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 formula_1 and the Minkowski inner product formula_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.
In physics, Lorentz transformations became known at the beginning of the 20th century, when it was discovered that they exhibit the symmetry of Maxwell's equations. Subsequently, they became fundamental to all of physics, because they formed the basis of special relativity in which they exhibit the symmetry of Minkowski spacetime, making the speed of light invariant between different inertial frames. They relate the spacetime coordinates of two arbitrary inertial frames of reference with constant relative speed "v". In one frame, the position of an event is given by "x,y,z" and time "t", while in the other frame the same event has coordinates "x′,y′,z′" and "t′".
Most general Lorentz transformations.
The general quadratic form "q(x)" with coefficients of a symmetric matrix A, the associated bilinear form "b(x,y)", and the linear transformations of "q(x)" and "b(x,y)" into "q(x′)" and "b(x′,y′)" using the transformation matrix g, can be written as
 & =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}'
\quad\left(A_{ij}=A_{ji}\right)\\
\hline \left.\begin{align}x_{i}^{\prime} & =\sum_{j=0}^{n}g_{ij}x_{j}=\mathbf{g}\cdot\mathbf{x}\\
x_{i} & =\sum_{j=0}^{n}g_{ij}^{(-1)}x_{j}^{\prime}=\mathbf{g}^{-1}\cdot\mathbf{x}'
\right|\mathbf{g}^\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}'
in which case "n=1" is the binary quadratic form, "n=2" is the ternary quadratic form, "n=3" is the quaternary quadratic form.
The general Lorentz transformation follows from () 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:
\cdot\mathbf{A} & =\mathbf{g}^{-1}\\
\mathbf{g}^\cdot\mathbf{A}\cdot\mathbf{g} & =\mathbf{A}\\
\mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} & =\mathbf{A}\\
\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\
1\quad & (j=k>0)\\
0\quad & (j\ne k)
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\
1\quad & (i=k>0)\\
0\quad & (i\ne k)
\right.
If formula_3 in () are interpreted as homogeneous coordinates, then the corresponding inhomogenous coordinates formula_4 follow by
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 ()):
\hline -x_{0}^{2}+\cdots+x_{n}^{2}=-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}=0 & \rightarrow & -1+u_{1}^{2}+\cdots+u_{n}^{2}=-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0
\end{matrix}\\
\hline \begin{align}u_{s}^{\prime} & =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\
\left|\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\
1\quad & (j=k>0)\\
0\quad & (j\ne k)
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\
1\quad & (i=k>0)\\
0\quad & (i\ne k)
\right.
\right.
Lorentz transformation via imaginary orthogonal transformation.
By using the imaginary quantities formula_6 in x as well as formula_7 "(s=1,2...n)" in g, the Lorentz transformation () 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:
\cdot\mathbf{x}'
\end{matrix}\left|\begin{align}\sum_{i=0}^{n}g_{ij}g_{ik} & =\left\{ \begin{align}1\quad & (j=k)\\
0\quad & (j\ne k)
\right.\\
\sum_{j=0}^{n}g_{ij}g_{kj} & =\left\{ \begin{align}1\quad & (i=k)\\
0\quad & (i\ne k)
\right.
\right.
A well known example of this orthogonal transformation is spatial rotation in terms of trigonometric functions, which become Lorentz transformations by using an imaginary angle formula_8, so that trigonometric functions become equivalent to hyperbolic functions:
or in exponential form using Euler's formula formula_9:
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 ():
which resembles precisely the relations of hyperbolic functions in terms of hyperbolic angle formula_10. Thus by adding an unchanged formula_11-axis, a Lorentz boost or hyperbolic rotation for "n=2" (being the same as a rotation around an imaginary angle formula_12 in () or a translation in the hyperbolic plane in terms of the hyperboloid model) is given by
 & =\cosh\eta & (d)\\
\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} & =\sinh\eta & (e)\\
\frac{\tanh q\pm\tanh\eta}{1\pm\tanh q\tanh\eta} & =\tanh\left(q\pm\eta\right) & (f)
in which the rapidity can be composed of arbitrary many rapidities formula_13 as per the angle sum laws of hyperbolic sines and cosines, so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between angle sum laws of circular trigonometry and spatial rotations. Alternatively, the hyperbolic angle sum laws "themselves" can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the unit hyperbola:
Finally, Lorentz boost () assumes a simple form by using squeeze mappings in analogy to Euler's formula in ():
In line with equation () one can use coordinates formula_14 inside the unit circle formula_15, thus the corresponding Lorentz transformations () obtain the form:
\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}=0 & \rightarrow & -1+u_{x}^{2}+u_{y}^{2}=-1+u_{x}^{\prime2}+u_{y}^{\prime2}=0
\end{matrix}\\
\hline {\scriptstyle \begin{align}\frac{\sinh\eta}{\cosh\eta} & =\tanh\eta=v\\
}\left|\begin{align} & (a) & & (b) & & (c)\\
u_{1}^{\prime} & =\frac{-\sinh\eta+u_{1}\cosh\eta}{\cosh\eta-u_{1}\sinh\eta} & & =\frac{u_{1}-\tanh\eta}{1-u_{1}\tanh\eta} & & =\frac{u_{1}-v}{1-u_{1}v}\\
u_{2}^{\prime} & =\frac{u_{2}}{\cosh\eta-u_{1}\sinh\eta} & & =\frac{u_{2}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} & & =\frac{u_{2}\sqrt{1-v^{2}}}{1-u_{1}v}\\
u_{1} & =\frac{\sinh\eta+u_{1}^{\prime}\cosh\eta}{\cosh\eta+u_{1}^{\prime}\sinh\eta} & & =\frac{u_{1}^{\prime}+\tanh\eta}{1+u_{1}^{\prime}\tanh\eta} & & =\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v}\\
\right.
