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 ${\displaystyle -x_{0}^{2}y_{0}^{2}+\cdots +x_{n}^{2}y_{n}^{2}}$.

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. eu 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 ${\displaystyle x,y,z}$ and time ${\displaystyle t}$ of an event, relative to a particular inertial frame of reference (the "rest system"), and the coordinates ${\displaystyle x',y',z'}$ and ${\displaystyle t'}$ of the same event relative to another coordinate system moving in the positive x-direction at a constant speed ${\displaystyle v}$, relative to the rest system.

Overview

Lorentz transformation via quadratic forms, Weierstrass coordinates, and Cayley absolute

The general quadratic form ${\displaystyle q(x)}$ with coefficients of a symmetric matrix ${\displaystyle \mathbf {A} }$, the associated bilinear form ${\displaystyle b(x,y)}$, and the linear transformations of ${\displaystyle q(x)}$ and ${\displaystyle b(x,y)}$ into ${\displaystyle q(x')}$ and ${\displaystyle b(x',y')}$ using the transformation matrix ${\displaystyle \mathbf {g} }$, 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 ${\displaystyle n=1}$ is the binary quadratic form introduced by Lagrange (1773) and Gauss (1798/1801), ${\displaystyle n=2}$ is the ternary quadratic form introduced by Gauss (1798/1801), ${\displaystyle n=3}$ is the quaternary quadratic form etc.

The Lorentz transformation follows from (Q1) by setting ${\displaystyle \mathbf {A} =\mathbf {A} '=\operatorname {diag} (-1,1,\dots ,1)}$ and ${\displaystyle \det \mathbf {g} =1}$. It forms an indefinite orthogonal group called the Lorentz group SO(n, 1), the quadratic form ${\displaystyle q(x)}$ becomes the Lorentz interval in terms of an indefinite quadratic form, and the associated bilinear form ${\displaystyle b(x)}$ 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}^{2}y_{0}^{2}+\cdots +x_{n}^{2}y_{n}^{2}&=-x_{0}^{\prime 2}y_{0}^{\prime 2}+\cdots +x_{n}^{\prime 2}y_{n}^{\prime 2}\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=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}}}

(1)

Such Lorentz transformations (1) for various dimensions were 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 in terms of Weierstrass coordinates of the hyperboloid model satisfying the relation ${\displaystyle -x_{0}^{2}+\cdots +x_{n}^{2}=-1}$.[M 1][4][5] Hyperbolic motions have also been studied by Klein (1871–1897) or Lindemann (1890/91) in terms of the Cayley–Klein metric and projective geometry using the "absolute" form ${\displaystyle -x_{0}^{2}+\cdots +x_{n}^{2}=0}$. In addition, infinitesimal transformations related to the Lie algebra of the group of hyperbolic motions leaving invariant the unit sphere ${\displaystyle x_{1}^{2}+\cdots +x_{n}^{2}-1=0}$ were given by Lie (1885-1893) and Werner (1889) and in terms of Weierstrass coordinates ${\displaystyle -x_{0}^{2}+\cdots +x_{n}^{2}=-1}$ by Killing (1888-1897).

By using the imaginary quantities ${\displaystyle [{\mathfrak {x}}_{0},\ {\mathfrak {x}}'_{0}]=\left[ix_{0},\ ix_{0}^{\prime }\right]}$, Lorentz transformation (1) assumes the form of an orthogonal transformation, the Lorentz interval becomes the Euclidean norm, and the Minkowski inner product becomes the dot product:[6]

{\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}^{2}{\mathfrak {y}}_{0}^{2}+x_{1}^{2}y_{1}^{2}+\cdots +x_{n}^{2}y_{n}^{2}&={\mathfrak {x}}_{0}^{\prime 2}{\mathfrak {y}}_{0}^{\prime 2}+x_{1}^{\prime 2}y_{1}^{\prime 2}+\cdots +x_{n}^{\prime 2}y_{n}^{\prime 2}\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}}}

(1b)

The cases ${\displaystyle n=1,2,3,4}$ of this quadratic form and its transformation was discussed by Euler (1771), its interpretation as a Lorentz transformation with ${\displaystyle n=3}$ using imaginary quantities was given by Minkowski (1907).

