History of Lorentz transformations#Voigt (1887)

Using the coefficients of a symmetric matrix A, the associated bilinear form, and a linear transformations in terms of transformation matrix g, the Lorentz transformation is given if the following conditions are satisfied:

:

-x_{0}y_{0}+\cdots+x_{n}y_{n} & =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime}

\end{align}

\\

\hline \begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\

\mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'

\end{matrix}\\

\hline \begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} & =\mathbf{g}^{-1}\\

\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} & =\mathbf{A}\\

\mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} & =\mathbf{A}

\end{align}

\end{matrix}\\

\hline \mathbf{A}={\rm diag}(-1,1,\dots,1)\\

\det \mathbf{g}=\pm1

\end{matrix}

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 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 becomes the Minkowski inner product.Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1Naimark (1964), 2 in four dimensions Long before the advent of special relativity it was used in topics such as the Cayley–Klein metric, hyperboloid model and other models of hyperbolic geometry, computations of elliptic functions and integrals, transformation of indefinite quadratic forms, squeeze mappings of the hyperbola, group theory, Möbius transformations, spherical wave transformation, transformation of the Sine-Gordon equation, Biquaternion algebra, split-complex numbers, Clifford algebra, and others.

In the special 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. Using the above conditions, the Lorentz transformation in 3+1 dimensions assume the form:

:$\backslash begin\{matrix\}-c^\{2\}t^\{2\}+x^\{2\}+y^\{2\}+z^\{2\}=-c^\{2\}t^\{\backslash prime2\}+x^\{\backslash prime2\}+y^\{\backslash prime2\}+z^\{\backslash prime2\}\backslash \backslash$

\hline \left.\begin{align}t' & =\gamma\left(t-x\frac{v}{c^{2}}\right)\\

x' & =\gamma(x-vt)\\

y' & =y\\

z' & =z

\end{align}

\right|\begin{align}t & =\gamma\left(t'+x\frac{v}{c^{2}}\right)\\

x & =\gamma(x'+vt')\\

y & =y'\\

z & =z'

\end{align}

\end{matrix}\Rightarrow\begin{align}(ct'+x') & =(ct+x)\sqrt{\frac{c+v}{c-v}}\\

(ct'-x') & =(ct-x)\sqrt{\frac{c-v}{c+v}}

\end{align}

In physics, analogous transformations have been introduced by Voigt (1887) related to an incompressible medium, and by Heaviside (1888), Thomson (1889), Searle (1896) and 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.Miller (1981), chapter 1 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.Miller (1981), chapter 4–7 Minkowski (1907–1908) used them to argue that space and time are inseparably connected as spacetime.

Regarding special representations of the Lorentz transformations: Minkowski (1907–1908) and Sommerfeld (1909) used imaginary trigonometric functions, Frank (1909) and Varićak (1910) used hyperbolic functions, Bateman and Cunningham (1909–1910) used spherical wave transformations, Herglotz (1909–10) used Möbius transformations, Plummer (1910) and Gruner (1921) used trigonometric Lorentz boosts, Ignatowski (1910) derived the transformations without light speed postulate, Noether (1910) and Klein (1910) as well Conway (1911) and Silberstein (1911) used Biquaternions, Ignatowski (1910/11), Herglotz (1911), and others used vector transformations valid in arbitrary directions, Borel (1913–14) used Cayley–Hermite parameter,

Woldemar Voigt (1887)Voigt (1887), p. 45 developed a transformation in connection with the Doppler effect and an incompressible medium, being in modern notation:Miller (1981), 114–115Pais (1982), Kap. 6b

:

\hline \left.\begin{align}\xi_{1} & =x_{1}-\varkappa t\\

\eta_{1} & =y_{1}q\\

\zeta_{1} & =z_{1}q\\

\tau & =t-\frac{\varkappa x_{1}}{\omega^{2}}\\

q & =\sqrt{1-\frac{\varkappa^{2}}{\omega^{2}}}

\end{align}

\right| & \begin{align}x^{\prime} & =x-vt\\

y^{\prime} & =\frac{y}{\gamma}\\

z^{\prime} & =\frac{z}{\gamma}\\

t^{\prime} & =t-\frac{vx}{c^{2}}\\

\frac{1}{\gamma} & =\sqrt{1-\frac{v^{2}}{c^{2}}}

\end{align}

\end{matrix}

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, and Lorentz invariant, so the combination is invariant too. For instance, Lorentz transformations can be extended by using factor $l$:Lorentz (1915/16), p. 197

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

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

Voigt sent his 1887 paper to Lorentz in 1908,{{cite arXiv | eprint=1411.2559 | last1=Heras | first1=Ricardo | title=A review of Voigt's transformations in the framework of special relativity | year=2014 | class=physics.hist-ph }} and that was acknowledged in 1909: {{Blockquote|In a paper "Über das Doppler'sche Princip", published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (7) (§ 3 of this book) [namely $\backslash Delta\backslash Psi-\backslash tfrac\{1\}\{c^\{2\}\}\backslash tfrac\{\backslash partial^\{2\}\backslash Psi\}\{\backslash partial\; t^\{2\}\}=0$] a transformation equivalent to the formulae (287) and (288) [namely $x^\{\backslash prime\}=\backslash gamma\; l\backslash left(x-vt\backslash right),\backslash \; y^\{\backslash prime\}=ly,\backslash \; z^\{\backslash prime\}=lz,\backslash \; t^\{\backslash prime\}=\backslash gamma\; l\backslash left(t-\backslash tfrac\{v\}\{c^\{2\}\}x\backslash right)$]. The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the free ether is contained in his paper.Lorentz (1915/16), p. 198}}

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.Bucherer (1908), p. 762

In 1888, Oliver HeavisideHeaviside (1888), p. 324 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:Brown (2003)

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

Consequently, Joseph John Thomson (1889)Thomson (1889), p. 12 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):

:

\hline \left.\begin{align}z & =\left\{ 1-\frac{\omega^{2}}{v^{2}}\right\} ^{\frac{1}{2}}z'\end{align}

\right| & \begin{align}z^{\ast}=z-vt & =\frac{z'}{\gamma}\end{align}

\end{matrix}

Thereby, inhomogeneous electromagnetic wave equations are transformed into a Poisson equation.Miller (1981), 98–99 Eventually, George Frederick Charles SearleSearle (1886), p. 333 noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called "Heaviside-Ellipsoid" of axial ratio

:

\hline \left.\begin{align} & \sqrt{\alpha}:1:1\\

\alpha= & 1-\frac{u^{2}}{v^{2}}

\end{align}

\right| & \begin{align} & \frac{1}{\gamma}:1:1\\

\frac{1}{\gamma^{2}} & =1-\frac{v^{2}}{c^{2}}

\end{align}

\end{matrix}

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)Lorentz (1892a), p. 141Miller (1982), 1.4 & 1.5

:

\hline \left.\begin{align}\mathfrak{x} & =\frac{V}{\sqrt{V^{2}-p^{2}}}x\\

t' & =t-\frac{\varepsilon}{V}\mathfrak{x}\\

\varepsilon & =\frac{p}{\sqrt{V^{2}-p^{2}}}

\end{align}

\right| & \begin{align}x^{\prime} & =\gamma x^{\ast}=\gamma(x-vt)\\

t^{\prime} & =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)\\

\gamma\frac{v}{c} & =\frac{v}{\sqrt{c^{2}-v^{2}}}

\end{align}

\end{matrix}

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)Lorentz (1892b), p. 141 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 had been made previously 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:Lorentz (1895), p. 37

:

\hline \left.\begin{align}x & =x^{\prime}\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}\\

y & =y^{\prime}\\

z & =z^{\prime}\\

t & =t^{\prime}

\end{align}

\right| & \begin{align}x^{\ast}=x-vt & =\frac{x^{\prime}}{\gamma}\\

y & =y^{\prime}\\

z & =z^{\prime}\\

t & =t^{\prime}

\end{align}

\end{matrix}

For solving optical problems Lorentz used the following transformation, in which the modified time variable was called "local time" ({{langx|de|Ortszeit}}) by him:Lorentz (1895), p. 49 for local time and p. 56 for spatial coordinates.

:

\hline \left.\begin{align}x & =\mathrm{x}-\mathfrak{p}_{x}t\\

y & =\mathrm{y}-\mathfrak{p}_{y}t\\

z & =\mathrm{z}-\mathfrak{p}_{z}t\\

t^{\prime} & =t-\frac{\mathfrak{p}_{x}}{V^{2}}x-\frac{\mathfrak{p}_{y}}{V^{2}}y-\frac{\mathfrak{p}_{z}}{V^{2}}z

\end{align}

\right| & \begin{align}x^{\prime} & =x-v_{x}t\\

y^{\prime} & =y-v_{y}t\\

z^{\prime} & =z-v_{z}t\\

t^{\prime} & =t-\frac{v_{x}}{c^{2}}x'-\frac{v_{y}}{c^{2}}y'-\frac{v_{z}}{c^{2}}z'

\end{align}

\end{matrix}

With this concept Lorentz could explain the Doppler effect, the aberration of light, and the Fizeau experiment.Janssen (1995), 3.1

In 1897, Larmor extended the work of Lorentz and derived the following transformationLarmor (1897), p. 229

:

\hline \left.\begin{align}x_{1} & =x\varepsilon^{\frac{1}{2}}\\

y_{1} & =y\\

z_{1} & =z\\

t^{\prime} & =t-vx/c^{2}\\

dt_{1} & =dt^{\prime}\varepsilon^{-\frac{1}{2}}\\

\varepsilon & =\left(1-v^{2}/c^{2}\right)^{-1}

\end{align}

\right| & \begin{align}x_{1} & =\gamma x^{\ast}=\gamma(x-vt)\\

y_{1} & =y\\

z_{1} & =z\\

t^{\prime} & =t-\frac{vx^{\ast}}{c^{2}}=t-\frac{v(x-vt)}{c^{2}}\\

dt_{1} & =\frac{dt^{\prime}}{\gamma}\\

\gamma^{2} & =\frac{1}{1-\frac{v^{2}}{c^{2}}}

\end{align}

\end{matrix}

Larmor noted that if it is assumed that the constitution of molecules is electrical then the FitzGerald–Lorentz contraction is a consequence of this transformation, explaining the Michelson–Morley experiment. It's notable that Larmor was the first who recognized that some sort of time dilation is a consequence of this transformation as well, because "individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ".Darrigol (2000), Chap. 8.5Macrossan (1986) 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:Larmor (1897/1929), p. 39

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

In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time t″=t′-εvx′/c² instead of the 1897 expression t′=t-vx/c² by replacing v/c² with εv/c², so that t″ is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the x′, y′, z′, t′ coordinates:Larmor (1900), p. 168

:

\hline \left.\begin{align}x^{\prime} & =x-vt\\

y^{\prime} & =y\\

z^{\prime} & =z\\

t^{\prime} & =t\\

t^{\prime\prime} & =t^{\prime}-\varepsilon vx^{\prime}/c^{2}

\end{align}

\right| & \begin{align}x^{\prime} & =x-vt\\

y^{\prime} & =y\\

z^{\prime} & =z\\

t^{\prime} & =t\\

t^{\prime\prime}=t^{\prime}-\frac{\gamma^{2}vx^{\prime}}{c^{2}} & =\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)

\end{align}

\end{matrix}

Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor (v/c)², and so he sought the transformations which were "accurate to second order" (as he put it). Thus he wrote the final transformations (where x′=x-vt and t″ as given above) as:Larmor (1900), p. 174

:

\hline \left.\begin{align}x_{1} & =\varepsilon^{\frac{1}{2}}x^{\prime}\\

y_{1} & =y^{\prime}\\

z_{1} & =z^{\prime}\\

dt_{1} & =\varepsilon^{-\frac{1}{2}}dt^{\prime\prime}=\varepsilon^{-\frac{1}{2}}\left(dt^{\prime}-\frac{v}{c^{2}}\varepsilon dx^{\prime}\right)\\

t_{1} & =\varepsilon^{-\frac{1}{2}}t^{\prime}-\frac{v}{c^{2}}\varepsilon^{\frac{1}{2}}x^{\prime}

\end{align}

\right| & \begin{align}x_{1} & =\gamma x^{\prime}=\gamma(x-vt)\\

y_{1} & =y'=y\\

z_{1} & =z'=z\\

dt_{1} & =\frac{dt^{\prime\prime}}{\gamma}=\frac{1}{\gamma}\left(dt^{\prime}-\frac{\gamma^{2}vdx^{\prime}}{c^{2}}\right)=\gamma\left(dt-\frac{vdx}{c^{2}}\right)\\

t_{1} & =\frac{t^{\prime}}{\gamma}-\frac{\gamma vx^{\prime}}{c^{2}}=\gamma\left(t-\frac{vx}{c^{2}}\right)

\end{align}

\end{matrix}

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:

{{Blockquote|p. 583: [..] Lorentz's transformation for passing from the field of activity of a stationary electrodynamic material system to that of one moving with uniform velocity of translation through the aether.
p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..]Larmor (1904a), p. 583, 585
p. 622: [..] the transformation first developed by Lorentz: namely, each point in space is to have its own origin from which time is measured, its "local time" in Lorentz's phraseology, and then the values of the electric and magnetic vectors [..] at all points in the aether between the molecules in the system at rest, are the same as those of the vectors [..] at the corresponding points in the convected system at the same local times.Larmor (1904b), p. 622}}

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):Lorentz (1899), p. 429

:

\hline \left.\begin{align}x^{\prime} & =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}x\\

y^{\prime} & =y\\

z^{\prime} & =z\\

t^{\prime} & =t-\frac{\mathfrak{p}_{x}}{V^{2}-\mathfrak{p}_{x}^{2}}x

\end{align}

\right| & \begin{align}x^{\prime} & =\gamma x^{\ast}=\gamma(x-vt)\\

y^{\prime} & =y\\

z^{\prime} & =z\\

t^{\prime} & =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)

\end{align}

\end{matrix}

Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows (where the above value of t′ has to be inserted):Lorentz (1899), p. 439

:

\hline \left.\begin{align}x & =\frac{\varepsilon}{k}x^{\prime\prime}\\

y & =\varepsilon y^{\prime\prime}\\

z & =\varepsilon x^{\prime\prime}\\

t^{\prime} & =k\varepsilon t^{\prime\prime}\\

k & =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}

\end{align}

\right| & \begin{align}x^{\ast}=x-vt & =\frac{\varepsilon}{\gamma}x^{\prime\prime}\\

y & =\varepsilon y^{\prime\prime}\\

z & =\varepsilon z^{\prime\prime}\\

t^{\prime}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right) & =\gamma\varepsilon t^{\prime\prime}\\

\gamma & =\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}

\end{align}

\end{matrix}

This is equivalent to the complete Lorentz transformation when solved for x″ and t″ and with ε=1. Like Larmor, Lorentz noticed in 1899Lorentz (1899), p. 442 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.Jannsen (1995), Kap. 3.3

In 1904 he rewrote the equations in the following form by setting l=1/ε (again, x* must be replaced by x-vt):Lorentz (1904), p. 812

:

\hline \left.\begin{align}x^{\prime} & =klx\\

y^{\prime} & =ly\\

z^{\prime} & =lz\\

t' & =\frac{l}{k}t-kl\frac{w}{c^{2}}x

\end{align}

\right| & \begin{align}x^{\prime} & =\gamma lx^{\ast}=\gamma l(x-vt)\\

y^{\prime} & =ly\\

z^{\prime} & =lz\\

t^{\prime} & =\frac{lt}{\gamma}-\frac{\gamma lvx^{\ast}}{c^{2}}=\gamma l\left(t-\frac{vx}{c^{2}}\right)

\end{align}

\end{matrix}

Under the assumption that l=1 when v=0, he demonstrated that l=1 must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor l to unity, Lorentz's transformations now assumed the same form as Larmor's and are now completed. Unlike Larmor, who restricted himself to show the covariance of Maxwell's equations to second order, Lorentz tried to widen its covariance to all orders in v/c. He also derived the correct formulas for the velocity dependence of electromagnetic mass, and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones.Lorentz (1904), p. 826 However, he didn't achieve full covariance of the transformation equations for charge density and velocity.Miller (1981), Chap. 1.12.2 When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:Jannsen (1995), Chap. 3.5.6

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

Lorentz's 1904 transformation was cited and used by Alfred Bucherer in July 1904:Bucherer, p. 129; Definition of s on p. 32

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

or by Wilhelm Wien in July 1904:Wien (1904), p. 394

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

or by Emil Cohn in November 1904 (setting the speed of light to unity):Cohn (1904a), pp. 1296-1297

:$x=\backslash frac\{x\_\{0\}\}\{k\},\backslash quad\; y=y\_\{0\},\backslash quad\; z=z\_\{0\},\backslash quad\; t=kt\_\{0\},\backslash quad\; t\_\{1\}=t\_\{0\}-w\backslash cdot\; r\_\{0\},\backslash quad\; k^\{2\}=\backslash frac\{1\}\{1-w^\{2\}\}$

or by Richard Gans in February 1905:Gans (1905), p. 169

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

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.Darrigol (2005), Kap. 4 He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed $c$ in both directions, which lead to what is nowadays called relativity of simultaneity, although Poincaré's calculation does not involve length contraction or time dilation.Poincaré (1900), pp. 272–273 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

:$\backslash delta\; t\_\{a\}=\backslash frac\{x^\{\backslash ast\}\}\{\backslash left(c-v\backslash right)\}$

and the time of flight back is

:$\backslash delta\; t\_\{b\}=\backslash frac\{x^\{\backslash ast\}\}\{\backslash left(c+v\backslash right)\}$.

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

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

identical to Lorentz (1892). By dropping the factor γ² under the assumption that $\backslash tfrac\{v^\{2\}\}\{c^\{2\}\}\backslash ll1$, 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)Cohn (1904b), p. 1408 and Max Abraham (1905).Abraham (1905), § 42

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:Poincaré (1905), p. 1505

:

y^{\prime} & =ly\\

z^{\prime} & =lz\\

t' & =kl(t+\varepsilon x)\\

k & =\frac{1}{\sqrt{1-\varepsilon^{2}}}

\end{align}

.

Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name "Lorentz transformation".Pais (1982), Chap. 6cKatzir (2005), 280–288 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.Miller (1981), Chap. 1.14

In July 1905 (published in January 1906)Poincaré (1905/06), pp. 129ff Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the principle of least action; he demonstrated in more detail the group characteristics of the transformation, which he called Lorentz group, and he showed that the combination x²+y²+z²-t² is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing $ct\backslash sqrt\{-1\}$ 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:Poincaré (1905/06), p. 144

:$\backslash xi\text{'}=\backslash frac\{\backslash xi+\backslash varepsilon\}\{1+\backslash xi\backslash varepsilon\},\backslash \; \backslash eta\text{'}=\backslash frac\{\backslash eta\}\{k(1+\backslash xi\backslash varepsilon)\}$.