By using the scalar product of formula_16, the resulting Lorentz transformation can be seen as equivalent to the hyperbolic law of cosines:
}{1+vu'\cos\alpha'}, & u'\sin\alpha' & =\frac{u\sin\alpha\sqrt{1-v^{2}}}{1-vu\cos\alpha}\\
\Rightarrow & u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(vu'\sin\alpha'\right){}^{2}}}{1+vu'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(vu\sin\alpha\right){}^{2}}}{1-vu\cos\alpha}\\
\Rightarrow & \frac{1}{\sqrt{1-u^{\prime2}}}=\frac{1}{\sqrt{1-v^{2}}}\frac{1}{\sqrt{1-u^{2}}}-\frac{v}{\sqrt{1-v^{2}}}\frac{u}{\sqrt{1-u^{2}}}\cos\alpha & (b)\\
\Rightarrow & \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow & \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha & (a)
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 formula_10 in () can be interpreted as the velocity related rapidity formula_18, so that formula_19 is the Lorentz factor, formula_20 the proper velocity, formula_21 the velocity of another object, formula_22 the velocity-addition formula, thus () becomes:
} & =\gamma & (d)\\
\frac{\beta}{\sqrt{1-\beta^{2}}} & =\beta\gamma & (e)\\
\frac{u'+v}{1+\frac{u'v}{c^{2}}} & =u & (f)
}\right.
Or in four dimensions and by setting formula_23 and adding an unchanged "z" the familiar form follows, using formula_24 as Doppler factor:
\right)\\
x' & =\gamma(x-vt)\\
y' & =y\\
z' & =z
\right|\begin{align}t & =\gamma\left(t'+x\frac{v}{c^{2}}\right)\\
x & =\gamma(x'+vt')\\
y & =y'\\
z & =z'
\end{matrix}\Rightarrow\begin{align}(ct'+x') & =(ct+x)\sqrt{\frac{c+v}{c-v}}\\
In physics, analogous transformations have been introduced by Voigt (1887) and by Lorentz (1892, 1895) who analyzed Maxwell's equations, they were completed by Larmor (1897, 1900) and Lorentz (1899, 1904), and brought into their modern form by Poincaré (1905) who gave the transformation the name of Lorentz. Eventually, Einstein (1905) showed in his development of special relativity that the transformations follow from the principle of relativity and constant light speed alone by modifying the traditional concepts of space and time, without requiring a mechanical aether in contradistinction to Lorentz and Poincaré. Minkowski (1907–1908) used them to argue that space and time are inseparably connected as spacetime. Minkowski (1907–1908) and Varićak (1910) showed the relation to imaginary and hyperbolic functions. Important contributions to the mathematical understanding of the Lorentz transformation were also made by other authors such as Herglotz (1909/10), Ignatowski (1910), Noether (1910) and Klein (1910), Borel (1913–14).
Also Lorentz boosts for arbitrary directions in line with () can be given as:
or in vector notation
Such transformations were formulated by Herglotz (1911) and Silberstein (1911) and others.
In line with equation (), one can substitute formula_26 in () or (), producing the Lorentz transformation of velocities (or velocity addition formula) in analogy to Beltrami coordinates of ():
\hline -c^{2}t^{2}+x^{2}+y^{2}=-c^{2}t^{\prime2}+x^{\prime2}+y^{\prime2}=0 & \rightarrow & -c^{2}+u_{x}^{2}+u_{y}^{2}=-c^{2}+u_{x}^{\prime2}+u_{y}^{\prime2}=0
\end{matrix}\\
\hline {\scriptstyle \begin{align}\frac{\sinh\eta}{\cosh\eta} & =\tanh\eta=\frac{v}{c}\\
}\left|\begin{align}u_{x}^{\prime} & =\frac{-c^{2}\sinh\eta+u_{x}c\cosh\eta}{c\cosh\eta-u_{x}\sinh\eta} & & =\frac{u_{x}-c\tanh\eta}{1-\frac{u_{x}}{c}\tanh\eta} & & =\frac{u_{x}-v}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{y}^{\prime} & =\frac{cu_{y}}{c\cosh\eta-u_{x}\sinh\eta} & & =\frac{u_{y}\sqrt{1-\tanh^{2}\eta}}{1-\frac{u_{x}}{c}\tanh\eta} & & =\frac{u_{y}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{x} & =\frac{c^{2}\sinh\eta+u_{x}^{\prime}c\cosh\eta}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} & & =\frac{u_{x}^{\prime}+c\tanh\eta}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} & & =\frac{u_{x}^{\prime}+v}{1+\frac{v}{c^{2}}u_{x}^{\prime}}\\
\right.
