of an orthogonal transformation is spatial rotation:

${\begin{matrix}{\mathfrak {x}}_{0}^{2}+x_{1}^{2}+x_{2}^{2}={\mathfrak {x}}_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline \left.{\begin{aligned}{\mathfrak {x}}_{0}&={\mathfrak {x}}_{0}^{\prime }\cos \phi +x_{1}^{\prime }\sin \phi \\x_{1}&=-{\mathfrak {x}}_{0}^{\prime }\sin \phi +x_{1}^{\prime }\cos \phi \\x_{2}&=x_{2}^{\prime }\end{aligned}}\right|{\begin{aligned}{\mathfrak {x}}_{0}^{\prime }&={\mathfrak {x}}_{0}\sin \phi +x_{1}\cos \phi \\x_{1}^{\prime }&={\mathfrak {x}}_{0}\cos \phi -x_{1}\sin \phi \\x_{2}^{\prime }&=x_{2}\end{aligned}}\end{matrix}}$

(2b)

This quadratic form with real numbers and its transformation was discussed by Euler (1771), its interpretation as a Lorentz transformation using one imaginary coordinate and imaginary angle was given by Minkowski (1907) and Sommerfeld (1909).

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):

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

(3a)

which resembles precisely the relations of hyperbolic functions by setting $g_{00}=g_{11}=\cosh \eta$ and $g_{01}=g_{10}=\sinh \eta$, with $\eta$ as the hyperbolic angle. Thus by adding an unchanged $x_{2}$-axis, a Lorentz boost for $n=2$ representing a translation in the hyperbolic plane in terms of Weierstrass coordinates of the hyperboloid model along one axis (being the same as a rotation around an imaginary angle $i\eta =\phi$ in (2b)) is given by

${\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline {\begin{aligned}x_{0}&=x_{0}^{\prime }\cosh \eta +x_{1}^{\prime }\sinh \eta \\x_{1}&=x_{0}^{\prime }\sinh \eta +x_{1}^{\prime }\cosh \eta \\x_{2}&=x_{2}^{\prime }\\\\x_{0}^{\prime }&=x_{0}\cosh \eta -x_{1}\sinh \eta \\x_{1}^{\prime }&=-x_{0}\sinh \eta +x_{1}\cosh \eta \\x_{2}^{\prime }&=x_{2}\end{aligned}}\left|{\scriptstyle {\begin{aligned}\sinh ^{2}\eta -\cosh ^{2}\eta &=-1&(a)\\\cosh ^{2}\eta -\sinh ^{2}\eta &=1&(b)\\{\frac {\sinh \eta }{\cosh \eta }}&=\tanh \eta &(c)\\{\frac {1}{\sqrt {1-\tanh ^{2}\eta }}}&=\cosh \eta &(d)\\{\frac {\tanh \eta }{\sqrt {1-\tanh ^{2}\eta }}}&=\sinh \eta &(e)\\{\frac {\tanh \eta +\tanh \zeta }{1+\tanh \eta \tanh \zeta }}&=\tanh \left(\eta +\zeta \right)&(f)\end{aligned}}}\right.\end{matrix}}$

(3b)

which can also be expressed in terms of exponential functions

${\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline {\begin{aligned}x_{0}^{\prime }+x_{1}^{\prime }&=e^{-\eta }\left(x_{0}+x_{1}\right)\\x_{0}^{\prime }-x_{1}^{\prime }&=e^{\eta }\left(x_{0}-x_{1}\right)\\x_{2}^{\prime }&=x_{2}\end{aligned}}\end{matrix}}$

(3c)

All hyperbolic relations (a,b,c,d,e,f) on the right of (3b) were given by Lambert (1768–1770). The Lorentz transformations (3b or 3c) were given by Cox (1882), Lindemann (1890/91), Gérard (1892), Killing (1893, 1897/98), Whitehead (1897/98), Woods (1903/05), Liebmann (1904/05), see § Historical formulas for Lorentz boosts.

In line with equation (1b) one can use coordinates $[u_{x},\ u_{y},\ 1]=\left[{\tfrac {x_{1}}{x_{0}}},\ {\tfrac {x_{2}}{x_{0}}},\ {\tfrac {x_{0}}{x_{0}}}\right]$, which in terms of hyperbolic geometry can be interpreted as changing the above Weierstrass coordinates into Beltrami coordinates inside the unit circle $u_{x}^{2}+u_{y}^{2}=1$, thus the corresponding Lorentz transformations (3b) obtain the form:

${\begin{matrix}u_{x}^{2}+u_{y}^{2}=u_{x}^{\prime 2}+u_{y}^{\prime 2}=1\\\hline \left.{\begin{aligned}u_{x}&={\frac {\sinh \eta +u'_{x}\cosh \eta }{\cosh \eta +u'_{x}\sinh \eta }}\\u_{y}&={\frac {u'_{y}}{\cosh \eta +u'_{x}\sinh \eta }}\\\\u'_{x}&={\frac {\sinh \eta -u_{x}\cosh \eta }{-\cosh \eta +u_{x}\sinh \eta }}\\u'_{y}&={\frac {-u_{y}}{-\cosh \eta +u_{x}\sinh \eta }}\end{aligned}}\right|{\scriptstyle {\begin{aligned}{\frac {\sinh \eta }{\cosh \eta }}&=\tanh \eta =v\\\cosh \eta &={\frac {1}{\sqrt {1-\tanh ^{2}\eta }}}\\u_{x}&=\tanh \zeta _{x}\\u_{y}&=\tanh \zeta _{y}\\u'_{x}&=\tanh \zeta '_{x}\\u'_{y}&=\tanh \zeta '_{y}\end{aligned}}}\left|{\begin{aligned}u_{x}&={\frac {\tanh \zeta '_{x}+\tanh \eta }{1+\tanh \zeta '_{x}\tanh \eta }}\\u_{y}&={\frac {\tanh \zeta '_{y}{\sqrt {1-\tanh ^{2}\eta }}}{1+\tanh \zeta '_{x}\tanh \eta }}\\\\u'_{x}&={\frac {\tanh \zeta {}_{x}-\tanh \eta }{1-\tanh \zeta {}_{x}\tanh \eta }}\\u'_{y}&={\frac {\tanh \zeta {}_{y}{\sqrt {1-\tanh ^{2}\eta }}}{1-\tanh \zeta {}_{x}\tanh \eta }}\end{aligned}}\right.\left|{\begin{aligned}u_{x}&={\frac {u'_{x}+v}{1+u'_{x}v}}\\u_{y}&={\frac {u'_{y}{\sqrt {1-v^{2}}}}{1+u'_{x}v}}\\\\u'_{x}&={\frac {u_{x}-v}{1-u_{x}v}}\\u'_{y}&={\frac {u_{y}{\sqrt {1-v^{2}}}}{1-u_{x}v}}\end{aligned}}\right.\end{matrix}}$

(3d)

These Lorentz transformations were given by Escherich (1874) (on the left) and Beltrami (1868) or Schur (1885/86, 1900/02) (on the right). By setting $u^{2}=u_{x}^{2}+u_{y}^{2}$ and $\tan \alpha ={\tfrac {u_{y}}{u_{x}}}$:

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

(3e)

the resulting Lorentz transformation (on the left) can be seen as equivalent to the hyperbolic law of cosines (on the right). 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 $\eta =\operatorname {atanh} \beta$ with $\beta =v/c$, so that $\gamma =\cosh \eta$ is the Lorentz factor, $\beta \gamma =\sinh \eta$ the proper velocity, $v=c\tanh \eta$ the relative velocity of two inertial frames, $u'=c\tanh \zeta$ the velocity of another object, $u=c\tanh \left(\eta +\zeta \right)$ the velocity-addition formula, thus (3b) becomes:

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

(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

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

(4b)

Similar transformations were 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. 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. 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).

Setting ${\scriptstyle [u_{x},\ u_{y},\ c^{2}]=\left[{\frac {x_{1}}{x_{0}}},\ {\frac {x_{2}}{x_{0}}},\ c^{2}{\frac {x_{0}}{x_{0}}}\right]}$ in (3b) or (4a), produces the Lorentz transformation of velocities (or velocity addition formula) in analogy to Beltrami coordinates of (3d):

${\begin{matrix}u_{x}^{2}+u_{y}^{2}=u_{x}^{\prime 2}+u_{y}^{\prime 2}=c^{2}\\\hline \left.{\begin{aligned}u_{x}&={\frac {c^{2}\sinh \eta +u'_{x}c\cosh \eta }{c\cosh \eta +u'_{x}\sinh \eta }}\\u_{y}&={\frac {cy'}{c\cosh \eta +u'_{x}\sinh \eta }}\\\\u'_{x}&={\frac {c^{2}\sinh \eta -u_{x}c\cosh \eta }{-c\cosh \eta +u_{x}\sinh \eta }}\\u'_{y}&={\frac {-cu_{y}}{-c\cosh \eta +u_{x}\sinh \eta }}\end{aligned}}\right|{\scriptstyle {\begin{aligned}{\frac {\sinh \eta }{\cosh \eta }}&=\tanh \eta ={\frac {v}{c}}\\\cosh \eta &={\frac {1}{\sqrt {1-\tanh ^{2}\eta }}}\\u_{x}&=c\tanh \zeta _{x}\\u_{y}&=c\tanh \zeta _{y}\\u'_{x}&=c\tanh \zeta '_{x}\\u'_{y}&=c\tanh \zeta '_{y}\end{aligned}}}\left|{\begin{aligned}u_{x}&={\frac {c\tanh \zeta '_{x}+c\tanh \eta }{1+\tanh \zeta '_{x}\tanh \eta }}\\u_{y}&={\frac {c\tanh \zeta '_{y}{\sqrt {1-\tanh ^{2}\eta }}}{1+\tanh \zeta '_{x}\tanh \eta }}\\\\u'_{x}&={\frac {c\tanh \zeta {}_{x}-c\tanh \eta }{1-\tanh \zeta {}_{x}\tanh \eta }}\\u'_{y}&={\frac {c\tanh \zeta {}_{y}{\sqrt {1-\tanh ^{2}\eta }}}{1-\tanh \zeta {}_{x}\tanh \eta }}\end{aligned}}\right.\left|{\begin{aligned}u_{x}&={\frac {u'_{x}+v}{1+{\frac {v}{c^{2}}}u'_{x}}}\\u_{y}&={\frac {u'_{y}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1+{\frac {v}{c^{2}}}u'_{x}}}\\\\u'_{x}&={\frac {u_{x}-v}{1-{\frac {v}{c^{2}}}u{}_{x}}}\\u'_{y}&={\frac {u_{y}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1-{\frac {v}{c^{2}}}u{}_{x}}}\end{aligned}}\right.\end{matrix}}$

(4c)

or more generally using the hyperbolic law of cosines in terms of (3e):

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

(4d)

The velocity addition formula was given by Einstein (1905), while the relations to trigonometric and hyperbolic functions were given by Sommerfeld (1909) and Varićak (1910).

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

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

or in vector notation

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

(4e)

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

Lorentz transformation via spherical wave transformation

Main articles: Spherical wave transformation and Lie sphere geometry

A general sphere transformation preserving the quadratic form $\lambda \left(-x_{0}^{2}+\dots +x_{n}^{2}\right)$ is the group Con(p,1) of spacetime conformal transformations in terms of inversions or special conformal transformations, which has the property of changing spheres into spheres. One can switch between the representations by using an imaginary radius coordinate $x_{0}=iR$ which gives $\lambda \left(x_{0}^{2}+\dots +x_{n}^{2}\right)$ (conformal transformation), or by using a real radius coordinate $x_{0}=R$ which gives $\lambda \left(-x_{0}^{2}+\dots +x_{n}^{2}\right)$ (spherical wave transformation). This group was studied by Lie (1871) and others in terms of contact transformations, in which $x_{0}$ is related to the radius $R$.

It turns out that Con(3,1) is isomorphic to the special orthogonal group SO(4,2), and contains the Lorentz group SO(3,1) as a subgroup by setting $\lambda =1$. More generally, Con(p,q) is isomorphic to SO(p+1,q+1) and contains SO(p,q) as subgroup. This implies that Con(p,0) is isomorphic to the Lorentz group of arbitrary dimensions SO(p+1,1). Consequently, the conformal group in the plane Con(2,0) – known as the group of Möbius transformations – is isomorphic to the Lorentz group SO(3,1). This can be seen using tetracyclical coordinates satisfying the form $-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0$, which were discussed by Pockels (1891), Klein (1893), Bôcher (1894).

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 introduced by Laguerre (1882) and discussed by Darboux (1887) leaving invariant $x^{2}+y^{2}+z^{2}-R^{2}$ with $R$ as radius, thus the Laguerre group is isomorphic to the Lorentz group

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 relation between Lie's sphere transformations and the Lorentz transformation was noted by Bateman & Cunningham (1909–1910) and others. Furthermore, 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

General transformations of arbitrary quadratic forms $q$ into themselves can also be given using independent parameters based on the Cayley transform $\mathbf {x} =(\mathbf {I} -\mathbf {T} )^{-1}\cdot (\mathbf {I} +\mathbf {T} )\cdot \mathbf {x}$ in the form:

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

(Q2)

where $\mathbf {A}$ is, as above, a symmetric matrix defining the quadratic form (there is no primed $\mathbf {A} '$ because the coefficients are assumed to be the same on both sides), $\mathbf {I}$ the identity matrix, and $\mathbf {T}$ an arbitrary antisymmetric matrix. 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). For instance, the choice $\mathbf {A} =\operatorname {diag} (1,1,1)$ gives an orthonormal transformation which can be used to describe spatial rotations in terms of the four Euler-Rodrigues parameters $[a,b,c,d]$ discovered by Euler (1771) and Rodrigues (1840) (which can be interpreted as the coefficients of quaternions).

Also the Lorentz interval and the Lorentz transformation can be produced by the Cayley–Hermite formalism. The Lorentz transformation in 2 dimensions follows from (Q2) by setting $\mathbf {A} =\operatorname {diag} (-1,1)$:

$\left.{\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}\\\hline \mathbf {x} '={\frac {1}{1-a^{2}}}\left[{\begin{matrix}1+a^{2}&-2a\\-2a&1+a^{2}\end{matrix}}\right]\cdot \mathbf {x} \end{matrix}}\right|\left\{\mathbf {T} ={\begin{vmatrix}0&a\\-a&0\end{vmatrix}}\right\}$

(5a)

or with $\mathbf {A} =\operatorname {diag} (-1,1,1)$:

${\scriptstyle \left.{\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}\\\hline \mathbf {x} '={\frac {1}{\kappa }}\left[{\begin{matrix}1+a^{2}+b^{2}+c^{2}&2(a-cb)&2(-ca-b)\\2(cb+a)&1-a^{2}-b^{2}+c^{2}&2(-c-ba)\\2(ca-b)&2(c-ba)&1-a^{2}+b^{2}-c^{2}\end{matrix}}\right]\cdot \mathbf {x} \\\left(\kappa =1+a^{2}-b^{2}-c^{2}\right)\end{matrix}}\right|\left\{\mathbf {T} ={\begin{vmatrix}0&a&-b\\-a&0&c\\b&-c&0\end{vmatrix}}\right\}}$

(5b)

or with $\mathbf {A} =\operatorname {diag} (-1,1,1,1)$:

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

(5c)

Equations containing the Lorentz transformations (5a, 5b, 5c) as special cases were given by Cayley (1855). In relativity, equations similar to (5b, 5c) were first employed by Borel (1913) to represent Lorentz transformations.

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

The previously mentioned Euler-Rodrigues parameters $a,b,c,1$ (i.e. Cayley-Hermite parameter in three Euclidean dimensions) are closely related to Cayley–Klein parameter $\alpha ,\beta ,\gamma ,\delta$ introduced by Helmholtz (1866/67), Cayley (1879) and Klein (1884) to connect Möbius transformations and rotations:

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

In modern publications, the Cayley–Klein parameters are related to a spin-matrix $\mathbf {D}$, 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 hyperbolic motions, the Hermitian matrix $\mathbf {u}$ associated with these transformations produces an invariant determinant $x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}$ 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". 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(2, 0) or projective special linear group PSL(2, C) is isomorphic to both the Lorentz group and the group of isometries of hyperbolic space.

In four dimensions, these Lorentz transformations can be represented as follows:

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

(6a)

or expressing $\mathbf {u} '$ in terms of $x_{0}^{\prime }\dots$ and $x_{0}\dots$ it follows:

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

(6b)

In the case of three dimensions it simplifies to:

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

(6c)

thus

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

(6d)

Lorentz Transformation (6d) was given by Gauss around 1800 (posthumously published 1863), Selling (1873), Bianchi (1888), Fricke (1891). The general transformation $\mathbf {u} '$ in (6c) was used in the theory of binary quadratic forms given by Lagrange (1773) and Gauss (1798/1801) as well as in relation to Fuchsian functions by Poincaré (1886), while $\mathbf {u} '$ in (6a) was given by Cayley (1854) and Klein (1884) in relation to surfaces of second degree and by Poincaré (1886) in relation to Fuchsian functions. The adaptation of (6a) to hyperbolic motions by which they become Lorentz transformations was provided by Klein (1889/90, 1896/97), Bianchi (1893), Fricke (1893, 1897) and Hausdorff (1899). In relativity, (6a) was first employed by Herglotz (1909/10).

Lorentz transformation via quaternions and hyperbolic numbers

The Lorentz transformations can also be expressed in terms of biquaternions having one real part $x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}$ and one purely imaginary part $ix_{0}$ (some authors use the opposite convention). Its general form (on the left) and the corresponding boost (on the right) are as follows (where the overline denotes Hamiltonian conjugation and * complex conjugation):

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

(7a)

Cayley (1854, 1855) derived quaternion transformations leaving invariant the sum of 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 Clifford's biquaternions $a+\omega b$ to hyperbolic geometry ($\omega ^{2}=-1$ for hyperbolic geometry, $\omega ^{2}=1$ elliptic, $\omega ^{2}=0$ parabolic). Stephanos (1883) divided Hamilton's biquaternions into one real and one imaginary part, and introduced a homography leaving invariant the equations of oriented spheres or oriented planes. 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 was given by Noether (1910), Klein (1910), Conway (1911), Silberstein (1911).

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

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

(7b)

After the trigonometric expression $e^{ix}=\cos x+i\sin x$ (Euler's formula) was given by Euler (1748) and the hyperbolic analogue $e^{\varepsilon \eta }$ 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 $\varepsilon ^{2}=1$, the elliptic one follows with $\varepsilon ^{2}=-1$, and parabolic with $\varepsilon ^{2}=0$ (this should not be confused with the expression $\omega ^{2}$ in Clifford's biquaternions also used by Cox, in which $\omega ^{2}=-1$ 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 $\varepsilon ^{2}=-1$ for elliptic, $\varepsilon ^{2}=0$ 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:

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

The Lorentz interval follows if the sign of one $i^{2}$ differs from all others. 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 $\varepsilon ^{2}=-1$ and parabolic motions $\varepsilon ^{2}=0$, all of which he also related to biquaternions.

Mathematics of the 19th century

Historical formulas for Lorentz boosts and velocity additions

A summary of historical Lorentz boost formulas consistent with (3a, 3b, 4a, 4b) and velocity additions consistent with (3d, 3e, 4c, 4d).

Lorentz boosts
Cox (1881/82)	${\begin{aligned}X&=x\cosh p-z\sinh p\\Z&=-x\sinh p+z\cosh p\end{aligned}}$ and ${\begin{aligned}x&=X\cosh p+Z\sinh p\\z&=X\sinh p+Z\cosh p\end{aligned}}$	$x^{2}-y^{2}-z^{2}$
Laguerre (1882)	$D'={\frac {D\left(1+\alpha ^{2}\right)-2\alpha R}{1-\alpha ^{2}}},\quad R'={\frac {2\alpha D-R\left(1+\alpha ^{2}\right)}{1-\alpha ^{2}}}$	$D^{2}-R^{2}$
Cox (1882/83)	${\begin{matrix}QP^{-1}=\cosh \theta +\iota \sinh \theta \\QP^{-1}=e^{\iota \theta }\\\left\{{\text{Lorentz boost with}}\ (Q,P)=\left(x_{1}^{\prime }+\iota x_{0}^{\prime },\ x_{1}+\iota x_{0}\right)\right\}\end{matrix}}\left(\iota ^{2}=1\right)$
Darboux (1887)	${\begin{aligned}x'&=x,&z'&={\frac {1+k^{2}}{1-k^{2}}}z-{\frac {2kR}{1-k^{2}}},\\y'&=y,&R'&={\frac {2kz}{1-k^{2}}}-{\frac {1+k^{2}}{1-k^{2}}}R,\end{aligned}}$	$X^{2}+Y^{2}+Z^{2}-R^{2}$
Lindemann (1890/91)	${\begin{matrix}{\begin{aligned}x_{2}&=\xi _{2}\cos \alpha +\xi _{3}\sin \alpha ,&x_{1}&=\xi _{1}\cos {\frac {a}{i}}+2ki\xi _{4}\sin {\frac {a}{i}},\\x_{3}&=-\xi _{2}\sin \alpha +\xi _{3}\cos \alpha ,&2kx_{4}&=i\xi _{1}\sin {\frac {a}{i}}+2k\xi _{4}\cos {\frac {a}{i}}.\end{aligned}}\\\left\{{\text{Lorentz boost with}}\ \alpha =0,\ 2k=1\right\}\end{matrix}}$	$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}$
Gérard (1892)	$\left.{\begin{aligned}X&=Z_{0}X'+X_{0}Z'\\Y&=Y'\\Z&=X_{0}X'+Z_{0}Z'\end{aligned}}\right|{\begin{aligned}X_{0}&=\operatorname {sh} OO'\\Z_{0}&=\operatorname {ch} OO'\end{aligned}}$	$X^{2}+Y^{2}-Z^{2}$
Killing (1893)	$y_{0}=x_{0}\operatorname {Ch} a+x_{1}\operatorname {Sh} a,\quad y_{1}=x_{0}\operatorname {Sh} a+x_{1}\operatorname {Ch} a,\quad y_{2}=x_{2}$	$x_{1}^{2}+x_{2}^{2}-x_{0}^{2}$
Whitehead (1897/98)	${\begin{matrix}{\begin{aligned}x'&=\left(\eta \cosh {\frac {\delta }{\gamma }}+\eta _{1}\sinh {\frac {\delta }{\gamma }}\right)e+\left(\eta \sinh {\frac {\delta }{\gamma }}+\eta _{1}\cosh {\frac {\delta }{\gamma }}\right)e_{1}\\&\qquad +\left(\eta _{2}\cos \alpha +\eta _{3}\sin \alpha \right)e_{2}+\left(\eta _{3}\cos \alpha -\eta _{2}\sin \alpha \right)e_{3}\end{aligned}}\\\left\{{\text{Lorentz boost with}}\ \alpha =0\right\}\end{matrix}}$	$\eta _{1}^{2}+\eta _{2}^{2}+\eta _{3}^{2}-\eta ^{2}$
Killing (1897/98)	${\begin{matrix}p'=p\operatorname {Ch} {\frac {\mu }{l}}+{\frac {u}{l}}\operatorname {Sh} {\frac {\mu }{l}},\quad u'=pl\operatorname {Sh} {\frac {\mu }{l}}+u\operatorname {Ch} {\frac {\mu }{l}},\quad v'=v\\{\text{or}}\\p'=p\operatorname {Ch} {\frac {\nu }{l}}+{\frac {v}{l}}\operatorname {Sh} {\frac {\nu }{l}},\quad u'=u,\quad v'=pl\operatorname {Sh} {\frac {\nu }{l}}+v\operatorname {Ch} {\frac {\nu }{l}}\end{matrix}}$	$u^{2}+v^{2}-l^{2}p^{2}$
Woods (1903/05)	${\begin{matrix}x_{1}^{\prime }=x_{1}\cos kl+x_{0}{\frac {\sin kl}{k}},\quad x_{2}^{\prime }=x_{2},\quad x_{2}^{\prime }=x_{3},\quad x_{0}^{\prime }=-x_{1}k\sin kl+x_{0}\cos kl\\\left\{{\text{Lorentz boost with}}\ k={\sqrt {-1}}\right\}\end{matrix}}$ ${\begin{matrix}{\begin{aligned}x_{1}^{\prime }&=x_{1}\cosh \alpha -x_{0}\sinh \alpha \\x_{2}^{\prime }&=x_{2}\cos \beta -x_{3}\sin \beta \\x_{3}^{\prime }&=x_{2}\sin \beta +x_{3}\cos \beta \\x_{0}^{\prime }&=-x_{1}\sinh \alpha +x_{0}\cosh \alpha \end{aligned}}\\\left\{{\text{Lorentz boost with}}\ \beta =0\right\}\end{matrix}}$	$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{0}^{2}$
Liebmann (1904/05)	$x_{1}^{\prime }=x'\operatorname {ch} a+p'\operatorname {sh} a,\quad y_{1}^{\prime }=y',\quad p_{1}^{\prime }=x'\operatorname {sh} a+p'\operatorname {ch} a$	$p^{\prime 2}-x^{\prime 2}-y^{\prime 2}$
Relativistic velocity addition
Lambert (1768)	${\begin{matrix}t\phi ''={\frac {t\phi +t\phi '}{1+t\phi \cdot t\phi '}}\\\left\{{\text{Velocity addition with}}\ {\scriptstyle \left(\left[t\phi ,\ t\phi ',\ t\phi ''\right]={\frac {1}{c}}\left[v,\ u_{x},\ u_{x}^{\prime }\right]\right)}\right\}\end{matrix}}$
Taurinus (1826)	${\begin{matrix}A=\operatorname {arccos} {\frac {\cos \left(\alpha {\sqrt {-1}}\right)-\cos \left(\beta {\sqrt {-1}}\right)\cos \left(\gamma {\sqrt {-1}}\right)}{\sin \left(\beta {\sqrt {-1}}\right)\sin \left(\gamma {\sqrt {-1}}\right)}},\\\left\{{\text{Velocity addition with}}\ {\scriptstyle \left(\left[\tanh \alpha ,\ \tanh \gamma ,\ \tanh \beta \right]={\frac {1}{c}}\left[{\sqrt {u_{x}^{\prime 2}+u_{y}^{\prime 2}}},\ {\sqrt {u_{x}^{2}+u_{y}^{2}}},\ v\right]\right)}\right\}\end{matrix}}$
Beltrami (1868)	${\begin{matrix}u''={\frac {aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'}},\ v''={\frac {a_{0}w_{0}v'}{a^{2}-r_{0}u'}},\\\left(r_{0}={\sqrt {u_{0}^{2}+v_{0}^{2}}},\ w_{0}={\sqrt {a^{2}-r_{0}^{2}}}\right)\\\left\{{\text{Velocity addition with}}\ {\scriptstyle \left[r_{0},u',v',u'',v''\right]=[v,u_{x},u_{y},u'_{x},u'_{y}],\ a=a_{0}=c}\right\}\end{matrix}}$	$u^{2}+v^{2}=a^{2}$
Beltrami (1868b)	${\begin{matrix}y_{1}={\frac {ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}}\ {\text{or}}\ x_{1}={\frac {a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}}},\ x_{r}=\pm {\frac {ay_{r}{\sqrt {a^{2}-a_{1}^{2}}}}{ab+a_{1}y_{1}}}\\(r=2,3,\dots ,n)\\\left\{{\text{Velocity addition with}}\ {\scriptstyle \left[a_{1},x_{1},y_{1},\dots ,x_{r},y_{r}\right]=\left[v,u_{x},u_{x}^{\prime },\dots ,u_{r},u_{r}^{\prime }\right],\ a=b=c}\right\}\end{matrix}}$	$x_{1}^{2}+\cdots +x_{n}^{2}=a^{2}$
Escherich (1874)	${\begin{matrix}x={\frac {\sinh {\frac {a}{k}}+x'\cosh {\frac {a}{k}}}{\cosh {\frac {a}{k}}+x'\sinh {\frac {a}{k}}}},\quad y={\frac {y'}{\cosh {\frac {a}{k}}+x'\sinh {\frac {a}{k}}}}\\\left\{{\text{Velocity addition with}}\ {\scriptstyle [x,\ y,\ x',\ y']={\frac {1}{c}}[u_{x},\ u_{y},\ u'_{x},\ u'_{y}]}\right\}\end{matrix}}$	$x^{2}+y^{2}=1$
Schur (1885/86)	${\begin{matrix}x_{1}=R^{2}{\frac {y_{1}+a_{1}}{R^{2}+a_{1}y_{1}}},\ x_{2}=R{\sqrt {R^{2}-a_{1}^{2}}}{\frac {y_{2}}{R^{2}+a_{1}y_{1}}},\dots ,\ x_{n}=R{\sqrt {R^{2}-a_{1}^{2}}}{\frac {y_{n}}{R^{2}+a_{1}y_{1}}}\\\left\{{\text{Velocity addition with}}\ {\scriptstyle [R,a_{1},x_{1},y_{1},\dots ,x_{n},y_{n}]=[c,v,u_{x}^{\prime },u_{y},\dots ,u_{n}^{\prime },u_{n}]}\right\}\end{matrix}}$	$x_{1}^{2}+\cdots +x_{n}^{2}=R^{2}$
Killing (1897/98)	${\begin{matrix}\xi '={\frac {\xi \operatorname {Ch} {\frac {\mu }{l}}+l\operatorname {Sh} {\frac {\mu }{l}}}{{\frac {\xi }{l}}\operatorname {Sh} {\frac {\mu }{l}}+\operatorname {Ch} {\frac {\mu }{l}}}},\ \eta '={\frac {\eta }{{\frac {\xi }{l}}\operatorname {Sh} {\frac {\mu }{l}}+\operatorname {Ch} {\frac {\mu }{l}}}}\\\left\{{\text{Velocity addition with}}\ {\scriptstyle [\xi ,\ \eta ,\ \xi ',\ \eta ']=[u_{x},\ u_{y},\ u'_{x},\ u'_{y}],\ l=c}\right\}\end{matrix}}$	$\xi ^{2}+\eta ^{2}=l^{2}$
Schur (1900/02)	${\begin{matrix}x'={\frac {x-a}{1-{\mathfrak {k}}ax}},\ y'={\frac {y{\sqrt {1-{\mathfrak {k}}a^{2}}}}{1-{\mathfrak {k}}ax}}\\\left\{{\text{Velocity addition with}}\ {\scriptstyle [a,\ x,\ y]=[v,\ u_{x},\ u_{y}],\ {\mathfrak {k}}={\frac {1}{c^{2}}}}\right\}\end{matrix}}$	$x^{2}+y^{2}={\frac {1}{\mathfrak {k}}}$