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 ${\displaystyle n=1}$ in (1):

{\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}}}

(2a)

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

(2b)

which can also be expressed in terms of exponential functions

{\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}^{\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}}}

(2c)

All hyperbolic relations (a,b,c,d,e,f) on the right of (2b) were given by Lambert (1768–1770). The Lorentz transformations (2b or 2c) 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.

By using the imaginary quantities ${\displaystyle [{\mathfrak {x}}_{0},\ {\mathfrak {x}}'_{0},\ \phi ]=\left[ix_{0},\ ix_{0}^{\prime },\ i\eta \right]}$, Lorentz transformation (2b) assumes the form of an orthogonal transformation representing spatial rotations:

{\displaystyle {\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}}}

(2d)

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

By inserting coordinates ${\displaystyle \scriptstyle [u_{x},\ u_{y},\ 1]=\left[{\frac {x_{1}}{x_{0}}},\ {\frac {x_{2}}{x_{0}}},\ {\frac {x_{0}}{x_{0}}}\right]}$ in (2b) in terms of the Beltrami–Klein model[7] of hyperbolic geometry inside the unit circle ${\displaystyle u_{x}^{2}+u_{y}^{2}=1}$, the corresponding Lorentz transformations in (2b) obtain the form:

{\displaystyle {\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}}}

(2e)

These Lorentz transformations were given by Escherich (1874) (on the left) and Schur (1900/02) (on the right).

By setting ${\displaystyle u^{2}=u_{x}^{2}+u_{y}^{2}}$ and ${\displaystyle \tan \alpha ={\tfrac {u_{y}}{u_{x}}}}$, the resulting Lorentz transformation (on the left)[8] can be seen as equivalent to the hyperbolic law of cosines (on the right):[R 1][9]

{\displaystyle {\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.}}

(2f)

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 ${\displaystyle c}$ as the speed of light, and a parameter ${\displaystyle v}$ as the relative velocity between two inertial reference frames. In particular, the hyperbolic angle ${\displaystyle \eta }$ in (2b) can be interpreted as the velocity related rapidity ${\displaystyle \eta =\operatorname {atanh} \beta }$ with ${\displaystyle \beta =v/c}$, so that ${\displaystyle \gamma =\cosh \eta }$ is the Lorentz factor, ${\displaystyle \beta \gamma =\sinh \eta }$ the proper velocity, ${\displaystyle v=c\tanh \eta }$ the relative velocity of two inertial frames, ${\displaystyle u'=c\tanh \zeta }$ the velocity of another object, ${\displaystyle u=c\tanh \left(\eta +\zeta \right)}$ the velocity-addition formula, thus (2b) 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}}}

(3a)

Or in four dimensions and by setting ${\displaystyle x_{0}=ct}$, ${\displaystyle x_{1}=x}$, ${\displaystyle x_{2}=y}$ and adding an unchanged ${\displaystyle z}$ the familiar form follows

{\displaystyle {\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}}}

(3b)

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.[10] 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.[11] 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 ${\displaystyle {\scriptstyle [u_{x},\ u_{y},\ c^{2}]=\left[{\frac {x_{1}}{x_{0}}},\ {\frac {x_{2}}{x_{0}}},\ c^{2}{\frac {x_{0}}{x_{0}}}\right]}}$ in (2b) or (3a), produces the Lorentz transformation of velocities (or velocity addition formula) in analogy to Beltrami coordinates of (2e):

{\displaystyle {\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}}}

(3c)

or more generally using the hyperbolic law of cosines in terms of (2f):[8][R 1][9]

{\displaystyle {\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.}}

(3d)

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 transformations for arbitrary directions in line with (1) can be given as:[12]