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 of 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 applied to the kinematics of moving frames.Miller (1981), Chap. 6Pais (1982), Kap. 7Darrigol (2005), Chap. 6

The notation for this transformation is equivalent to Poincaré's of 1905, except that Einstein didn't set the speed of light to unity:Einstein (1905), p. 902

:

\xi & =\beta(x-vt)\\

\eta & =y\\

\zeta & =z\\

\beta & =\frac{1}{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}

\end{align}

Einstein also defined the velocity addition formula:Einstein (1905), § 5 and § 9

:$\backslash begin\{matrix\}x=\backslash frac\{w\_\{\backslash xi\}+v\}\{1+\backslash frac\{vw\_\{\backslash xi\}\}\{V^\{2\}\}\}t,\backslash \; y=\backslash frac\{\backslash sqrt\{1-\backslash left(\backslash frac\{v\}\{V\}\backslash right)^\{2\}\}\}\{1+\backslash frac\{vw\_\{\backslash xi\}\}\{V^\{2\}\}\}w\_\{\backslash eta\}t\backslash \backslash$

U^{2}=\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2},\ w^{2}=w_{\xi}^{2}+w_{\eta}^{2},\ \alpha=\operatorname{arctg}\frac{w_{y}}{w_{x}}\\

U=\frac{\sqrt{\left(v^{2}+w^{2}+2vw\cos\alpha\right)-\left(\frac{vw\sin\alpha}{V}\right)^{2}}}{1+\frac{vw\cos\alpha}{V^{2}}}

\end{matrix}\left|\begin{matrix}\frac{u_{x}-v}{1-\frac{u_{x}v}{V^{2}}}=u_{\xi}\\

\frac{u_{y}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\eta}\\

\frac{u_{z}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\zeta}

\end{matrix}\right.

and the light aberration formula:Einstein (1905), § 7

:$\backslash cos\backslash varphi\text{'}=\backslash frac\{\backslash cos\backslash varphi-\backslash frac\{v\}\{V\}\}\{1-\backslash frac\{v\}\{V\}\backslash cos\backslash varphi\}$

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 (1907/15), pp. 927ffMinkowski (1907/08), pp. 53ff Minkowski particularly reformulated electrodynamics in a four-dimensional way (Minkowski spacetime).Walter (1999a) 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 the form (with c=1):Minkowski (1907/08), p. 59

:

x'_{2} & =x_{2}\\

x'_{3} & =x_{3}\cos i\psi+x_{4}\sin i\psi\\

x'_{4} & =-x_{3}\sin i\psi+x_{4}\cos i\psi\\

\cos i\psi & =\frac{1}{\sqrt{1-q^{2}}}

\end{align}

Even though Minkowski used the imaginary number iψ, he for once directly used the tangens hyperbolicus in the equation for velocity

:$-i\backslash tan\; i\backslash psi=\backslash frac\{e^\{\backslash psi\}-e^\{-\backslash psi\}\}\{e^\{\backslash psi\}+e^\{-\backslash psi\}\}=q$ with $\backslash psi=\backslash frac\{1\}\{2\}\backslash ln\backslash frac\{1+q\}\{1-q\}$.

Minkowski's expression can also by written as ψ=atanh(q) and was later called rapidity. He also wrote the Lorentz transformation in matrix form:Minkowski (1907/08), pp. 65–66, 81–82

:$\backslash begin\{matrix\}x\_\{1\}^\{2\}+x\_\{2\}^\{2\}+x\_\{3\}^\{2\}+x\_\{4\}^\{2\}=x\_\{1\}^\{\backslash prime2\}+x\_\{2\}^\{\backslash prime2\}+x\_\{3\}^\{\backslash prime2\}+x\_\{4\}^\{\backslash prime2\}\backslash \backslash$

\left(x_{1}^{\prime}=x',\ x_{2}^{\prime}=y',\ x_{3}^{\prime}=z',\ x_{4}^{\prime}=it'\right)\\

-x^{2}-y^{2}-z^{2}+t^{2}=-x^{\prime2}-y^{\prime2}-z^{\prime2}+t^{\prime2}\\

\hline x_{h}=\alpha_{h1}x_{1}^{\prime}+\alpha_{h2}x_{2}^{\prime}+\alpha_{h3}x_{3}^{\prime}+\alpha_{h4}x_{4}^{\prime}\\

\mathrm{A}=\mathrm{\left|\begin{matrix}\alpha_{11}, & \alpha_{12}, & \alpha_{13}, & \alpha_{14}\\

\alpha_{21}, & \alpha_{22}, & \alpha_{23}, & \alpha_{24}\\

\alpha_{31}, & \alpha_{32}, & \alpha_{33}, & \alpha_{34}\\

\alpha_{41}, & \alpha_{42}, & \alpha_{43}, & \alpha_{44}

\end{matrix}\right|,\ \begin{align}\bar{\mathrm{A}}\mathrm{A} & =1\\

\left(\det \mathrm{A}\right)^{2} & =1\\

\det \mathrm{A} & =1\\

\alpha_{44} & >0

\end{align}

}

\end{matrix}

As a graphical representation of the Lorentz transformation he introduced the Minkowski diagram, which became a standard tool in textbooks and research articles on relativity:Minkowski (1908/09), p. 77

File:Minkowski1.png

Using an imaginary rapidity such as Minkowski, Arnold Sommerfeld (1909) formulated the Lorentz boost and the relativistic velocity addition in terms of trigonometric functions and the spherical law of cosines:Sommerfeld (1909), p. 826ff.

:$\backslash begin\{matrix\}\backslash left.\backslash begin\{array\}\{lrl\}$

x'= & x\ \cos\varphi+l\ \sin\varphi, & y'=y\\

l'= & -x\ \sin\varphi+l\ \cos\varphi, & z'=z

\end{array}\right\} \\

\left(\operatorname{tg}\varphi=i\beta,\ \cos\varphi=\frac{1}{\sqrt{1-\beta^{2}}},\ \sin\varphi=\frac{i\beta}{\sqrt{1-\beta^{2}}}\right)\\

\hline \beta=\frac{1}{i}\operatorname{tg}\left(\varphi_{1}+\varphi_{2}\right)=\frac{1}{i}\frac{\operatorname{tg}\varphi_{1}+\operatorname{tg}\varphi_{2}}{1-\operatorname{tg}\varphi_{1}\operatorname{tg}\varphi_{2}}=\frac{\beta_{1}+\beta_{2}}{1+\beta_{1}\beta_{2}}\\

\cos\varphi=\cos\varphi_{1}\cos\varphi_{2}-\sin\varphi_{1}\sin\varphi_{2}\cos\alpha\\

v^{2}=\frac{v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos\alpha-\frac{1}{c^{2}}v_{1}^{2}v_{2}^{2}\sin^{2}\alpha}{\left(1+\frac{1}{c^{2}}v_{1}v_{2}\cos\alpha\right)^{2}}

\end{matrix}

Hyperbolic functions were used by Philipp Frank (1909), who derived the Lorentz transformation using ψ as rapidity:Frank (1909), pp. 423-425

:$\backslash begin\{matrix\}x\text{'}=x\backslash varphi(a)\backslash ,\{\backslash rm\; ch\}\backslash ,\backslash psi+t\backslash varphi(a)\backslash ,\{\backslash rm\; sh\}\backslash ,\backslash psi\backslash \backslash$

t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\

\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\

\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}

\end{matrix}

In line with Sophus 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 $\backslash lambda\backslash left(dx^\{2\}+dy^\{2\}+dz^\{2\}+du^\{2\}\backslash right)$, but also Maxwells equations are covariant with respect to these transformations, irrespective of the choice of λ. These variants of conformal or Lie sphere transformations were called spherical wave transformations by Bateman.Bateman (1909/10), pp. 223ffCunningham (1909/10), pp. 77ff 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.Klein (1910) In particular, by setting λ=1 the Lorentz group {{nowrap|SO(1,3)}} can be seen as a 10-parameter subgroup of the 15-parameter spacetime conformal group {{nowrap|Con(1,3)}}.

Bateman (1910–12)Bateman (1910/12), pp. 358–359 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),Cartan (1912), p. 23 Henri Poincaré (1912–21)Poincaré (1912/21), p. 145 and others.

Following Felix 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) and the hyperbolic case equivalent to Lorentz transformations or squeeze mappings are as follows:Herglotz (1909/10), pp. 404-408

:$\backslash left.\backslash begin\{matrix\}z\_\{1\}^\{2\}+z\_\{2\}^\{2\}+z\_\{3\}^\{2\}-z\_\{4\}^\{2\}=0\backslash \backslash$

z_{1}=x,\ z_{2}=y,\ z_{3}=z,\ z_{4}=t\\

Z=\frac{z_{1}+iz_{2}}{z_{4}-z_{3}}=\frac{x+iy}{t-z},\ Z'=\frac{x'+iy'}{t'-z'}\\

Z=\frac{\alpha Z'+\beta}{\gamma Z'+\delta}

\end{matrix}\right|\begin{matrix}Z=Z'e^{\vartheta}\\

\begin{align}x & =x', & t-z & =(t'-z')e^{\vartheta}\\

y & =y', & t+z & =(t'+z')e^{-\vartheta}

\end{align}

\end{matrix}

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:Varićak (1910), p. 93

:

x' & =x\operatorname{ch}u-l\operatorname{sh}u,\\

y' & =y,\quad z'=z,\\

\operatorname{ch}u & =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}

\end{align}

and showed the relation of rapidity to the Gudermannian function and the angle of parallelism:

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

He also related the velocity addition to the hyperbolic law of cosines:Varićak (1910), p. 94

:$\backslash begin\{matrix\}\backslash operatorname\{ch\}\{u\}=\backslash operatorname\{ch\}\{u\_\{1\}\}\backslash operatorname\; ch\{u\_\{2\}\}+\backslash operatorname\{sh\}\{u\_\{1\}\}\backslash operatorname\{sh\}\{u\_\{2\}\}\backslash cos\backslash alpha\backslash \backslash$

\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\

v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)

\end{matrix}

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

w:Henry Crozier Keating Plummer (1910) defined the Lorentz boost in terms of trigonometric functionsPlummer (1910), p. 256

:$\backslash begin\{matrix\}\backslash tau=t\backslash sec\backslash beta-x\backslash tan\backslash beta/U\backslash \backslash$

\xi=x\sec\beta-Ut\tan\beta\\

\eta=y,\ \zeta=z,\\

\hline \sin\beta=v/U

\end{matrix}

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:Ignatowski (1910), pp. 973–974Ignatowski (1910/11), p. 13

:

dt' & =-pqn\ dx+p\ dt\\

p & =\frac{1}{\sqrt{1-q^{2}n}}

\end{align}

The variable n can be seen as a space-time constant whose value has to be determined by experiment or taken from a known physical law such as electrodynamics. For that purpose, Ignatowski used the above-mentioned Heaviside ellipsoid representing a contraction of electrostatic fields by x/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when n=1/c², resulting in p=γ and the Lorentz transformation. 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),Frank & Rothe (1911), pp. 825ff; (1912), p. 750ff. with various authors developing similar methods in subsequent years.Baccetti (2011), see references 1–25 therein.