Lambert (1768–1770) – hyperbolic functions

After Vincenzo Riccati introduced hyperbolic functions in 1757, Johann Heinrich Lambert (read 1767, published 1768) gave the following relations, in which $tang\ \phi$ or abbreviated $t\phi$ was equated by Lambert to the tangens hyperbolicus $\scriptstyle {\frac {e^{u}-e^{-u}}{e^{u}+e^{-u}}}$ of a variable $u$, or in modern notation $\tan \phi =\tanh u$:

$\left.{\begin{aligned}\xi \xi -1&=\eta \eta &(a)\\1+\eta \eta &=\xi \xi &(b)\\{\frac {\eta }{\xi }}&=tang\ \phi =t\phi &(c)\\\xi &={\frac {1}{\sqrt {1-t\phi ^{2}}}}&(d)\\\eta &={\frac {t\phi }{\sqrt {1-t\phi ^{2}}}}&(e)\\t\phi ''&={\frac {t\phi +t\phi '}{1+t\phi \cdot t\phi '}}&(f)\\t\phi '&={\frac {t\phi ''-t\phi }{1-t\phi \cdot t\phi ''}}&(g)\end{aligned}}\right|{\begin{aligned}2u&=\log {\frac {1+t\phi }{1-t\phi }}\\\xi &={\frac {e^{u}+e^{-u}}{2}}\\\eta &={\frac {e^{u}-e^{-u}}{2}}\\t\phi &={\frac {e^{u}-e^{-u}}{e^{u}+e^{-u}}}\\e^{u}&=\xi +\eta \\e^{-u}&=\xi -\eta \end{aligned}}$

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

${\begin{aligned}t(y+z)&=(ty+tz):(1+ty\cdot tz)&(f)\\t(y-z)&=(ty-tz):(1-ty\cdot tz)&(g)\end{aligned}}$

The hyperbolic relations (a,b,c,d,e) are equivalent to the hyperbolic relations on the right of (3b). In addition, by setting $tang\ \phi ={\tfrac {v}{c}}$, (c) becomes the relative velocity between two frames, (d) the Lorentz factor, (e) the proper velocity, (f) or (g) becomes the Lorentz transformation of velocity (or relativistic velocity addition formula) for collinear velocities in (4a).

Euler (1771) – orthogonal transformation

Leonhard Euler (1771) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as orthogonal transformation, as well as under rotations using Euler angles. The case of two dimensions is given by

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

or three dimensions

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

These coefficiens $A,B,C,D,E,F,G,H,I$ were related by Euler to four parameter $p,q,r,s$, which where rediscovered by Olinde Rodrigues (1840) who related them to rotation angles now called Euler–Rodrigues parameters:

${\begin{matrix}{\begin{aligned}A&={\frac {pp+qq-rr-ss}{u}}&B&={\frac {2pq+2ps}{u}}&C&={\frac {2qs-2pr}{u}}\\D&={\frac {2qr-2ps}{u}}&E&={\frac {pp-qq+rr-ss}{u}}&F&={\frac {2pq+2rs}{u}}\\G&={\frac {2qs+2pr}{u}}&H&={\frac {2rs-2pq}{u}}&I&={\frac {pp-qq-rr+ss}{u}}\end{aligned}}\\(u=pp+qq+rr+ss)\end{matrix}}$

The orthogonal transformation in four dimensions was given by him as

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

As shown by Minkowski (1907), the orthogonal transformation can be directly used as Lorentz transformation (2b) by making one of the variables imaginary.

Gauss (1798–1818)

Binary quadratic forms

After the invariance of the sum of squares under linear substitutions was discussed by Euler (1771), the general expressions of a binary quadratic form and its transformation was formulated by Lagrange (1773) as follows

${\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\\hline {\begin{aligned}y&=Ms+Nx\\z&=ms+nx\end{aligned}}\left|{\begin{matrix}{\begin{aligned}P&=pM^{2}+2qMm+rm^{2}\\Q&=pMN+q(Mn+Nm)+rmn\\R&=pN^{2}+2qNn+rn^{2}\end{aligned}}\\\downarrow \\PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2}\end{matrix}}\right.\end{matrix}}$

which is equivalent to (Q1)$(n=1)$. The theory of binary quadratic forms was considerably expanded by Carl Friedrich Gauss (1798, published 1801) in his Disquisitiones Arithmeticae. He rewrote Lagrange's formalism as follows using integer coefficients $\alpha \beta \gamma \delta$:

${\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\F'=a'x^{\prime 2}+2b'x'y'+c'y^{\prime 2}=(a',b',c')\\\hline {\begin{aligned}x&=\alpha x'+\beta y'\\y&=\gamma x'+\delta y'\\\\x'&=\delta x-\beta y\\y'&=-\gamma x+\alpha y\end{aligned}}\left|{\begin{matrix}{\begin{aligned}a'&=a\alpha ^{2}+2b\alpha \gamma +c\gamma ^{2}\\b'&=a\alpha \beta +b(\alpha \delta +\beta \gamma )+c\gamma \delta \\c'&=a\beta ^{2}+2b\beta \delta +c\delta ^{2}\end{aligned}}\\\downarrow \\b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha \delta -\beta \gamma )^{2}\end{matrix}}\right.\end{matrix}}$

which is equivalent to (Q1)$(n=1)$. As pointed out by Gauss, $F$ and $F'$ are called "proper equivalent" if $\alpha \delta -\beta \gamma =1$, so that $F$ is contained in $F'$ as well as $F'$ is contained in $F$. In addition, if another form $F''$ is contained by the same procedure in $F'$ it is also contained in $F$ and so forth.

The Lorentz interval $-x_{0}^{2}+x_{1}^{2}$ and the Lorentz transformation (1a)$(n=1)$ are a special case of the binary quadratic form of Lagrange and Gauss by setting $(a,b,c)=(a',b',c')=(1,0,-1)$.

Alternatively, the transformation of coefficients $(a,b,c)$ corresponds to Lorentz transformation $\mathbf {u} '$ in (6c) by setting

${\begin{aligned}(a,b,c)&=\left(x_{0}+x_{2},\ x_{1},\ x_{0}-x_{2}\right)\\(a',b',c')&=\left(x_{0}^{\prime }+x_{2}^{\prime },\ x_{1}^{\prime },\ x_{0}^{\prime }-x_{2}^{\prime }\right)\end{aligned}}$.

Ternary quadratic forms

Gauss (1798/1801) also discussed ternary quadratic forms with the general expression

${\begin{matrix}f=ax^{2}+a'x^{\prime 2}+a''x^{\prime \prime 2}+2bx'x''+2b'xx''+2b''xx'=\left({\begin{matrix}a,&a',&a''\\b,&b',&b''\end{matrix}}\right)\\g=my^{2}+m'y^{\prime 2}+m''y^{\prime \prime 2}+2ny'y''+2n'yy''+2n''yy'=\left({\begin{matrix}m,&m',&m''\\n,&n',&n''\end{matrix}}\right)\\\hline {\begin{aligned}x&=\alpha y+\beta y'+\gamma y''\\x'&=\alpha 'y+\beta 'y'+\gamma 'y''\\x''&=\alpha ''y+\beta ''y'+\gamma ''y''\end{aligned}}\end{matrix}}$

which is equivalent to (Q1)$(n=2)$. Gauss called these forms definite when they have the same sign such as $x^{2}+y^{2}+z^{2}$, or indefinite in the case of different signs such as $x^{2}+y^{2}-z^{2}$. While discussing the classification of ternary quadratic forms ${\scriptstyle \left({\begin{matrix}a,&a',&a''\\b,&b',&b''\end{matrix}}\right)}$, Gauss (1801) presented twenty special cases, among them these six variants:

${\begin{matrix}\left({\begin{matrix}1,&-1,&1\\0,&0,&0\end{matrix}}\right),\ \left({\begin{matrix}-1,&1,&1\\0,&0,&0\end{matrix}}\right),\ \left({\begin{matrix}1,&1,&-1\\0,&0,&0\end{matrix}}\right),\\\left({\begin{matrix}1,&-1,&-1\\0,&0,&0\end{matrix}}\right),\ \left({\begin{matrix}-1,&1,&-1\\0,&0,&0\end{matrix}}\right),\ \left({\begin{matrix}-1,&-1,&1\\0,&0,&0\end{matrix}}\right)\end{matrix}}$

These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval $x^{2}+x^{\prime 2}-x^{\prime \prime 2}$ and the Lorentz transformation (1a)$(n=2)$ is an indefinite ternary quadratic form, which follows from the general ternary form by setting:

$\left({\begin{matrix}a,&a',&a''\\b,&b',&b''\end{matrix}}\right)=\left({\begin{matrix}m,&m',&m''\\n,&n',&n''\end{matrix}}\right)=\left({\begin{matrix}1,&1,&-1\\0,&0,&0\end{matrix}}\right)$

Cayley–Klein parameter

The determination of all transformations of the Lorentz interval (as a special case of an integer ternary quadratic form) into itself was explicitly worked out by Gauss around 1800 (posthumously published in 1863), for which he provided a coefficient system $\alpha \beta \gamma \delta$ in term os an :

${\begin{matrix}\left({\begin{matrix}1,&1,&-1\\0,&0,&0\end{matrix}}\right)\\\hline {\begin{matrix}\alpha \delta +\beta \gamma &\alpha \beta -\gamma \delta &\alpha \beta +\gamma \delta \\\alpha \gamma -\beta \delta &{\frac {1}{2}}(\alpha \alpha +\delta \delta -\beta \beta -\gamma \gamma )&{\frac {1}{2}}(\alpha \alpha +\gamma \gamma -\beta \beta -\delta \delta )\\\alpha \gamma +\beta \delta &{\frac {1}{2}}(\alpha \alpha +\beta \beta -\gamma \gamma -\delta \delta )&{\frac {1}{2}}(\alpha \alpha +\beta \beta +\gamma \gamma +\delta \delta )\end{matrix}}\\(\alpha \delta -\beta \gamma =1)\end{matrix}}$

Gauss' result was cited by Selling (1873), Bachmann (1911), Dickson (1923). The parameters $\alpha \beta \gamma \delta$, when applied to spatial rotations, were later called Cayley–Klein parameters.

This is equivalent to Lorentz transformation (6d).

Homogeneous coordinates

Gauss (1818) discussed planetary motions together with formulating elliptic functions. In order to simplify the integration, he transformed the expression

$(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E$

into

$G+G'\cos T^{2}+G''\sin T^{2}$

in which $\cos T,\sin T$ are related to $\cos E,\sin E$ by the following transformation including an arbitrary constant $k$, which Gauss then rewrote by setting $k=1$:

${\begin{matrix}\left(\alpha +\alpha '\cos T+\alpha ''\sin T\right)^{2}+\left(\beta +\beta '\cos T+\beta ''\sin T\right)^{2}-\left(\gamma +\gamma '\cos T+\gamma ''\sin T\right)^{2}=0\\k\left(\cos ^{2}T+\sin ^{2}T-1\right)\\\hline {\begin{aligned}\cos E&={\frac {\alpha +\alpha '\cos T+\alpha ''\sin T}{\gamma +\gamma '\cos T+\gamma ''\sin T}}\\\sin E&={\frac {\beta +\beta '\cos T+\beta ''\sin T}{\gamma +\gamma '\cos T+\gamma ''\sin T}}\end{aligned}}\left|{\begin{aligned}-\alpha \alpha -\beta \beta +\gamma \gamma &=k&\alpha \alpha -\alpha '\alpha '-\alpha ''\alpha ''&=-k\\-\alpha '\alpha '-\beta '\beta '+\gamma '\gamma '&=-k&\beta \beta -\beta '\beta '-\beta ''\beta ''&=-k\\-\alpha ''\alpha ''-\beta ''\beta ''+\gamma ''\gamma ''&=-k&\gamma \gamma -\gamma '\gamma '-\gamma ''\gamma ''&=+k\\-\alpha '\alpha ''-\beta '\beta ''+\gamma '\gamma ''&=0&\beta \gamma -\beta '\gamma '-\beta ''\gamma ''&=0\\-\alpha ''\alpha -\beta ''\beta +\gamma ''\gamma &=0&\gamma \alpha -\gamma '\alpha '-\gamma ''\alpha ''&=0\\-\alpha \alpha '-\beta \beta '+\gamma \gamma '&=0&\alpha \beta -\alpha '\beta '-\alpha ''\beta ''&=0\end{aligned}}\right.\\\hline k=1\\{\begin{aligned}t\cos E&=\alpha +\alpha '\cos T+\alpha ''\sin T\\t\sin E&=\beta +\beta '\cos T+\beta ''\sin T\\t&=\gamma +\gamma '\cos T+\gamma ''\sin T\end{aligned}}\left|{\begin{aligned}-\alpha \alpha -\beta \beta +\gamma \gamma &=1\\-\alpha '\alpha '-\beta '\beta '+\gamma '\gamma '&=-1\\-\alpha ''\alpha ''-\beta ''\beta ''+\gamma ''\gamma ''&=-1\\-\alpha '\alpha ''-\beta '\beta ''+\gamma '\gamma ''&=0\\-\alpha ''\alpha -\beta ''\beta +\gamma ''\gamma &=0\\-\alpha \alpha '-\beta \beta '+\gamma \gamma '&=0\end{aligned}}\right.\end{matrix}}{\text{ }}$

Gauss' case $k=1$ is equivalent to the coefficient system in Lorentz transformations (1a) and (1b).

Further setting $[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime },\ u_{2}^{\prime }\right]$ it becomes Lorentz transformation (1b) $(n=2)$.

Subsequently he showed that these relations can be reformulated using three variables $x,y,z$ and $u,u',u''$, so that

$aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz$

can be transformed into

$Guu+G'u'u'+G''u''u''$,

in which $x,y,z$ and $u,u',u''$ are related by the transformation:

${\begin{aligned}x&=\alpha u+\alpha 'u'+\alpha ''u''\\y&=\beta u+\beta 'u'+\beta ''u''\\z&=\gamma u+\gamma 'u'+\gamma ''u''\\\\u&=-\alpha x-\beta y+\gamma z\\u'&=\alpha 'x+\beta 'y-\gamma 'z\\u''&=\alpha ''x+\beta ''y-\gamma ''z\end{aligned}}\left|{\begin{aligned}-\alpha \alpha -\beta \beta +\gamma \gamma &=1\\-\alpha '\alpha '-\beta '\beta '+\gamma '\gamma '&=-1\\-\alpha ''\alpha ''-\beta ''\beta ''+\gamma ''\gamma ''&=-1\\-\alpha '\alpha ''-\beta '\beta ''+\gamma '\gamma ''&=0\\-\alpha ''\alpha -\beta ''\beta +\gamma ''\gamma &=0\\-\alpha \alpha '-\beta \beta '+\gamma \gamma '&=0\end{aligned}}\right.$

This is equivalent to Lorentz transformation (1a) $(n=2)$ satisfying $x^{2}+y^{2}-z^{2}=u^{2}+u^{\prime 2}-u^{\prime \prime 2}$, and can be related to Gauss' previous equations in terms of homogeneous coordinates $\left[\cos T,\sin T,\cos E,\sin E\right]=\left[{\frac {x}{z}},\ {\frac {y}{z}},\ {\frac {u}{u''}},\ {\frac {u'}{u''}}\right]$.

Taurinus (1826) – Hyperbolic law of cosines

After the addition theorem for the tangens hyperbolicus was given by Lambert (1868), hyperbolic geometry was used by Franz Taurinus (1826), and later by Nikolai Lobachevsky (1829/30) and others, to formulate the hyperbolic law of cosines:

$A=\operatorname {arccos} {\frac {\cos \left(\alpha {\sqrt {-1}}\right)-\cos \left(\beta {\sqrt {-1}}\right)\cos \left(\gamma {\sqrt {-1}}\right)}{\sin \left(\beta {\sqrt {-1}}\right)\sin \left(\gamma {\sqrt {-1}}\right)}}$

When solved for $\cos \left(\alpha {\sqrt {-1}}\right)$ it corresponds to the Lorentz transformation in Beltrami coordinates (3e), and by defining the rapidities ${\scriptstyle \left(\left[{\frac {U}{c}},\ {\frac {v}{c}},\ {\frac {u}{c}}\right]=\left[\tanh \alpha ,\ \tanh \beta ,\ \tanh \gamma \right]\right)}$ it corresponds to the relativistic velocity addition formula (4d)

.

Jacobi (1827, 1833/34) – Homogeneous coordinates

Following Gauss (1818), Carl Gustav Jacob Jacobi (1827 and 1833/34) formulated Gauss' transformation in 3 dimensions:

${\begin{matrix}1827:&{\begin{matrix}\cos P^{2}+\sin P^{2}\cos \vartheta ^{2}+\sin P^{2}\sin \vartheta ^{2}-1\\=k\left(\cos \psi ^{2}+\sin \psi ^{2}\cos \varphi ^{2}+\sin \psi ^{2}\sin \varphi ^{2}-1\right)\\\hline {\scriptstyle {\begin{aligned}\cos P&={\frac {\alpha +\alpha '\cos \psi +\alpha ''\sin \psi \cos \varphi +\alpha '''\sin \psi \sin \varphi }{\delta +\delta '\cos \psi +\delta ''\sin \psi \cos \varphi +\delta '''\sin \psi \sin \varphi }}\\\sin P\cos \vartheta &={\frac {\beta +\beta '\cos \psi +\beta ''\sin \psi \cos \varphi +\beta '''\sin \psi \sin \varphi }{\delta +\delta '\cos \psi +\delta ''\sin \psi \cos \varphi +\delta '''\sin \psi \sin \varphi }}\\\sin P\sin \vartheta &={\frac {\gamma +\beta '\cos \psi +\gamma ''\sin \psi \cos \varphi +\gamma '''\sin \psi \sin \varphi }{\delta +\delta '\cos \psi +\delta ''\sin \psi \cos \varphi +\delta '''\sin \psi \sin \varphi }}\\\\\cos \psi &={\frac {-\delta '+\alpha '\cos P+\beta '\sin P\cos \vartheta +\gamma '\sin P\sin \vartheta }{\delta -\alpha \cos P-\beta \sin P\cos \vartheta -\gamma \sin P\sin \vartheta }}\\\sin \psi \cos \varphi &={\frac {-\delta ''+\alpha ''\cos P+\beta ''\sin P\cos \vartheta +\gamma ''\sin P\sin \vartheta }{\delta -\alpha \cos P-\beta \sin P\cos \vartheta -\gamma \sin P\sin \vartheta }}\\\sin \psi \sin \varphi &={\frac {-\delta '''+\alpha '''\cos P+\beta '''\sin P\cos \vartheta +\gamma '''\sin P\sin \vartheta }{\delta -\alpha \cos P-\beta \sin P\cos \vartheta -\gamma \sin P\sin \vartheta }}\end{aligned}}\left|{\begin{aligned}\alpha \alpha +\beta \beta +\gamma \gamma -\delta \delta &=-k\\\alpha '\alpha '+\beta '\beta '+\gamma '\gamma '-\delta '\delta '&=k\\\alpha ''\alpha ''+\beta ''\beta ''+\gamma ''\gamma ''-\delta ''\delta ''&=k\\\alpha '''\alpha '''+\beta '''\beta '''+\gamma '''\gamma '''-\delta '''\delta '''&=k\\\alpha \alpha '+\beta \beta '+\gamma \gamma '-\delta \delta '&=0\\\alpha \alpha ''+\beta \beta ''+\gamma \gamma ''-\delta \delta ''&=0\\\alpha \alpha '''+\beta \beta '''+\gamma \gamma '''-\delta \delta '''&=0\\\alpha ''\alpha '''+\beta ''\beta '''+\gamma ''\gamma '''-\delta ''\delta '''&=0\\\alpha '''\alpha '+\beta '''\beta '+\gamma '''\gamma '-\delta '''\delta '&=0\\\alpha '\alpha ''+\beta '\beta ''+\gamma '\gamma ''-\delta '\delta ''&=0\end{aligned}}\right.}\end{matrix}}\\\hline 1833:&{\begin{matrix}xx+yy+zz=uu+vv+ww=1\\\hline {\begin{aligned}u&={\frac {\alpha +\alpha 'x+\alpha ''y+\alpha '''z}{\delta +\delta 'x+\delta ''y+\delta '''z}}\\v&={\frac {\beta +\beta 'x+\beta ''y+\beta '''z}{\delta +\delta 'x+\delta ''y+\delta '''z}}\\w&={\frac {\gamma +\gamma 'x+\gamma ''y+\gamma '''z}{\delta +\delta 'x+\delta ''y+\delta '''z}}\end{aligned}}\end{matrix}}\end{matrix}}$

By setting ${\scriptstyle {\begin{aligned}[][\cos P,\ \sin P\cos \varphi ,\ \sin P\sin \varphi ]&=\left[u_{1},\ u_{2},\ u_{3}\right]\\{}[\cos \psi ,\ \sin \psi \cos \vartheta ,\ \sin \psi \sin \vartheta ]&=\left[u_{1}^{\prime },\ u_{2}^{\prime },\ u_{3}^{\prime }\right]\end{aligned}}}$ and $k=1$, the 1827 version is equivalent to Lorentz transformation (1b) $(n=2)$, while the 1833 version is identical in the first place.