or using trigonometric and hyperbolic identities it becomes the hyperbolic law of cosines in terms of ():
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}\\
\Rightarrow & u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(\frac{vu'\sin\alpha'}{c}\right){}^{2}}}{1+\frac{v}{c^{2}}u'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(\frac{vu\sin\alpha}{c}\right){}^{2}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\Rightarrow & \frac{1}{\sqrt{1-\frac{u^{\prime2}}{c^{2}}}}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}-\frac{v/c}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{u/c}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\cos\alpha\\
\Rightarrow & \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow & \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha
and by further setting "u=u′=c" the relativistic aberration of light follows:
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). 
Lorentz transformation via conformal, spherical wave, and Laguerre transformation.
If one only requires the invariance of the light cone represented by the differential equation formula_27, 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
One can switch between two representations of this group by using an imaginary sphere radius coordinate "x0=iR" with the interval formula_29 related to conformal transformations, or by using a real radius coordinate "x0=R" with the interval formula_30 related to spherical wave transformations in terms of contact transformations preserving circles and spheres. It turns out that Con(1,3) is isomorphic to the special orthogonal group SO(2,4), and contains the Lorentz group SO(1,3) as a subgroup by setting λ=1. More generally, Con(q,p) is isomorphic to SO(q+1,p+1) and contains SO(q,p) as subgroup. 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). This can be seen using tetracyclical coordinates satisfying the form formula_31.
A special case of Lie's geometry of oriented spheres is the Laguerre group, transforming oriented planes and lines into each other. It's generated by the Laguerre inversion leaving invariant formula_32 with "R" as radius, thus the Laguerre group is isomorphic to the Lorentz group.
Lorentz transformation via Cayley–Hermite transformation.
The general transformation () 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):
\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}'
For instance, the choice A=diag(1,1,1) gives an orthogonal transformation which can be used to describe spatial rotations corresponding to the Euler-Rodrigues parameters "[a,b,c,d]" which can be interpreted as the coefficients of quaternions. Setting "d=1", the equations have the form:
\hline x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\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)\\
\end{matrix}\right]\cdot\mathbf{x}\\
\left(\kappa=1+a^{2}+b^{2}+c^{2}\right)
Also the Lorentz interval and the general Lorentz transformation in any dimension can be produced by the Cayley–Hermite formalism. For instance, Lorentz transformation () with "n"=1 follows from () with:
\hline -x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{1-a^{2}}\left[\begin{matrix}1+a^{2} & -2a\\
\end{matrix}\Rightarrow\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}x_{0} & =x_{0}^{\prime}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}}+x_{1}^{\prime}\frac{2\beta_{0}}{1-\beta_{0}^{2}} & = & \frac{x_{0}^{\prime}\left(1+\beta_{0}^{2}\right)+x_{1}^{\prime}2\beta_{0}}{1-\beta_{0}^{2}}\\
x_{1} & =x_{0}^{\prime}\frac{2\beta_{0}}{1-\beta_{0}^{2}}+x_{1}^{\prime}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} & = & \frac{x_{0}^{\prime}2\beta_{0}+x_{1}^{\prime}\left(1+\beta_{0}^{2}\right)}{1-\beta_{0}^{2}}\\
x_{0}^{\prime} & =x_{0}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}}-x_{1}\frac{2\beta_{0}}{1-\beta_{0}^{2}} & = & \frac{x_{0}\left(1+\beta_{0}^{2}\right)-x_{1}2\beta_{0}}{1-\beta_{0}^{2}}\\
\right|{\scriptstyle \begin{align}\frac{2\beta_{0}}{1+\beta_{0}^{2}} & =\beta\\
\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} & =\gamma\\
\frac{2\beta_{0}}{1-\beta_{0}^{2}} & =\beta\gamma
This becomes Lorentz boost ( or ) by setting formula_33, which is equivalent to the relation formula_34 known from Loedel diagrams, thus () can be interpreted as a Lorentz boost from the viewpoint of a "median frame" in which two other inertial frames are moving with equal speed formula_35 in opposite directions.