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.Klein (1908), p. 165

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 $\backslash omega=\backslash sqrt\{-1\}$, which he also related to the speed of light by setting ω²=-c². He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations:Noether (1910), pp. 939–943

:$\backslash begin\{matrix\}V=\backslash frac\{Q\_\{1\}vQ\_\{2\}\}\{T\_\{1\}T\_\{2\}\}\backslash \backslash$

\hline X^{2}+Y^{2}+Z^{2}+\omega^{2}S^{2}=x^{2}+y^{2}+z^{2}+\omega^{2}s^{2}\\

\hline \begin{align}V & =Xi+Yj+Zk+\omega S\\

v & =xi+yj+zk+\omega s\\

Q_{1} & =(+Ai+Bj+Ck+D)+\omega(A'i+B'j+C'k+D')\\

Q_{2} & =(-Ai-Bj-Ck+D)+\omega(A'i+B'j+C'k-D')\\

T_{1}T_{2} & =T_{1}^{2}=T_{2}^{2}=A^{2}+B^{2}+C^{2}+D^{2}+\omega^{2}\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)

\end{align}

\end{matrix}

Besides citing quaternion related standard works by Arthur Cayley (1854), Noether referred to the entries in Klein's encyclopedia by Eduard Study (1899) and the French version by Élie Cartan (1908).Cartan & Study (1908), sections 35–36 Cartan's version contains a description of Study's dual numbers, Clifford's biquaternions (including the choice $\backslash omega=\backslash sqrt\{-1\}$ for hyperbolic geometry), and Clifford algebra, with references to Stephanos (1883), Buchheim (1884–85), Vahlen (1901–02) and others.

Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:Klein (1910), p. 300

:

& \quad-\left(i_{1}x_{0}+i_{2}y_{0}+i_{3}z_{0}+ict_{0}\right)

\end{align}

=\frac{\left[\begin{align} & \left(i_{1}(A+iA')+i_{2}(B+iB')+i_{3}(C+iC')+i_{4}(D+iD')\right)\\

& \quad\cdot\left(i_{1}x+i_{2}y+i_{3}z+ict\right)\\

& \quad\quad\cdot\left(i_{1}(A-iA')+i_{2}(B-iB')+i_{3}(C-iC')-(D-iD')\right)

\end{align}

\right]}{\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)-\left(A^{2}+B^{2}+C^{2}+D^{2}\right)}\\

\hline \text{where}\\

AA'+BB'+CC'+DD'=0\\

A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}

\end{matrix}

or in March 1911Klein (1911), pp. 602ff.

:$\backslash begin\{matrix\}g\text{'}=\backslash frac\{pg\backslash pi\}\{M\}\backslash \backslash$

\hline \begin{align}g & =\sqrt{-1}ct+ix+jy+kz\\

g' & =\sqrt{-1}ct'+ix'+jy'+kz'\\

p & =(D+\sqrt{-1}D')+i(A+\sqrt{-1}A')+j(B+\sqrt{-1}B')+k(C+\sqrt{-1}C')\\

\pi & =(D-\sqrt{-1}D')-i(A-\sqrt{-1}A')-j(B-\sqrt{-1}B')-k(C-\sqrt{-1}C')\\

M & =\left(A^{2}+B^{2}+C^{2}+D^{2}\right)-\left(A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}\right)\\

& AA'+BB'+CC'+DD'=0\\

& A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime2}+B^{\prime2}+C^{\prime2}+D^{\prime2}

\end{align}

\end{matrix}

Arthur W. Conway in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:Conway (1911), p. 8

:

\mathtt{\sigma} & =\mathbf{a}\mathtt{\sigma}'\mathbf{a}^{-1}

\end{align}

\\

e=\mathbf{a}^{-1}e'\mathbf{a}^{-1}\\

\hline a=\left(1-hc^{-1}\lambda\right)^{\frac{1}{2}}\left(1+c^{-2}\lambda^{2}\right)^{-\frac{1}{4}}

\end{matrix}

Also Ludwik Silberstein in November 1911Silberstein (1911/12), p. 793 as well as in 1914,Silberstein (1914), p. 156 formulated the Lorentz transformation in terms of velocity v:

:$\backslash begin\{matrix\}q\text{'}=QqQ\backslash \backslash$

\hline \begin{align}q & =\mathbf{r}+l=xi+yj+zk+\iota ct\\

q & '=\mathbf{r}'+l'=x'i+y'j+z'k+\iota ct'\\

Q & =\frac{1}{\sqrt{2}}\left(\sqrt{1+\gamma}+\mathrm{u}\sqrt{1-\gamma}\right)\\

& =\cos\alpha+\mathrm{u}\sin\alpha=e^{\alpha\mathrm{u}}\\

& \left\{ \gamma=\left(1-v^{2}/c^{2}\right)^{-1/2},\ 2\alpha=\operatorname{arctg}\ \left(\iota\frac{v}{c}\right)\right\}

\end{align}

\end{matrix}

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

{{Further|Lorentz transformation#Vector transformations}}

Vladimir Ignatowski (1910, published 1911) showed how to reformulate the Lorentz transformation in order to allow for arbitrary velocities and coordinates:Ignatowski (1910/11a), p. 23; (1910/11b), p. 22

:

b' & =pb-pqn\mathfrak{A}\mathfrak{c}_{0}\\

\\

\mathfrak{A} & =\mathfrak{A}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}'+pqb'\mathfrak{c}_{0}\\

b & =pb'+pqn\mathfrak{A}'\mathfrak{c}_{0}

\end{align}

\right.\end{matrix}\\

\left[\mathfrak{v}=\mathbf{u},\ \mathfrak{A}=\mathbf{x},\ b=t,\ \mathfrak{c}_{0}=\frac{\mathbf{v}}{v},\ p=\gamma,\ n=\frac{1}{c^{2}}\right]

\end{matrix}

Gustav Herglotz (1911)Herglotz (1911), p. 497 also showed how to formulate the transformation in order to allow for arbitrary velocities and coordinates v=(v_x, v_y, v_z) and r=(x, y, z):

:

\hline \left.\begin{align}x^{0} & =x+\alpha u(ux+vy+wz)-\beta ut\\

y^{0} & =y+\alpha v(ux+vy+wz)-\beta vt\\

z^{0} & =z+\alpha w(ux+vy+wz)-\beta wt\\

t^{0} & =-\beta(ux+vy+wz)+\beta t\\

& \alpha=\frac{1}{\sqrt{1-s^{2}}\left(1+\sqrt{1-s^{2}}\right)},\ \beta=\frac{1}{\sqrt{1-s^{2}}}

\end{align}

\right| & \begin{align}x' & =x+\alpha v_{x}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{x}t\\

y' & =y+\alpha v_{y}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{y}t\\

z' & =z+\alpha v_{z}\left(v_{x}x+v_{y}y+v_{z}z\right)-\gamma v_{z}t\\

t' & =-\gamma\left(v_{x}x+v_{y}y+v_{z}z\right)+\gamma t\\

& \alpha=\frac{\gamma^{2}}{\gamma+1},\ \gamma=\frac{1}{\sqrt{1-v^{2}}}

\end{align}

\end{matrix}

This was simplified using vector notation by Ludwik Silberstein (1911 on the left, 1914 on the right):Silberstein (1911/12), p. 792; (1914), p. 123

:$\backslash begin\{array\}\{c|c\}$

\begin{align}\mathbf{r}' & =\mathbf{r}+(\gamma-1)(\mathbf{ru})\mathbf{u}+i\beta\gamma lu\\

l' & =\gamma\left[l-i\beta(\mathbf{ru})\right]

\end{align}

& \begin{align}\mathbf{r}' & =\mathbf{r}+\left[\frac{\gamma-1}{v^{2}}(\mathbf{vr})-\gamma t\right]\mathbf{v}\\

t' & =\gamma\left[t-\frac{1}{c^{2}}(\mathbf{vr})\right]

\end{align}

\end{array}

Equivalent formulas were also given by Wolfgang Pauli (1921),Pauli (1921), p. 555 with Erwin Madelung (1922) providing the matrix formMadelung (1921), p. 207