After Cauchy (1829) formulated the orthogonal transformation for arbitrary dimensions, Jacobi (1833/34) used this result to extend his previous formulas:

${\begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots +x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots +y_{n}y_{n}\\\hline {\scriptstyle {\begin{aligned}y_{\varkappa }&=\alpha _{1}^{(\varkappa )}x_{1}+\alpha _{2}^{(\varkappa )}x_{2}+\dots +\alpha _{n}^{(\varkappa )}x_{n}\\\\x_{\varkappa }&=\alpha _{\varkappa }^{\prime }y_{1}+\alpha _{\varkappa }^{\prime \prime }y_{2}+\dots +\alpha _{\varkappa }^{(n)}y_{n}\end{aligned}}\left|{\begin{aligned}\alpha _{\varkappa }^{\prime }\alpha _{\lambda }^{\prime }+\alpha _{\varkappa }^{\prime \prime }\alpha _{\lambda }^{\prime \prime }+\dots +\alpha _{\varkappa }^{(n)}\alpha _{\lambda }^{(n)}&=0\\\alpha _{\varkappa }^{\prime }\alpha _{\varkappa }^{\prime }+\alpha _{\varkappa }^{\prime \prime }\alpha _{\varkappa }^{\prime \prime }+\dots +\alpha _{\varkappa }^{(n)}\alpha _{\varkappa }^{(n)}&=1\\\\\alpha _{1}^{(\varkappa )}\alpha _{1}^{(\lambda )}+\alpha _{2}^{(\varkappa )}\alpha _{2}^{(\lambda )}+\dots +\alpha _{n}^{(\varkappa )}\alpha _{n}^{(\lambda )}&=0\\\alpha _{1}^{(\varkappa )}\alpha _{1}^{(\varkappa )}+\alpha _{2}^{(\varkappa )}\alpha _{2}^{(\varkappa )}+\dots +\alpha _{n}^{(\varkappa )}\alpha _{n}^{(\varkappa )}&=1\end{aligned}}\right.}\\\hline {\frac {x_{\varkappa }}{x_{n}}}=-i\xi _{\varkappa },\ {\frac {y_{\varkappa }}{y_{n}}}=i\nu _{\varkappa }\\1-\xi _{1}\xi _{1}-\xi _{2}\xi _{2}-\dots -\xi _{n-1}\xi _{n-1}={\frac {y_{n}y_{n}}{x_{n}x_{n}}}\left(1-\nu _{1}\nu _{1}-\nu _{2}\nu _{2}-\dots -\nu _{n-1}\nu _{n-1}\right)\\\alpha _{n}^{(\varkappa )}=i\alpha ^{(\varkappa )},\ \alpha _{\varkappa }^{(n)}=-i\alpha _{\varkappa },\ \alpha _{n}^{(n)}=\alpha \\{\begin{aligned}{\frac {y_{\varkappa }}{y_{n}}}&={\frac {\alpha _{1}^{(\varkappa )}x_{1}+\alpha _{2}^{(\varkappa )}x_{2}+\dots +\alpha _{n}^{(\varkappa )}x_{n}}{\alpha _{1}^{(n)}x_{1}+\alpha _{2}^{(n)}x_{2}+\dots +\alpha _{n}^{(n)}x_{n}}}\\\\{\frac {x_{\varkappa }}{x_{n}}}&={\frac {\alpha _{\varkappa }^{\prime }y_{1}+\alpha _{\varkappa }^{\prime \prime }y_{2}+\dots +\alpha _{\varkappa }^{(n)}y_{n}}{\alpha _{1}^{(n)}x_{1}+\alpha _{2}^{(n)}x_{2}+\dots +\alpha _{n}^{(n)}x_{n}}}\end{aligned}}\left|{\begin{aligned}\nu _{\varkappa }&={\frac {\alpha ^{(\varkappa )}-\alpha _{1}^{(\varkappa )}\xi _{1}-\alpha _{2}^{(\varkappa )}\xi _{2}\dots -\alpha _{n-1}^{(\varkappa )}\xi _{n-1}}{\alpha -\alpha _{1}\xi _{1}-\alpha _{2}\xi _{2}\dots -\alpha _{n-1}\xi _{n-1}}}\\\\\xi _{\varkappa }&={\frac {\alpha _{\varkappa }-\alpha _{\varkappa }^{\prime }\nu _{1}-\alpha _{2}^{\prime \prime }\nu _{2}\dots -\alpha _{\varkappa }^{(n-1)}\nu _{n-1}}{\alpha -\alpha ^{\prime }\nu _{1}-\alpha ^{\prime \prime }\nu _{2}\dots -\alpha ^{(n-1)}\nu _{n-1}}}\end{aligned}}\right.\\\xi _{1}\xi _{1}-\xi _{2}\xi _{2}-\dots -\xi _{n-1}\xi _{n-1}=1\\\nu _{1}\nu _{1}-\nu _{2}\nu _{2}-\dots -\nu _{n-1}\nu _{n-1}=1\end{matrix}}$

The second transformation system is equivalent to Lorentz transformation (1b) up to a sign change.

He also stated the following linear substitution leaving invariant the Lorentz interval:

${\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots -u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots -w_{n-1}w_{n-1}\\\hline {\scriptstyle {\begin{aligned}u&=\alpha w-\alpha 'w_{1}-\alpha ''w_{2}-\dots -\alpha ^{(n-1)}w_{n-1}\\u_{1}&=\alpha _{1}w-\alpha _{1}^{\prime }w_{1}-\alpha _{1}^{\prime \prime }w_{2}-\dots -\alpha _{1}^{(n-1)}w_{n-1}\\&\dots \\u_{n-1}&=\alpha _{n-1}w-\alpha _{n-1}^{\prime }w_{1}-\alpha _{n-1}^{\prime \prime }w_{2}-\dots -\alpha _{n-1}^{(n-1)}w_{n-1}\\\\w&=\alpha u-\alpha _{1}u_{1}-\alpha _{2}^{\prime \prime }u_{2}-\dots -\alpha _{n-1}u_{n-1}\\w_{1}&=\alpha 'u-\alpha _{1}^{\prime }u_{1}-\alpha _{2}^{\prime }u_{2}-\dots -\alpha _{n-1}^{\prime }u_{n-1}\\&\dots \\w_{n-1}&=\alpha ^{(n-1)}u-\alpha _{1}^{(n-1)}u_{1}-\alpha _{2}^{(n-1)}u_{2}-\dots -\alpha _{n-1}^{(n-1)}u_{n-1}\end{aligned}}\left|{\begin{aligned}\alpha \alpha -\alpha '\alpha '-\alpha ''\alpha ''\dots -\alpha ^{(n-1)}\alpha ^{(n-1)}&=+1\\\alpha _{\varkappa }\alpha _{\varkappa }-\alpha _{\varkappa }^{\prime }\alpha _{\varkappa }^{\prime }-\alpha _{\varkappa }^{\prime \prime }\alpha _{\varkappa }^{\prime \prime }\dots -\alpha _{\varkappa }^{(n-1)}\alpha _{\varkappa }^{(n-1)}&=-1\\\alpha \alpha _{\varkappa }-\alpha ^{\prime }\alpha _{\varkappa }^{\prime }-\alpha ^{\prime \prime }\alpha _{\varkappa }^{\prime \prime }\dots -\alpha ^{(n-1)}\alpha _{\varkappa }^{(n-1)}&=0\\\alpha _{\varkappa }\alpha _{\lambda }-\alpha _{\varkappa }^{\prime }\alpha _{\lambda }^{\prime }-\alpha _{\varkappa }^{\prime \prime }\alpha _{\lambda }^{\prime \prime }\dots -\alpha _{\varkappa }^{(n-1)}\alpha _{\lambda }^{(n-1)}&=0\\\\\alpha \alpha -\alpha _{1}\alpha _{1}-\alpha _{2}\alpha _{2}\dots -\alpha _{n-1}\alpha _{n-1}&=+1\\\alpha _{\varkappa }\alpha _{\varkappa }-\alpha _{1}^{\varkappa }\alpha _{1}^{\varkappa }-\alpha _{2}^{\prime \prime }\alpha _{2}^{\prime \prime }\dots -\alpha _{n-1}^{(\varkappa )}\alpha _{n-1}^{(\varkappa )}&=-1\\\alpha \alpha ^{(\varkappa )}-\alpha _{1}\alpha _{1}^{(\varkappa )}-\alpha _{2}\alpha _{2}^{(\varkappa )}\dots -\alpha _{n-1}\alpha _{n-1}^{(\varkappa )}&=0\\\alpha ^{(\varkappa )}\alpha ^{(\lambda )}-\alpha _{1}^{(\varkappa )}\alpha _{1}^{\lambda l)}-\alpha _{2}^{(\varkappa )}\alpha _{2}^{(\lambda )}\dots -\alpha _{n-1}^{(\varkappa )}\alpha _{n-1}^{(\lambda )}&=0\end{aligned}}{\text{ }}\right.}\end{matrix}}$

This is equivalent to Lorentz transformation (1a) up to a sign change.

Cauchy (1829) – Orthogonal transformation

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

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

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

Lebesgue (1837) – Homogeneous coordinates

Victor-Amédée Lebesgue (1837) summarized the previous work of Gauss (1818), Jacobi (1827, 1833), Cauchy 1829. He started with the orthogonal transformation

${\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n}^{2}\ (9)\\\hline {\scriptstyle {\begin{aligned}x_{1}&=a_{1,1}y_{1}+a_{1,2}y_{2}+\dots +a_{1,n}y_{n}\\x_{2}&=a_{2,1}y_{1}+a_{2,2}y_{2}+\dots +a_{2,n}y_{n}\\\dots \\x_{n}&=a_{n,1}x_{1}+a_{n,2}x_{2}+\dots +a_{n,n}x_{n}\\\\y_{1}&=a_{1,1}x_{1}+a_{2,1}x_{2}+\dots +a_{n,1}x_{n}\\y_{2}&=a_{1,2}x_{1}+a_{2,2}x_{2}+\dots +a_{n,2}x_{n}&(12)\\\dots \\y_{n}&=a_{1,n}x_{1}+a_{2,n}x_{2}+\dots +a_{n,n}x_{n}\end{aligned}}\left|{\begin{aligned}a_{1,\alpha }^{2}+a_{2,\alpha }^{2}+\dots +a_{n,\alpha }^{2}&=1&(10)\\a_{1,\alpha }a_{1,\beta }+a_{2,\alpha }a_{2,\beta }+\dots +a_{n,\alpha }a_{n,\beta }&=0&(11)\\a_{\alpha ,1}^{2}+a_{\alpha ,2}^{2}+\dots +a_{\alpha ,n}^{2}&=1&(13)\\a_{\alpha ,1}a_{\beta ,1}+a_{\alpha ,2}a_{\beta ,2}+\dots +a_{\alpha ,n}a_{\beta ,n}&=0&(14)\end{aligned}}\right.}\end{matrix}}$

He modified these equations in order show the invariance of the Lorentz interval

$x_{1}^{2}+x_{2}^{2}+\dots +x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n-1}^{2}-y_{n}^{2}$

by giving the following instructions: In equation (9) change the sign of the last term of each member. In the first $n-1$ equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies $\alpha =n$ change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the $n$-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the $n$-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign.

These instructions give Lorentz transformation (1a) in the form:

${\scriptstyle {\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots +x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n-1}^{2}-y_{n}^{2}\\\hline {\begin{aligned}x_{1}&=a_{1,1}y_{1}+a_{1,2}y_{2}+\dots +a_{1,n}y_{n}\\x_{2}&=a_{2,1}y_{1}+a_{2,2}y_{2}+\dots +a_{2,n}y_{n}\\\dots \\x_{n}&=a_{n,1}x_{1}+a_{n,2}x_{2}+\dots +a_{n,n}x_{n}\\\\y_{1}&=a_{1,1}x_{1}+a_{2,1}x_{2}+\dots +a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\y_{2}&=a_{1,2}x_{1}+a_{2,2}x_{2}+\dots +a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\\dots \\-y_{n}&=a_{1,n}x_{1}+a_{2,n}x_{2}+\dots +a_{n-1,n}x_{n-1}-a_{n,n}x_{n}\end{aligned}}\left|{\begin{aligned}a_{1,\alpha }^{2}+a_{2,\alpha }^{2}+\dots +a_{n-1,\alpha }^{2}-a_{n,\alpha }^{2}&=1\\a_{1,n}^{2}+a_{2,n}^{2}+\dots +a_{n-1,n}^{2}-a_{n,n}^{2}&=-1\\a_{1,\alpha }a_{1,\beta }+a_{2,\alpha }a_{2,\beta }+\dots +a_{n-1,\alpha }a_{n-1,\beta }-a_{n,\alpha }a_{n,\beta }&=0\\a_{\alpha ,1}^{2}+a_{\alpha ,2}^{2}+\dots +a_{\alpha ,n-1}^{2}-a_{\alpha ,n}^{2}&=1\\a_{n,1}^{2}+a_{n,2}^{2}+\dots +a_{n,n-1}^{2}-a_{n,n}^{2}&=-1\\a_{\alpha ,1}a_{\beta ,1}+a_{\alpha ,2}a_{\beta ,2}+\dots +a_{\alpha ,n-1}a_{\beta ,n-1}-a_{\alpha ,n}a_{\beta ,n}&=0\end{aligned}}\right.\end{matrix}}}$

He went on to redefine the variables of the Lorentz interval and its transformation:

${\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots +x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n-1}^{2}-y_{n}^{2}\\\downarrow \\{\begin{aligned}x_{1}&=x_{n}\cos \theta _{1},&x_{2}&=x_{n}\cos \theta _{2},\dots &x_{n-1}&=x_{n}\cos \theta _{n-1}\\y_{1}&=y_{n}\cos \phi _{1},&y_{2}&=y_{n}\cos \phi _{2},\dots &y_{n-1}&=y_{n}\cos \phi _{n-1}\end{aligned}}\\\downarrow \\\cos ^{2}\theta _{1}+\cos ^{2}\theta _{2}+\dots +\cos ^{2}\theta _{n-1}=1\\\cos ^{2}\phi _{1}+\cos ^{2}\phi _{2}+\dots +\cos ^{2}\phi _{n-1}=1\\\hline \\\cos \theta _{i}={\frac {a_{i,1}\cos \phi _{1}+a_{i,2}\cos \phi _{2}+\dots +a_{i,n-1}\cos \phi _{n-1}+a_{i,n}}{a_{n,1}\cos \phi _{1}+a_{n,2}\cos \phi _{2}+\dots +a_{n,n-1}\cos \phi _{n-1}+a_{n,n}}}\\(i=1,2,3\dots n)\end{matrix}}$

Setting $[\cos \theta _{i},\ \cos \phi _{i}]=\left[u_{s},\ u_{s}^{\prime }\right]$ it is equivalent to Lorentz transformation (1b).

Cayley (1846–1858)

Euler–Rodrigues parameter and Cayley–Hermite transformation

The Euler–Rodrigues parameters discovered by Euler (1871) and Rodrigues (1840) leaving invariant $x_{0}^{2}+x_{1}^{2}+x_{2}^{2}$ were extended to $x_{0}^{2}+\dots +x_{n}^{2}$ by Arthur Cayley (1846) as a byproduct of what is now called the Cayley transform using the method of skew–symmetric coefficients. Following Cayley's methods, a general transformation for quadratic forms into themselves in three (1853) and arbitrary (1854) dimensions was provided by Hermite (1853, 1854). Hermite's formula was simplified and brought into matrix form equivalent to (Q2) by Cayley (1855a)

${\scriptstyle (\mathrm {x,y,z} \dots )=\left(\left|{\begin{matrix}a,&h,&g&\dots \\h,&b,&f&\dots \\g,&f,&c&\dots \\\dots &\dots &\dots &\dots \end{matrix}}\right|^{-1}\left|{\begin{matrix}a,&h-\nu ,&g+\mu &\dots \\h+\nu ,&b,&f-\lambda &\dots \\g-\mu ,&f+\lambda ,&c&\dots \\\dots &\dots &\dots &\dots \end{matrix}}\right|\left|{\begin{matrix}a,&h+\nu ,&g-\mu &\dots \\h-\nu ,&b,&f+\lambda &\dots \\g+\mu ,&f-\lambda ,&c&\dots \\\dots &\dots &\dots &\dots \end{matrix}}\right|^{-1}\left|{\begin{matrix}a,&h,&g&\dots \\h,&b,&f&\dots \\g,&f,&c&\dots \\\dots &\dots &\dots &\dots \end{matrix}}\right|\right)\ ^{\frown }(x,y,z\dots )}$

which he abbreviated in 1858, where $\Upsilon$ is any skew-symmetric matrix:

$(\mathrm {x,y,z} )=\left(\Omega ^{-1}(\Omega -\Upsilon )(\Omega +\Upsilon )^{-1}\Omega \right)(x,y,z)$

The Cayley–Hermite transformation becomes equivalent to the Lorentz transformation (5a) by setting $\Omega =\operatorname {diag} \left(-1,1\right)$ and (5b) by setting $a\Omega =\operatorname {diag} \left(-1,1,1\right)$ and (5c) by setting $\Omega =\operatorname {diag} \left(-1,1,1,1\right)$.

Using the parameters of (1855a), Cayley in a subsequent paper (1855b) particularly discussed several special cases, such as:

${\begin{matrix}a\mathrm {x} ^{2}+b\mathrm {y} ^{2}=ax^{2}+by^{2}\\\hline (\mathrm {x,y} )={\frac {1}{ab+\nu ^{2}}}\cdot \left[{\begin{matrix}ab-\nu ^{2},&-2\nu b\\2\nu a,&ab-\nu ^{2}\end{matrix}}\right](x,y)\end{matrix}}$

This becomes equivalent to the Lorentz transformation (5a) in 1+1 dimensions by setting $(a,b)=(-1,1)$.

or:

${\scriptstyle {\begin{matrix}a\mathrm {x} ^{2}+b\mathrm {y} ^{2}+c\mathrm {z} ^{2}=ax^{2}+by^{2}+cz^{2}\\\hline (\mathrm {x,y,z} )={\frac {1}{abc+a\lambda ^{2}+b\mu ^{2}+c\nu ^{2}}}\times \left[{\begin{matrix}abc+a\lambda ^{2}-b\mu ^{2}-c\nu ^{2},&2(\lambda \mu -c\nu )b&2(\nu \lambda +b\mu )c\\2(\lambda \mu +c\nu )a,&abc-a\lambda ^{2}+b\mu ^{2}-c\nu ^{2}&2(\mu \nu -a\lambda )c\\2(\nu \lambda -b\mu )a&2(\mu \nu +a\lambda )b&abc-a\lambda ^{2}-b\mu ^{2}-c\nu ^{2}\end{matrix}}\right](x,y,z)\end{matrix}}}$

This becomes equivalent to the Lorentz transformation (5b) by setting $(a,b,c)=(-1,1,1)$.

or:

${\scriptstyle {\begin{matrix}a\mathrm {x} ^{2}+b\mathrm {y} ^{2}+c\mathrm {z} ^{2}+d\mathrm {w} ^{2}=ax^{2}+by^{2}+cz^{2}+dw^{2}\\\hline (\mathrm {x,y,z,w} )={\frac {1}{k}}\cdot \left[{\begin{aligned}&abcd-bc\rho ^{2}+ca\sigma ^{2}+ab\tau ^{2}+ad\lambda ^{2}&&2b\left(-cd\nu -\tau \phi +d\lambda \mu -c\rho \sigma \right),\\&\quad -bd\mu ^{2}-cd\nu ^{2}-\phi ^{2},&&abcd+bc\rho ^{2}-ca\sigma ^{2}+ab\tau ^{2}-ad\lambda ^{2}\\&2a\left(cd\nu +\tau \phi +d\lambda \mu -c\rho \sigma \right),&&\quad +bd\mu ^{2}-cd\nu ^{2}-\phi ^{2},\\&2a\left(-bd\mu -\sigma \phi +d\lambda \nu -b\rho \tau \right),&&2b\left(ad\lambda +\rho \phi +d\mu \nu -a\sigma \tau \right),\\&2a\left(bc\rho +\lambda \phi +c\nu \sigma -b\mu \tau \right),&&2b\left(ac\sigma +\mu \phi -c\nu \rho +a\lambda \tau \right),\\\\&\quad 2c\left(bd\mu +\sigma \phi +d\lambda \nu -b\rho \tau \right),&&\quad 2d\left(-bc\rho -\lambda \phi +c\nu \sigma -b\mu \tau \right)\\&\quad 2c\left(-ad\lambda -\rho \phi +d\mu \nu -a\sigma \tau \right),&&\quad 2d\left(-ac\sigma -\mu \phi -c\nu \rho +a\lambda \tau \right)\\&\quad abcd+bc\rho ^{2}+ca\sigma ^{2}-ab\tau ^{2}-ad\lambda ^{2}&&\quad 2d\left(-ab\tau -\nu \phi +b\mu \rho -a\lambda \sigma \right)\\&\quad \quad -bd\mu ^{2}+cd\nu ^{2}-\phi ^{2},&&\quad abcd-bc\rho ^{2}-ca\sigma ^{2}-ab\tau ^{2}+ad\lambda ^{2}\\&\quad 2c\left(ab\tau +\nu \phi +b\mu \rho -a\lambda \sigma \right),&&\quad \quad +bd\mu ^{2}+cd\nu ^{2}-\phi ^{2},\end{aligned}}\right]\cdot (x,y,z,w)\\\left({\begin{aligned}k&=abcd+bc\rho ^{2}+ca\sigma ^{2}+ab\tau ^{2}+ad\lambda ^{2}+bd\mu ^{2}+cd\nu ^{2}+\phi ^{2}\\\phi &=\lambda \rho +\mu \sigma +\nu \tau \end{aligned}}\right)\end{matrix}}}$

This becomes equivalent to the Lorentz transformation (5c) by setting $(a,b,c,d)=(-1,1,1,1)$.

Cayley–Klein parameter

Already in 1854, Cayley published an alternative method of transforming quadratic forms by using certain parameters $\alpha \beta \gamma \delta$ in relation to a homographic transformation of a surface of second order into itself:

${\begin{matrix}xy-zw=0\\x_{2}y_{2}-z_{2}w_{2}=x_{1}y_{1}-z_{1}w_{1}\\\hline \left.{\begin{aligned}MM'x_{2}&=\gamma '\delta x_{1}+\alpha \alpha 'y_{1}-\alpha '\delta z_{1}-\alpha \gamma 'w_{1}\\MM'y_{2}&=\beta \beta 'x_{1}+\gamma \delta 'y_{1}-\beta \delta 'z_{1}-\beta '\gamma w_{1}\\MM'z_{2}&=\beta \gamma 'x_{1}+\gamma \alpha 'y_{1}-\beta \alpha 'z_{1}-\gamma \gamma 'w_{1}\\MM'w_{2}&=\beta '\delta x_{1}+\alpha \delta 'y_{1}-\delta \delta 'z_{1}-\alpha \beta 'w_{1}\end{aligned}}\right|{\begin{aligned}M^{2}&=\alpha \beta -\gamma \delta \\M^{\prime 2}&=\alpha '\beta '-\gamma '\delta '\end{aligned}}\end{matrix}}$

In the same paper, Cayley also introduced four different parameters $M^{2}=a^{2}+b^{2}+c^{2}+d^{2}$ in order to demonstrate the invariance of $x_{1}^{2}+y_{1}^{2}+z_{1}^{2}+w_{1}^{2}$, and subsequently showed the relation to quaternions. Fricke & Klein (1897) credited Cayley by calling the above transformation the most general (real or complex) space collineation of first kind of an absolute surface of second kind into itself. Parameters $\alpha ,\beta ,\gamma ,\delta$ are similar to what was later called Cayley–Klein parameters in relation to spatial rotations (which was done by Cayley in 1879 and before by Hermann von Helmholtz (1866/67)).