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

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}}}

(3e)

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

Lorentz transformation via spherical wave transformation

A general sphere transformation preserving the quadratic form ${\displaystyle \lambda \left(-x_{0}^{2}+\dots +x_{n}^{2}\right)}$ 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 ${\displaystyle x_{0}=iR}$ which gives ${\displaystyle \lambda \left(x_{0}^{2}+\dots +x_{n}^{2}\right)}$ (conformal transformation), or by using a real radius coordinate ${\displaystyle x_{0}=R}$ which gives ${\displaystyle \lambda \left(-x_{0}^{2}+\dots +x_{n}^{2}\right)}$ (spherical wave transformation). This group was studied by Lie (1871) and others in terms of contact transformations, in which ${\displaystyle x_{0}}$ is related to the radius ${\displaystyle R}$.

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 ${\displaystyle \lambda =1}$. More generally, Con(p,q) is isomorphic to SO(p+1,q+1) and contains SO(p,q) as subgroup.[13] 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).[14][15] 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).

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 ${\displaystyle x^{2}+y^{2}+z^{2}-R^{2}}$ with ${\displaystyle R}$ 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.[16][17]

Lorentz transformation via Cayley–Hermite transformation

General transformations of arbitrary quadratic forms ${\displaystyle q}$ into themselves can also be given using independent parameters based on the Cayley transform ${\displaystyle \mathbf {x} =(\mathbf {I} -\mathbf {T} )^{-1}\cdot (\mathbf {I} +\mathbf {T} )\cdot \mathbf {x} }$ in the form:[18][19]

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

(Q2)

where ${\displaystyle \mathbf {A} }$ is, as above, a symmetric matrix defining the quadratic form (there is no primed ${\displaystyle \mathbf {A} '}$ because the coefficients are assumed to be the same on both sides), ${\displaystyle \mathbf {I} }$ the identity matrix, and ${\displaystyle \mathbf {T} }$ 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 ${\displaystyle \mathbf {A} =\operatorname {diag} (1,1,1)}$ gives the spatial rotation in terms of Euler-Rodrigues parameters 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 this formalism.[R 2][R 3][20] The Lorentz transformation in 2 dimensions follows from (Q2) by setting ${\displaystyle \mathbf {A} =\operatorname {diag} (-1,1)}$:

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

(4a)

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

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

(4b)

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

{\displaystyle {\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}}}}

(4c)

Equations containing the Lorentz transformations (4a, 4b, 4c) as special cases were given by Cayley (1855). In relativity, equations similar to (4b, 4c) were first employed by Borel (1913) to represent Lorentz transformations.

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

Lorentz transformations can also be formulated by using Cayley–Klein parameter ${\displaystyle \alpha \beta \gamma \delta }$, which were used by Helmholtz (1866/67), Cayley (1879) and Klein (1884) to connect Möbius transformations and rotations. There is also a close relation between Cayley–Klein parameters for ordinary 3d rotations and the corresponding Euler-Rodrigues parameters as discussed above:[M 2]

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

In modern publications, the Cayley–Klein parameters are related to a spin-matrix ${\displaystyle \mathbf {D} }$, 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 hyperbolic motions, the Hermitian matrix ${\displaystyle \mathbf {u} }$ associated with these transformations produces an invariant determinant ${\displaystyle x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}}$ 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".[21] 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:[22][21][23][24]

{\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}}}

(5a)

or expressing ${\displaystyle \mathbf {u} '}$ in terms of ${\displaystyle x_{0}^{\prime }\dots }$ and ${\displaystyle x_{0}\dots }$ it follows:[25]

{\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}}}

(5b)

In the case of three dimensions it simplifies to:[26][24]

{\displaystyle {\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}}}

(5c)

thus

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

(5d)