:$\backslash begin\{array\}\{c|c|c|c|c\}$

& x & y & z & t\\

\hline x' & 1-\frac{v_{x}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) & -\frac{v_{x}v_{y}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) & -\frac{v_{x}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) & \frac{-v_{x}}{\sqrt{1-\beta^{2}}}\\

y' & -\frac{v_{x}v_{y}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) & 1-\frac{v_{y}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) & -\frac{v_{y}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) & \frac{-v_{y}}{\sqrt{1-\beta^{2}}}\\

z' & -\frac{v_{x}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) & -\frac{v_{y}v_{z}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) & 1-\frac{v_{z}^{2}}{v^{2}}\left(1-\frac{1}{\sqrt{1-\beta^{2}}}\right) & \frac{-v_{z}}{\sqrt{1-\beta^{2}}}\\

t' & \frac{-v_{x}}{c^{2}\sqrt{1-\beta^{2}}} & \frac{-v_{y}}{c^{2}\sqrt{1-\beta^{2}}} & \frac{-v_{z}}{c^{2}\sqrt{1-\beta^{2}}} & \frac{1}{\sqrt{1-\beta^{2}}}

\end{array}

These formulas were called "general Lorentz transformation without rotation" by Christian Møller (1952),Møller (1952/55), pp. 41–43 who in addition gave an even more general Lorentz transformation in which the Cartesian axes have different orientations, using a rotation operator $\backslash mathfrak\{D\}$. In this case, v′=(v′_x, v′_y, v′_z) is not equal to -v=(-v_x, -v_y, -v_z), but the relation $\backslash mathbf\{v\}\text{'}=-\backslash mathfrak\{D\}\backslash mathbf\{v\}$ holds instead, with the result

:$\backslash begin\{array\}\{c\}$

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

t' & =\gamma\left(t-(\mathbf{v}\cdot\mathbf{x})/c^{2}\right)

\end{align}

\end{array}

Émile 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:Borel (1913/14), p. 39

:$\backslash begin\{matrix\}x^\{2\}+y^\{2\}-z^\{2\}-1=0\backslash \backslash$

\hline {\scriptstyle \begin{align}\delta a & =\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, & \delta b & =2(\lambda\mu+\nu\rho), & \delta c & =-2(\lambda\nu+\mu\rho),\\

\delta a' & =2(\lambda\mu-\nu\rho), & \delta b' & =-\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, & \delta c' & =2(\lambda\rho-\mu\nu),\\

\delta a & =2(\lambda\nu-\mu\rho), & \delta b & =2(\lambda\rho+\mu\nu), & \delta c'' & =-\left(\lambda^{2}+\mu^{2}+\nu^{2}+\rho^{2}\right),

\end{align}

}\\

\left(\delta=\lambda^{2}+\mu^{2}-\rho^{2}-\nu^{2}\right)\\

\lambda=\nu=0\rightarrow\text{Hyperbolic rotation}

\end{matrix}

In four dimensions:Borel (1913/14), p. 41

:$\backslash begin\{matrix\}F=\backslash left(x\_\{1\}-x\_\{2\}\backslash right)^\{2\}+\backslash left(y\_\{1\}-y\_\{2\}\backslash right)^\{2\}+\backslash left(z\_\{1\}-z\_\{2\}\backslash right)^\{2\}-\backslash left(t\_\{1\}-t\_\{2\}\backslash right)^\{2\}\backslash \backslash$

\hline {\scriptstyle \begin{align} & \left(\mu^{2}+\nu^{2}-\alpha^{2}\right)\cos\varphi+\left(\lambda^{2}-\beta^{2}-\gamma^{2}\right)\operatorname{ch}{\theta} & & -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi-\gamma\operatorname{sh}{\theta}\\

& -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi+\gamma\operatorname{sh}{\theta} & & \left(\mu^{2}+\nu^{2}-\beta^{2}\right)\cos\varphi+\left(\mu^{2}-\alpha^{2}-\gamma^{2}\right)\operatorname{ch}{\theta}\\

& -(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi-\beta\operatorname{sh}{\theta} & & -(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\lambda\sin\varphi+\alpha\operatorname{sh}{\theta}\\

& (\gamma\mu-\beta\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta} & & -(\alpha\nu-\lambda\gamma)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\

\\

& \quad-(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi+\beta\operatorname{sh}{\theta} & & \quad(\beta\nu-\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta}\\

& \quad-(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})-\lambda\sin\varphi-\alpha\operatorname{sh}{\theta} & & \quad(\lambda\gamma-\alpha\nu)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\

& \quad\left(\lambda^{2}+\mu^{2}-\gamma^{2}\right)\cos\varphi+\left(\nu^{2}-\alpha^{2}-\beta^{2}\right)\operatorname{ch}{\theta} & & \quad(\alpha\mu-\beta\lambda)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta}\\

& \quad(\beta\gamma-\alpha\mu)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta} & & \quad-\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\cos\varphi+\left(\lambda^{2}+\mu^{2}+\nu^{2}\right)\operatorname{ch}{\theta}

\end{align}

}\\

\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\lambda^{2}-\mu^{2}-\nu^{2}=-1\right)

\end{matrix}

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:Gruner (1921a),

:$\backslash begin\{matrix\}v=\backslash alpha\backslash cdot\; c;\backslash quad\backslash beta=\backslash frac\{1\}\{\backslash sqrt\{1-\backslash alpha^\{2\}\}\}\backslash \backslash$

\sin\varphi=\alpha;\quad\beta=\frac{1}{\cos\varphi};\quad\alpha\beta=\tan\varphi\\

\hline x'=\frac{x}{\cos\varphi}-t\cdot\tan\varphi,\quad t'=\frac{t}{\cos\varphi}-x\cdot\tan\varphi

\end{matrix}

In another paper Gruner used the alternative relations:Gruner (1921b)

:$\backslash begin\{matrix\}\backslash alpha=\backslash frac\{v\}\{c\};\backslash \; \backslash beta=\backslash frac\{1\}\{\backslash sqrt\{1-\backslash alpha^\{2\}\}\};\backslash \backslash$

\cos\theta=\alpha=\frac{v}{c};\ \sin\theta=\frac{1}{\beta};\ \cot\theta=\alpha\cdot\beta\\

\hline x'=\frac{x}{\sin\theta}-t\cdot\cot\theta,\quad t'=\frac{t}{\sin\theta}-x\cdot\cot\theta