Furthermore, Lorentz transformation () with "n"=2 is given by:
\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\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)\\
\end{matrix}\right]\cdot\mathbf{x}\\
\left(\kappa=1-a^{2}-b^{2}+c^{2}\right)
or using "n"=3:
\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[{\scriptstyle \begin{align} & 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}\\
}\right]\cdot\mathbf{x}\\
\left(\begin{align}\kappa & =1-a^{2}-b^{2}-c^{2}+d^{2}+e^{2}+f^{2}-p^{2}\\
p & =af+be+cd
\right)
As described in equation (), the Lorentz interval is closely connected to the alternative form formula_36, which in terms of the Cayley–Hermite parameters is invariant under the transformation:
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 () with "d=1") are closely related to Cayley–Klein parameter α,β,γ,δ in order to connect Möbius transformations formula_37 and rotations:
thus () becomes:
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 formula_39 (the overline denotes complex conjugate), and the Möbius transformation of formula_40. When defined in terms of isometries of hyperbolic space (hyperbolic motions), the Hermitian matrix u associated with these Möbius transformations produces an invariant determinant formula_41 identical to the Lorentz interval. Therefore, these transformations were described by John Lighton Synge as being a "factory for the mass production of Lorentz transformations". It also turns out that the related spin group Spin(3, 1) or special linear group SL(2, C) acts as the double cover of the Lorentz group (one Lorentz transformation corresponds to two spin transformations of different sign), while the Möbius group Con(0,2) or projective special linear group PSL(2, C) is isomorphic to both the Lorentz group and the group of isometries of hyperbolic space.
In space, the Möbius/Spin/Lorentz transformations can be written as:
\eta' & =\gamma\xi+\delta\eta
\right.\\
\hline \left.\begin{matrix}\mathbf{u}=\left(\begin{matrix}X_{1} & X_{2}\\
\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}\\
\end{matrix}\right)\\
\end{matrix}\right|\begin{matrix}\mathbf{D}=\left(\begin{matrix}\alpha & \beta\\
\gamma & \delta
\end{matrix}\right)\\
\end{matrix}\\
\hline \mathbf{u}'=\mathbf{D}\cdot\mathbf{u}\cdot\bar{\mathbf{D}}^{\mathrm{T}}=\begin{align}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}\\
\hline \begin{align}X_{3}^{\prime}X_{2}^{\prime}-X_{1}^{\prime}X_{4}^{\prime} & =X_{3}X_{2}-X_{1}X_{4}=0\\
thus:
or in line with equation () one can substitute formula_42 so that the Möbius/Lorentz transformations become related to the unit sphere:
In the plane, the transformations can be written as:
\eta' & =\gamma\xi+\delta\eta
\right.\\
\hline \left.\begin{matrix}\mathbf{u}=\left(\begin{matrix}X_{1} & X_{2}\\
\end{matrix}\right)=\left(\begin{matrix}\xi^{2} & \xi\eta\\
\end{matrix}\right)=\left(\begin{matrix}x_{0}+x_{2} & x_{1}\\
\end{matrix}\right)\\
\end{matrix}\right|\begin{matrix}\mathbf{D}=\left(\begin{matrix}\alpha & \beta\\
\gamma & \delta
\end{matrix}\right)\\
\end{matrix}\\
\hline \mathbf{u}'=\mathbf{D}\cdot\mathbf{u}\cdot\mathbf{D}^{\mathrm{T}}=\begin{align}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\\
\hline \begin{align}X_{2}^{\prime2}-X_{1}^{\prime}X_{3}^{\prime} & =X_{2}^{2}-X_{1}X_{3}=0\\
thus
which includes the special case formula_43 implying formula_44, reducing the transformation to a Lorentz boost in 1+1 dimensions:
\quad\Rightarrow\quad\begin{align}x_{0} & =\frac{x_{0}^{\prime}\left(\alpha^{4}+1\right)+x_{2}^{\prime}\left(\alpha^{4}-1\right)}{2\alpha^{2}}\\
x_{1} & =x_{1}^{\prime}\\
Finally, by using the Lorentz interval related to a hyperboloid, the Möbius/Lorentz transformations can be written
Lorentz transformation via quaternions and hyperbolic numbers.
The Lorentz transformations can also be expressed in terms of biquaternions: A Minkowskian quaternion (or minquat) "q" having one real part and one purely imaginary part is multiplied by biquaternion "a" applied as pre- and postfactor. Using an overline to denote quaternion conjugation and * for complex conjugation, its general form (on the left) and the corresponding boost (on the right) are as follows:
Often connected with quaternionic systems is the hyperbolic number formula_45, which also allows to formulate the Lorentz transformations:
More extended forms of complex and (bi-)quaternionic systems in terms of Clifford algebra can also be used to express the Lorentz transformations. For instance, using a system "a" of Clifford numbers one can transform the following general quadratic form into itself, in which the individual values of formula_46 can be set to +1 or -1 at will, while the Lorentz interval follows if the sign of one formula_47 differs from all others.:
Lorentz transformation via trigonometric functions.