Cayley's transformation becomes the Lorentz transformation $\mathbf {u} '$ in (6a) by setting $\alpha \beta -\gamma \delta =1$ and:

$(x_{1},\ y_{1},\ z_{1},\ w_{1})=\left(x_{0}+x_{3},\ x_{0}-x_{3},\ x_{1}+ix_{2},\ x_{1}-ix_{2}\right)$

Subsequently solved for $x_{0},x_{1},x_{2},x_{3}$ it becomes Lorentz transformation (6b).

Quaternions

In 1845, Cayley showed that the Euler-Rodrigues parameters representing rotations can be related to quaternions using a pre- and a postfactor

${\begin{matrix}x_{\prime }^{2}+y_{\prime }^{2}+z_{\prime }^{2}=x^{2}+y^{2}+z^{2}\\\hline (1+\lambda i+\mu j+\nu k)^{-1}(ix+jy+kz)(1+\lambda i+\mu j+\nu k)=\\{\scriptstyle ={\frac {1}{1+\lambda ^{2}+\mu ^{2}+\nu ^{2}}}\left\{{\begin{matrix}i\left[x\left(\lambda ^{2}+\mu ^{2}-\nu ^{2}\right)+2y(\mu \nu +\lambda )+2z(\lambda \nu -\mu )\right]\\+j\left[2x\left(\lambda \mu -\nu \right)+y\left(1-\lambda ^{2}+\mu ^{2}-\nu ^{2}\right)+2z(\mu \nu +\lambda )\right]\\+k\left[2x\left(\lambda \nu +\mu \right)+2y\left(\mu \nu -\lambda \right)+z\left(1-\lambda ^{2}-\mu ^{2}+\nu ^{2}\right)\right]\end{matrix}}={\begin{aligned}i(\alpha x+\alpha 'y+\alpha ''z)\\+j(\beta x+\beta 'y+\beta ''z)\\+k(\gamma x+\gamma 'y+\gamma ''z)\end{aligned}}\right\}}\\\downarrow \\{\begin{aligned}x_{\prime }=&i(\alpha x+\alpha 'y+\alpha ''z)\\y_{\prime }=&j(\beta x+\beta 'y+\beta ''z)\\z_{\prime }=&k(\gamma x+\gamma 'y+\gamma ''z)\end{aligned}}\end{matrix}}$

and in 1848 he used the abbreviated form

$\Pi _{1}=\Lambda \Pi \Lambda ^{-1}\quad \ \left[{\begin{matrix}\Lambda =1+\lambda i+\mu j+\nu k\\\Pi _{1}=ix_{1}+jy_{1}+kz_{1}\end{matrix}}\right]$

In 1854 he showed how to transform the sum of four squares into itself:

${\begin{matrix}x_{2}^{2}+y_{2}^{2}+z_{2}^{2}+w_{2}^{2}=x^{2}+y^{2}+z^{2}+w^{2}\\\hline MM'\left(w_{2}+ix_{2}+jy_{2}+kz_{2}\right)=(d+ia+jb+kc)(w+ix+jy+kz)(d'+ia'+jb'+kc')\\\left({\begin{aligned}M^{2}&=a^{2}+b^{2}+c^{2}+d^{2}\\M^{\prime 2}&=a^{\prime 2}+b^{\prime 2}+c^{\prime 2}+d^{\prime 2}\end{aligned}}\right)\end{matrix}}$

or in 1855:

${\begin{matrix}\mathrm {x} ^{2}+\mathrm {y} ^{2}+\mathrm {z} ^{2}+\mathrm {w} ^{2}=x^{2}+y^{2}+z^{2}+w^{2}\\\hline \left(\mathrm {x} i+\mathrm {y} j+\mathrm {z} k+\mathrm {w} \right)=-{\frac {1}{\sqrt {MM'}}}(\alpha i+\beta j+\gamma k+\delta )(xi+yj+zk+w)(\alpha 'i+\beta 'j+\gamma 'k+\delta ')\\\left({\begin{aligned}M^{2}&=\alpha ^{2}+\beta ^{2}+\gamma ^{2}+\delta ^{2}\\M^{\prime 2}&=\alpha ^{\prime 2}+\beta ^{\prime 2}+\gamma ^{\prime 2}+\delta ^{\prime 2}\end{aligned}}\right)\end{matrix}}$

Cayley's quaternion transformation of the sum of four squares, abbreviated $QqQ^{\ast }$, served as a role model for the representation of the Lorentz transformation by Noether (1910), Klein (1910), Silberstein (1911), in which the scalar part is imaginary.

Cayley absolute

In 1859, Cayley found out that a quadratic form or projective quadric can be used as an "absolute", serving as the basis of a projective metric (the Cayley–Klein metric). For instance, using the absolute $x^{2}+y^{2}+z^{2}=0$, he defined the distance of two points as follows

$\cos ^{-1}{\frac {xx'+yy'+zz'}{{\sqrt {x^{2}+y^{2}+z^{2}}}{\sqrt {x^{\prime 2}+y^{\prime 2}+z^{\prime 2}}}}}$

and he also alluded to the case of the unit sphere $x^{2}+y^{2}+z^{2}=1$. In the hands of Klein (1871), all of this became essential for the discussion of non-Euclidean geometry (in particular the Cayley–Klein or Beltrami–Klein model of hyperbolic geometry) and associated quadratic forms and transformations, including the Lorentz interval and Lorentz transformation.

Hermite (1853, 1854) – Cayley–Hermite transformation

Using Cayley's (1846) method of skew–symmetric coefficients, Charles Hermite (1853) derived a transformation leaving invariant not only definite ternary forms that can be reduced to $\pm \left(x^{2}+y^{2}+z^{2}\right)$, but also indefinite ternary forms that can be reduced to the Lorentz interval $\pm \left(x^{2}+y^{2}-z^{2}\right)$, as well as almost all kinds of ternary quadratic forms. This was generalized by him in 1854 to all dimensions, so Hermite arrived at a transformation scheme leaving invariant almost all quadratic forms:

$x_{r}=\xi _{r}+{\frac {1}{2}}\sum _{s=1}^{n}\lambda _{r,s}{\frac {df}{d\xi _{s}}},\ \left(\lambda _{r,s}=-\lambda _{s,r},\ \lambda _{r,r}=0\right)$

This result was subsequently expressed in matrix form by Cayley (1855). Later, Ferdinand Georg Frobenius (1877) added some modifications in order to include some special cases of quadratic forms that cannot be dealt with by the Cayley–Hermite transformation.

This is equivalent to equation (Q2), and becomes the Lorentz transformation by setting the coefficients of the quadratic form $f$ to $\operatorname {diag} (-1,1,\dots 1)$.

Bour (1856) – Homogeneous coordinates

Following Gauss (1818), Edmond Bour (1856) wrote the transformations:

${\begin{matrix}\cos ^{2}E+\sin ^{2}E-1=k\left(\cos ^{2}T+\sin ^{2}T-1\right)\\\hline {\begin{matrix}{\begin{aligned}\cos E&={\frac {\alpha +\alpha '\cos T+\alpha ''\sin T}{\gamma +\gamma '\cos T+\gamma ''\sin T}}\\\sin E&={\frac {\beta +\beta '\cos T+\beta ''\sin T}{\gamma +\gamma '\cos T+\gamma ''\sin T}}\end{aligned}}\\\hline \\k=+1\\t=\gamma +\gamma '\cos T+\gamma ''\sin T,\\1=u,\ \cos T=u',\ \sin T=u',\\t=z,\ t\cos E=x,\ t\sin E=y\\\downarrow \\{\begin{aligned}x&=\alpha u+\alpha 'u'+\alpha ''u''\\y&=\beta u+\beta 'u'+\beta ''u''\\z&=\gamma u+\gamma 'u'+\gamma ''u''\\\\u&=\gamma z-\alpha x-\beta y\\u'&=\alpha 'x+\beta 'y'-\gamma 'z\\u''&=\alpha ''x+\beta ''y-\gamma ''z\end{aligned}}\end{matrix}}\left|{\begin{aligned}-\alpha ^{2}-\beta ^{2}+\gamma ^{2}&=k\\-\alpha ^{\prime 2}-\beta ^{\prime 2}+\gamma ^{\prime 2}&=-k\\-\alpha ^{\prime \prime 2}-\beta ^{\prime \prime 2}+\gamma ^{\prime \prime 2}&=-k\\\alpha \alpha '+\beta \beta '-\gamma \gamma '&=0\\\alpha \alpha ''+\beta \beta ''-\gamma \gamma ''&=0\\\alpha '\alpha ''+\beta '\beta ''-\gamma '\gamma ''&=0\\\\\alpha ^{2}-\alpha ^{\prime 2}-\alpha ^{\prime \prime 2}&=-k\\\beta ^{2}-\beta ^{\prime 2}-\beta ^{\prime \prime 2}&=-k\\\gamma ^{2}-\gamma ^{\prime 2}-\gamma ^{\prime \prime 2}&=k\\\beta \gamma -\beta '\gamma '-\beta ''\gamma ''&=0\\\alpha \gamma -\alpha '\gamma '-\alpha ''\gamma ''&=0\\\alpha \beta -\alpha '\beta '-\alpha ''\beta ''&=0\end{aligned}}\right.\end{matrix}}$

Setting $[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime },u_{2}^{\prime }\right]$, the first transformation system becomes Lorentz transformation (1b) $(n=2)$.

The second transformation system is equivalent to Lorentz transformation (1a) $(n=2)$, implying $x^{2}+y^{2}-z^{2}=u^{2}+u^{\prime 2}-u^{\prime \prime 2}$

Somov (1863) – Homogeneous coordinates

Following Gauss (1818), Jacobi (1827, 1833), and Bour (1856), Osip Ivanovich Somov (1863) wrote the transformation systems:

${\begin{matrix}{\begin{aligned}\cos \phi &={\frac {m\cos \psi +n\sin \psi +s}{m''\cos \psi +n''\sin \psi +s''}}\\\sin \phi &={\frac {m'\cos \psi +n'\sin \psi +s'}{m''\cos \psi +n''\sin \psi +s''}}\end{aligned}}\\\cos ^{2}\phi +\cos ^{2}\phi =1,\ \cos ^{2}\psi +\cos ^{2}\psi =1\\{\begin{aligned}\cos \phi &=x,&\sin \phi &=y\\\cos \psi &=x',&\sin \psi &=y'\end{aligned}}\left|{\begin{aligned}x&={\frac {mx'+ny'+s}{m''x'+n''y'+s''}}\\y&={\frac {m'x'+n'y'+s'}{m''x'+n''y'+s''}}\end{aligned}}\right.\\x^{2}+y^{2}=1,\ x^{\prime 2}+y^{\prime 2}=1\\\hline {\begin{aligned}\cos \phi &={\frac {x}{z}},&\sin \phi &={\frac {y}{z}}\\\cos \psi &={\frac {x'}{z'}},&\sin \psi &={\frac {y'}{z'}}\end{aligned}}\left|{\begin{aligned}{\frac {x}{z}}&={\frac {mx'+ny'+sz'}{m''x'+n''y'+s''z'}}\\{\frac {y}{z}}&={\frac {m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'}}\end{aligned}}\right.\\x^{2}+y^{2}=z^{2},\ x^{\prime 2}+y^{\prime 2}=z^{\prime 2}\\\downarrow \\{\begin{aligned}x&=mx'+ny'+sz'\\y&=m'x'+n'y'+s'z'\\z&=m''x'+n''y'+s''z'\\\\x'&=mx+m'y-m''z\\y'&=nx+n'y-n''z\\z'&=-sx-s'y+s''z\end{aligned}}\left|{\begin{aligned}m^{2}+m^{\prime 2}-m^{\prime \prime 2}&=1\\n^{2}+n^{\prime 2}-n^{\prime \prime 2}&=1\\-s^{2}-s^{\prime 2}+s^{\prime \prime 2}&=1\\ns+n's'-n''s''&=0\\sm+s'm'-s''m''&=0\\mn+m'n'-m''n''&=0\end{aligned}}\right.{\begin{aligned}m^{2}+n^{2}-s^{2}&=1\\m^{\prime 2}+n^{\prime 2}-s^{\prime 2}&=1\\-m^{\prime \prime 2}-n^{\prime \prime 2}+s^{\prime \prime 2}&=1\\-m'm''-n'n''+s's''&=0\\-m''m-n''n+s''s&=0\\mm'+nn'-ss'&=0\end{aligned}}\\x^{\prime 2}+y^{\prime 2}-z^{\prime 2},\ x^{2}+y^{2}-z^{2}\\dx^{2}+dy^{2}-dz^{2}=dx^{\prime 2}+dy^{\prime 2}-dz^{\prime 2}\end{matrix}}$

The first transformation system is equivalent to Lorentz transformation (1b) $(n=2)$.

The second transformation system is equivalent to Lorentz transformation (1a) $(n=2)$.

Beltrami (1868) – Beltrami coordinates

Eugenio Beltrami (1868a) introduced coordinates of the Beltrami–Klein model of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:

${\begin{matrix}ds^{2}=R^{2}{\frac {\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}}\\u^{2}+v^{2}=a^{2}\\\hline u''={\frac {aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'}},\ v''={\frac {a_{0}w_{0}v'}{a^{2}-r_{0}u'}},\\\left(r_{0}={\sqrt {u_{0}^{2}+v_{0}^{2}}},\ w_{0}={\sqrt {a^{2}-r_{0}^{2}}}\right)\\\hline ds^{2}=R^{2}{\frac {\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}}\\(R=R{\sqrt {-1}},\ a=a{\sqrt {-1}})\end{matrix}}$

(where a and R are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)

${\begin{matrix}ds=R{\frac {\sqrt {dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots +dx_{n}^{2}}}{x}}\\x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=a^{2}\\\hline y_{1}={\frac {ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}}\ {\text{or}}\ x_{1}={\frac {a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}}},\ x_{r}=\pm {\frac {ay_{r}{\sqrt {a^{2}-a_{1}^{2}}}}{ab+a_{1}y_{1}}}\ (r=2,3,\dots ,n)\\\hline ds=R{\frac {\sqrt {dx_{1}^{2}+dx_{2}^{2}+\cdots +dx_{n}^{2}-dx^{2}}}{x}}\\x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\\\left(R=R{\sqrt {-1}},\ x=x{\sqrt {-1}},\ a=a{\sqrt {-1}}\right)\end{matrix}}$

Setting $a=a_{0}=c$ and $\left[r_{0},u',u'',v',v''\right]=\left[v,u_{x},u_{x}^{\prime },u_{y},u_{y}^{\prime }\right]$, Beltrami's (1868a) formulas become the relativistic velocity addition formulas (3d or 4c), being special cases of the general velocity addition (1b). In his (1868b) formulas, one sets $a=b=c$ and $\left[a_{1},x_{1},y_{1},x_{r},y_{r},(r=2,3)\right]=\left[v,u_{x},u_{x}^{\prime },u_{y},u_{y}^{\prime },u_{z},u_{z}^{\prime }\right]$

Klein (1871–1897)

Cayley absolute and non-Euclidean geometry

Elaborating on Cayley's (1859) definition of an "absolute" (Cayley–Klein metric), Felix Klein (1871) defined a "fundamental conic section" in order to discuss motions such as rotation and translation in the non-Euclidean plane, and another fundamental form by using homogeneous coordinates $x,y$ related to a circle with radius $2c$ with measure of curvature $-{\tfrac {1}{4c^{2}}}$. When $c$ is positive, the measure of curvature is negative and the fundamental conic section is real, thus the geometry becomes hyperbolic (Beltrami–Klein model):

$x_{1}x_{2}-x_{3}^{2}=0$

$x^{2}+y^{2}-4c^{2}=0$

In (1873) he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to an ellipsoid or two-sheet hyperboloid.

Using positive $c$ in $-{\tfrac {1}{4c^{2}}}$ in line with hyperbolic geometry or directly by setting $-{\tfrac {1}{4c^{2}}}=-x_{0}$, Klein's two quadratic forms can be related to expressions $X_{2}^{2}-X_{1}X_{3}$ and $x_{0}^{2}-x_{1}^{2}-x_{2}^{2}$ for the Lorentz interval in (6c).

Möbius transformation, spin transformation, Cayley–Klein parameter

In (1872) while devising the Erlangen program, Klein discussed the general relation between projective metrics, binary forms and conformal geometry transforming a sphere into itself in terms of linear transformations of the complex variable $x+iy$. Following Klein, these relations were discussed by Ludwig Wedekind (1875) using $z'={\tfrac {\alpha z+\beta }{\gamma z+\delta }}$. Klein (1875) then showed that all finite groups of motions follow by determining all finite groups of such linear transformations of $x+iy$ into itself. In (1878), Klein classified the substitutions of $\omega '={\tfrac {\alpha \omega +\beta }{\gamma \omega +\delta }}$ with $\alpha \delta -\beta \gamma =1$ into hyperbolic, elliptic, parabolic, and in (1882) he added the loxodromic substitution as the combination of elliptic and hyperbolic ones. (In 1890, Robert Fricke in his edition of Klein's lectures of elliptic functions and Modular forms, referred to the analogy of this treatment to the theory of quadratic forms as given by Gauss and in particular Dirichlet.)

In (1884) Klein related the linear fractional transformations (interpreted as rotations around the $x+iy$-sphere) to Cayley–Klein parameter $\alpha ,\beta ,\gamma ,\delta$, to Euler–Rodrigues parameters $a,b,c,d$, and to the unit sphere by means of stereographic projection, and also discussed transformations preserving surfaces of second degree equivalent to the transformation given by Cayley (1854):

${\begin{matrix}\left.{\begin{matrix}z'={\frac {\alpha z+\beta }{\gamma z+\delta }}\rightarrow z=z_{1}:z_{2}\rightarrow {\begin{aligned}z_{1}^{\prime }&=\alpha z_{1}+\beta z_{2}\\z_{2}^{\prime }&=\gamma z_{1}+\delta z_{2}\end{aligned}}\\\xi ^{2}+\eta ^{2}+\zeta ^{2}=1\\z=x+iy={\frac {\xi +i\eta }{1-\zeta }}\\z'={\frac {(d+ic)z-(b-ia)}{(b+ia)z+(d-ic)}}\\\left(a^{2}+b^{2}+c^{2}+d^{2}=1\right)\end{matrix}}\right|&{\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\\lambda '={\frac {a\lambda +b}{c\lambda +d}},\ \mu '={\frac {a'\mu +b'}{c'\mu +d'}}\\\lambda =\lambda _{1}:\lambda _{2},\ \mu =\mu _{1}:\mu _{2}\\X_{1}:X_{2}:X_{3}:X_{4}=\lambda _{1}\mu _{1}:-\lambda _{2}\mu _{1}:\lambda _{1}\mu _{2}:\lambda _{2}\mu _{2}\end{matrix}}\end{matrix}}$

The formulas on the left related to the unit sphere become equivalent to Lorentz transformations (6a) by relating them to homogeneous coordinates

${\begin{aligned}(\xi ,\ \eta ,\ \zeta ,\ 1)&=\left({\frac {x_{1}}{x_{0}}},\ {\frac {x_{2}}{x_{0}}},\ {\frac {x_{3}}{x_{0}}},\ {\frac {x_{0}}{x_{0}}}\right)\\&\downarrow \\\xi ^{2}+\eta ^{2}+\zeta ^{2}-1&=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{0}^{2}=0\end{aligned}}$.

The formulas on the right can be related to those on the left by setting

$(X_{1},\ X_{2},\ X_{3},\ X_{4})=\left(\xi +i\eta ,\ 1-\zeta ,\ -(1+\eta ),\ \xi -i\eta \right)$

and become equivalent to Lorentz transformation (6a) by setting

$(X_{1},\ X_{2},\ X_{3},\ X_{4})=\left(x_{0}+ix_{3},\ x_{2}+x_{0},\ x_{2}-x_{0},\ x_{1}-ix_{3}\right)$

and subsequently solved for $x_{1}\dots$ it becomes Lorentz transformation (6b).

In his lecture in the winter semester of 1889/90 (published 1892–93), he discussed the hyperbolic plane by using (as in 1871) the Lorentz interval in terms of a circle with radius $2k$ as the basis of hyperbolic geometry, as well as the modified conic section to discuss the "kinematics of hyperbolic geometry", consisting of motions and congruent displacements of the hyperbolic plane into itself:

${\begin{matrix}{\begin{matrix}x^{2}+y^{2}-4k^{2}t^{2}=0\\x_{1}x_{3}-x_{2}^{2}=0\end{matrix}}&\left|{\begin{matrix}{\frac {x_{1}}{x_{2}}}={\frac {x_{2}}{x_{3}}}=\lambda ={\frac {\lambda _{1}}{\lambda _{2}}}\\\lambda '={\frac {\alpha \lambda +\beta }{\gamma \lambda +\delta }}\rightarrow {\begin{aligned}\lambda _{1}^{\prime }&=\alpha \lambda _{1}+\beta \lambda _{2}\\\lambda _{2}^{\prime }&=\gamma \lambda _{1}+\delta \lambda _{2}\end{aligned}}\\\left(\alpha \delta -\beta \gamma =1\right)\\{\begin{aligned}x_{1}:x_{2}:x_{3}&=\lambda ^{2}:\lambda :1=\lambda _{1}^{2}:\lambda _{1}\lambda _{2}:\lambda _{2}^{2}\\&=\lambda ^{\prime 2}:\lambda ':1=\lambda _{1}^{\prime 2}:\lambda _{1}^{\prime }\lambda _{2}^{\prime }:\lambda _{2}^{\prime 2};\end{aligned}}\end{matrix}}\right.\end{matrix}}$

Klein's Lorentz interval $x^{2}+y^{2}-4k^{2}t^{2}$ can be connected with the other interval $x_{1}x_{3}-x_{2}^{2}$ by setting

$(x_{1},\ x_{2},\ x_{3})=\left(x+iy,\ 2kt,\ x-iy\right)$,

by which it becomes equivalent to Lorentz transformation (6c) with $2k=1$, and subsequently solved for $x_{1}\dots$ its become equivalent to Lorentz transformation (6d).

In his lecture in the summer semester of 1890 (published 1892–93), he discussed general surfaces of second degree, including an "oval" surface corresponding to hyperbolic space and its motions:

$\left.{\begin{matrix}{\text{General surfaces of second degree}}:\\{\begin{aligned}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}&{\text{(no real parts,elliptic)}}\\z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}&{\text{(oval,hyperbolic)}}\\z_{1}^{2}+z_{2}^{2}-z_{3}^{2}-z_{4}^{2}&{\text{(ring)}}\\z_{1}^{2}-z_{2}^{2}-z_{3}^{2}-z_{4}^{2}&{\text{(oval,hyperbolic)}}\\-z_{1}^{2}-z_{2}^{2}-z_{3}^{2}-z_{4}^{2}&{\text{(no real parts,elliptic)}}\end{aligned}}\\{\text{all of which can be brought into the form:}}\\y_{1}y_{3}+y_{2}y_{4}=0\\{\text{Transformation:}}\\{\begin{aligned}\varrho y_{1}&=\lambda _{1}\mu _{1},&\varrho y_{1}^{\prime }&=\lambda _{1}^{\prime }\mu _{1}^{\prime }\\\varrho y_{2}&=\lambda _{2}\mu _{1},&\varrho y_{2}^{\prime }&=\lambda _{2}^{\prime }\mu _{1}^{\prime }\\\varrho y_{3}&=\lambda _{2}\mu _{2},&\varrho y_{3}^{\prime }&=-\lambda _{2}^{\prime }\mu _{2}^{\prime }\\\varrho y_{4}&=\lambda _{1}\mu _{2},&\varrho y_{4}^{\prime }&=\lambda _{1}^{\prime }\mu _{2}^{\prime }\end{aligned}}\end{matrix}}\right|{\begin{matrix}{\text{Oval (=hyperbolic motions in space):}}\\x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0\\=\left(x_{1}+ix_{3}\right)\left(x_{1}-ix_{3}\right)+\left(x_{2}+x_{4}\right)\left(x_{2}-x_{4}\right)=0\\=y_{1}y_{3}+y_{2}y_{4}=0\\\\x^{2}+y^{2}+z^{2}-1=0\\\hline \lambda ={\frac {x+iy}{1-z}},\ \lambda '={\frac {\alpha \lambda +\beta }{\gamma \lambda +\delta }},\ \mu '={\frac {{\bar {\alpha }}\mu +{\bar {\beta }}}{{\bar {\gamma }}\mu +{\bar {\delta }}}}\\{\begin{aligned}\lambda _{1}^{\prime }&=\alpha \lambda _{1}+\beta \lambda _{2}\\\lambda _{2}^{\prime }&=\gamma \lambda _{1}+\delta \lambda _{2}\end{aligned}},\ {\begin{aligned}\mu _{1}^{\prime }&={\bar {\alpha }}\mu _{1}+{\bar {\beta }}\mu _{2}\\\mu _{2}^{\prime }&={\bar {\gamma }}\mu _{1}+{\bar {\delta }}\mu _{2}\end{aligned}}\end{matrix}}$

Plugging the values for $\lambda ,\mu ,\lambda ',\mu ',\dots$ from the right into the transformations on the left, leads to Lorentz transformation (6a), which becomes clear when the implicit relation of Klein's two Lorentz intervals $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}$ and $x^{2}+y^{2}+z^{2}-1$ is considered:

$(x,\ y,\ z,\ 1)=\left({\frac {x_{1}}{x_{4}}},\ {\frac {x_{2}}{x_{4}}},\ {\frac {x_{3}}{x_{4}}},\ {\frac {x_{4}}{x_{4}}}\right)$.