\end{matrix}

• Derivations of the Lorentz transformations
• History of special relativity

{{Wikiversity inline|History of Topics in Special Relativity/mathsource}}

{{Reflist|3|group=R}}

• {{Cite book | author=Abraham, M.| year=1905 | chapter=§ 42. Die Lichtzeit in einem gleichförmig bewegten System|title= Theorie der Elektrizität: Elektromagnetische Theorie der Strahlung | publisher=Teubner | location=Leipzig}}
• {{Cite journal|author=Bateman, Harry|year=1910|orig-year=1909|title=The Transformation of the Electrodynamical Equations |journal=Proceedings of the London Mathematical Society|volume=8|pages=223–264|doi=10.1112/plms/s2-8.1.223|title-link=s:en:The Transformation of the Electrodynamical Equations}}
• {{Cite journal|author=Bateman, Harry|year=1912|orig-year=1910|title=Some geometrical theorems connected with Laplace's equation and the equation of wave motion |journal=American Journal of Mathematics|volume=34|issue=3|doi=10.2307/2370223|pages=325–360|url=https://archive.org/details/jstor-2370223|jstor=2370223}}
• {{Cite book|author=Borel, Émile |year=1914|title=Introduction Geometrique à quelques Théories Physiques|publisher=Gauthier-Villars|location=Paris|url=http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=04710001}}
• {{Cite journal |author=Brill, J.|year=1925 |journal=Mathematical Proceedings of the Cambridge Philosophical Society|title= Note on the Lorentz group|pages=630–632|volume=22|issue=5 |doi=10.1017/S030500410000949X|bibcode=1925PCPS...22..630B|s2cid=121117536 }}
• {{Cite book |author=Bucherer, A. H. |year=1904 |title=Mathematische Einführung in die Elektronentheorie |publisher=Teubner |location=Leipzig |url=https://archive.org/details/mathematischeei02buchgoog}}
• {{Citation |author=Bucherer, A. H. |year=1908 |title=Messungen an Becquerelstrahlen. Die experimentelle Bestätigung der Lorentz-Einsteinschen Theorie. (Measurements of Becquerel rays. The Experimental Confirmation of the Lorentz-Einstein Theory) |journal=Physikalische Zeitschrift |volume=9 |issue=22 |pages=758–762}}. For Minkowski's and Voigt's statements see p. 762.
• {{Cite journal|author=Cartan, Élie|year=1912|journal=Société de Mathématique the France – Comptes Rendus des Séances|title=Sur les groupes de transformation de contact et la Cinématique nouvelle|pages=23|url=https://archive.org/stream/bulletinsocit40soci#page/422/mode/2up}}
• {{Citation |author=Cohn, Emil |year=1904a |title=Zur Elektrodynamik bewegter Systeme I |trans-title=On the Electrodynamics of Moving Systems I |journal= Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften |volume=1904/2 |issue=40 |pages =1294–1303|url=https://archive.org/details/sitzungsberichte1904deut}}
• {{Citation |author=Cohn, Emil |year=1904b |title=Zur Elektrodynamik bewegter Systeme II |trans-title=On the Electrodynamics of Moving Systems II |journal= Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften |volume=1904/2 |issue=43 |pages =1404–1416|url=https://archive.org/details/sitzungsberichte1904deut}}
• {{Cite journal|author=Conway, A. W.|year=1911|title=On the application of quaternions to some recent developments of electrical theory|journal=Proceedings of the Royal Irish Academy, Section A|volume=29|pages=1–9|url= https://archive.org/download/proceedingsofro29roya}}
• {{Cite journal|author=Cunningham, Ebenezer|year=1910|orig-year=1909|title=The principle of Relativity in Electrodynamics and an Extension Thereof|journal=Proceedings of the London Mathematical Society |volume=8|pages=77–98|doi=10.1112/plms/s2-8.1.77|title-link=s:en:The principle of Relativity in Electrodynamics and an Extension Thereof}}
• {{Citation |author=Einstein, Albert |year=1905 |title=Zur Elektrodynamik bewegter Körper |journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_891-921.pdf |doi=10.1002/andp.19053221004|bibcode = 1905AnP...322..891E |doi-access=free }}. See also: [http://www.fourmilab.ch/etexts/einstein/specrel/ English translation].
• {{cite journal|author=Frank, Philipp |year=1909 |title=Die Stellung des Relativitätsprinzips im System der Mechanik und Elektrodynamik|journal=Wiener Sitzungsberichte IIA |volume=118 |pages=373–446|hdl=2027/mdp.39015073682224 |url=http://hdl.handle.net/2027/mdp.39015073682224}}
• {{Cite journal|author1=Frank, Philipp |author2=Rothe, Hermann|year=1911|title=Über die Transformation der Raum-Zeitkoordinaten von ruhenden auf bewegte Systeme|journal=Annalen der Physik|volume=339|issue=5|pages=825–855|url=http://gallica.bnf.fr/ark:/12148/bpt6k15337j/f845.table|doi=10.1002/andp.19113390502|bibcode = 1911AnP...339..825F }}
• {{Cite journal|author1=Frank, Philipp |author2=Rothe, Hermann|title=Zur Herleitung der Lorentztransformation|journal=Physikalische Zeitschrift|volume=13|year=1912|pages=750–753}}
• {{Citation |author=Gans, Richard |year=1905 |title=H. A. Lorentz. Elektromagnetische Vorgänge |trans-title=H.A. Lorentz: Electromagnetic Phenomena |journal= Beiblätter zu den Annalen der Physik |volume=29 |issue=4 |pages =168–170|url=https://archive.org/details/beibltterzudena18pockgoog}}
• {{cite journal|author1=Gruner, Paul |author2=Sauter, Josef |name-list-style=amp |title=Représentation géométrique élémentaire des formules de la théorie de la relativité|journal=Archives des sciences physiques et naturelles|series=5|volume=3|pages=295–296|year=1921a|url=http://gallica.bnf.fr/ark:/12148/bpt6k2991536/f295.image|trans-title=Elementary geometric representation of the formulas of the special theory of relativity}}
• {{cite journal|author=Gruner, Paul|title=Eine elementare geometrische Darstellung der Transformationsformeln der speziellen Relativitätstheorie|journal=Physikalische Zeitschrift|volume=22|pages=384–385|year=1921b|trans-title=An elementary geometrical representation of the transformation formulas of the special theory of relativity}}
• {{Citation |author=Heaviside, Oliver |year=1889 |title=On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric |journal=Philosophical Magazine |series=5 |volume=27 |issue=167 |pages=324–339 |doi=10.1080/14786448908628362 |url=https://zenodo.org/record/1431195 }}
• {{Citation|author=Herglotz, Gustav|year=1910|orig-year=1909|title=Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper|trans-title=Wikisource translation: On bodies that are to be designated as "rigid" from the standpoint of the relativity principle|journal=Annalen der Physik|volume=336|issue=2 |pages=393–415|doi=10.1002/andp.19103360208|bibcode = 1910AnP...336..393H|url=https://zenodo.org/record/1424161}}
• {{Cite journal|author=Herglotz, G.|year=1911|title=Über die Mechanik des deformierbaren Körpers vom Standpunkte der Relativitätstheorie|journal=Annalen der Physik|volume=341|issue=13|pages=493–533|doi=10.1002/andp.19113411303|url=http://gallica.bnf.fr/ark:/12148/bpt6k153397.image.f509|bibcode=1911AnP...341..493H}}; English translation by David Delphenich: [http://www.neo-classical-physics.info/uploads/3/0/6/5/3065888/herglotz_-_rel._cont._mech..pdf On the mechanics of deformable bodies from the standpoint of relativity theory].
• {{Cite journal|author=Ignatowsky, W. v.|title=Einige allgemeine Bemerkungen über das Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=11|year=1910|pages=972–976|title-link=s:de:Einige allgemeine Bemerkungen über das Relativitätsprinzip}}
• {{Cite journal|author=Ignatowski, W. v.|title=Das Relativitätsprinzip|journal=Archiv der Mathematik und Physik|volume=18|year=1911|orig-year=1910|pages=17–40|title-link=s:de:Das Relativitätsprinzip (Ignatowski)}}
• {{Cite journal|author=Ignatowski, W. v.|title=Eine Bemerkung zu meiner Arbeit: "Einige allgemeine Bemerkungen zum Relativitätsprinzip"|journal=Physikalische Zeitschrift|volume=12|year=1911|pages=779|title-link=s:de:Eine Bemerkung zu meiner Arbeit: "Einige allgemeine Bemerkungen zum Relativitätsprinzip"}}
• {{Cite book|author=Klein, F.|editor=Hellinger, E.|year=1908|title=Elementarmethematik vom höheren Standpunkte aus. Teil I. Vorlesung gehalten während des Wintersemesters 1907-08|publisher=Teubner|location=Leipzig|url=https://archive.org/details/elementarmathem00kleigoog}}
• {{Cite book|author=Klein, Felix|title=Gesammelte Mathematische Abhandlungen |year=1921|orig-year=1910|chapter=Über die geometrischen Grundlagen der Lorentzgruppe|volume=1|pages=533–552|doi=10.1007/978-3-642-51960-4_31|title-link=s:de:Über die geometrischen Grundlagen der Lorentzgruppe|doi-broken-date=1 November 2024 |isbn=978-3-642-51898-0}}
• {{Cite book|author=Klein, F. |author2=Sommerfeld A.|editor=Noether, Fr.|year=1910|title=Über die Theorie des Kreisels. Heft IV|location=Leipzig|publisher=Teuber|url=https://archive.org/details/fkleinundasommer019696mbp}}
• {{Cite book|author=Klein, F.|editor=Hellinger, E.|year=1911|title=Elementarmethematik vom höheren Standpunkte aus. Teil I (Second Edition). Vorlesung gehalten während des Wintersemesters 1907-08|publisher=Teubner|location=Leipzig|hdl=2027/mdp.39015068187817}}