The following general relation connects the speed of light and the relative velocity to hyperbolic and trigonometric functions, where formula_10 is the rapidity in (), formula_49 is equivalent to the Gudermannian function formula_50, and formula_51 is equivalent to the Lobachevskian angle of parallelism formula_52:
a) Using formula_54 one obtains the relations formula_55 and formula_56, and the Lorentz boost takes the form:
x_{2}^{\prime} & =x_{2}\\
x_{0} & =x_{0}^{\prime}\sec\theta+x_{1}^{\prime}\tan\theta & & =\frac{x_{0}^{\prime}+x_{1}^{\prime}\sin\theta}{\cos\theta}\\
x_{1} & =x_{0}^{\prime}\tan\theta+x_{1}^{\prime}\sec\theta & & =\frac{x_{0}^{\prime}\sin\theta+x_{1}^{\prime}}{\cos\theta}\\
\right|{\scriptstyle \begin{align}\tan^{2}\theta-\sec^{2}\theta & =-1\\
\frac{\tan\theta}{\sec\theta} & =\sin\theta\\
\frac{1}{\sqrt{1-\sin^{2}\theta}} & =\sec\theta\\
\frac{\sin\theta}{\sqrt{1-\sin^{2}\theta}} & =\tan\theta
b) Using formula_57 one obtains the relations formula_58 and formula_59, and the Lorentz boost takes the form:
x_{2}^{\prime} & =x_{2}\\
x_{0} & =x_{0}^{\prime}\csc\vartheta+x_{1}^{\prime}\cot\vartheta & & =\frac{x_{0}^{\prime}+x_{1}^{\prime}\cos\vartheta}{\sin\vartheta}\\
x_{1} & =x_{0}^{\prime}\cot\vartheta+x_{1}^{\prime}\csc\vartheta & & =\frac{x_{0}^{\prime}\cos\vartheta+x_{1}^{\prime}}{\sin\vartheta}\\
\right|{\scriptstyle \begin{align}\cot^{2}\vartheta-\csc^{2}\vartheta & =-1\\
\frac{\cot\vartheta}{\csc\vartheta} & =\cos\vartheta\\
\frac{1}{\sqrt{1-\cos^{2}\vartheta}} & =\csc\vartheta\\
\frac{\cos\vartheta}{\sqrt{1-\cos^{2}\vartheta}} & =\cot\vartheta
Lorentz transformation via squeeze mappings.
As already indicated in equations () in exponential form or () in terms of Cayley–Klein parameter, Lorentz boosts in terms of hyperbolic rotations can be expressed as squeeze mappings. Using asymptotic coordinates of a hyperbola ("u,v"), they have the general form (some authors alternatively add a factor of 2 or formula_60):
That this equation system indeed represents a Lorentz boost can be seen by plugging (1) into (2) and solving for the individual variables:
Variables "u, v" in () can be rearranged to produce another form of squeeze mapping, resulting in Lorentz transformation () in terms of Cayley-Hermite parameter:
x_{1}^{\prime} & =-x_{0}\frac{2a}{1-a^{2}}+x_{1}\frac{1+a^{2}}{1-a^{2}} & & =\frac{-x_{0}2a+x_{1}\left(1+a^{2}\right)}{1-a^{2}}\\
x_{0} & =x_{0}^{\prime}\frac{1+a^{2}}{1-a^{2}}+x_{1}^{\prime}\frac{2a}{1-a^{2}} & & =\frac{x_{0}^{\prime}\left(1+a^{2}\right)+x_{1}^{\prime}2a}{1-a^{2}}\\
On the basis of factors "k" or "a", all previous Lorentz boosts (, , , ) can be expressed as squeeze mappings as well:
 & \frac{1+\sin\theta}{\cos\theta} & \frac{1+\cos\vartheta}{\sin\vartheta}=\cot\frac{\vartheta}{2}\\
\hline \frac{k-1}{k+1} & a & \tanh\frac{\eta}{2} & \frac{\gamma-1}{\beta\gamma} & \frac{1-\cos\theta}{\sin\theta}=\tan\frac{\theta}{2} & \frac{1-\sin\vartheta}{\cos\vartheta}\\
\hline \frac{k^{2}-1}{k^{2}+1} & \frac{2a}{1+a^{2}} & \tanh\eta & \beta & \sin\theta & \cos\vartheta\\
\hline \frac{k^{2}+1}{2k} & \frac{1+a^{2}}{1-a^{2}} & \cosh\eta & \gamma & \sec\theta & \csc\vartheta\\
\hline \frac{k^{2}-1}{2k} & \frac{2a}{1-a^{2}} & \sinh\eta & \beta\gamma & \tan\theta & \cot\vartheta
Electrodynamics and special relativity.
Voigt (1887).
Woldemar Voigt (1887) developed a transformation in connection with the Doppler effect and an incompressible medium, being in modern notation:
If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation (). In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Optical phenomena in free space are scale, conformal (using the factor λ discussed above), and Lorentz invariant, so the combination is invariant too. For instance, Lorentz transformations can be extended by using formula_62:
"l"=1/γ gives the Voigt transformation, "l"=1 the Lorentz transformation. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a principle of relativity in general. It was demonstrated by Poincaré and Einstein that one has to set "l"=1 in order to make the above transformation symmetric and to form a group as required by the relativity principle, therefore the Lorentz transformation is the only viable choice.