Subsequently solved for $x_{1}\dots$ it becomes Lorentz transformation (6b).

In (1896/97), Klein again defined hyperbolic motions and explicitly used $t$ as time coordinate:

$\left.{\begin{matrix}x^{2}+y^{2}+z^{2}-t^{2}=0\\=(x+iy)(x-iy)+(z+t)(z-t)=0\\x+iy:x-iy:z+t:t-z=\zeta _{1}\zeta _{2}^{\prime }:\zeta _{2}\zeta _{1}^{\prime }:\zeta _{1}\zeta _{1}^{\prime }:\zeta _{2}\zeta _{2}^{\prime }\\{\frac {\zeta _{1}}{\zeta _{2}}}=\zeta \rightarrow \zeta ={\frac {x+iy}{t-z}}={\frac {t+z}{x-iy}};\\X^{2}+Y^{2}+Z^{2}-T^{2}=0=\ {\text{etc.}}\\\zeta ={\frac {\alpha Z+\beta }{\gamma Z+\delta }}\rightarrow {\begin{aligned}\zeta _{1}&=\alpha Z_{1}+\beta Z_{2}\\\zeta _{2}&=\gamma Z_{1}+\delta Z_{2}\end{aligned}},\ {\begin{aligned}\zeta _{1}^{\prime }&={\bar {\alpha }}Z_{1}^{\prime }+{\bar {\beta }}Z_{2}^{\prime }\\\zeta _{2}^{\prime }&={\bar {\gamma }}Z_{1}^{\prime }+{\bar {\delta }}Z_{2}^{\prime }{\text{ }}\end{aligned}}\\(\alpha \delta -\beta \gamma =1)\end{matrix}}\right|{\begin{array}{c|c|c|c|c}&X+iY&X-iY&T+Z&T-Z\\\hline x+iy&\alpha {\bar {\delta }}&\beta {\bar {\gamma }}&\alpha {\bar {\gamma }}&\beta {\bar {\delta }}\\\hline x-iy&\gamma {\bar {\beta }}&\delta {\bar {\alpha }}&\gamma {\bar {\alpha }}&\delta {\bar {\beta }}\\\hline t+z&\alpha {\bar {\beta }}&\beta {\bar {\alpha }}&\alpha {\bar {\alpha }}&\beta {\bar {\beta }}\\\hline t-z&\gamma {\bar {\delta }}&\delta {\bar {\gamma }}&\gamma {\bar {\gamma }}&\delta {\bar {\delta }}\end{array}}$

This is equivalent to Lorentz transformation (6a).

Klein's work was summarized and extended by Robert Fricke (supported by Klein), see Fricke (1893, 1897).

Spiral transformation

In 1890 Klein discussed a general type of Euclidean or Non-Euclidean motion in relation to a problem posed by Helmholtz (1868), with the following transformation

$x'+iy'=e^{(\alpha +i\beta )t}(x+iy)+a+ib$

In 1893 he called the special case with $a=b=0$ a "spiral transformation":

${\begin{aligned}x'+iy'&=e^{(\alpha +i\beta )t}(x+iy)&{\text{(1)}}\\x'-iy'&=e^{(\alpha -i\beta )t}(x-iy)&{\text{(2)}}\end{aligned}}$

The Lorentz boost (3c) can be formulated by setting ${\begin{aligned}[][\beta ,\ iy,\ iy']&=[0,\ y,\ y']\ {\text{(1,2)}}\\t&=-t\ {\text{(1)}}\end{aligned}}$

Conformal transformation and polyspherical coordinates

In relation to line geometry, Klein (1871/72) used coordinates satisfying the condition $s_{1}^{2}+s_{2}^{2}+s_{2}^{2}+s_{2}^{2}+s_{5}^{2}=0$. They were introduced in 1868 (belatedly published in 1872/73) by Gaston Darboux as a system of five coordinates in $R_{3}$ (later called "pentaspherical" coordinates) in which the last coordinate is imaginary. Sophus Lie (1871) more generally used $n+2$ coordinates in $R_{n}$ (later called "polyspherical" coordinates) satisfying $\scriptstyle \sum _{i=1}^{i=n+2}x_{i}^{2}=0$ in which the last coordinate is imaginary, as a means to discuss conformal transformations generated by inversions. These simultaneous publications can be explained by the fact that Darboux, Lie, and Klein corresponded with each other by letter.

When the last coordinate is defined as real, the corresponding polyspherical coordinates satisfy the form of a sphere. Initiated by lectures of Klein between 1889–1890, his student Friedrich Pockels (1891) used such real coordinates, emphasizing that all of these coordinate systems remain invariant under conformal transformations generated by inversions:

$x_{1}^{2}+x_{2}^{2}+\cdots +x_{n+1}^{2}-x_{n+2}^{2}=0{\text{ or }}\sum _{1}^{n+1}x_{h}^{2}-x_{n+2}^{2}=0$

Special cases were described by Klein (1893):

$y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}-y_{5}^{2}=0$ (pentaspherical).

$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0$ (tetracyclical).

Both systems were also described by Maxime Bôcher (1894) in an expanded version of a thesis supervised by Klein.

Polyspherical coordinates indicate that the conformal group Con(p,0) is isomorphic to the Lorentz group SO(p+1,1). For instance, Con(2,0) – known as Möbius group – is related to tetracyclical coordinates satisfying $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0$, which is nothing other than the Lorentz interval invariant under the Lorentz group SO(3,1).

Lie (1871–1893)

Conformal, spherical, and orthogonal transformations

In several papers between 1847 and 1850 it was shown by Joseph Liouville that the relation $\lambda \left(\delta x^{2}+\delta y^{2}+\delta z^{2}\right)$ is invariant under the group of conformal transformations generated by "transformation by reciprocal radii" transforming spheres into spheres, which can be related to spherical wave transformations, special conformal transformations or Möbius transformations. (The conformal nature of the linear fractional transformation ${\tfrac {a+bz}{c+dz}}$ of a complex variable $z$ was already discussed by Euler (1777)).

Liouville's theorem was extended to all dimensions by Sophus Lie (1871a). In addition, Lie described a manifold whose elements can be represented by spheres, where the last imaginary coordinate $iy_{n+1}$ represents the radius:

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

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

Eventually, Lie (1871/72) pointed out that the conformal point transformations of $R_{n}$ consist of motions (such as rigid transformations and orthogonal transformations), similarity transformations, and inversions.

The Lorentz group SO(3,1) is a subgroup of the conformal group Con(3,1) in terms of Lie sphere geometry in which the radius indicates the fourth coordinate.

Lie group, hyperbolic motions, and infinitesimal transformations

In (1885/86), Lie identified the projective group of a general surface of second degree $\sum f_{ik}x_{i}'x_{k}'=0$ with the group of non-Euclidean motions. In a thesis guided by Lie, Hermann Werner (1889) discussed this projective group by using the equation of a unit sphere as the surface of second degree, and also gave the corresponding infinitesimal projective transformations:

${\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}=1\\\hline x_{i}p_{\varkappa }-x_{\varkappa }p_{i},\quad p_{i}-x_{i}\sum _{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad (i,\varkappa =1,\dots ,n)\\{\text{where}}\\\left(Q_{i},Q_{\varkappa }\right)=R_{i,\varkappa };\ \left(Q_{i},Q_{j,\varkappa }\right)=\varepsilon _{i,j}Q_{\varkappa }-\varepsilon _{i,\varkappa }Q_{j};\\\left(R_{i,\varkappa },R_{\mu ,\nu }\right)=\varepsilon _{\varkappa ,\mu }R_{i,\nu }-\varepsilon _{\varkappa ,\nu }R_{i,\mu }-\varepsilon _{,\mu }R_{\varkappa ,\nu }+\varepsilon _{i,\nu }R_{\varkappa ,\mu }\\\left[\varepsilon _{i,\varkappa }\equiv 0\ {\text{for}}\ i\neq \varkappa ;\ \varepsilon _{i,i}=1\right]\end{matrix}}$

More generally, Lie (1890) defined non-Euclidean motions in terms of two forms $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm 1=0$ in which the imaginary form with $+1$ denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:

${\begin{matrix}x_{1}^{2}+\dots +x_{n}^{2}-1=0\\\hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad (i,k=1\dots n)\end{matrix}}$

Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form:

$x^{2}+y^{2}-1=0$ or $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0$ or $x_{1}^{2}+\dots +x_{n}^{2}-1=0$.

The group of hyperbolic motions is isomorphic to the Lorentz group. The interval $x_{1}^{2}+\dots +x_{n}^{2}-1=0$ becomes the Lorentz interval $x_{1}^{2}+\dots +x_{n}^{2}-x_{0}^{2}=0$ by setting

$(x_{1},\dots ,\ x_{n},\ 1)=\left({\frac {x_{1}}{x_{0}}},\dots ,\ {\frac {x_{n}}{x_{0}}},\ {\frac {x_{0}}{x_{0}}}\right)$

Selling (1873–74) – Quadratic forms

Continuing the work of Gauss (1801) on ternary quadratic forms, and Hermite (1853) on the reduction of indefinite ternary forms, Eduard Selling (1873) used the auxiliary coefficients $\xi \eta \zeta$ by which a definite form ${\mathfrak {f}}$ and an indefinite form $f$ can be rewritten in terms of three squares:

${\scriptstyle {\begin{aligned}{\mathfrak {f}}&={\mathfrak {a}}x^{2}+{\mathfrak {b}}y^{2}+{\mathfrak {c}}z^{2}+2{\mathfrak {g}}yz+2{\mathfrak {h}}zx+2{\mathfrak {k}}xy\\&=\left(\xi x+\eta y+\zeta z\right)^{2}+\left(\xi _{1}x+\eta _{1}y+\zeta _{1}z\right)^{2}+\left(\xi _{2}x+\eta _{2}y+\zeta _{2}z\right)^{2}\\\\f&=ax^{2}+by^{2}+cz^{2}+2gyz+2hzx+2kxy\\&=\left(\xi x+\eta y+\zeta z\right)^{2}-\left(\xi _{1}x+\eta _{1}y+\zeta _{1}z\right)^{2}-\left(\xi _{2}x+\eta _{2}y+\zeta _{2}z\right)^{2}\end{aligned}}\left|{\begin{aligned}\xi ^{2}+\xi _{1}^{2}+\xi _{2}^{2}&={\mathfrak {a}}\\\eta ^{2}+\eta _{1}^{2}+\eta _{2}^{2}&={\mathfrak {b}}\\\zeta ^{2}+\zeta _{1}^{2}+\zeta _{2}^{2}&={\mathfrak {c}}\\\eta \zeta +\eta _{1}\zeta _{1}+\eta _{2}\zeta _{2}&={\mathfrak {g}}\\\zeta \xi +\zeta _{1}\xi _{1}+\zeta _{2}\xi _{2}&={\mathfrak {h}}\\\xi \eta +\xi _{1}\eta _{1}+\xi _{2}\eta _{2}&={\mathfrak {k}}\end{aligned}}\right|{\begin{aligned}\xi ^{2}-\xi _{1}^{2}-\xi _{2}^{2}&=a\\\eta ^{2}-\eta _{1}^{2}-\eta _{2}^{2}&=b\\\zeta ^{2}-\zeta _{1}^{2}-\zeta _{2}^{2}&=c\\\eta \zeta -\eta _{1}\zeta _{1}-\eta _{2}\zeta _{2}&=g\\\zeta \xi -\zeta _{1}\xi _{1}-\zeta _{2}\xi _{2}&=h\\\xi \eta -\xi _{1}\eta _{1}-\xi _{2}\eta _{2}&=k\end{aligned}}}$

In addition, Selling showed that auxiliary coefficients $\xi \eta \zeta$ can be geometrically interpreted as point coordinates which are in motion upon one sheet of a two-sheet hyperboloid, which is related to Selling's formalism for the reduction of indefinite forms by using definite forms.

Selling also reproduced the Lorentz transformation given by Gauss (1800/63), to whom he gave full credit, and called it the only example of a particular indefinite ternary form known to him that has ever been discussed:

${\begin{matrix}\left({\begin{matrix}1,&-1,&-1\\0,&0,&0\end{matrix}}\right)\\\hline W={\begin{vmatrix}{\frac {1}{2}}\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}+\delta ^{2}\right)&{\frac {1}{2}}\left(\alpha ^{2}+\beta ^{2}-\gamma ^{2}-\delta ^{2}\right)&\alpha \gamma +\beta \delta \\{\frac {1}{2}}\left(\alpha ^{2}-\beta ^{2}+\gamma ^{2}-\delta ^{2}\right)&{\frac {1}{2}}\left(\alpha ^{2}-\beta ^{2}-\gamma ^{2}+\delta ^{2}\right)&\alpha \gamma -\beta \delta \\\alpha \beta +\gamma \delta &\alpha \beta -\gamma \delta &\alpha \delta +\beta \gamma \end{vmatrix}}\\\left({\begin{vmatrix}\alpha &\beta \\\gamma &\delta \end{vmatrix}}=1\right)\end{matrix}}$

This is equivalent to Lorentz transformation (6d).

Escherich (1874) – Beltrami coordinates

Gustav von Escherich (1874) discussed the plane of constant negative curvature based on the Beltrami–Klein model of hyperbolic geometry by Beltrami (1868), as well as Christoph Gudermann's (1830) rectangular coordinates $x=\tan a$ and $y=\tan b$ and coordinate transformations using trigonometric functions in the cases of rotation and translation related to sphere geometry. By using hyperbolic functions $x=\tanh(a/k)$ and $y=\tanh(b/k)$, Escherich gave the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation has the form:

$x={\frac {\sinh {\frac {a}{k}}+x'\cosh {\frac {a}{k}}}{\cosh {\frac {a}{k}}+x'\sinh {\frac {a}{k}}}}$ and $y={\frac {y'}{\cosh {\frac {a}{k}}+x'\sinh {\frac {a}{k}}}}$

This is equivalent to Lorentz transformation (3d), also equivalent to the relativistic velocity addition (4c) by setting ${\frac {a}{k}}=\operatorname {atanh} \beta$ and $[x,\ y,\ x',\ y']=[u_{x},\ u_{y},\ u'_{x},\ u'_{y}]$, and equivalent to Lorentz boost (3b) by setting $\scriptstyle (x,\ y,\ x',\ y')=\left({\frac {x_{1}}{x_{0}}},\ {\frac {x_{2}}{x_{0}}},\ {\frac {x_{1}^{\prime }}{x_{0}^{\prime }}},\ {\frac {x_{2}^{\prime }}{x_{0}^{\prime }}}\right)$. This is the relation between the Beltrami coordinates in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the hyperboloid model introduced by Killing (1878–1893), Poincaré (1881), and Cox (1881). Both coordinate systems were compared by Cox (1881).

Killing (1878–1893)

Weierstrass coordinates

Wilhelm Killing (1878–1880) described non-Euclidean geometry by using Weierstrass coordinates (named after Karl Weierstrass who described them in lectures in 1872 which Killing attended) obeying the form

$k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}$ with $ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}$

or

$k^{2}x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}=k^{2}$

where $k$ is the reciprocal measure of curvature, $k^{2}=\infty$ denotes Euclidean geometry, $k^{2}>0$ elliptic geometry, and $k^{2}<0$ hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form. In (1879/80) he wrote the corresponding transformations as a general rotation matrix

${\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\\hline {\begin{matrix}\cos \eta \tau +\lambda ^{2}{\frac {1-\cos \eta \tau }{\eta ^{2}}},&\nu {\frac {\sin \eta \tau }{\eta }}+\lambda \mu {\frac {1-\cos \eta \tau }{\eta ^{2}}},&-\mu \sin {\frac {\eta \tau }{\eta }}+\nu \lambda {\frac {1-\cos \eta \tau }{\eta ^{2}}}\\-k^{2}\nu {\frac {\sin \eta \tau }{\eta }}+k^{2}\lambda \mu {\frac {1-\cos \eta \tau }{\eta ^{2}}},&\cos \eta \tau +\mu ^{2}{\frac {1-\cos \eta \tau }{\eta ^{2}}},&\lambda {\frac {\sin \eta \tau }{\eta }}+k^{2}\mu \nu {\frac {1-\cos \eta \tau }{\eta ^{2}}}\\k^{2}\mu {\frac {\sin \eta \tau }{\eta }}+k^{2}\nu \lambda {\frac {1-\cos \eta \tau }{\eta ^{2}}},&-\lambda {\frac {\sin \eta \tau }{\eta }}+k^{2}\mu \nu {\frac {1-\cos \eta \tau }{\eta ^{2}}},&\cos \eta \tau +\nu ^{2}{\frac {1-\cos \eta \tau }{\eta ^{2}}}\end{matrix}}\\\left(\lambda ^{2}+k^{2}\mu ^{2}+k^{2}\nu ^{2}=\eta ^{2}\right)\end{matrix}}$

In (1885) he wrote the Weierstrass coordinates and their transformation as follows:

${\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime 2}+x^{\prime 2}+y^{\prime 2}\\ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\\hline {\begin{aligned}k^{2}p'&=k^{2}wp+w'x+w''y\\x'&=ap+a'x+a''y\\y'&=bp+b'x+b''y\\\\k^{2}p&=k^{2}wp'+ax'+by'\\x&=w'p'+a'x+b'y'\\y&=w''p'+a''x'+b''y'\end{aligned}}\left|{\begin{aligned}k^{2}w^{2}+w^{\prime 2}+w^{\prime \prime 2}&=k^{2}\\{\frac {a^{2}}{k^{2}}}+a^{\prime 2}+a^{\prime \prime 2}&=1\\{\frac {b^{2}}{k^{2}}}+b^{\prime 2}+b^{\prime \prime 2}&=1\\aw+a'w'+a''w''&=0\\bw+b'w'+b''w''&=0\\{\frac {ab}{k^{2}}}+a'b'+a''b''&=0\end{aligned}}\right.\end{matrix}}$

This is similar to Lorentz transformation (1a)$(n=2)$ with $k^{2}=-1$

In (1885) he also gave the transformation for $n$ dimensions:

${\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}=k^{2}\\ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots +dx_{n}^{2}\\\hline \left.{\begin{aligned}k^{2}\xi _{0}&=k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots +a_{0n}x_{0}\\\xi _{\varkappa }&=a_{\varkappa 0}x_{0}+a_{\varkappa 1}x_{1}+\dots +a_{\varkappa n}x_{n}\\\\k^{2}x_{0}&=a_{00}k^{2}\xi _{0}+a_{10}\xi _{1}+\dots +a_{n0}\xi _{n}\\x_{\varkappa }&=a_{0\varkappa }\xi _{0}+a_{1\varkappa }\xi _{1}+\dots +a_{n\varkappa }\xi _{n}\end{aligned}}\right|{\begin{aligned}k^{2}a_{00}^{2}+a_{10}^{2}+\dots +a_{n0}^{2}&=k^{2}\\a_{00}a_{0\varkappa }+a_{10}a_{1\varkappa }+\dots +a_{n0}a_{n\varkappa }&=0\\{\frac {a_{0\iota }a_{0\varkappa }}{k^{2}}}+a_{0\iota }a_{1\varkappa }+\dots +a_{n\iota }a_{n\varkappa }=\delta _{\iota \kappa }&=1\ {\text{or}}\ 0\end{aligned}}\end{matrix}}$

This is similar to Lorentz transformation (1a) with $k^{2}=-1$

In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force. Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting $k^{2}=-1$ and using $p,x,y$ as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid $p^{2}-x^{2}-y^{2}=1$ (known as hyperboloid model), by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space. Finally, in (1893) he wrote:

${\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\\hline {\begin{aligned}t'&=at+bu+cv\\u'&=a't+b'u+c'v\\v'&=a''t+b''u+c''v\end{aligned}}\left|{\begin{aligned}k^{2}a^{2}+a^{\prime 2}+a^{\prime \prime 2}&=k^{2}\\k^{2}b^{2}+b^{\prime 2}+b^{\prime \prime 2}&=1\\k^{2}c^{2}+b^{\prime 2}+c^{\prime \prime 2}&=1\\k^{2}ab+a'b'+a''b''&=0\\k^{2}ac+a'c'+a''c''&=0\\k^{2}bc+b'c'+b''c''&=0\end{aligned}}\right.\end{matrix}}$

This is equivalent to Lorentz transformation (1a)$(n=2)$ with $k^{2}=-1$

and for $n$ dimensions

${\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}=k^{2}\\k^{2}y_{0}y_{0}^{\prime }+y_{1}y_{1}^{\prime }+\cdots +y_{n}y_{n}^{\prime }=k^{2}x_{0}x_{0}^{\prime }+x_{1}x_{1}^{\prime }+\cdots +x_{n}x_{n}^{\prime }\\ds^{2}=k^{2}dx_{0}^{2}+\dots +dx_{n}^{2}\\\hline {\begin{aligned}y_{0}&=a_{00}x_{0}+a_{01}x_{1}+\dots +a_{0n}x_{n}\\y_{1}&=a_{10}x_{0}+a_{11}x_{1}+\dots +a_{1n}x_{n}\\&\,\,\,\vdots \\y_{n}&=a_{n0}x_{0}+a_{n1}x_{1}+\dots +a_{nn}x_{n}\end{aligned}}\left|{\begin{aligned}k^{2}a_{00}^{2}+a_{10}^{2}+\dots +a_{n0}^{2}&=k^{2}\\k^{2}a_{0\varkappa }^{2}+a_{1\varkappa }^{2}+\dots +a_{n\varkappa }^{2}&=1\\k^{2}a_{00}a_{0\varkappa }+a_{10}a_{1\varkappa }+\dots +a_{n0}a_{n\varkappa }&=0\\k^{2}a_{0\varkappa }a_{0\lambda }+a_{1\varkappa }a_{1\lambda }+\dots +a_{n\varkappa }a_{n\lambda }&=0\\(\varkappa ,\lambda =1,\dots ,n,\ \lambda \lessgtr \varkappa )\end{aligned}}\right.\end{matrix}}$

This is equivalent to Lorentz transformation (1a) with $k^{2}=-1$

Translation in the hyperbolic plane

The case of translation was given by Killing (1893) in the form

$y_{0}=x_{0}\operatorname {Ch} a+x_{1}\operatorname {Sh} a,\quad y_{1}=x_{0}\operatorname {Sh} a+x_{1}\operatorname {Ch} a,\quad y_{2}=x_{2}$

This is equivalent to Lorentz boost (3b).

In 1898, Killing wrote that relation in a form similar to Escherich (1874), and derived the corresponding Lorentz transformation for the two cases were $v$ is unchanged or $u$ is unchanged:

${\begin{matrix}\xi '={\frac {\xi \operatorname {Ch} {\frac {\mu }{l}}+l\operatorname {Sh} {\frac {\mu }{l}}}{{\frac {\xi }{l}}\operatorname {Sh} {\frac {\mu }{l}}+\operatorname {Ch} {\frac {\mu }{l}}}},\ \eta '={\frac {\eta }{{\frac {\xi }{l}}\operatorname {Sh} {\frac {\mu }{l}}+\operatorname {Ch} {\frac {\mu }{l}}}}\\\hline {\frac {u}{p}}=\xi ,\ {\frac {v}{p}}=\eta \\\hline p'=p\operatorname {Ch} {\frac {\mu }{l}}+{\frac {u}{l}}\operatorname {Sh} {\frac {\mu }{l}},\quad u'=pl\operatorname {Sh} {\frac {\mu }{l}}+u\operatorname {Ch} {\frac {\mu }{l}},\quad v'=v\\{\text{or}}\\p'=p\operatorname {Ch} {\frac {\nu }{l}}+{\frac {v}{l}}\operatorname {Sh} {\frac {\nu }{l}},\quad u'=u,\quad v'=pl\operatorname {Sh} {\frac {\nu }{l}}+v\operatorname {Ch} {\frac {\nu }{l}}\end{matrix}}$

This is equivalent to Lorentz boost (3b).