Lorentz Transformation (5d) was given by Gauss around 1800 (posthumously published 1863). The general transformation ${\displaystyle \mathbf {u} '}$ in (5c) was used in the theory of binary quadratic forms given by Lagrange (1773) and Gauss (1798/1801), while ${\displaystyle \mathbf {u} '}$ in (5a) was given by Cayley (1854) and Klein (1884) in relation to surfaces of second degree. The adaptation of (5a) to hyperbolic motions by which they become Lorentz transformations was provided by Klein (1889/90, 1896/97), Fricke & Klein (1897) and Hausdorff (1899). In relativity, (5a) 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 ${\displaystyle x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}}$ and one purely imaginary part ${\displaystyle ix_{0}}$ (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):[27][28]

{\displaystyle \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}}}

(6)

Cayley (1854, 1855) derived quaternion transformations leaving invariant the sum of squares ${\displaystyle x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{2}^{2}}$. Cox (1882/83) discussed the Lorentz interval in terms of Weierstrass coordinates ${\displaystyle x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{2}^{2}=1}$ in the course of adapting Clifford's biquaternions ${\displaystyle a+\omega b}$ to hyperbolic geometry (${\displaystyle \omega ^{2}=-1}$ for hyperbolic geometry, ${\displaystyle \omega ^{2}=1}$ elliptic, ${\displaystyle \omega ^{2}=0}$ 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 ${\displaystyle x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{2}^{2}=0}$ and adapted Clifford's biquaternions to hyperbolic geometry similar to Cox by using all three values of ${\displaystyle \omega ^{2}}$. 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 ${\displaystyle \varepsilon ^{2}=1}$, which also allows to formulate the Lorentz transformations:[29][30]

{\displaystyle {\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}}}

(7)

After the trigonometric expression ${\displaystyle e^{ix}=\cos x+i\sin x}$ (Euler's formula) was given by Euler (1748) and the hyperbolic analogue ${\displaystyle e^{\varepsilon \eta }}$ by Cockle (1848) in the framework of tessarines, it was shown by Cox (1882/83) that one can identify ${\displaystyle ww^{\prime -1}=e^{\varepsilon \eta }}$ with associative quaternion multiplication. Here, ${\displaystyle e^{\varepsilon \eta }}$ is the hyperbolic versor with ${\displaystyle \varepsilon ^{2}=1}$, the elliptic one follows with ${\displaystyle \varepsilon ^{2}=-1}$, and parabolic with ${\displaystyle \varepsilon ^{2}=0}$ (this should not be confused with the expression ${\displaystyle \omega ^{2}}$ in Clifford's biquaternions also used by Cox, in which ${\displaystyle \omega ^{2}=-1}$ is hyperbolic). The hyperbolic versor was also discussed by Macfarlane (1892, 1894, 1900) in terms of hyperbolic quaternions. The expression ${\displaystyle \varepsilon ^{2}=1}$ for hyperbolic motions (and ${\displaystyle \varepsilon ^{2}=-1}$ for elliptic, ${\displaystyle \varepsilon ^{2}=0}$ 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 ${\displaystyle a}$ of Clifford numbers one can transform the following general quadratic form into itself, in which the individual values of ${\displaystyle i_{1}^{2},i_{2}^{2},\dots }$ can be set to +1 or -1 at will:[31][32]

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

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

Mathematics of the 19th century

Historical formulas for Lorentz boosts

A summary of historical Lorentz boost formulas consistent with (2a, 2b, 2c):