Also Hermann Minkowski said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in 1887. Voigt responded in the same paper by saying that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.
Heaviside (1888), Thomson (1889), Searle (1896).
In 1888, Oliver Heaviside investigated the properties of charges in motion according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:
Consequently, Joseph John Thomson (1889) found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the Galilean transformation "z-vt" in his equation):
Thereby, inhomogeneous electromagnetic wave equations are transformed into a Poisson equation. Eventually, George Frederick Charles Searle noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called "Heaviside-Ellipsoid" of axial ratio
Lorentz (1892, 1895).
In order to explain the aberration of light and the result of the Fizeau experiment in accordance with Maxwell's equations, Lorentz in 1892 developed a model ("Lorentz ether theory") in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson)
where "x*" is the Galilean transformation "x-vt". Except the additional γ in the time transformation, this is the complete Lorentz transformation (). While "t" is the "true" time for observers resting in the aether, "t′" is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in two steps. At first an implicit Galilean transformation, and later the expansion into the "fictitious" electromagnetic system with the aid of the Lorentz transformation. In order to explain the negative result of the Michelson–Morley experiment, he (1892b) introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced length contraction in his theory (without proof as he admitted). The same hypothesis was already made by George FitzGerald in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.
In 1895, Lorentz further elaborated on his theory and introduced the "theorem of corresponding states". This theorem states that a moving observer (relative to the ether) in his "fictitious" field makes the same observations as a resting observers in his "real" field for velocities to first order in "v/c". Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:
For solving optical problems Lorentz used the following transformation, in which the modified time variable was called "local time" () by him:
With this concept Lorentz could explain the Doppler effect, the aberration of light, and the Fizeau experiment.
Larmor (1897, 1900).
In 1897, Larmor extended the work of Lorentz and derived the following transformation
Larmor noted that if it is assumed that the constitution of molecules is electrical then the FitzGerald–Lorentz contraction is a consequence of this transformation, explaining the Michelson–Morley experiment. It's notable that Larmor was the first who recognized that some sort of time dilation is a consequence of this transformation as well, because "individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ". Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than "(v/c)"2 – when his 1897 paper was reprinted in 1929, Larmor added the following comment in which he described how they can be made valid to all orders of "v/c":
In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time "t″=t′-εvx′/c2" instead of the 1897 expression "t′=t-vx/c2" by replacing "v/c"2 with "εv/c"2, so that "t″" is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the "x′, y′, z′, t′" coordinates:
Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor "(v/c)"2, and so he sought the transformations which were "accurate to second order" (as he put it). Thus he wrote the final transformations (where "x′=x-vt" and "t″" as given above) as:
by which he arrived at the complete Lorentz transformation (). Larmor showed that Maxwell's equations were invariant under this two-step transformation, "to second order in "v/c"" – it was later shown by Lorentz (1904) and Poincaré (1905) that they are indeed invariant under this transformation to all orders in "v/c".
Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term "Lorentz transformation" for Lorentz's first order transformations of coordinates and field configurations:
Lorentz (1899, 1904).
Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 (again, "x"* must be replaced by "x-vt"):
Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows (where the above value of "t′" has to be inserted):
This is equivalent to the complete Lorentz transformation () when solved for "x″" and "t″" and with ε=1. Like Larmor, Lorentz noticed in 1899 also some sort of time dilation effect in relation to the frequency of oscillating electrons "that in "S" the time of vibrations be "kε" times as great as in "S0"", where "S0" is the aether frame.
In 1904 he rewrote the equations in the following form by setting "l"=1/ε (again, "x"* must be replaced by "x-vt"):
Under the assumption that "l=1" when "v"=0, he demonstrated that "l=1" must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor "l" to unity, Lorentz's transformations now assumed the same form as Larmor's and are now completed. Unlike Larmor, who restricted himself to show the covariance of Maxwell's equations to second order, Lorentz tried to widen its covariance to all orders in "v/c". He also derived the correct formulas for the velocity dependence of electromagnetic mass, and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones. However, he didn't achieve full covariance of the transformation equations for charge density and velocity. When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:
Lorentz's 1904 transformation was cited and used by Alfred Bucherer in July 1904:
or by Wilhelm Wien in July 1904:
or by Emil Cohn in November 1904 (setting the speed of light to unity):
or by Richard Gans in February 1905:
Poincaré (1900, 1905).
Local time.
Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, Henri Poincaré in 1900 commented on the origin of Lorentz's "wonderful invention" of local time. He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed formula_81 in both directions, which lead to what is nowadays called relativity of simultaneity, although Poincaré's calculation does not involve length contraction or time dilation. In order to synchronise the clocks here on Earth (the "x*, t"* frame) a light signal from one clock (at the origin) is sent to another (at "x"*), and is sent back. It's supposed that the Earth is moving with speed "v" in the "x"-direction (= "x"*-direction) in some rest system ("x, t") ("i.e." the luminiferous aether system for Lorentz and Larmor). The time of flight outwards is
and the time of flight back is
The elapsed time on the clock when the signal is returned is "δta+δtb" and the time "t*=(δta+δtb)/2" is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time "t=δta" is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus
identical to Lorentz (1892). By dropping the factor γ2 under the assumption that formula_85, Poincaré gave the result "t*=t-vx*/c2", which is the form used by Lorentz in 1895.