Infinitesimal transformations and Lie group

Killing (1887/88) defined the infinitesimal projective transformations in relation to the following quadratic form of second degree by:

${\begin{matrix}x_{1}^{2}+\dots +x_{m+1}^{2}=1\\\hline X_{\iota \varkappa }f=x_{i}{\frac {\partial f}{\partial x_{\varkappa }}}-x_{\varkappa }{\frac {\partial f}{\partial x_{\iota }}}\\{\text{where}}\\\left(X_{\iota \varkappa },X_{\iota \lambda }\right)=X_{\varkappa \lambda };\ \left(X_{\iota \varkappa },X_{\lambda \mu }\right)=0;\\\left[\iota \neq \varkappa \neq \lambda \neq \mu \right]\end{matrix}}$

and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:

${\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}=k^{2}\\\hline X_{\iota \varkappa }=x_{\iota }p_{\varkappa }-x_{\varkappa }p_{\iota },\quad X_{\iota }=x_{0}p_{\iota }-{\frac {x_{\iota }p_{0}}{k^{2}}}\\{\text{where}}\\\left(X_{\iota }X_{\iota \varkappa }\right)=X_{\varkappa }f;\ \left(X_{\iota }X_{\varkappa \lambda }\right)=0;\ \left(X_{\iota }X_{\varkappa }\right)=-{\frac {1}{k^{2}}}X_{\iota \varkappa }f;\end{matrix}}$

In (1897/98) he pointed out (see the following formulas on the left) that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to form (1) and transitive when related to form (2), and he also showed (on the right) the relation of Weierstrass coordinates to the notation of Killing (1887/88) and Werner (1889), Lie (1890):

${\begin{matrix}\left.{\begin{matrix}{\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}&(1)\\k^{2}x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}=k^{2}&(2)\end{matrix}}\\\hline V_{\varkappa }=k^{2}x_{0}p_{\varkappa }-x_{\varkappa }p_{0},\quad U_{\iota \varkappa }=p_{\iota }x_{\varkappa }-p_{\varkappa }x_{\iota }\\{\text{where}}\\\left(V_{\iota },V_{\varkappa }\right)=k^{2}U_{\iota \varkappa },\ \left(V_{\iota },U_{\iota \varkappa }\right)=-V_{\varkappa },\ \left(V_{\iota },U_{\varkappa \lambda }\right)=0,\\\left(U_{\iota \varkappa },U_{\iota \lambda }\right)=U_{\varkappa \lambda },\ \left(U_{\iota \varkappa },U_{\lambda \mu }\right)=0\\\left[\iota ,\varkappa ,\lambda ,\mu =1,2,\dots n\right]\end{matrix}}\right|&{\begin{matrix}y_{1}={\frac {x_{1}}{x_{0}}},\ y_{2}={\frac {x_{2}}{x_{0}}},\dots y_{n}={\frac {x_{n}}{x_{0}}}\\\downarrow \\k^{2}+y_{1}^{2}+y_{2}^{2}+\dots +y_{n}^{2}=0\\\hline q_{\varkappa }+{\frac {y_{\varkappa }}{k^{2}}}\sum _{\varrho }y_{y}q_{\varrho },\quad q_{\iota }y_{\varkappa }-q_{\varkappa }y_{\iota }\end{matrix}}\end{matrix}}$

Setting $k^{2}=-1$ denotes the group of hyperbolic motions and thus the Lorentz group.

Poincaré (1881 – 1887)

Weierstrass coordinates

Henri Poincaré (1881) published a work which connected the work of Hermite and Selling (1873/74) on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880). He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:

${\begin{matrix}{\begin{aligned}F&=(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\&=\xi ^{2}+\eta ^{2}-\zeta ^{2}=-1\\F&=(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\&=\xi ^{\prime 2}+\eta ^{\prime 2}-\zeta ^{\prime 2}=-1\end{aligned}}\\\hline {\begin{aligned}\xi '&=\alpha \xi +\beta \eta +\gamma \zeta \\\eta '&=\alpha '\xi +\beta '\eta +\gamma '\zeta \\\zeta '&=\alpha ''\xi +\beta ''\eta +\gamma ''\zeta \end{aligned}}\left|{\begin{aligned}\alpha ^{2}+\alpha ^{\prime 2}-\alpha ^{\prime \prime 2}&=1\\\beta ^{2}+\beta ^{\prime 2}-\beta ^{\prime \prime 2}&=1\\\gamma ^{2}+\gamma ^{\prime 2}-\gamma ^{\prime \prime 2}&=-1\\\alpha \beta +\alpha '\beta '-\alpha ''\beta ''&=0\\\alpha \gamma +\alpha '\gamma '-\alpha ''\gamma ''&=0\\\beta \gamma +\beta '\gamma '-\beta ''\gamma ''&=0\end{aligned}}\right.\end{matrix}}$

He went on to describe the properties of "hyperbolic coordinates". Poincaré mentioned the hyperboloid model also in (1887).

This is equivalent to Lorentz transformation (1a)$(n=2)$.

Möbius transformation

After Poincare used the Möbius transformation ${\tfrac {az+b}{cz+d}}$ in relation to Fuchsian functions in 1881, he discussed and extended Klein's (1878-1882) study on the relation between Möbius transformations and hyperbolic, elliptic, parabolic, and loxodromic substitutions, arriving at the following transformation in 1883:

${\begin{matrix}\left(z,\ {\frac {\alpha z+\beta }{\gamma z+\delta }}\right),\ \left(z_{0},\ {\frac {\alpha _{0}z_{0}+\beta _{0}}{\gamma _{0}z_{0}+\delta _{0}}}\right)\\\hline z=\xi +i\eta ,\ z_{0}=\xi -i\eta ,\ \rho ^{2}=\xi ^{2}+\eta ^{2}+\zeta ^{2}\\\hline {\begin{aligned}\rho ^{\prime 2}&={\frac {\rho ^{2}\alpha \alpha _{0}+z\alpha \beta _{0}+z_{0}\beta \alpha _{0}+\beta \beta _{0}}{\rho ^{2}\gamma \gamma _{0}+z\gamma \delta _{0}+z_{0}\delta \gamma _{0}+\delta \delta _{0}}}\\z^{\prime }&={\frac {\rho ^{2}\alpha \gamma _{0}+z\alpha \delta _{0}+z_{0}\beta \gamma _{0}+\beta \delta _{0}}{\rho ^{2}\gamma \gamma _{0}+z\gamma \delta _{0}+z_{0}\delta \gamma _{0}+\delta \delta _{0}}}\\z_{0}^{\prime }&={\frac {\rho ^{2}\gamma \alpha _{0}+z\gamma \beta _{0}+z_{0}\delta \alpha _{0}+\delta \beta _{0}}{\rho ^{2}\gamma \gamma _{0}+z\gamma \delta _{0}+z_{0}\delta \gamma _{0}+\delta \delta _{0}}}\end{aligned}}\end{matrix}}$

Setting $[\rho ^{2},\ \xi ,\ \zeta ]=\left[{\tfrac {X_{1}}{X_{4}}},\ {\tfrac {X_{2}}{X_{4}}},\ {\tfrac {X_{3}}{X_{4}}}\right]$ this becomes transformation $\mathbf {u'}$ in (6a) and becomes the complete Lorentz transformation by setting ${\scriptstyle \left[{\begin{matrix}X_{1}&X_{2}\\X_{3}&X_{4}\end{matrix}}\right]=\left[{\begin{matrix}x_{0}+x_{3}&x_{1}-ix_{2}\\x_{1}+ix_{2}&x_{0}-x_{3}\end{matrix}}\right]}$.

In 1886, Poincaré investigated the relation between indefinite ternary quadratic forms and Fuchsian functions:

${\begin{matrix}\left(z,\ {\frac {\alpha z+\beta }{\gamma z+\delta }}\right)\\\hline Y^{\prime 2}-X'Z'=Y^{2}-XZ\\\hline {\begin{aligned}X'&=\alpha ^{2}X+2\alpha \gamma Y+\gamma ^{2}Z\\Y'&=\alpha \beta X+(\alpha \delta +\beta \gamma )Y+\gamma \delta Z\\Z'&=\beta ^{2}X+2\beta \gamma Y+\delta ^{2}Z\end{aligned}}\\\left[{\begin{aligned}X=&ax+by+cz,&Y&=a'x+b'y+c'z,&Z&=a''x+b''y+c''z,\\X'=&ax'+by'+cz',&Y'&=a'x'+b'y'+c'z',&Z'&=a''x'+b''y'+c''z',\end{aligned}}\right]\end{matrix}}$

This is equivalent to transformation $\mathbf {u'}$ in (6c) and becomes the complete Lorentz transformation by suitibly choosing the coefficients $a,b,c,\dots$ so that $[X,Y,Z]=[x+z,\ y,\ -x+z]$.

Cox (1881–1883)

Weierstrass coordinates

Homersham Cox (1881/82) – referring to similar rectangular coordinates used by Gudermann (1830) and George Salmon (1862) on a sphere, and to Escherich (1874) as reported by Johannes Frischauf (1876) in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:

${\begin{matrix}z^{2}-x^{2}-y^{2}=1\\z^{2}-y^{2}-x^{2}=Z^{2}-Y^{2}-X^{2}\\\hline {\begin{aligned}x&=l_{1}X+l_{2}Y+l_{3}Z\\y&=m_{1}X+m_{2}Y+m_{3}Z\\z&=n_{1}X+n_{2}Y+n_{3}Z\\\\X&=l_{1}x+m_{1}y-n_{1}z\\Y&=l_{2}x+m_{2}y-n_{2}z\\Z&=l_{3}x+m_{3}y-n_{3}z\end{aligned}}\left|{\begin{aligned}l_{1}^{2}+m_{1}^{2}-n_{1}^{2}&=1&l_{1}^{2}+l_{2}^{2}-l_{3}^{2}&=1\\l_{2}^{2}+m_{2}^{2}-n_{2}^{2}&=1&m_{1}^{2}+m_{2}^{2}-m_{3}^{2}&=1\\l_{3}^{2}+m_{3}^{2}-n_{3}^{2}&=1&n_{1}^{2}+n_{2}^{2}-n_{3}^{2}&=1\\l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2}&=0&l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3}&=0\\l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3}&=0&m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3}&=0\\l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1}&=0&n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3}&=0\end{aligned}}\right.\end{matrix}}$

This is equivalent to Lorentz transformation (1a)$(n=2)$ up to a sign change.

Cox also gave the Weierstrass coordinates in their transformation in hyperbolic space:

${\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime 2}-x^{\prime 2}-y^{\prime 2}-z^{\prime 2}\\\hline {\begin{aligned}x&=l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\y&=m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\z&=n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\w&=r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\\\x'&=l_{1}x+m_{1}y+n_{1}z-r_{1}w\\y'&=l_{2}x+m_{2}y+n_{2}z-r_{2}w\\z'&=l_{3}x+m_{3}y+n_{3}z-r_{3}w\\w'&=l_{4}x+m_{4}y+n_{4}z-r_{4}w\end{aligned}}\left|{\begin{aligned}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2}&=1\\l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2}&=1\\l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2}&=1\\l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}&=1\\l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3}&=0\\l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1}&=0\\l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4}&=0\\l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4}&=0\\l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4}&=0\end{aligned}}\right.\end{matrix}}$

This is equivalent to Lorentz transformation (1a)$(n=3)$ up to a sign change.

The case of translation was also given by him, where the y-axis remains unchanged:

${\begin{aligned}X&=x\cosh p-z\sinh p\\Z&=-x\sinh p+z\cosh p\end{aligned}}$ and ${\begin{aligned}x&=X\cosh p+Z\sinh p\\z&=X\sinh p+Z\cosh p\end{aligned}}$

This is equivalent to Lorentz boost (3b).

Quaternions

Subsequently, Cox (1882/83) also described hyperbolic geometry in terms of an analogue to quaternions and Hermann Grassmann's exterior algebra. To that end, he used hyperbolic numbers, which were first introduced by James Cockle (1848) in the framework of his tessarine algebra as follows:

$\varepsilon ^{jy}=1+jy+{\frac {y^{2}}{2}}+j{\frac {y^{3}}{2.3}}+{\frac {y^{4}}{2.3.4}}+\&c.,\ \left(j^{2}=1\right)$.

In the hands of Cox (who doesn't mention Cockle) this expression becomes a means to transfer point P to point Q in the hyperbolic plane, which he wrote in the form:

${\begin{matrix}QP^{-1}=\cosh \theta +\iota \sinh \theta \\QP^{-1}=e^{\iota \theta }\end{matrix}}\left(\iota ^{2}=1\right)$

In (1882/83a) he showed the equivalence of $PQ=-\cosh \theta +\iota \sinh \theta$ with "quaternion multiplication", and in (1882/83b) he described $QP^{-1}=\cosh \theta +\iota \sinh \theta$ as being "associative quaternion multiplication". He also showed that the position of point P in the hyperbolic plane may be determined by three quantities in terms of Weierstrass coordinates obeying the relation $z^{2}-x^{2}-y^{2}=1$.

Cox's associative quaternion multiplication using the hyperbolic versor is equivalent to the Lorentz boost (7b) by setting $Q=x_{1}^{\prime }+\iota x_{0}^{\prime }$ and $P=x_{1}+\iota x_{0}$.

Cox went on to develop an algebra for hyperbolic space analogous to Clifford's biquaternions. While Clifford (1873) used biquaternions of the form $a+\omega b$ in which $\omega ^{2}=0$ denotes parabolic space and $\omega ^{2}=1$ elliptic space, Cox discussed hyperbolic space using the imaginary quantity ${\sqrt {-1}}$ and therefore $\omega ^{2}=-1$. He also obtained relations of quaternion multiplication in terms of Weierstrass coordinates:

${\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\\hline {\begin{aligned}PO^{-1}&=\cosh \theta +(li+mj+nk)\sinh \theta \\&=w+xi+yj+zk\\\\OP^{-1}&=\cosh \theta -(li+mj+nk)\sinh \theta \\&=w-xi-yj-zk\end{aligned}}\end{matrix}}$

Hill (1882) – Homogeneous coordinates

Following Gauss (1818), George William Hill (1882) formulated the equations

${\begin{matrix}k\left(\sin ^{2}T+\cos ^{2}T-1\right)\\k\left(\sin ^{2}E+\cos ^{2}E-1\right)\\\hline {\begin{aligned}\cos E'&={\frac {\alpha +\alpha '\sin T+\alpha ''\cos T}{\gamma +\gamma '\sin T+\gamma ''\cos T}}\\\sin E'&={\frac {\beta +\beta '\sin T+\beta ''\cos T}{\gamma +\gamma '\sin T+\gamma ''\cos T}}\\\hline \\x&=\alpha u+\alpha 'u'+\alpha ''u''\\y&=\beta u+\beta 'u'+\beta ''u''\\z&=\gamma u+\gamma 'u'+\gamma ''u''\\\\u&=\alpha x-\beta y+\gamma z\\u'&=\alpha 'x+\beta 'y'-\gamma 'z\\u''&=\alpha ''x+\beta ''y-\gamma ''z\end{aligned}}\left|{\begin{aligned}\alpha ^{2}+\beta ^{2}-\gamma ^{2}&=-1&\alpha \alpha '+\beta \beta '-\gamma \gamma '&=0\\\alpha ^{\prime 2}+\beta ^{\prime 2}-\gamma ^{\prime 2}&=1&\alpha \alpha ''+\beta \beta ''-\gamma \gamma ''&=0\\\alpha ^{\prime \prime 2}+\beta ^{\prime \prime 2}-\gamma ^{\prime \prime 2}&=1&\alpha '\alpha ''+\beta '\beta ''-\gamma '\gamma ''&=0\\\\&\left(k=-1\right)\\\alpha ^{2}-\alpha ^{\prime 2}-\alpha ^{\prime \prime 2}&=k&\alpha \beta -\alpha '\beta '-\alpha ''\beta ''&=0\\\beta ^{2}-\beta ^{\prime 2}-\beta ^{\prime \prime 2}&=k&\alpha \gamma -\alpha '\gamma '-\alpha ''\gamma ''&=0\\\gamma ^{2}-\gamma ^{\prime 2}-\gamma ^{\prime \prime 2}&=-k&\beta \gamma -\beta '\gamma '-\beta ''\gamma ''&=0\end{aligned}}\right.\end{matrix}}$

The first transformation is equivalent to Lorentz transformation (1b) $(n=2)$ with $[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime },u_{2}^{\prime }\right]$.

The second transformation system is equivalent to Lorentz transformation (1a) $(n=2)$ .

Laguerre (1882) – Laguerre inversion

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

$\left.{\begin{aligned}D'&={\frac {D\left(1+\alpha ^{2}\right)-2\alpha R}{1-\alpha ^{2}}}\\R'&={\frac {2\alpha D-R\left(1+\alpha ^{2}\right)}{1-\alpha ^{2}}}\end{aligned}}\right|{\begin{aligned}D^{2}-D^{\prime 2}&=R^{2}-R^{\prime 2}\\D-D'&=\alpha (R-R')\\D+D'&={\frac {1}{\alpha }}(R+R')\end{aligned}}$

This is in agreement with Lorentz boost (3a) because $\scriptstyle \left({\frac {1+\alpha ^{2}}{1-\alpha ^{2}}}\right)^{2}-\left({\frac {2\alpha }{1-\alpha ^{2}}}\right)^{2}=1$, thus $\scriptstyle {\frac {1+\alpha ^{2}}{1-\alpha ^{2}}}=g_{00}=g_{11}$ and $\scriptstyle {\frac {2\alpha }{1-\alpha ^{2}}}=g_{01}=g_{10}$.

Stephanos (1883) – Biquaternions

Cyparissos Stephanos (1883) showed that Hamilton's biquaternion $a_{0}+a_{1}\iota _{1}+a_{2}\iota _{2}+a_{3}\iota _{3}$ can be interpreted as an oriented sphere in terms of Lie's sphere geometry (1871), having the vector $a_{1}\iota _{1}+a_{2}\iota _{2}+a_{3}\iota _{3}$ as its center and the scalar $a_{0}{\sqrt {-1}}$ as its radius. Its norm $a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}$ is thus equal to the power of a point of the corresponding sphere. In particular, the norm of two quaternions $N\left(Q_{1}-Q_{2}\right)$ (the corresponding spheres are in contact with $N\left(Q_{1}-Q_{2}\right)=0$) is equal to the tangential distance between two spheres. The general contact transformation between two spheres then can be given by a homography using 4 arbitrary quaternions $A,B,C,D$ and two variable quaternions $X,Y$:

$XAY+XB+CY+D=0$ (or $X=-{\frac {CY+D}{AY+B}}$).

Stephanos pointed out that the special case $A=0$ denotes transformations of oriented planes (see Laguerre's transformation of oriented planes (1882)).

The Lorentz group SO(3,1) is a subgroup of the conformal group Con(3,1) in terms of Lie's transformations of orientied spheres in which the radius indicates the fourth coordinate. The Lorentz group is isomorphic to the group of Laguerre's transformation of oriented planes.

Buchheim (1884–85) – Biquaternions

Arthur Buchheim (1884, published 1885) applied Clifford's biquaternions and their operator $\omega$ to different forms of geometries (Buchheim mentions Cox (1882) as well). He defined the scalar $\omega ^{2}=e^{2}$ which in the case $-1$ denotes hyperbolic space, $1$ elliptic space, and $0$ parabolic space. He derived the following relations consistent with the Cayley–Klein absolute:

${\begin{matrix}\delta ^{2}+e^{2}\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)=0\\\hline P=\delta +\omega \rho =\delta +\omega (\alpha i+\beta j+\gamma k)\\P'=\delta '+\omega \rho '=\delta '+\omega (\alpha 'i+\beta 'j+\gamma 'k)\\{\begin{aligned}PKP'&=(\delta +\omega \rho )(\delta '+\omega \rho ')\\&=\delta \delta '-e^{2}\rho \rho '+\omega (\rho \delta '-\rho '\delta )\\PKP&=\delta ^{2}-e^{2}\rho ^{2}\\\cos(PP')&={\frac {\delta \delta '+e^{2}\left(\alpha \alpha '+\beta \beta '+\gamma \gamma '\right)}{\left\{\delta ^{2}+e^{2}\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)\right\}^{\frac {1}{2}}\cdot \left\{\delta ^{\prime 2}+e^{2}\left(\alpha ^{\prime 2}+\beta ^{\prime 2}+\gamma ^{\prime 2}\right)\right\}^{\frac {1}{2}}}}\end{aligned}}\end{matrix}}$

By choosing $e^{2}=-1$ for hyperbolic space, the Cayley absolute becomes the Lorentz interval.

Callandreau (1885) – Homogeneous coordinates

Following Gauss (1818) and Hill (1882), Octave Callandreau (1885) formulated the equations

${\begin{matrix}k\left(\sin ^{2}T+\cos ^{2}T-1\right)=\\(\alpha +\alpha '\sin T+\alpha ''\cos T)^{2}+(\beta +\beta '\sin T+\beta ''\cos T)^{2}-(\gamma +\gamma '\sin T+\gamma ''\cos T)^{2}\\\hline {\begin{aligned}\cos \varepsilon '&={\frac {\alpha +\alpha '\sin T+\alpha ''\cos T}{\gamma +\gamma '\sin T+\gamma ''\cos T}}\\\sin \varepsilon '&={\frac {\beta +\beta '\sin T+\beta ''\cos T}{\gamma +\gamma '\sin T+\gamma ''\cos T}}\end{aligned}}\left|{\begin{aligned}&\left(k=1\right)\\\alpha ^{2}+\beta ^{2}-\gamma ^{2}&=-k&\alpha \alpha '+\beta \beta '-\gamma \gamma '&=0\\\alpha ^{\prime 2}+\beta ^{\prime 2}-\gamma ^{\prime 2}&=+k&\alpha \alpha ''+\beta \beta ''-\gamma \gamma ''&=0\\\alpha ^{\prime \prime 2}+\beta ^{\prime \prime 2}-\gamma ^{\prime \prime 2}&=+k&\alpha '\alpha ''+\beta '\beta ''-\gamma '\gamma ''&=0\\\\\alpha ^{2}-\alpha ^{\prime 2}-\alpha ^{\prime \prime 2}&=-1&\alpha \beta -\alpha '\beta '-\alpha ''\beta ''&=0\\\beta ^{2}-\beta ^{\prime 2}-\beta ^{\prime \prime 2}&=-1&\alpha \gamma -\alpha '\gamma '-\alpha ''\gamma ''&=0\\\gamma ^{2}-\gamma ^{\prime 2}-\gamma ^{\prime \prime 2}&=+1&\beta \gamma -\beta '\gamma '-\beta ''\gamma ''&=0\end{aligned}}\right.\end{matrix}}$

The transformation system is equivalent to Lorentz transformation (1b) $(n=2)$ with $[\cos T,\sin T,\cos \varepsilon ',\sin \varepsilon ']=\left[u_{1},u_{2},u_{1}^{\prime },u_{2}^{\prime }\right]$.

Lipschitz (1885–86) – Clifford algebra

Clifford algebra (which includes quaternions as special cases) was used by Rudolf Lipschitz (1885/86) who introduced the orthogonal transformation $\Lambda X=Y\Lambda _{1}$ of a definite quadratic form as a sum or squares $x_{1}^{2}+x_{2}^{2}\dots +x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n}^{2}$ into itself, which he discussed both for real as well as imaginary expressions. He then discussed the general indefinite form and its transformation by using $m$ real and $n-m$ imaginary quantities:

${\begin{matrix}{\mathfrak {x}}_{1}^{2}+\dots +{\mathfrak {x}}_{m}^{2}-{\mathfrak {x}}_{m+1}^{2}-\dots -{\mathfrak {x}}_{n}^{2}={\mathfrak {y}}_{1}^{2}+\dots +{\mathfrak {y}}_{m}^{2}-{\mathfrak {y}}_{m+1}^{2}-\dots -{\mathfrak {y}}_{n}^{2}\\\hline P^{(m+1,\dots n)}{\mathfrak {\bar {X}}}={\mathfrak {\bar {Y}}}P_{1}^{(m+1,\dots n)}\\\left({\begin{aligned}{\mathfrak {\bar {X}}}&={\mathfrak {x}}_{1}+k_{1}k_{2}{\mathfrak {x}}_{2}+\dots +k_{1}k_{m}{\mathfrak {x}}_{m}-k_{1}k_{m+1}h{\mathfrak {x}}_{m+1}-\dots -k_{1}k_{n}h{\mathfrak {x}}_{n}\\{\mathfrak {\bar {Y}}}&={\mathfrak {y}}_{1}+k_{1}k_{2}{\mathfrak {y}}_{2}+\dots +k_{1}k_{m}{\mathfrak {y}}_{m}-k_{1}k_{m+1}h{\mathfrak {y}}_{m+1}-\dots -k_{1}k_{n}h{\mathfrak {y}}_{n}\\P&=\varrho _{0}+k_{1}k_{2}\varrho _{12}+\dots +k_{1}k_{2}k_{3}k_{4}\varrho _{1234}+\cdots ,\\P_{1}&=\varrho _{0}-k_{1}k_{2}\varrho _{12}+\dots -k_{1}k_{2}k_{3}k_{4}\varrho _{1234}+\cdots ,\end{aligned}}\right)\\\left(h={\sqrt {-1}}\right)\end{matrix}}$

By setting $m=n-1$, the Lorentz interval ${\mathfrak {x}}_{1}^{2}+\dots +{\mathfrak {x}}_{m}^{2}-{\mathfrak {x}}_{n}^{2}={\mathfrak {y}}_{1}^{2}+\dots +{\mathfrak {y}}_{m}^{2}-{\mathfrak {y}}_{n}^{2}$ and the Lorentz transformation follows

Schur (1885/86, 1900/02) – Beltrami coordinates

Friedrich Schur (1885/86) discussed spaces of constant Riemann curvature, and by following Beltrami (1868) he used the transformation

$x_{1}=R^{2}{\frac {y_{1}+a_{1}}{R^{2}+a_{1}y_{1}}},\ x_{2}=R{\sqrt {R^{2}-a_{1}^{2}}}{\frac {y_{2}}{R^{2}+a_{1}y_{1}}},\dots ,\ x_{n}=R{\sqrt {R^{2}-a_{1}^{2}}}{\frac {y_{n}}{R^{2}+a_{1}y_{1}}}$

This is equivalent to Lorentz transformation (3d) and therefore also equivalent to the relativistic velocity addition (4c) by setting $[R,a_{1},x_{1},y_{1},\dots ,x_{n},y_{n}]=[c,v,u_{x}^{\prime },u_{y},\dots ,u_{n}^{\prime },u_{n}]$.

