# 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 ${\displaystyle -x_{0}^{2}+\cdots +x_{n}^{2}}$ and the Minkowski inner product ${\displaystyle -x_{0}y_{0}+\cdots +x_{n}y_{n}}$.

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

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

## Overview

### Most general Lorentz transformations

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

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

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

If xi, x′i in (1a) are interpreted as homogeneous coordinates, then the corresponding inhomogenous coordinates us, u′s follow by

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

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

{\displaystyle {\begin{matrix}{\begin{matrix}-x_{0}^{2}+\cdots +x_{n}^{2}=-x_{0}^{\prime 2}+\dots +x_{n}^{\prime 2}&\rightarrow &{\begin{aligned}-1+u_{1}^{2}+\cdots +u_{n}^{2}&={\scriptstyle {\frac {-1+u_{1}^{\prime 2}+\cdots +u_{n}^{\prime 2}}{\left(g_{00}+g_{01}u_{1}^{\prime }+\dots +g_{0n}u_{n}^{\prime }\right)^{2}}}}\\{\scriptstyle {\frac {-1+u_{1}^{2}+\cdots +u_{n}^{2}}{\left(g_{00}-g_{10}u_{1}-\dots -g_{n0}u_{n}\right)^{2}}}}&=-1+u_{1}^{\prime 2}+\cdots +u_{n}^{\prime 2}\end{aligned}}\\\hline -x_{0}^{2}+\cdots +x_{n}^{2}=-x_{0}^{\prime 2}+\dots +x_{n}^{\prime 2}=0&\rightarrow &-1+u_{1}^{2}+\cdots +u_{n}^{2}=-1+u_{1}^{\prime 2}+\cdots +u_{n}^{\prime 2}=0\end{matrix}}\\\hline {\begin{aligned}u_{s}&={\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 for various dimensions were used by Gauss (1818), Jacobi (1827–1833), Lebesgue (1837), Bour (1856), Somov (1863), Hill (1882), Callandreau (1885) in order to simplify computations of elliptic functions and integrals, by Picard (1882-1884) in relation to Hermitian quadratic forms, or by Woods (1901, 1903) in terms of the Beltrami–Klein model of hyperbolic geometry. In addition, infinitesimal transformations in terms of the Lie algebra of the group of hyperbolic motions leaving invariant the unit sphere ${\displaystyle -1+u_{1}^{\prime 2}+\cdots +u_{n}^{\prime 2}=0}$ 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:

### Lorentz transformation via imaginary orthogonal transformation

By using the imaginary quantities ${\displaystyle [{\mathfrak {x}}_{0},\ {\mathfrak {x}}'_{0}]=\left[ix_{0},\ ix_{0}^{\prime }\right]}$ in x as well as ${\displaystyle [{\mathfrak {g}}_{0s},\ {\mathfrak {g}}_{s0}]=\left[ig_{0s},\ ig_{s0}\right]}$ (s=1,2...n) in g, the Lorentz transformation (1a) assumes the form of an orthogonal transformation of Euclidean space forming the orthogonal group O(n) if det g=±1 or the special orthogonal group SO(n) if det g=+1, the Lorentz interval becomes the Euclidean norm, and the Minkowski inner product becomes the dot product:[9]

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

A well known example of this orthogonal transformation is spatial rotation in terms of trigonometric functions, which can be used as Lorentz transformation by using imaginary quantities as well as hyperbolic functions:

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

(2b)

or in exponential form using Euler's formula e=cos(φ)+i·sin(φ):

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

(2c)

Defining ${\displaystyle [{\mathfrak {x}}_{0},\ {\mathfrak {x}}'_{0},\ \phi ]}$ as real, spatial rotation in the form (2b-1) was introduced by Euler (1771) and in the form (2c-1) by Wessel (1799). The interpretation of (2b) as Lorentz boost (i.e. Lorentz transformation without spatial rotation) in which ${\displaystyle [{\mathfrak {x}}_{0},\ {\mathfrak {x}}'_{0},\ \phi ]}$ correspond to the imaginary quantities ${\displaystyle [ix_{0},\ ix'_{0},\ i\eta ]}$ was given by Minkowski (1907) and Sommerfeld (1909). As shown in the next section using hyperbolic functions, (2b) becomes (3b) while (2c) becomes (3c).