Similar physical interpretations of local time were later given by Emil Cohn (1904) and Max Abraham (1905).
Lorentz transformation.
On June 5, 1905 (published June 9) Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form ():
Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name "Lorentz transformation". Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by setting "l"=1, and modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity, "i.e." making them fully Lorentz covariant.
In July 1905 (published in January 1906) Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the principle of least action; he demonstrated in more detail the group characteristics of the transformation, which he called Lorentz group, and he showed that the combination "x2+y2+z2-t2" is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing formula_87 as a fourth imaginary coordinate, and he used an early form of four-vectors. He also formulated the velocity addition formula (), which he had already derived in unpublished letters to Lorentz from May 1905:
Einstein (1905) – Special relativity.
On June 30, 1905 (published September 1905) Einstein published what is now called special relativity and gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. While Lorentz considered "local time" to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in "v/c" this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations concern the nature of space and time.
The notation for this transformation is equivalent to Poincaré's of 1905 and (), except that Einstein didn't set the speed of light to unity:
Einstein also defined the velocity addition formula (, ):
and the light aberration formula ():
Minkowski (1907–1908) – Spacetime.
The work on the principle of relativity by Lorentz, Einstein, Planck, together with Poincaré's four-dimensional approach, were further elaborated and combined with the hyperboloid model by Hermann Minkowski in 1907 and 1908. Minkowski particularly reformulated electrodynamics in a four-dimensional way (Minkowski spacetime). For instance, he wrote "x, y, z, it" in the form "x1, x2, x3, x4". By defining ψ as the angle of rotation around the "z"-axis, the Lorentz transformation assumes a form (with "c"=1) in agreement with ():
Even though Minkowski used the imaginary number iψ, he for once directly used the tangens hyperbolicus in the equation for velocity
Minkowski's expression can also by written as ψ=atanh(q) and was later called rapidity. He also wrote the Lorentz transformation in matrix form equivalent to () ("n"=3):
As a graphical representation of the Lorentz transformation he introduced the Minkowski diagram, which became a standard tool in textbooks and research articles on relativity:
Sommerfeld (1909) – Spherical trigonometry.
Using an imaginary rapidity such as Minkowski, Arnold Sommerfeld (1909) formulated a transformation equivalent to Lorentz boost (), and the relativistic velocity addition () in terms of trigonometric functions and the spherical law of cosines:
Bateman and Cunningham (1909–1910) – Spherical wave transformation.
In line with Lie's (1871) research on the relation between sphere transformations with an imaginary radius coordinate and 4D conformal transformations, it was pointed out by Bateman and Cunningham (1909–1910), that by setting "u=ict" as the imaginary fourth coordinates one can produce spacetime conformal transformations. Not only the quadratic form formula_97, but also Maxwells equations are covariant with respect to these transformations, irrespective of the choice of λ. These variants of conformal or Lie sphere transformations were called spherical wave transformations by Bateman. However, this covariance is restricted to certain areas such as electrodynamics, whereas the totality of natural laws in inertial frames is covariant under the Lorentz group. In particular, by setting λ=1 the Lorentz group can be seen as a 10-parameter subgroup of the 15-parameter spacetime conformal group .
Bateman (1910/12) also alluded to the identity between the Laguerre inversion and the Lorentz transformations. In general, the isomorphism between the Laguerre group and the Lorentz group was pointed out by Élie Cartan (1912, 1915/55), Henri Poincaré (1912/21) and others.
Herglotz (1909/10) – Möbius transformation.
Following Klein (1889–1897) and Fricke & Klein (1897) concerning the Cayley absolute, hyperbolic motion and its transformation, Gustav Herglotz (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic. The general case (on the left) equivalent to Lorentz transformation () and the hyperbolic case (on the right) equivalent to Lorentz transformation () or squeeze mapping () are as follows:
Varićak (1910) – Hyperbolic functions.
Following Sommerfeld (1909), hyperbolic functions were used by Vladimir Varićak in several papers starting from 1910, who represented the equations of special relativity on the basis of hyperbolic geometry in terms of Weierstrass coordinates. For instance, by setting "l=ct" and "v/c=tanh(u)" with "u" as rapidity he wrote the Lorentz transformation in agreement with ():
and showed the relation of rapidity to the Gudermannian function and the angle of parallelism:
He also related the velocity addition to the hyperbolic law of cosines:
Subsequently, other authors such as E. T. Whittaker (1910) or Alfred Robb (1911, who coined the name rapidity) used similar expressions, which are still used in modern textbooks.
Ignatowski (1910).