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:

$x'={\frac {x-a}{1-{\mathfrak {k}}ax}},\quad y'={\frac {y{\sqrt {1-{\mathfrak {k}}a^{2}}}}{1-{\mathfrak {k}}ax}}$

where ${\mathfrak {k}}$ can have values $>0$, $<0$ or $\infty$.

This is equivalent to Lorentz transformation (3d) and therefore also equivalent to the relativistic velocity addition (4c) by setting $[{\mathfrak {k}},\ a,\ x,\ y]=\left[{\tfrac {1}{c^{2}}},\ v,\ u_{x},\ u_{y}\right]$.

He also defined the triangle

${\frac {1}{\sqrt {1-{\mathfrak {k}}c^{2}}}}={\frac {1}{\sqrt {1-{\mathfrak {k}}a^{2}}}}\cdot {\frac {1}{\sqrt {1-{\mathfrak {k}}b^{2}}}}-{\frac {a}{\sqrt {1-{\mathfrak {k}}a^{2}}}}\cdot {\frac {b}{\sqrt {1-{\mathfrak {k}}b^{2}}}}\cos \gamma$

This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition (3e, b) or (4d) by setting $[{\mathfrak {k}},\ c,\ a,\ b]=\left[{\tfrac {1}{c^{2}}},\ {\sqrt {u_{x}^{\prime 2}+u_{y}^{\prime 2}}},\ v,\ {\sqrt {u_{x}^{2}+u_{y}^{2}}}\right]$.

Darboux (1887) – Laguerre inversion

Following Laguerre (1882), Gaston Darboux (1887) presented the Laguerre inversions in four dimensions using coordinates $x,y,z,R$:

${\begin{matrix}x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-R^{\prime 2}=x^{2}+y^{2}+z^{2}-R^{2}\\\hline {\begin{aligned}x'&=x,&z'&={\frac {1+k^{2}}{1-k^{2}}}z-{\frac {2kR}{1-k^{2}}},\\y'&=y,&R'&={\frac {2kz}{1-k^{2}}}-{\frac {1+k^{2}}{1-k^{2}}}R,\end{aligned}}\end{matrix}}$

This is in agreement with Lorentz boost (3a) because $\scriptstyle \left({\frac {1+k^{2}}{1-k^{2}}}\right)^{2}-\left({\frac {2k}{1-k^{2}}}\right)^{2}=1$, thus $\scriptstyle {\frac {1+k^{2}}{1-k^{2}}}=g_{00}=g_{11}$ and $\scriptstyle {\frac {2k}{1-k^{2}}}=g_{01}=g_{10}$.

Bianchi (1888, 1893) – Möbius and spin transformations

Related to Klein's (1871) and Poincaré's (1881-1887) work on non-Euclidean geometry and indefinite quadratic forms, Luigi Bianchi (1888) analyzed the differential Lorentz interval $ds^{2}=dx^{2}+dy^{2}-dz^{2}$, and alluded to the Möbius transformations and its parameters $\alpha ,\beta ,\gamma ,\delta$ in order preserve the Lorentz interval, for which he gave credit to Gauss (1800/63):

${\begin{matrix}\omega ={\frac {\alpha \omega '+\beta }{\gamma \omega '+\delta }}\ (\alpha \delta -\beta \gamma =1)\\\hline x^{2}+y^{2}-z^{2}\\\left({\begin{matrix}{\frac {\alpha ^{2}-\beta ^{2}-\gamma ^{2}+\delta ^{2}}{2}},&\gamma \delta -\alpha \beta ,&{\frac {-\alpha ^{2}-\beta ^{2}+\gamma ^{2}+\delta ^{2}}{2}}\\\beta \delta -\alpha \gamma ,&\alpha \delta +\beta \gamma ,&\beta \delta +\alpha \gamma \\{\frac {-\alpha ^{2}+\beta ^{2}-\gamma ^{2}+\delta ^{2}}{2}},&\alpha \beta +\gamma \delta ,&{\frac {\alpha ^{2}+\beta ^{2}+\gamma ^{2}+\delta ^{2}}{2}}\end{matrix}}\right)\end{matrix}}$

This is equivalent to Lorentz transformation (6d)

In 1893, Bianchi gave the coefficients in the case of four dimensions:

${\begin{matrix}{\begin{aligned}z&={\frac {\alpha z'+\beta }{\gamma z'+\delta }}\\&(\alpha \delta -\beta \gamma =1)\end{aligned}}\rightarrow {\begin{aligned}z&={\frac {\xi }{\eta }}\\z'&={\frac {\xi '}{\eta '}}\end{aligned}}\rightarrow {\begin{aligned}\xi &=\alpha \xi '+\beta \eta '\\\eta &=\gamma \xi '+\delta \eta '\\\\\xi _{0}&=\alpha _{0}\xi '_{0}+\beta _{0}\eta '_{0}\\\eta _{0}&=\gamma _{0}\xi '_{0}+\delta _{0}\eta '_{0}\end{aligned}}\\\hline {\scriptstyle F=\left(u_{1}+u{}_{4}\right)\xi \xi _{0}+\left(u_{2}+iu{}_{3}\right)\xi \eta _{0}+\left(u_{2}-iu{}_{3}\right)\xi _{0}\eta +\left(u_{4}-u{}_{1}\right)\eta \eta _{0}}\\{\scriptstyle F'=\left(u'_{1}+u'{}_{4}\right)\xi '\xi '_{0}+\left(u'_{2}+iu'{}_{3}\right)\xi '\eta '_{0}+\left(u'_{2}-iu'{}_{3}\right)\xi '_{0}\eta '+\left(u'_{4}-u'{}_{1}\right)\eta '\eta '_{0}}\\{\scriptstyle \left(u_{2}+iu{}_{3}\right)\left(u_{2}-iu{}_{3}\right)+\left(u_{1}-u{}_{4}\right)\left(u_{1}+u{}_{4}\right)=\left(u'_{2}+iu'{}_{3}\right)\left(u'_{2}-iu'{}_{3}\right)+\left(u'_{1}-u'{}_{4}\right)\left(u'_{1}+u'{}_{4}\right)}\\\hline {\scriptstyle {\begin{aligned}u'_{1}+u'_{4}&=\alpha \alpha _{0}\left(u_{1}+u{}_{4}\right)+\alpha \gamma _{0}\left(u_{2}+iu{}_{3}\right)+\alpha _{0}\gamma \left(u_{2}-iu{}_{3}\right)+\gamma \gamma _{0}\left(u_{4}-u{}_{1}\right)\\u'_{2}+iu'_{3}&=\alpha \beta _{0}\left(u_{1}+u{}_{4}\right)+\alpha \delta _{0}\left(u_{2}+iu{}_{3}\right)+\beta _{0}\gamma \left(u_{2}-iu{}_{3}\right)+\gamma \delta _{0}\left(u_{4}-u{}_{1}\right)\\u'_{2}-iu'_{3}&=\alpha _{0}\beta \left(u_{1}+u{}_{4}\right)+\alpha _{0}\delta \left(u_{2}-iu{}_{3}\right)+\beta \gamma _{0}\left(u_{2}+iu{}_{3}\right)+\gamma _{0}\delta \left(u_{4}-u{}_{1}\right)\\u'_{4}-u'_{1}&=\beta \beta _{0}\left(u_{1}+u{}_{4}\right)+\beta \delta _{0}\left(u_{2}+iu{}_{3}\right)+\beta _{0}\delta \left(u_{2}-iu{}_{3}\right)+\delta \delta _{0}\left(u_{4}-u{}_{1}\right)\end{aligned}}}\end{matrix}}$

This is equivalent to Lorentz transformation (6a)

Solving for $u'_{1}\dots$ Bianchi obtained:

${\begin{matrix}u_{1}^{2}+u_{2}^{2}+u_{3}^{2}-u_{4}^{2}=u_{1}^{\prime 2}+u_{2}^{\prime 2}+u_{3}^{\prime 2}-u_{4}^{\prime 2}\\\hline {\scriptstyle {\begin{aligned}u'_{1}&={\frac {1}{2}}\left(\alpha \alpha _{0}-\beta \beta _{0}-\gamma \gamma _{0}+\delta \delta _{0}\right)u_{1}+{\frac {1}{2}}\left(\alpha \gamma _{0}+\alpha _{0}\gamma -\beta \delta _{0}-\beta _{0}\delta \right)u_{2}+\\&+{\frac {i}{2}}\left(\alpha \gamma _{0}-\alpha _{0}\gamma +\beta _{0}\delta -\beta \delta _{0}\right)u_{3}+{\frac {1}{2}}\left(\alpha \alpha _{0}-\beta \beta _{0}+\gamma \gamma _{0}-\delta \delta _{0}\right)u_{4}\\u'_{2}&={\frac {1}{2}}\left(\alpha \beta _{0}+\alpha _{0}\beta -\gamma \delta _{0}-\gamma _{0}\delta \right)u_{1}+{\frac {1}{2}}\left(\alpha \delta _{0}+\alpha _{0}\delta +\beta \gamma _{0}+\beta _{0}\gamma \right)u_{2}+\\&+{\frac {i}{2}}\left(\alpha \delta _{0}-\alpha _{0}\delta +\beta \gamma _{0}-\beta _{0}\gamma \right)u_{3}+{\frac {1}{2}}\left(\alpha \beta _{0}+\alpha _{0}\beta +\gamma \delta _{0}+\gamma _{0}\delta \right)u_{4}\\u'_{3}&={\frac {i}{2}}\left(\alpha _{0}\beta -\alpha \beta _{0}+\gamma \delta _{0}-\gamma _{0}\delta \right)u_{1}+{\frac {i}{2}}\left(\alpha _{0}\delta -\alpha \delta _{0}+\beta \gamma _{0}-\beta _{0}\gamma \right)u_{2}+\\&+{\frac {1}{2}}\left(\alpha \delta _{0}+\alpha _{0}\delta -\beta \gamma _{0}-\beta _{0}\gamma \right)u_{3}+{\frac {i}{2}}\left(\alpha _{0}\beta -\alpha \beta _{0}+\gamma _{0}\delta -\gamma \delta _{0}\right)u_{4}\\u'_{4}&={\frac {1}{2}}\left(\alpha \alpha _{0}+\beta \beta _{0}-\gamma \gamma _{0}-\delta \delta _{0}\right)u_{1}+{\frac {1}{2}}\left(\alpha \gamma _{0}+\alpha _{0}\gamma +\beta \delta _{0}+\beta _{0}\delta \right)u_{2}+\\&+{\frac {i}{2}}\left(\alpha \gamma _{0}-\alpha _{0}\gamma +\beta \delta _{0}-\beta _{0}\delta \right)u_{3}+{\frac {1}{2}}\left(\alpha \alpha _{0}+\beta \beta _{0}+\gamma \gamma _{0}+\delta \delta _{0}\right)u_{4}\end{aligned}}}\end{matrix}}$

This is equivalent to Lorentz transformation (6b)

Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute

Ferdinand von Lindemann discussed hyperbolic geometry in his (1890/91) edition of the lectures on geometry of Alfred Clebsch. Citing Killing (1885) and Poincaré (1887) in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set

${\begin{matrix}\Omega _{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ {\text{and}}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\\Omega _{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ {\text{and}}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}\end{matrix}}$

In addition, following Klein (1871) he employed the Cayley absolute related to surfaces of second degree, by using the following quadratic form and its transformation

${\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\X_{1}X_{4}+X_{2}X_{3}=\Xi _{1}\Xi _{4}+\Xi _{2}\Xi _{3}\\\hline {\begin{aligned}X_{1}&=\left(\lambda +\lambda _{1}\right)U_{4}&\Xi _{1}&=\left(\lambda -\lambda _{1}\right)U_{4}&X_{1}&={\frac {\lambda +\lambda _{1}}{\lambda -\lambda _{1}}}\Xi _{1}\\X_{2}&=\left(\lambda +\lambda _{3}\right)U_{4}&\Xi _{2}&=\left(\lambda -\lambda _{3}\right)U_{4}&X_{2}&={\frac {\lambda +\lambda _{3}}{\lambda -\lambda _{3}}}\Xi _{2}\\X_{3}&=\left(\lambda -\lambda _{3}\right)U_{2}&\Xi _{3}&=\left(\lambda +\lambda _{3}\right)U_{2}&X_{3}&={\frac {\lambda -\lambda _{3}}{\lambda +\lambda _{3}}}\Xi _{3}\\X_{4}&=\left(\lambda -\lambda _{1}\right)U_{1}&\Xi _{4}&=\left(\lambda +\lambda _{1}\right)U_{1}&X_{4}&={\frac {\lambda -\lambda _{1}}{\lambda +\lambda _{1}}}\Xi _{4}\end{aligned}}\end{matrix}}$

into which he put

${\begin{aligned}X_{1}&=x_{1}+2kx_{4},&X_{2}&=x_{2}+ix_{3},&\lambda +\lambda _{1}&=\left(\lambda -\lambda _{1}\right)e^{a},\\X_{4}&=x_{1}-2kx_{4},&X_{3}&=x_{2}-ix_{3},&\lambda +\lambda _{3}&=\left(\lambda -\lambda _{3}\right)e^{ai},\end{aligned}}$

he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations ($a=0$) or translations ($\alpha =0$):

${\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\\hline {\begin{aligned}x_{2}&=\xi _{2}\cos \alpha +\xi _{3}\sin \alpha ,&x_{1}&=\xi _{1}\cos {\frac {a}{i}}+2ki\xi _{4}\sin {\frac {a}{i}},\\x_{3}&=-\xi _{2}\sin \alpha +\xi _{3}\cos \alpha ,&2kx_{4}&=i\xi _{1}\sin {\frac {a}{i}}+2k\xi _{4}\cos {\frac {a}{i}}.\end{aligned}}\end{matrix}}$

This is equivalent to Lorentz boost (3b) with $\alpha =0$ and $2k=1$.

Fricke (1891–1897) – Möbius and spin transformations

Robert Fricke (1891) – following the work of his teacher Klein (1878–1882) as well as Poincaré (1881–1887) on automorphic functions and group theory, obtained the following transformation for an integer ternary quadratic form

${\begin{matrix}\omega '={\frac {\delta \omega +\beta }{\gamma \omega +\alpha }}\ (\alpha \delta -\beta \gamma =1),\ \omega ={\frac {\eta }{\xi }},\\\hline {\begin{aligned}\xi '&=\xi \alpha ^{2}+2\eta \alpha \gamma +\zeta \gamma ^{2}\\\eta '&=\xi \alpha \beta +\eta (\alpha \delta +\beta \gamma )+\zeta \gamma \delta \\\zeta '&=\xi \beta ^{2}+2\eta \beta \delta +\zeta \delta ^{2}\end{aligned}}\\\hline \xi '\zeta '-\eta '^{2}=(\alpha \delta -\beta \gamma )^{2}\left(\xi \zeta -\eta ^{2}\right)\\\xi =x{\sqrt {q}}-y,\ \eta =z,\ \zeta =x{\sqrt {q}}+y\\\hline qx^{\prime 2}-y^{\prime 2}-z^{\prime 2}=qx^{2}-y^{2}-z^{2}\\\hline \left({\begin{matrix}{\frac {1}{2}}\left(+\alpha ^{2}+\beta ^{2}+\gamma ^{2}+\delta ^{2}\right)&{\frac {1}{2{\sqrt {q}}}}\left(-\alpha ^{2}-\beta ^{2}+\gamma ^{2}+\delta ^{2}\right)&{\frac {1}{\sqrt {q}}}(\alpha \gamma +\beta \delta )\\{\frac {1}{2}}{\sqrt {q}}\left(-\alpha ^{2}+\beta ^{2}-\gamma ^{2}+\delta ^{2}\right)&{\frac {1}{2}}\left(+\alpha ^{2}-\beta ^{2}-\gamma ^{2}+\delta ^{2}\right)&(-\alpha \gamma +\beta \delta )\\{\sqrt {q}}(\alpha \beta +\gamma \delta )&(-\alpha \beta +\gamma \delta )&(\alpha \delta +\beta \gamma )\end{matrix}}\right)\end{matrix}}$

By setting $q=1$, this is equivalent to Lorentz transformation (6c) and (6d)

And the general case of four dimensions in 1893:

${\begin{matrix}y'_{2}y'_{3}-y'_{1}y'_{4}=y_{2}y_{3}-y_{1}y_{4}\\\hline {\begin{aligned}y_{1}^{\prime }&=\alpha {\bar {\alpha }}y_{1}+\alpha {\bar {\beta }}y_{2}+\beta {\bar {\alpha }}y_{3}+\beta {\bar {\beta }}y_{4}\\y_{2}^{\prime }&=\alpha {\bar {\gamma }}y_{1}+\alpha {\bar {\delta }}y_{2}+\beta {\bar {\gamma }}y_{3}+\beta {\bar {\delta }}y_{4}\\y_{3}^{\prime }&=\gamma {\bar {\alpha }}y_{1}+\gamma {\bar {\beta }}y_{2}+\delta {\bar {\alpha }}y_{3}+\delta {\bar {\beta }}y_{4}\\y_{4}^{\prime }&=\gamma {\bar {\gamma }}y_{1}+\gamma {\bar {\delta }}y_{2}+\delta {\bar {\gamma }}y_{3}+\delta {\bar {\delta }}y_{4}\end{aligned}}\\\hline {\begin{aligned}y_{1}&=z_{4}{\sqrt {s}}+z_{3}{\sqrt {r}},&y_{2}&=z_{1}{\sqrt {p}}+iz_{2}{\sqrt {q}}\\y_{3}&=z_{1}{\sqrt {p}}-iz_{2}{\sqrt {q}},&y_{4}&=z_{4}{\sqrt {s}}-z_{3}{\sqrt {r}}\end{aligned}}\\\hline pz_{1}^{\prime 2}+qz{}_{2}^{\prime 2}+rz{}_{3}^{\prime 2}-sz{}_{4}^{\prime 2}=pz_{1}^{2}+qz_{2}^{2}+rz_{3}^{2}-sz_{4}^{2}\\\hline z'_{i}=\alpha _{i1}z_{1}+\alpha _{i2}z_{2}+\alpha _{i3}z_{3}+\alpha _{i4}z_{4}\\{\begin{aligned}2\alpha _{11}\ {\text{or}}\ 2\alpha _{22}&=\alpha {\bar {\delta }}+\delta {\bar {\alpha }}\pm \beta {\bar {\gamma }}\pm \gamma {\bar {\beta }},&2\alpha _{33}\ {\text{or}}\ 2\alpha _{44}&=\alpha {\bar {\alpha }}+\delta {\bar {\delta }}\pm \beta {\bar {\beta }}\pm \gamma {\bar {\gamma }}\\{\frac {2\alpha _{12}{\sqrt {p}}}{i{\sqrt {p}}}}\ {\text{or}}\ {\frac {2\alpha _{21}i{\sqrt {p}}}{\sqrt {p}}}&=\alpha {\bar {\delta }}-{\bar {\delta }}\alpha \mp \beta {\bar {\gamma }}\pm \gamma {\bar {\beta }},&{\frac {2\alpha _{34}{\sqrt {r}}}{\sqrt {s}}}\ {\text{or}}\ {\frac {2\alpha _{43}{\sqrt {s}}}{\sqrt {r}}}&=\alpha {\bar {\alpha }}-\delta {\bar {\delta }}\pm \beta {\bar {\beta }}\pm \gamma {\bar {\gamma }}\\{\frac {2\alpha _{13}{\sqrt {p}}}{\sqrt {r}}}\ {\text{or}}\ {\frac {2\alpha _{24}i{\sqrt {p}}}{\sqrt {s}}}&=\alpha {\bar {\gamma }}-\delta {\bar {\beta }}\pm \gamma {\bar {\alpha }}\pm \beta {\bar {\delta }},&{\frac {2\alpha _{14}{\sqrt {p}}}{\sqrt {s}}}\ {\text{or}}\ {\frac {2\alpha _{23}i{\sqrt {q}}}{\sqrt {r}}}&=\alpha {\bar {\gamma }}+\delta {\bar {\beta }}\pm \gamma {\bar {\alpha }}\pm \beta {\bar {\delta }}\\{\frac {2\alpha _{31}{\sqrt {r}}}{\sqrt {p}}}\ {\text{or}}\ {\frac {2\alpha _{43}{\sqrt {s}}}{i{\sqrt {q}}}}&=\alpha {\bar {\beta }}-\delta {\bar {\gamma }}\pm \beta {\bar {\alpha }}\mp \gamma {\bar {\delta }},&{\frac {2\alpha _{41}{\sqrt {s}}}{\sqrt {p}}}\ {\text{or}}\ {\frac {2\alpha _{32}{\sqrt {r}}}{i{\sqrt {q}}}}&=\alpha {\bar {\beta }}+\delta {\bar {\gamma }}\pm \beta {\bar {\alpha }}\pm \gamma {\bar {\delta }}\end{aligned}}\end{matrix}}$

By setting $p=q=r=s=1$, this is equivalent to Lorentz transformation (6a) and (6b)

Supported by Felix Klein, Fricke summarized his and Klein's work in a treatise concerning automorphic functions (1897). Using a sphere as the absolute, in which the interior of the sphere is denoted as hyperbolic space, they defined hyperbolic motions, and stressed that any hyperbolic motion corresponds to "circle relations" (now called Möbius transformations):

$\left.{\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\=(z_{4}+z_{3})(z_{4}-z_{3})-(z_{1}+iz_{2})(z_{1}-iz_{2})=0\\=y_{1}y_{4}-y_{2}y_{3}=0\\\left(y_{1}=z_{4}+z_{3},\ y_{2}=z_{1}+iz_{2},\ y_{3}=z_{1}-iz_{2},\ y_{4}=z_{4}-z_{3}\right)\\\zeta ={\frac {z_{1}+iz_{2}}{z_{4}-z_{3}}},\ {\bar {\zeta }}={\frac {z_{1}-iz_{2}}{z_{4}-z_{3}}}\\\zeta '={\frac {\alpha \zeta +\beta }{\gamma \zeta +\delta }},\ {\bar {\zeta }}'={\frac {{\overline {\alpha \zeta }}+{\bar {\beta }}}{{\overline {\gamma \zeta }}+{\bar {\delta }}}}\quad (\alpha \delta -\beta \gamma \neq 0)\\z_{1}:z_{2}:z_{3}z_{4}=(\zeta +{\bar {\zeta }}):-i(\zeta -{\bar {\zeta }}):(\zeta {\bar {\zeta }}-1):(\zeta {\bar {\zeta }}+1)\\y_{1}:y_{2}:y_{3}y_{4}=\zeta {\bar {\zeta }}:\zeta :{\bar {\zeta }}:1=\zeta _{1}{\bar {\zeta }}_{1}:\zeta _{1}{\bar {\zeta }}_{2}:\zeta _{2}{\bar {\zeta }}_{1}:\zeta _{2}{\bar {\zeta }}_{2}\\\left(\zeta =\zeta _{1}:\zeta _{2},\ {\bar {\zeta }}={\bar {\zeta }}_{1}:{\bar {\zeta }}_{2}\right)\end{matrix}}\right|{\begin{aligned}y_{1}^{\prime }&=\alpha {\bar {\alpha }}y_{1}+\alpha {\bar {\beta }}y_{2}+\beta {\bar {\alpha }}y_{3}+\beta {\bar {\beta }}y_{4}\\y_{2}^{\prime }&=\alpha {\bar {\gamma }}y_{1}+\alpha {\bar {\delta }}y_{2}+\beta {\bar {\gamma }}y_{3}+\beta {\bar {\delta }}y_{4}\\y_{3}^{\prime }&=\gamma {\bar {\alpha }}y_{1}+\gamma {\bar {\beta }}y_{2}+\delta {\bar {\alpha }}y_{3}+\delta {\bar {\beta }}y_{4}\\y_{4}^{\prime }&=\gamma {\bar {\gamma }}y_{1}+\gamma {\bar {\delta }}y_{2}+\delta {\bar {\gamma }}y_{3}+\delta {\bar {\delta }}y_{4}\end{aligned}}$

This is equivalent to Lorentz transformation (6a).

Gérard (1892) – Weierstrass coordinates

Louis Gérard (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to (1a)$(n=2)$:

${\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\X^{2}+Y^{2}-Z^{2}=X^{\prime 2}+Y^{\prime 2}-Z^{\prime 2}\\\hline {\begin{aligned}X&=aX'+a'Y'+a''Z'\\Y&=bX'+b'Y'+b''Z'\\Z&=cX'+c'Y'+c''Z'\\\\X'&=aX+bY-cZ\\Y'&=a'X+b'Y-c'Z\\Z'&=-a''X-b''Y+c''Z\end{aligned}}\left|{\begin{aligned}a^{2}+b^{2}-c^{2}&=1\\a^{\prime 2}+b^{\prime 2}-c^{\prime 2}&=1\\a^{\prime \prime 2}+b^{\prime \prime 2}-c^{\prime \prime 2}&=-1\\aa'+bb'-cc'&=0\\a'a''+b'b''-c'c''&=0\\a''a+b''b-c''c&=0\end{aligned}}\right.\end{matrix}}$

This is equivalent to Lorentz transformation (1a)$(n=2)$.