• {{Citation |author=Larmor, Joseph |year=1897 |title=On a Dynamical Theory of the Electric and Luminiferous Medium, Part 3, Relations with material media |journal=Philosophical Transactions of the Royal Society |volume=190 |pages=205–300 |doi=10.1098/rsta.1897.0020|bibcode = 1897RSPTA.190..205L |title-link=s:Dynamical Theory of the Electric and Luminiferous Medium III |doi-access=free }}
• {{Citation |author=Larmor, Joseph |year=1929 |orig-year=1897|title=Mathematical and Physical Papers: Volume II |chapter=On a Dynamical Theory of the Electric and Luminiferous Medium. Part 3: Relations with material media|pages=2–132|publisher=Cambridge University Press|isbn=978-1-107-53640-1}} (Reprint of Larmor (1897) with new annotations by Larmor.)
• {{Citation |author=Larmor, Joseph |year=1900 |title=Aether and Matter |publisher=Cambridge University Press|title-link=s:Aether and Matter }}
• {{Cite journal|author=Larmor, Joseph|title=On the intensity of the natural radiation from moving bodies and its mechanical reaction|journal=Philosophical Magazine|year=1904a|volume=7|issue=41|pages=[https://archive.org/details/londonedinburgh671904lond/page/578 578]–586|url=https://archive.org/details/londonedinburgh671904lond|doi=10.1080/14786440409463149}}
• {{Cite journal|author=Larmor, Joseph|title=On the ascertained Absence of Effects of Motion through the Aether, in relation to the Constitution of Matter, and on the FitzGerald-Lorentz Hypothesis|journal=Philosophical Magazine|volume=7|issue=42|year=1904b|pages=621–625|doi=10.1080/14786440409463156|title-link=s:en:Absence of Effects of Motion through the Aether}}
• {{Citation |author=Lorentz, Hendrik Antoon |year=1892a |title=La Théorie electromagnétique de Maxwell et son application aux corps mouvants |journal=Archives Néerlandaises des Sciences Exactes et Naturelles |volume=25 |pages=363–552 |url=https://archive.org/details/lathorielectrom00loregoog}}
• {{Citation |last=Lorentz |first=Hendrik Antoon |year=1892b |title=De relatieve beweging van de aarde en den aether |trans-title=The Relative Motion of the Earth and the Aether |journal=Zittingsverlag Akad. V. Wet. |pages=74–79 |volume=1}}
• {{Citation |author=Lorentz, Hendrik Antoon |year=1895 |title=Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern |trans-title=Attempt of a Theory of Electrical and Optical Phenomena in Moving Bodies |location=Leiden |publisher=E.J. Brill|title-link=s:de:Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern }}
• {{Citation |author=Lorentz, Hendrik Antoon |year=1899 |title=Simplified Theory of Electrical and Optical Phenomena in Moving Systems |journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences |volume=1 |pages=427–442|title-link=s:Simplified Theory of Electrical and Optical Phenomena in Moving Systems |bibcode=1898KNAB....1..427L }}
• {{Citation |author=Lorentz, Hendrik Antoon |year=1904 |title=Electromagnetic phenomena in a system moving with any velocity smaller than that of light |journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences |volume=6 |pages=809–831|title-link=s:Electromagnetic phenomena |bibcode=1903KNAB....6..809L }}
• {{Citation |author=Lorentz, Hendrik Antoon |year=1916|orig-year=1915|title=The theory of electrons and its applications to the phenomena of light and radiant heat |url=https://archive.org/details/electronstheory00lorerich |place=Leipzig & Berlin |publisher=B.G. Teubner}}
• {{Citation |doi=10.1002/andp.19153521505 |author=Minkowski, Hermann |orig-year=1907|year=1915 |title=Das Relativitätsprinzip |journal=Annalen der Physik |volume=352 |issue=15 |pages=927–938|bibcode = 1915AnP...352..927M |title-link=s:de:Das Relativitätsprinzip (Minkowski) }}
• {{Citation |author=Minkowski, Hermann |year=1908 |orig-year=1907 |title=Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern |trans-title=The Fundamental Equations for Electromagnetic Processes in Moving Bodies |journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse |pages=53–111|title-link=s:Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern }}
• {{Citation |author=Minkowski, Hermann |year=1909 |orig-year=1908 |title=Space and Time |journal=Physikalische Zeitschrift |volume=10 |pages=75–88|title-link=s:Space and Time }}
• {{Cite journal|author=Müller, Hans Robert|author-link=Hans Robert Müller|year=1948|journal=Monatshefte für Mathematik und Physik|title=Zyklographische Betrachtung der Kinematik der speziellen Relativitätstheorie|volume=52| issue=4 |pages=337–353|doi=10.1007/BF01525338|s2cid=120150204|url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN00246988X}}
• {{citation|author=Plummer, H.C.K.|year=1910 |title=On the Theory of Aberration and the Principle of Relativity|journal=Monthly Notices of the Royal Astronomical Society|volume=40 |issue=3 |pages=252–266|doi=10.1093/mnras/70.3.252 |bibcode=1910MNRAS..70..252P|doi-access=free}}
• {{Citation |author=Poincaré, Henri |year=1900 |title=La théorie de Lorentz et le principe de réaction |journal=Archives Néerlandaises des Sciences Exactes et Naturelles |volume=5 |pages=252–278|title-link=s:fr:La théorie de Lorentz et le principe de réaction }}. See also the [http://www.physicsinsights.org/poincare-1900.pdf English translation].
• {{Citation |author=Poincaré, Henri |year=1906 |orig-year=1904 |chapter=The Principles of Mathematical Physics |title=Congress of arts and science, universal exposition, St. Louis, 1904 |volume=1 |pages=604–622 |publisher=Houghton, Mifflin and Company |location=Boston and New York}}
• {{Citation |author=Poincaré, Henri |year=1905 |title=Sur la dynamique de l'électron |trans-title=On the Dynamics of the Electron |journal=Comptes Rendus |volume=140 |pages=1504–1508|title-link=s:fr:Sur la dynamique de l'électron (juin) }}
• {{Citation |author=Poincaré, Henri |year=1906 |orig-year=1905 |title=Sur la dynamique de l'électron |trans-title=On the Dynamics of the Electron |journal=Rendiconti del Circolo Matematico di Palermo |volume=21 |pages=129–176 |doi=10.1007/BF03013466|bibcode=1906RCMP...21..129P |title-link=s:fr:Sur la dynamique de l'électron (juillet) |hdl=2027/uiug.30112063899089 |s2cid=120211823 |hdl-access=free }}
• {{Cite journal|author=Poincaré, Henri|year=1921|orig-year=1912|title=Rapport sur les travaux de M. Cartan (fait à la Faculté des sciences de l'Université de Paris)|journal=Acta Mathematica|volume=38|issue=1|pages=137–145|doi=10.1007/bf02392064 |url=https://archive.org/stream/actamathematica38upps#page/n153/mode/2up|doi-access=free}} Written by Poincaré in 1912, printed in Acta Mathematica in 1914 though belatedly published in 1921.
• {{Citation |author=Searle, George Frederick Charles |year=1897 |title=On the Steady Motion of an Electrified Ellipsoid |journal=Philosophical Magazine |series=5 |volume=44 |issue=269 |pages=329–341 |doi=10.1080/14786449708621072|title-link=s:On the Steady Motion of an Electrified Ellipsoid }}
• {{Citation | author=Silberstein, L. | year=1912 | orig-year=1911|title=Quaternionic form of relativity| journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science | volume =23 | issue=137|pages =790–809|url=https://archive.org/details/londonedinburg6231912lond | doi=10.1080/14786440508637276}}
• {{Citation|author=Sommerfeld, A.|year=1909|title=Über die Zusammensetzung der Geschwindigkeiten in der Relativtheorie|trans-title=Wikisource translation: On the Composition of Velocities in the Theory of Relativity|journal=Verh. Dtsch. Phys. Ges.|volume=21|pages=577–582}}
• {{Citation |author=Thomson, Joseph John |year=1889 |title=On the Magnetic Effects produced by Motion in the Electric Field |journal=Philosophical Magazine |volume=28 |series=5 |issue=170 |pages=1–14 |doi=10.1080/14786448908619821|title-link=s:On the Magnetic Effects produced by Motion in the Electric Field }}
• {{Citation | author=Varićak, V. | year=1910 | title=Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie | journal = Physikalische Zeitschrift| volume =11| pages =93–6|trans-title=Application of Lobachevskian Geometry in the Theory of Relativity}}
• {{Citation | author=Varičak, V. | year=1912 | title=Über die nichteuklidische Interpretation der Relativtheorie |trans-title=On the Non-Euclidean Interpretation of the Theory of Relativity | journal=Jahresbericht der Deutschen Mathematiker-Vereinigung | volume =21 | pages =103–127| title-link=s:de:Über die nichteuklidische Interpretation der Relativtheorie }}
• {{Citation |author=Voigt, Woldemar |year=1887 |title=Ueber das Doppler'sche Princip |trans-title=On the Principle of Doppler |journal=Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen |issue=2 |pages=41–51|title-link=s:de:Ueber das Doppler'sche Princip }}
• {{Cite journal |last1=Wien |first1=Wilhelm |year=1904 |title=Zur Elektronentheorie |journal=Physikalische Zeitschrift |volume=5 |issue=14 |pages=393–395 |title-link=s:de:Zur Elektronentheorie }}