 Escherich (1874) ${\displaystyle {\begin{matrix}x={\frac {\sinh {\frac {a}{k}}+x'\cosh {\frac {a}{k}}}{\cosh {\frac {a}{k}}+x'\sinh {\frac {a}{k}}}},\quad y={\frac {y'}{\cosh {\frac {a}{k}}+x'\sinh {\frac {a}{k}}}}\\\left\{{\text{Lorentz boost with}}\ {\scriptstyle (x,\ y,\ x',\ y')=\left({\frac {x_{1}}{x_{0}}},\ {\frac {x_{2}}{x_{0}}},\ {\frac {x_{1}^{\prime }}{x_{0}^{\prime }}},\ {\frac {x_{2}^{\prime }}{x_{0}^{\prime }}}\right)}\right\}\end{matrix}}}$ Cox (1881/82) {\displaystyle {\begin{aligned}X&=x\cosh p-z\sinh p\\Z&=-x\sinh p+z\cosh p\end{aligned}}} and {\displaystyle {\begin{aligned}x&=X\cosh p+Z\sinh p\\z&=X\sinh p+Z\cosh p\end{aligned}}} Laguerre (1882) ${\displaystyle D'={\frac {D\left(1+\alpha ^{2}\right)-2\alpha R}{1-\alpha ^{2}}},\quad R'={\frac {2\alpha D-R\left(1+\alpha ^{2}\right)}{1-\alpha ^{2}}}}$ Cox (1882/83) ${\displaystyle {\begin{matrix}QP^{-1}=\cosh \theta +\iota \sinh \theta \\QP^{-1}=e^{\iota \theta }\\\left\{{\text{Lorentz boost with}}\ (Q,P)=\left(x_{1}^{\prime }+\iota x_{0}^{\prime },\ x_{1}+\iota x_{0}\right)\right\}\end{matrix}}\left(\iota ^{2}=1\right)}$ Darboux (1887) {\displaystyle {\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}}} Lindemann (1890/91) {\displaystyle {\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}}} Gérard (1892) {\displaystyle \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}}} Killing (1893) ${\displaystyle y_{0}=x_{0}\operatorname {Ch} a+x_{1}\operatorname {Sh} a,\quad y_{1}=x_{0}\operatorname {Sh} a+x_{1}\operatorname {Ch} a,\quad y_{2}=x_{2}}$ Whitehead (1897/98) {\displaystyle {\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}}} Killing (1897/98) ${\displaystyle {\begin{matrix}\xi '={\frac {\xi \operatorname {Ch} {\frac {\mu }{l}}+l\operatorname {Sh} {\frac {\mu }{l}}}{{\frac {\xi }{l}}\operatorname {Sh} {\frac {\mu }{l}}+\operatorname {Ch} {\frac {\mu }{l}}}},\ \eta '={\frac {\eta }{{\frac {\xi }{l}}\operatorname {Sh} {\frac {\mu }{l}}+\operatorname {Ch} {\frac {\mu }{l}}}}\\\hline {\frac {u}{p}}=\xi ,\ {\frac {v}{p}}=\eta \\\hline p'=p\operatorname {Ch} {\frac {\mu }{l}}+{\frac {u}{l}}\operatorname {Sh} {\frac {\mu }{l}},\quad u'=pl\operatorname {Sh} {\frac {\mu }{l}}+u\operatorname {Ch} {\frac {\mu }{l}},\quad v'=v\\{\text{or}}\\p'=p\operatorname {Ch} {\frac {\nu }{l}}+{\frac {v}{l}}\operatorname {Sh} {\frac {\nu }{l}},\quad u'=u,\quad v'=pl\operatorname {Sh} {\frac {\nu }{l}}+v\operatorname {Ch} {\frac {\nu }{l}}\end{matrix}}}$ Schur (1900/02) ${\displaystyle {\begin{matrix}x'={\frac {x-a}{1-{\mathfrak {k}}ax}},\ y'={\frac {y{\sqrt {1-{\mathfrak {k}}a^{2}}}}{1-{\mathfrak {k}}ax}}\\\left\{{\text{Lorentz boost with}}\ {\scriptstyle \left[x,\ y,\ x',\ y',\ a\right]=\left[{\frac {x_{1}}{x_{0}}},\ {\frac {x_{2}}{x_{0}}},\ {\frac {x_{1}^{\prime }}{x_{0}^{\prime }}},\ {\frac {x_{2}^{\prime }}{x_{0}^{\prime }}},\ \tanh \eta ={\frac {\sinh \eta }{\cosh \eta }}\right]},\ {\mathfrak {k}}=1\right\}\end{matrix}}}$ Woods (1903/05) ${\displaystyle {\begin{matrix}x_{1}^{\prime }=x_{1}\cos kl+x_{0}{\frac {\sin kl}{k}},\quad x_{2}^{\prime }=x_{2},\quad x_{2}^{\prime }=x_{3},\quad x_{0}^{\prime }=-x_{1}k\sin kl+x_{0}\cos kl\\\left\{{\text{Lorentz boost with}}\ k={\sqrt {-1}}\right\}\end{matrix}}}$ {\displaystyle {\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}}} Liebmann (1904/05) ${\displaystyle x_{1}^{\prime }=x'\operatorname {ch} a+p'\operatorname {sh} a,\quad y_{1}^{\prime }=y',\quad p_{1}^{\prime }=x'\operatorname {sh} a+p'\operatorname {ch} a}$