He gave the case of translation as follows:

${\begin{aligned}X&=Z_{0}X'+X_{0}Z'\\Y&=Y'\\Z&=X_{0}X'+Z_{0}Z'\end{aligned}}$ with ${\begin{aligned}X_{0}&=\operatorname {sh} OO'\\Z_{0}&=\operatorname {ch} OO'\end{aligned}}$

This is equivalent to Lorentz boost (3b).

Macfarlane (1892–1900) – Hyperbolic quaternions

Alexander Macfarlane (1892, 1893) – similar to Cockle (1848) and Cox (1882/83) – defined the hyperbolic versor in terms of hyperbolic numbers

$\cosh A+\sinh A\cdot \alpha ^{\frac {\pi }{2}},\ \left(\alpha ^{\frac {\pi }{2}}\alpha ^{\frac {\pi }{2}}=1\right)$

and in 1894 he defined the "exspherical" versor

$\beta ^{iu}=\cosh u+i\sinh u\cdot \beta ^{\frac {\pi }{2}}$

and used them to analyze hyperboloids of one- or two sheets. This was further extended by him in (1900) in order to express trigonometry in terms of hyperbolic quaternions $re^{b\beta }$, with $\beta ^{2}=+1$ and $r={\sqrt {x^{2}-y^{2}}}$, the hyperbolic number $x+y\beta$, and the hyperbolic versor $e^{b\beta }$.

The hyperbolic versor is the basis of Lorentz boost (7b).

Whitehead (1897/98) – Universal algebra

Alfred North Whitehead (1898) discussed the kinematics of hyperbolic space as part of his study of universal algebra, and obtained the following transformation:

${\begin{aligned}x'&=\left(\eta \cosh {\frac {\delta }{\gamma }}+\eta _{1}\sinh {\frac {\delta }{\gamma }}\right)e+\left(\eta \sinh {\frac {\delta }{\gamma }}+\eta _{1}\cosh {\frac {\delta }{\gamma }}\right)e_{1}\\&\qquad +\left(\eta _{2}\cos \alpha +\eta _{3}\sin \alpha \right)e_{2}+\left(\eta _{3}\cos \alpha -\eta _{2}\sin \alpha \right)e_{3}\end{aligned}}$

This is equivalent to Lorentz boost (3b) with $\alpha =0$.

Hausdorff (1899)

Weierstrass coordinates

Felix Hausdorff (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:

${\begin{matrix}p^{2}-x^{2}-y^{2}=1\\\hline {\begin{aligned}x&=a_{1}x'+a_{2}y'+x_{0}p'\\y&=b_{1}x'+b{}_{2}y'+y_{0}p'\\p&=e_{1}x'+e_{2}y'+p_{0}p'\\\\x'&=a_{1}x+b_{1}y-e_{1}p\\y'&=a_{2}x+b_{2}y-e_{2}p\\-p'&=x_{0}x+y_{0}y-p_{0}p\end{aligned}}\left|{\begin{aligned}a_{1}^{2}+b_{1}^{2}-e_{1}^{2}&=1&a_{1}^{2}+a_{2}^{2}-x_{0}^{2}&=1\\a_{2}^{2}+b_{2}^{2}-e_{2}^{2}&=1&b_{1}^{2}+b_{2}^{2}-y_{0}^{2}&=1\\-x_{0}^{2}-y_{0}^{2}+p_{0}^{2}&=1&-e_{1}^{2}-e_{2}^{2}+p_{0}^{2}&=1\\a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0}&=0&b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0}&=0\\a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0}&=0&a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0}&=0\\a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2}&=0&a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0}&=0\end{aligned}}\right.\end{matrix}}$

This is equivalent to Lorentz transformation (1a)$(n=2)$.

Möbius transformation

Hausdorff (1899) also discussed the relation of the above coordinates to conformal Möbius transformations:

${\begin{matrix}\nu ={\frac {x+iy}{p+1}}={\frac {p-1}{x-iy}},\ {\bar {\nu }}={\frac {x-iy}{p+1}}={\frac {p-1}{x+iy}}\\x={\frac {\nu +{\bar {\nu }}}{1-\nu {\bar {\nu }}}},\ y=i{\frac {{\bar {\nu }}-\nu }{1-\nu {\bar {\nu }}}},\ p={\frac {1+\nu {\bar {\nu }}}{1-\nu {\bar {\nu }}}}\\x'+iy'=\left(a_{1}+ia_{2}\right)x+\left(b_{1}+ib_{2}\right)y-\left(e_{1}+ie_{2}\right)p\\p'+1=-x_{0}x-y_{0}y+p_{0}p+1\\\nu '=e^{i\chi }{\frac {\nu -\nu _{0}}{1-\nu \nu _{0}}},\ e^{i\chi }={\frac {e_{1}+ie_{2}}{x_{0}+iy_{0}}}={\frac {x_{0}-iy_{0}}{e_{1}-ie_{2}}}={\frac {\alpha }{\bar {\alpha }}},\ \nu _{0}=-{\frac {\beta }{\alpha }},\ {\bar {\nu }}_{0}=-{\frac {\bar {\beta }}{\bar {\alpha }}}\\\nu '={\frac {\alpha \nu +\beta }{{\bar {\beta }}\nu +{\bar {\alpha }}}},\ \alpha {\bar {\alpha }}-\beta {\bar {\beta }}>0\end{matrix}}$

This is similar to Lorentz transformation (6a) with $\scriptstyle (p,\ x,\ y,\ 1)=\left({\frac {x_{0}}{x_{3}}},\ {\frac {x_{1}}{x_{3}}},\ {\frac {x_{2}}{x_{3}}},\ {\frac {x_{3}}{x_{3}}}\right)$.

Vahlen (1901/02) – Clifford algebra and Möbius transformation

Modifying Lipschitz's (1885/86) variant of Clifford numbers, Theodor Vahlen (1901/02) formulated Möbius transformations (which he called vector transformations) and biquaternions in order to discuss motions in n-dimensional non-Euclidean space, leaving the following quadratic form invariant (where $j^{2}=+1$ represents hyperbolic motions, $j^{2}=-1$ elliptic motions, $j^{2}=0$ parabolic motions):

${\begin{matrix}x_{0}^{2}-i_{i}^{2}x_{1}^{2}-\dots -i_{p}^{2}x_{p}^{2}=j^{2}\quad \left(i_{\alpha }^{2}=1,\ 0,\ {\text{or}}-1\right)\\\hline ax=ya'\\{\begin{aligned}a&=a_{0}+a_{1}i_{1}+\dots +a_{12}i_{1}i_{2}\dots +a_{12\dots p}i_{1}i_{2}\dots i_{p}\\a'&=a_{0}-a_{1}i_{1}-\dots +a_{12}i_{1}i_{2}\dots +(-1)^{p}a_{12\dots p}i_{1}i_{2}\dots i_{p}\\x&=x_{0}+x_{1}i_{1}+\dots +x_{p}i_{p}\end{aligned}}\\\hline {\text{Vector transformation}}\\y={\frac {ax+b'}{j^{2}bx+a'}}\\\left[z={\frac {cy+d'}{j^{2}dy+c'}},\quad y={\frac {ax+b'}{j^{2}bx+a'}}\right]\rightarrow z={\frac {Ax+B'}{j^{2}Bx+A'}}\\\hline {\text{Biquaternion}}\\(c+d'j)(a+b'j)=A+B'j\quad \left(i_{\alpha }j+ji_{\alpha }=0\right)\end{matrix}}$

The group of hyperbolic motions or the Möbius group are isomorphic to the Lorentz group.

Woods (1901–1905) – Weierstrass coordinates

Frederick S. Woods (1901/02) defined the following invariant quadratic form and its projective transformation (he pointed out that this can be connected to hyperbolic geometry by setting $k={\sqrt {-1}}R$ with $R$ as real quantity):

${\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\\hline {\begin{aligned}u'&={\frac {\alpha _{1}u+\alpha _{2}v+\alpha _{3}w+\alpha _{4}}{\delta _{1}u+\delta _{2}v+\delta _{3}w+\delta _{4}}}\\v'&={\frac {\beta _{1}u+\beta _{2}v+\beta _{3}w+\beta _{4}}{\delta _{1}u+\delta _{2}v+\delta _{3}w+\delta _{4}}}\\w'&={\frac {\gamma _{1}u+\gamma _{2}v+\gamma _{3}w+\gamma _{4}}{\delta _{1}u+\delta _{2}v+\delta _{3}w+\delta _{4}}}\end{aligned}}\left|{\begin{aligned}k^{2}\left(\alpha _{i}^{2}+\beta _{i}^{2}+\gamma _{i}^{2}\right)+\delta _{i}^{2}&=k^{2}\\(i=1,2,3)\\k^{2}\left(\alpha _{4}^{2}+\beta _{4}^{2}+\gamma _{4}^{2}\right)+\delta _{4}^{2}&=1\\\alpha _{i}\alpha _{h}+\beta _{i}\beta _{h}+\gamma _{i}\gamma _{h}+\delta _{i}\delta _{h}&=0\\(i,h=1,2,3,4;\ i\neq h)\end{aligned}}\right.\end{matrix}}$

This is equivalent to Lorentz transformation (1b)$(n=2)$ with $k={\sqrt {-1}}$.

Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with $k={\sqrt {-1}}k$ for hyperbolic space):

${\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\ds^{2}={\frac {1}{k^{2}}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\\hline {\begin{aligned}x_{1}^{\prime }&=\alpha _{1}x_{1}+\alpha _{2}x_{2}+\alpha _{3}x_{3}+\alpha _{0}x_{0}\\x_{2}^{\prime }&=\beta _{1}x_{1}+\beta _{2}x_{2}+\beta _{3}x_{3}+\beta _{0}x_{0}\\x_{3}^{\prime }&=\gamma _{1}x_{1}+\gamma _{2}x_{2}+\gamma _{3}x_{3}+\gamma _{0}x_{0}\\x_{0}^{\prime }&=\delta _{1}x_{1}+\delta _{2}x_{2}+\delta _{3}x_{3}+\delta _{0}x_{0}\end{aligned}}\left|{\begin{aligned}\delta _{0}^{2}+k^{2}\left(\alpha _{0}^{2}+\beta _{0}^{2}+\gamma _{0}^{2}\right)&=1\\\delta _{i}^{2}+k^{2}\left(\alpha _{i}^{2}+\beta _{i}^{2}+\gamma _{i}^{2}\right)&=k^{2}\\(i=1,2,3)\\\delta _{i}\delta _{h}+k^{2}\left(\alpha _{i}\alpha _{h}+\beta _{i}\beta _{h}+\gamma _{i}\gamma _{h}\right)&=0\\(i,h=0,1,2,3;\ i\neq h)\end{aligned}}\right.\end{matrix}}$

This is equivalent to Lorentz transformation (1a)$(n=3)$ with $k^{2}=-1$.

and the case of translation:

$x_{1}^{\prime }=x_{1}\cos kl+x_{0}{\frac {\sin kl}{k}},\quad x_{2}^{\prime }=x_{2},\quad x_{2}^{\prime }=x_{3},\quad x_{0}^{\prime }=-x_{1}k\sin kl+x_{0}\cos kl$

This is equivalent to Lorentz boost (3b) with $k^{2}=-1$.

and the loxodromic substitution for hyperbolic space:

${\begin{matrix}{\begin{aligned}x_{1}^{\prime }&=x_{1}\cosh \alpha -x_{0}\sinh \alpha \\x_{2}^{\prime }&=x_{2}\cos \beta -x_{3}\sin \beta \\x_{3}^{\prime }&=x_{2}\sin \beta +x_{3}\cos \beta \\x_{0}^{\prime }&=-x_{1}\sinh \alpha +x_{0}\cosh \alpha \end{aligned}}\end{matrix}}$

This is equivalent to Lorentz boost (3b) with $\beta =0$.

Liebmann (1904–05) – Weierstrass coordinates

Heinrich Liebmann (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to (1a)$(n=2)$

${\begin{matrix}p^{\prime 2}-x^{\prime 2}-y^{\prime 2}=1\\\hline {\begin{aligned}x_{1}&=\alpha _{11}x+\alpha _{12}y+\alpha _{13}p\\y_{1}&=\alpha _{21}x+\alpha _{22}y+\alpha _{23}p\\x_{1}&=\alpha _{31}x+\alpha _{32}y+\alpha _{33}p\\\\x&=\alpha _{11}x_{1}+\alpha _{21}y_{1}-\alpha _{31}p_{1}\\y&=\alpha _{12}x_{1}+\alpha _{22}y_{1}-\alpha _{32}p_{1}\\p&=-\alpha _{13}x_{1}-\alpha _{23}y_{1}+\alpha _{33}p_{1}\end{aligned}}\left|{\begin{aligned}\alpha _{33}^{2}-\alpha _{13}^{2}-\alpha _{23}^{2}&=1\\-\alpha _{31}^{2}+\alpha _{11}^{2}+\alpha _{21}^{2}&=1\\-\alpha _{32}^{2}+\alpha _{12}^{2}+\alpha _{22}^{2}&=1\\\alpha _{31}\alpha _{32}-\alpha _{11}\alpha _{12}-\alpha _{21}\alpha _{22}&=0\\\alpha _{32}\alpha _{33}-\alpha _{12}\alpha _{13}-\alpha _{22}\alpha _{23}&=0\\\alpha _{33}\alpha _{31}-\alpha _{23}\alpha _{11}-\alpha _{23}\alpha _{21}&=0\end{aligned}}\right.\end{matrix}}$

This is equivalent to Lorentz transformation (1a)$(n=2)$.

and the case of translation:

$x_{1}^{\prime }=x'\operatorname {ch} a+p'\operatorname {sh} a,\quad y_{1}^{\prime }=y',\quad p_{1}^{\prime }=x'\operatorname {sh} a+p'\operatorname {ch} a$

This is equivalent to Lorentz boost (3b).

Special relativity

Voigt (1887)

Woldemar Voigt (1887) developed a transformation in connection with the Doppler effect and an incompressible medium, being in modern notation:

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

If the right-hand sides of his equations are multiplied by $\gamma$ 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 $\lambda$ discussed above), and Lorentz invariant, so the combination is invariant too. For instance, Lorentz transformations can be extended by using $l={\sqrt {\lambda }}$:

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

$l=1/\gamma$ 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, 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.

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.

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

In 1888, Oliver Heaviside 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:

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

Consequently, Joseph John Thomson (1889) 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):

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

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

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

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)

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

where x^* is the Galilean transformation x-vt. Except the additional $\gamma$ in the time transformation, this is the complete Lorentz transformation (4b). 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) 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:

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

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

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

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

Larmor (1897, 1900)

In 1897, Larmor extended the work of Lorentz and derived the following transformation

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

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 $1/\gamma$". 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 $(v/c)$:

Nothing need be neglected: the transformation is exact if $v/c^{2}$ is replaced by $\varepsilon v/c^{2}$ 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 $t''=t'-\varepsilon vx'/c^{2}$ instead of the 1897 expression $t^{\prime }=t-vx/c^{2}$ by replacing $v/c^{2}$ with $\varepsilon v/c^{2}$, 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:

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

Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor $v^{2}/c^{2}$, 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:

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

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 [..]
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.

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, $x^{\ast }$ must be replaced by $x-vt$):

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

Then he introduced a factor $\varepsilon$ 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):

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

This is equivalent to the complete Lorentz transformation (4b) when solved for $x''$ and $t''$ and with $\varepsilon =1$. Like Larmor, Lorentz noticed in 1899 also some sort of time dilation effect in relation to the frequency of oscillating electrons "that in $S$ the time of vibrations be $k\varepsilon$ times as great as in $S_{0}$", where $S_{0}$ is the aether frame.

In 1904 he rewrote the equations in the following form by setting $l=1/\varepsilon$ (again, $x^{\ast }$ must be replaced by $x-vt$):

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

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. However, he didn't achieve full covariance of the transformation equations for charge density and velocity. When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:

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:

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

or by Wilhelm Wien in July 1904:

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

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

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

or by Richard Gans in February 1905:

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

Poincaré (1900, 1905)

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. 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. In order to synchronise the clocks here on Earth (the $x^{\ast }$, $t^{\ast }$ frame) a light signal from one clock (at the origin) is sent to another (at $x^{\ast }$), and is sent back. It's supposed that the Earth is moving with speed $v$ in the $x$-direction (= $x^{\ast }$-direction) in some rest system ($x$, $t$) (i.e. the luminiferous aether system for Lorentz and Larmor). The time of flight outwards is

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

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 $\delta t_{o}+\delta t_{b}$ and the time $t^{\ast }=\left(\delta t_{o}+\delta t_{b}\right)/2$ is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time $t=\delta t_{o}$ 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 $\gamma ^{2}$ under the assumption that ${\frac {v^{2}}{c^{2}}}\ll 1$, Poincaré gave the result $t^{\ast }=t-vx^{\ast }/c^{2}$, which is the form used by Lorentz in 1895.

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

Lorentz transformation

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

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

Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name "Lorentz transformation". 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.

In July 1905 (published in January 1906) 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 $x^{2}+y^{2}+z^{2}-c^{2}t^{2}$ 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.

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.

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:

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

Einstein also defined the velocity addition formula (4c) (which also has been done by Poincaré in May 1905 in unpublished letters to Lorentz):

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

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. Minkowski particularly reformulated electrodynamics in a four-dimensional way (Minkowski spacetime). For instance, he wrote $x,y,z,it$ in the form $x_{1},x_{2},x_{3},x_{4}$. By defining $\psi$ as the angle of rotation around the $z$-axis, the Lorentz transformation assumes a form (with $c=1$) in agreement with (2b):

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

Even though Minkowski used the imaginary number $i\psi$, he for once 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 $\psi =\operatorname {artanh} (q)$ and was later called rapidity. He also wrote the Lorentz transformation in matrix form equivalent to (2a)$(n=3)$:

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

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:

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 relativistc velocity addition (4c) in terms of trigonometric functions and the spherical law of cosines:

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

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

It was pointed out by Bateman and Cunningham (1909–1910), that by setting $u=ict$ as the fourth coordinates in 4D conformal transformation, one can produce spacetime conformal transformations. Not only the quadratic form $\lambda \left(\delta x^{2}+\delta y^{2}+\delta z^{2}+\delta u^{2}\right)$, but also Maxwells equations are covariant with respect to these transformations, irrespective of the choice of $\lambda$. These variants of conformal or Lie's sphere transformations were called spherical wave transformations by Bateman. 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. In particular, by setting $\lambda =1$ the Lorentz group can be seen as a 10-parameter subgroup of the 15-parameter spacetime conformal group.

Bateman (1910/12) 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), Henri Poincaré (1912/21) 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 (3c) are as follows:

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

Varićak (1910) – Weierstrass coordinates

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):

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

Varićak also related the velocity addition to the hyperbolic law of cosines:

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

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.

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:

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

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 $x/\gamma$ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when $n=1/c^{2}$, resulting in $p=\gamma$ 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), with various authors developing similar methods in subsequent years.

Noether (1910), Klein (1910), Conway (1911), Silberstein (1911) – Quaternions

In an appedix 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 $\omega ^{2}=-c^{2}$. He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations equivalent to (7a):

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

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). 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.

Already in 1908, while describing "Drehstreckungen" (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor) by using Cayley's (1854) quaternion multiplication formalism, Felix Klein 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. Citing Noether, in August 1910 Klein published the following quaternion substitutions forming the group of Lorentz transformations:

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

or in March 1911

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

Independently, also Arthur W. Conway in February 1911 succeeded in combining quaternions and relativity (where $\sigma$ is the force and $e$ the charge)

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

Also Ludwik Silberstein in November 1911 as well as in 1914, succeeded in combining quaternions and relativity

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

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

Further information: Lorentz transformation § Vector transformations

Gustav Herglotz (1911) showed how to formulate the transformation equivalent to (4e) in order to allow for arbitrary velocities and coordinates $\mathbf {v} =\left(v_{x},\ v_{y},\ v_{z}\right)$ and $\mathbf {r} =\left(x,\ y,\ z\right)$:

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

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

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

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

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

These formulas were called "general Lorentz transformation without rotation" by Christian Møller (1952), 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, $\mathbf {v} '=\left(v_{x}^{\prime },\ v_{y}^{\prime },\ v_{z}^{\prime }\right)$ is not equal to $-\mathbf {v} =\left(-v_{x},\ -v_{y},\ -v_{z}\right)$, 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):

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

In four dimensions equivalent to (5c):

${\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}}$

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: exponetiation 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.

See also

• History of special relativity

References

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• Abraham, M. (1905). "§ 42. Die Lichtzeit in einem gleichförmig bewegten System" . Theorie der Elektrizität: Elektromagnetische Theorie der Strahlung. Leipzig: Teubner.
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• Borel, Émile (1914). Introduction Geometrique à quelques Théories Physiques. Paris.{{cite book}}: CS1 maint: location missing publisher (link)
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• Frank, Philipp; Rothe, Hermann (1912). "Zur Herleitung der Lorentztransformation". Physikalische Zeitschrift. 13: 750–753.
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• Larmor, Joseph (1900), Aether and Matter , Cambridge University Press
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• Poincaré, Henri (1921) [1912]. "Rapport sur les travaux de M. Cartan (fait à la Faculté des sciences de l'Université de Paris)". Acta Mathematica. 38 (1): 137–145. doi:10.1007/bf02392064. Written by Poincaré in 1912, printed in Acta Mathematica in 1914 though belatedly published in 1921.
• Searle, George Frederick Charles (1897), "On the Steady Motion of an Electrified Ellipsoid" , Philosophical Magazine, 5, 44 (269): 329–341, doi:10.1080/14786449708621072
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• Thomson, Joseph John (1889), "On the Magnetic Effects produced by Motion in the Electric Field" , Philosophical Magazine, 5, 28 (170): 1–14, doi:10.1080/14786448908619821
• Varičak, Vladimir (1912), "Über die nichteuklidische Interpretation der Relativtheorie"  [On the Non-Euclidean Interpretation of the Theory of Relativity], Jahresbericht der Deutschen Mathematiker-Vereinigung, 21: 103–127
• Voigt, Woldemar (1887), "Ueber das Doppler'sche Princip"  [On the Principle of Doppler], Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen (2): 41–51
• Wien, Wilhelm (1904). "Zur Elektronentheorie" . Physikalische Zeitschrift. 5 (14): 393–395.

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• Hawkins, Thomas (2013). "The Cayley–Hermite problem and matrix algebra". The Mathematics of Frobenius in Context: A Journey Through 18th to 20th Century Mathematics. Springer. ISBN 1461463335.
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• Pauli, Wolfgang (1921), "Die Relativitätstheorie", Encyclopädie der mathematischen Wissenschaften, 5 (2): 539–776
  In English: Pauli, W. (1981) [1921]. Theory of Relativity. Vol. 165. Dover Publications. ISBN 0-486-64152-X. {{cite book}}: |journal= ignored (help)
• Pais, Abraham (1982), Subtle is the Lord: The Science and the Life of Albert Einstein, New York: Oxford University Press, ISBN 0-19-520438-7
• Penrose, R.; Rindler W. (1984), Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields, Cambridge University Press, ISBN 0521337070{{citation}}: CS1 maint: multiple names: authors list (link)
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• Rosenfeld, B.A. (1988). A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. New York: Springer. ISBN 1441986804.
• Rothe, H. (1916). "Systeme geometrischer Analyse". Encyclopädie der mathematischen Wissenschaften. 3.1.1: 1282–1425.
• Schottenloher, M. (2008). A Mathematical Introduction to Conformal Field Theory. Springer. ISBN 3540706909.
• Silberstein, L. (1914). The Theory of Relativity. London: Macmillan.
• Sobczyk, G. (1995). "The Hyperbolic Number Plane". The College Mathematics Journal. 26 (4): 268–280. doi:10.2307/2687027.
• Sommerville, D. M. L. Y. (1911). Bibliography of non-Euclidean geometry. London.{{cite book}}: CS1 maint: location missing publisher (link)
• Synge, J. L. (1956), Relativity: The Special Theory, North Holland
• Synge, J.L. (1972). "Quaternions, Lorentz transformations, and the Conway–Dirac–Eddington matrices". Communications of the Dublin Institute for Advanced Studies. 21.
• Walter, Scott (1999a). "Minkowski, mathematicians, and the mathematical theory of relativity". In H. Goenner; J. Renn; J. Ritter; T. Sauer (eds.). The Expanding Worlds of General Relativity. Vol. 7. Birkhäuser. pp. 45–86. ISBN 978-0-8176-4060-6. {{cite book}}: |journal= ignored (help)
• Walter, Scott (1999b). "The non-Euclidean style of Minkowskian relativity". In J. Gray (ed.). The Symbolic Universe: Geometry and Physics. Oxford: University Press. pp. 91–127. {{cite book}}: External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help)
• Walter, Scott (2012). "Figures of light in the early history of relativity". In Rowe D., Sauer T., Walter S. (ed.). Beyond Einstein. Vol. 14. Birkhäuser. doi:10.1007/978-1-4939-7708-6_1. ISBN 978-1-4939-7708-6. {{cite book}}: |journal= ignored (help); External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help)CS1 maint: multiple names: editors list (link)

External links

• Mathpages: 1.4 The Relativity of Light