### 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):

{\displaystyle {\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 g00=g11=cosh(η) and g01=g10=sinh(η), with η as the hyperbolic angle. Thus by adding an unchanged x2-axis, a Lorentz boost or hyperbolic rotation for n=2 (being the same as a rotation around an imaginary angle iη=φ in (2b) or a translation in the hyperbolic plane in terms of the hyperboloid model) is given by

{\displaystyle {\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)

or in exponential form as squeeze mappings in analogy to Euler's formula in (2c):[10]

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

(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, see § Historical formulas for Lorentz boosts) were given by Cox (1882), Lindemann (1890/91), Gérard (1892), Killing (1893, 1897/98), Whitehead (1897/98), Woods (1903/05) and Liebmann (1904/05) in terms of Weierstrass coordinates of the hyperboloid model. Lorentz transformations (3c-1) were given by Lindemann (1890/91) and Herglotz (1909), while formulas equivalent to (3c-2) by Klein (1871).

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

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

(3d)

These Lorentz transformations were given by Escherich (1874) and Killing (1898) (on the left), as well as Beltrami (1868) and Schur (1885/86, 1900/02) (on the right) in terms of Beltrami coordinates[11] of hyperbolic geometry. By using the scalar product of [u1, u2], the resulting Lorentz transformation can be seen as equivalent to the hyperbolic law of cosines:[12][R 1][13]

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

(3e)

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 η in (3b) can be interpreted as the velocity related rapidity η=atanh(β) with β=v/c, so that γ=cosh(η) is the Lorentz factor, βγ=sinh(η) the proper velocity, v=c·tanh(η) the relative velocity of two inertial frames, u′=c·tanh(ζ) the velocity of another object, u=c·tanh(η+ζ) the velocity-addition formula, thus (3b) becomes:

{\displaystyle {\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 x0=ct, x1=x, x2=y and adding an unchanged z the familiar form follows

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

(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.[14] Eventually, Einstein (1905) showed in his development of special relativity that the transformations follow from the principle of relativity and constant light speed alone by modifying the traditional concepts of space and time, without requiring a mechanical aether in contradistinction to Lorentz and Poincaré.[15] Minkowski (1907–1908) used them to argue that space and time are inseparably connected as spacetime. Minkowski (1907–1908) and Varićak (1910) showed the relation to imaginary and hyperbolic functions. Important contributions to the mathematical understanding of the Lorentz transformation were also made by other authors such as Herglotz (1909/10), Ignatowski (1910), Noether (1910) and Klein (1910), Borel (1913–14).

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

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

or in vector notation

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

(4c)

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

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

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

(4d)

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

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

(4e)

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

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

(4f)

The velocity addition formulas were given by Einstein (1905) and Poincaré (1905/06), the aberration formula for cos(α) by Einstein (1905), while the relations to the spherical and hyperbolic law of cosines were given by Sommerfeld (1909) and Varićak (1910). These formulas resemble the equations of an ellipse of eccentricity v/c, eccentric anomaly α' and true anomaly α, first geometrically formulated by Kepler (1609) and explicitly written down by Euler (1735, 1748), Lagrange (1770) and many others in relation to planetary motions.[18][19]

### Lorentz transformation via trigonometric functions

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

${\displaystyle {\frac {v}{c}}=\tanh \eta =\tan \varphi =\sin \theta =\cos \vartheta }$

This relation was first defined by Varićak (1910).

a) Using ${\displaystyle \sin \theta ={\tfrac {v}{c}}}$ one obtains the relations ${\displaystyle \sec \theta =\gamma }$ and ${\displaystyle \tan \theta =\beta \gamma }$, and the Lorentz boost takes the form:[20]

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

(5a)

This Lorentz transformation was first used by Scheffers (1899) in terms of contact transformations in the plane (Laguerre geometry), by Gruner (1921) while developing Loedel diagrams, and by Vladimir Karapetoff in the 1920s.

b) Using ${\displaystyle \cos \vartheta ={\tfrac {v}{c}}}$ one obtains the relations ${\displaystyle \csc \vartheta =\gamma }$ and ${\displaystyle \cot \vartheta =\beta \gamma }$, and the Lorentz boost takes the form:[20]

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

(5b)

This Lorentz transformation was first used by Gruner (1921) while developing Loedel diagrams.

### Lorentz transformation via conformal, spherical wave, and Laguerre transformation

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

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

One can switch between two representations of this group by using an imaginary sphere radius coordinate x0=iR with the interval ${\displaystyle dx_{0}^{2}+\dots +dx_{n}^{2}}$ related to conformal transformations, or by using a real radius coordinate x0=R with the interval ${\displaystyle -dx_{0}^{2}+\dots +dx_{n}^{2}}$ related to spherical wave transformations. Both representations were studied by Lie (1871) and others. It was shown by Bateman & Cunningham (1909–1910), that the group Con(1,3) is the most general one leaving invariant the equations of Maxwell's electrodynamics.