While earlier derivations and formulations of the Lorentz transformation relied from the outset on optics, electrodynamics, or the invariance of the speed of light, Vladimir Ignatowski (1910) showed that it is possible to use the principle of relativity (and related group theoretical principles) alone, in order to derive the following transformation between two inertial frames:
The variable "n" can be seen as a space-time constant whose value has to be determined by experiment or taken from a known physical law such as electrodynamics. For that purpose, Ignatowski used the above-mentioned Heaviside ellipsoid representing a contraction of electrostatic fields by "x"/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when "n=1/c"2, resulting in "p"=γ and the Lorentz transformation (). With "n"=0, no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved by Philipp Frank and Hermann Rothe (1911, 1912), with various authors developing similar methods in subsequent years.
Noether (1910), Klein (1910) – Quaternions.
Felix Klein (1908) described Cayley's (1854) 4D quaternion multiplications as "Drehstreckungen" (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor), and pointed out that the modern principle of relativity as provided by Minkowski is essentially only the consequent application of such Drehstreckungen, even though he didn't provide details.
In an appendix to Klein's and Sommerfeld's "Theory of the top" (1910), Fritz Noether showed how to formulate hyperbolic rotations using biquaternions with formula_103, which he also related to the speed of light by setting ω2=-"c"2. He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations equivalent to ():
Besides citing quaternion related standard works such as Cayley (1854), Noether referred to the entries in Klein's encyclopedia by Eduard Study (1899) and the French version by Élie Cartan (1908). Cartan's version contains a description of Study's dual numbers, Clifford's biquaternions (including the choice formula_103 for hyperbolic geometry), and Clifford algebra, with references to Stephanos (1883), Buchheim (1884/85), Vahlen (1901/02) and others.
Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:
or in March 1911
Conway (1911), Silberstein (1911) – Quaternions.
Arthur W. Conway in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:
Also Ludwik Silberstein in November 1911 as well as in 1914, formulated the Lorentz transformation in terms of velocity "v":
Silberstein cites Cayley (1854, 1855) and Study's encyclopedia entry (in the extended French version of Cartan in 1908), as well as the appendix of Klein's and Sommerfeld's book.
Herglotz (1911), Silberstein (1911) – Vector transformation.
Gustav Herglotz (1911) showed how to formulate the transformation equivalent to () in order to allow for arbitrary velocities and coordinates v="(vx, vy, vz)" and r="(x, y, z)":
This was simplified using vector notation by Ludwik Silberstein (1911 on the left, 1914 on the right):
Equivalent formulas were also given by Wolfgang Pauli (1921), with Erwin Madelung (1922) providing the matrix form
These formulas were called "general Lorentz transformation without rotation" by Christian Møller (1952), who in addition gave an even more general Lorentz transformation in which the Cartesian axes have different orientations, using a rotation operator formula_113. In this case, v′="(v′x, v′y, v′z)" is not equal to -v="(-vx, -vy, -vz)", but the relation formula_114 holds instead, with the result
Borel (1913–14) – Cayley–Hermite parameter.
Borel (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and Cayley's (1846) parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions equivalent to ():
In four dimensions equivalent to ():
Gruner (1921) – Trigonometric Lorentz boosts.
In order to simplify the graphical representation of Minkowski space, Paul Gruner (1921) (with the aid of Josef Sauter) developed what is now called Loedel diagrams, using the following relations:
In another paper Gruner used the alternative relations:
Euler's gap.
In pursuing the history in years before Lorentz enunciated his expressions, one looks to the essence of the concept. In mathematical terms, Lorentz transformations are squeeze mappings, the linear transformations that turn a square into a rectangles of the same area. Before Euler, the squeezing was studied as quadrature of the hyperbola and led to the hyperbolic logarithm. In 1748 Euler issued his precalculus textbook where the number e is exploited for trigonometry in the unit circle. The first volume of Introduction to the Analysis of the Infinite had no diagrams, allowing teachers and students to draw their own illustrations.
There is a gap in Euler's text where Lorentz transformations arise. A feature of natural logarithm is its interpretation as area in hyperbolic sectors. In relativity the classical concept of velocity is replaced with rapidity, a hyperbolic angle concept built on hyperbolic sectors. A Lorentz transformation is a hyperbolic rotation which preserves differences of rapidity, just as the circular sector area is preserved with a circular rotation. Euler's gap is the lack of hyperbolic angle and hyperbolic functions, later developed by Johann H. Lambert. Euler described some transcendental functions: exponentiation and circular functions. He used the exponential series formula_120 With the imaginary unit i2 = – 1, and splitting the series into even and odd terms, he obtained
This development misses the alternative:
Here Euler could have noted split-complex numbers along with complex numbers.
For physics, one space dimension is insufficient. But to extend split-complex arithmetic to four dimensions leads to hyperbolic quaternions, and opens the door to abstract algebra's 
hypercomplex numbers. Reviewing the expressions of Lorentz and Einstein, one observes that the Lorentz factor is an algebraic function of velocity. For readers uncomfortable with transcendental functions cosh and sinh, algebraic functions may be more to their liking.