Lambert (1768–1770) – hyperbolic functions

After Vincenzo Riccati introduced hyperbolic functions in 1757,[M 3] Johann Heinrich Lambert (read 1767, published 1768) gave the following relations, in which ${\displaystyle tang\ \phi }$ or abbreviated ${\displaystyle t\phi }$ was equated by Lambert to the tangens hyperbolicus ${\displaystyle \scriptstyle {\frac {e^{u}-e^{-u}}{e^{u}+e^{-u}}}}$ of a variable ${\displaystyle u}$, or in modern notation ${\displaystyle \tan \phi =\tanh u}$:[M 4][33]

{\displaystyle \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:[M 5]

{\displaystyle {\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 (2b). In addition, by setting ${\displaystyle tang\ \phi ={\tfrac {v}{c}}}$, (c) becomes the relative velocity between two frames, (d) the Lorentz factor, (e) the proper velocity, (f) or (g) the relativistic velocity addition formula for collinear velocities in (3a).

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[M 6]

{\displaystyle {\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[M 7]

{\displaystyle {\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 ${\displaystyle A,B,C,D,E,F,G,H,I}$ were related by Euler to four parameter ${\displaystyle p,q,r,s}$, which where rediscovered by Olinde Rodrigues (1840) who related them to rotation angles[M 8] now called Euler–Rodrigues parameters:[M 9]

{\displaystyle {\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[M 10]

{\displaystyle {\begin{matrix}V^{2}+X^{2}+Y^{2}+Z^{2}=v^{2}+x^{2}+y^{2}+z^{2}\\\hline {\begin{aligned}X&=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 (2d) by making one of the variables imaginary.

Gauss (1798–1801)

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 discussed by Lagrange (1773).[M 11]

{\displaystyle {\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)${\displaystyle (n=1)}$. 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 ${\displaystyle \alpha \beta \gamma \delta }$:[M 12]

{\displaystyle {\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)${\displaystyle (n=1)}$. As pointed out by Gauss, ${\displaystyle F}$ and ${\displaystyle F'}$ are called "proper equivalent" if ${\displaystyle \alpha \delta -\beta \gamma =1}$, so that ${\displaystyle F}$ is contained in ${\displaystyle F'}$ as well as ${\displaystyle F'}$ is contained in ${\displaystyle F}$. In addition, if another form ${\displaystyle F''}$ is contained by the same procedure in ${\displaystyle F'}$ it is also contained in ${\displaystyle F}$ and so forth.[M 13]

 The Lorentz interval ${\displaystyle -x_{0}^{2}+x_{1}^{2}}$ and the Lorentz transformation (1)${\displaystyle (n=1)}$ are a special case of the binary quadratic form of Lagrange and Gauss by setting ${\displaystyle (a,b,c)=(a',b',c')=(1,0,-1)}$. Alternatively, the transformation of coefficients ${\displaystyle (a,b,c)}$ corresponds to Lorentz transformation ${\displaystyle \mathbf {u} '}$ in (5c) by setting {\displaystyle {\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}}}.