It turns out that Con(1,3) is isomorphic to the special orthogonal group SO(2,4), and contains the Lorentz group SO(1,3) as a subgroup by setting λ=1. More generally, Con(q,p) is isomorphic to SO(q+1,p+1) and contains SO(q,p) as subgroup.[21] This implies that Con(0,p) is isomorphic to the Lorentz group of arbitrary dimensions SO(1,p+1). Consequently, the conformal group in the plane Con(0,2) – known as the group of Möbius transformations – is isomorphic to the Lorentz group SO(1,3).[22][23] This can be seen using tetracyclical coordinates satisfying the form ${\displaystyle -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0}$, which were discussed by Pockels (1891), Klein (1893), Bôcher (1894). The relation between Con(1,3) and the Lorentz group was noted by Bateman & Cunningham (1909–1910) and others.

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 x2+y2+z2-R2 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 group isomorphism between the Laguerre group and Lorentz group was pointed out by Bateman (1910), Cartan (1912, 1915/55), Poincaré (1912/21) and others.[24][25]

### Lorentz transformation via Cayley–Hermite transformation

The general transformation (Q1) of any quadratic form into itself can also be given using arbitrary parameters based on the Cayley transform (I-T)−1·(I+T), where I is the identity matrix, T an arbitrary antisymmetric matrix, and by adding A as symmetric matrix defining the quadratic form (there is no primed A' because the coefficients are assumed to be the same on both sides):[26][27]

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

(Q2)

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 A=diag(1,1,1) gives an orthogonal transformation which can be used to describe spatial rotations corresponding to the Euler-Rodrigues parameters [a,b,c,d] discovered by Euler (1771) and Rodrigues (1840), which can be interpreted as the coefficients of quaternions. Setting d=1, the equations have the form:

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

(Q3)

Also the Lorentz interval and the general Lorentz transformation in any dimension can be produced by the Cayley–Hermite formalism.[R 2][R 3][28][29] For instance, Lorentz transformation (1a) with n=1 follows from (Q2) with:

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

(6a)

which becomes Lorentz boost (4a or 4b) by setting ${\displaystyle {\tfrac {2a}{1+a^{2}}}={\tfrac {v}{c}}}$ with c as speed of light and v as velocity between inertial frames. Lorentz transformation (1a) with n=2 is given by:

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

(6b)

or using n=3:

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

(6c)

Equations containing the Lorentz transformations (6a, 6b, 6c) as special cases were given by Cayley (1855), Lorentz transformation (6a) was given by (up to a sign change) Laguerre (1882), Darboux (1887), and Smith (1887) in relation to Laguerre geometry, and Lorentz transformation (6b) was given by Bachmann (1869). In relativity, equations similar to (6b, 6c) were first employed by Borel (1913) to represent Lorentz transformations.

As described in equation (3c), the Lorentz interval is closely connected to the alternative form ${\displaystyle X_{2}^{2}-X_{1}X_{3}}$,[30] which in terms of the Cayley–Hermite parameters is invariant under the transformation:[M 2]

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

(6d)

This transformation was given by Cayley (1884), even though he didn't relate it to the Lorentz interval but rather to ${\displaystyle x_{0}^{2}+x_{1}^{2}+x_{2}^{2}}$. As shown in the next section in equation (7d), many authors (some before Cayley) expressed the invariance of ${\displaystyle X_{2}^{2}-X_{1}X_{3}}$ and its relation to the Lorentz interval by using the alternative Cayley–Klein parameters and Möbius transformations.

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

The previously mentioned Euler-Rodrigues parameter a,b,c,d (i.e. Cayley-Hermite parameter in equation (Q3) with d=1) are closely related to Cayley–Klein parameter α,β,γ,δ introduced by Helmholtz (1866/67), Cayley (1879) and Klein (1884) to connect Möbius transformations ${\displaystyle {\tfrac {\alpha \zeta +\beta }{\gamma \zeta +\delta }}}$ and rotations:[M 3]

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

thus (Q3) becomes:

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

(Q4)

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

In space, the Möbius/Lorentz transformations can be written as:[32][31][33][34]

{\displaystyle {\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}}}

(7a)

thus:[35]

 {\displaystyle {\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}}}