{{Reflist|3}}

• {{Cite journal|author1=Baccetti, Valentina |author2=Tate, Kyle |author3=Visser, Matt |year=2012|title=Inertial frames without the relativity principle|journal=Journal of High Energy Physics|volume=2012 |issue=5 |pages=119|doi=10.1007/JHEP05(2012)119 |arxiv=1112.1466|bibcode = 2012JHEP...05..119B |s2cid=118695037 }}
• {{Cite book|author=Bachmann, P.|year=1898|title=Die Arithmetik der quadratischen Formen. Erste Abtheilung|location=Leipzig|url=https://gallica.bnf.fr/ark:/12148/bpt6k994828|publisher=B.G. Teubner}}
• {{Cite book|author=Bachmann, P.|year=1923|title=Die Arithmetik der quadratischen Formen. Zweite Abtheilung|location=Leipzig|url=https://archive.org/details/arithdquadrachen02bachrich|publisher=B.G. Teubner}}
• {{Cite journal|author=Barnett, J. H. |year=2004|journal=Mathematics Magazine|volume=77|issue=1|title=Enter, stage center: The early drama of the hyperbolic functions|pages=15–30|url=https://www.maa.org/sites/default/files/pdf/cms_upload/321922717729.pdf.bannered.pdf|doi=10.1080/0025570x.2004.11953223|s2cid=121088132}}
• {{Cite book|author=Bôcher, Maxim|year=1907|title=Introduction to higher algebra|chapter=Quadratic forms|publisher=Macmillan|location=New York|chapter-url=https://archive.org/details/cu31924002936536}}
• {{Cite book|title=Relativity and Common Sense|last=Bondi|first=Hermann|publisher=Doubleday & Company|year=1964|location=New York|url=https://archive.org/details/RelativityCommonSense}}
• {{Cite book|author=Bonola, R.|title=Non-Euclidean geometry: A critical and historical study of its development|year=1912|location=Chicago|publisher=Open Court|url=https://archive.org/details/noneuclideangeom00bono}}
• {{Citation |author=Brown, Harvey R. |title= The origins of length contraction: I. The FitzGerald-Lorentz deformation hypothesis |journal=American Journal of Physics |year=2001 |volume=69 |issue=10 |pages=1044–1054 |url=http://philsci-archive.pitt.edu/archive/00000218/ |doi=10.1119/1.1379733|arxiv = gr-qc/0104032 |bibcode = 2001AmJPh..69.1044B|s2cid= 2945585 }} See also "Michelson, FitzGerald and Lorentz: the origins of relativity revisited", [http://philsci-archive.pitt.edu/987/ Online].
• {{Cite journal |author=Cartan, É. |author2=Study, E. |year=1908 |title=Nombres complexes|journal= Encyclopédie des Sciences Mathématiques Pures et Appliquées|volume=1|issue=1 |pages=328–468|url=http://gallica.bnf.fr/ark:/12148/bpt6k2440f/f173.image}}
• {{Cite journal|author=Cartan, É. |author2=Fano, G.|year=1955|orig-year=1915|journal=Encyclopédie des Sciences Mathématiques Pures et Appliquées|volume=3|title=La théorie des groupes continus et la géométrie|issue=1|pages=39–43|url=http://gallica.bnf.fr/ark:/12148/bpt6k29100t/f194.image}} (Only pages 1–21 were published in 1915, the entire article including pp. 39–43 concerning the groups of Laguerre and Lorentz was posthumously published in 1955 in Cartan's collected papers, and was reprinted in the Encyclopédie in 1991.)
• {{Cite book|author=Coolidge, Julian|author-link=Julian Coolidge|year=1916|title=A treatise on the circle and the sphere|publisher=Clarendon Press|location=Oxford|title-link=A Treatise on the Circle and the Sphere}}
• {{Citation |author=Darrigol, Olivier |year=2000 |title=Electrodynamics from Ampère to Einstein |isbn=978-0-19-850594-5 |publisher=Oxford Univ. Press |location=Oxford |url-access=registration |url=https://archive.org/details/electrodynamicsf0000darr }}
• {{Citation |author=Darrigol, Olivier |title=The Genesis of the theory of relativity |year=2005 |journal=Séminaire Poincaré |volume=1 |pages=1–22 |url=http://www.bourbaphy.fr/darrigol2.pdf |doi=10.1007/3-7643-7436-5_1|isbn=978-3-7643-7435-8 |bibcode=2006eins.book....1D }}
• {{Cite book|author=Dickson, L.E.|year=1923|title=History of the theory of numbers, Volume III, Quadratic and higher forms|location=Washington|url=https://archive.org/details/historyoftheoryo03dickuoft|publisher=Washington Carnegie Institution of Washington}}
• {{Cite journal |author=Fjelstad, P.|year=1986 |journal=American Journal of Physics |title= Extending special relativity via the perplex numbers|pages=416–422|volume=54|issue=5 |doi=10.1119/1.14605|bibcode=1986AmJPh..54..416F}}
• {{Cite journal |author=Girard, P. R.|year=1984 |journal=European Journal of Physics |title= The quaternion group and modern physics|pages=25–32|volume=5|issue=1|doi=10.1088/0143-0807/5/1/007|bibcode=1984EJPh....5...25G|s2cid=250775753 }}
• {{Cite journal|author=Gray, J.|title=Non-euclidean geometry—A re-interpretation|year=1979|journal=Historia Mathematica|volume=6|issue=3|pages=236–258|doi=10.1016/0315-0860(79)90124-1|doi-access=free}}
• {{Cite book|author=Gray, J. |author2=Scott W.|year=1997|title=Trois suppléments sur la découverte des fonctions fuchsiennes|chapter=Introduction|pages=7–28|location=Berlin|chapter-url=http://scottwalter.free.fr/papers/3supintro.pdf|url=http://henripoincarepapers.univ-lorraine.fr/chp/hp-pdf/hp1997tsa.pdf }}
• {{Cite book|author=Hawkins, Thomas|year=2013|title=The Mathematics of Frobenius in Context: A Journey Through 18th to 20th Century Mathematics|chapter=The Cayley–Hermite problem and matrix algebra|publisher=Springer|isbn=978-1461463337}}
• {{Citation |author=Janssen, Michel |year=1995 |title=A Comparison between Lorentz's Ether Theory and Special Relativity in the Light of the Experiments of Trouton and Noble (Thesis) |url=http://www.mpiwg-berlin.mpg.de/litserv/diss/janssen_diss/}}
• {{Cite journal|author=Kastrup, H. A.|title=On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics|journal=Annalen der Physik|volume=520|issue=9–10|year=2008|pages=631–690|arxiv=0808.2730|doi=10.1002/andp.200810324|bibcode = 2008AnP...520..631K |s2cid=12020510}}
• {{Citation |author=Katzir, Shaul |year=2005 |journal=Physics in Perspective |title=Poincaré's Relativistic Physics: Its Origins and Nature |pages=268–292 |volume=7 |doi=10.1007/s00016-004-0234-y |issue=3|bibcode = 2005PhP.....7..268K |s2cid=14751280 }}
• {{Cite book|author=Klein, F.|year=1897|orig-year=1896|title=The Mathematical Theory of the Top|publisher=Scribner|location=New York|url=https://archive.org/details/mathematicaltheo00kleiuoft}}
• {{Cite book|author1=Klein, Felix |author2=Blaschke, Wilhelm |year=1926|title=Vorlesungen über höhere Geometrie|publisher=Springer|location=Berlin|url=http://resolver.sub.uni-goettingen.de/purl?PPN373601816}}
• {{Cite book|author=von Laue, M.|year=1921|title=Die Relativitätstheorie, Band 1|edition=fourth edition of "Das Relativitätsprinzip"|publisher=Vieweg|url=https://archive.org/details/dierelativitts01laueuoft}}; First edition 1911, second expanded edition 1913, third expanded edition 1919.
• {{Cite journal |author=Lorente, M.|year=2003 |journal=Symmetries in Science |title= Representations of classical groups on the lattice and its application to the field theory on discrete space-time|pages=437–454|volume=VI |arxiv=hep-lat/0312042|bibcode=2003hep.lat..12042L}}
• {{Citation |author=Macrossan, M. N. |year=1986 |journal=The British Journal for the Philosophy of Science |title=A Note on Relativity Before Einstein |pages=232–234 |volume=37 |issue=2 |url=http://espace.library.uq.edu.au/view/UQ:9560 |doi=10.1093/bjps/37.2.232|citeseerx=10.1.1.679.5898 }}
• {{Cite book|author=Madelung, E.|title=Die mathematischen Hilfsmittel des Physikers|year=1922|publisher=Springer|location=Berlin|url=https://archive.org/details/diemathematisch00madegoog}}
• {{Cite journal|author=Majerník, V.|year=1986|journal=American Journal of Physics|title=Representation of relativistic quantities by trigonometric functions|volume=54|issue=6|pages=536–538|doi=10.1119/1.14557|bibcode=1986AmJPh..54..536M}}
• {{Cite journal |author=Meyer, W.F.|year=1899 |journal=Encyclopädie der Mathematischen Wissenschaften|title= Invariantentheorie|pages=322–455|volume=1 |issue=1 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360504671}}
• {{Citation |author=Miller, Arthur I. |year=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3 |url-access=registration |url=https://archive.org/details/alberteinsteinss0000mill }}
• {{Cite book|author=Møller, C.|title=The theory of relativity|year=1955|orig-year=1952|publisher=Oxford Clarendon Press|url=https://archive.org/details/theoryofrelativi029229mbp}}
• {{Cite journal|author=Müller, Emil|author-link=Emil Müller (mathematician)|year=1910|journal=Encyclopädie der Mathematischen Wissenschaften|volume=3.1.1|title=Die verschiedenen Koordinatensysteme|pages=596–770|url=http://resolver.sub.uni-goettingen.de/purl?PPN360609635}}
• {{Cite journal |author=Musen, P. |year=1970|title=A Discussion of Hill's Method of Secular Perturbations...|journal=Celestial Mechanics|volume=2|issue=1 |pages=41–59|bibcode=1970CeMec...2...41M|doi=10.1007/BF01230449|hdl=2060/19700018328|s2cid=122335532|hdl-access=free}}
• {{Cite book|author=Naimark, M. A. |year=2014|orig-year=1964|title=Linear Representations of the Lorentz Group|location=Oxford|isbn=978-1483184982 }}
• {{Cite journal|author=Pacheco, R.|year=2008|journal=Geometriae Dedicata|volume=146|issue=1|title=Bianchi–Bäcklund transforms and dressing actions, revisited.|pages=85–99|doi=10.1007/s10711-009-9427-5|arxiv=0808.4138|s2cid=14356965}}
• {{Citation |author=Pais, Abraham |year=1982 |title= Subtle is the Lord: The Science and the Life of Albert Einstein |place = New York |publisher=Oxford University Press |isbn=978-0-19-520438-4}}
• {{Citation |author=Pauli, Wolfgang|author-link=Wolfgang Pauli|year=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title= Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}
  In English: {{cite book|author=Pauli, W.|title=Theory of Relativity|volume=165|publisher=Dover Publications|year=1981|orig-year=1921|isbn=978-0-486-64152-2}}
• {{Citation |author=Penrose, R. |author2=Rindler W. |year=1984 |title=Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields |publisher=Cambridge University Press|isbn=978-0521337076}}
• {{Citation |author=Plummer, H. C. |year=1910 |title=On the Theory of Aberration and the Principle of Relativity |journal=Monthly Notices of the Royal Astronomical Society|volume=70|issue=3 |pages=252–266|doi=10.1093/mnras/70.3.252|doi-access=free |bibcode=1910MNRAS..70..252P}}
• {{Cite book|author=Ratcliffe, J. G.|year=1994|title=Foundations of Hyperbolic Manifolds|chapter=Hyperbolic geometry|pages=[https://archive.org/details/foundationsofhyp0000ratc/page/56 56–104]|location=New York|isbn=978-0387943480|chapter-url=https://archive.org/details/foundationsofhyp0000ratc/page/56}}
• {{Cite journal|author=Reynolds, W. F.|year=1993|title=Hyperbolic geometry on a hyperboloid|journal=The American Mathematical Monthly|volume=100|issue=5|pages=442–455|jstor=2324297|doi=10.1080/00029890.1993.11990430|s2cid=124088818 }}
• {{Cite book|author=Rindler, W.|year=2013|orig-year=1969|title=Essential Relativity: Special, General, and Cosmological|publisher=Springer|isbn=978-1475711356}}
• {{Cite book|author=Robinson, E.A.|year=1990|title=Einstein's relativity in metaphor and mathematics|publisher=Prentice Hall|isbn=9780132464970}}
• {{Cite book|author=Rosenfeld, B.A.|year=1988|title=A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space|publisher=Springer|location=New York|isbn=978-1441986801}}
• {{Cite journal|author=Rothe, H.|year=1916|journal=Encyclopädie der Mathematischen Wissenschaften|volume=3.1.1|title=Systeme geometrischer Analyse|pages=1282–1425|url=http://resolver.sub.uni-goettingen.de/purl?PPN360609767}}
• {{Cite book|author=Schottenloher, M.|year=2008|title=A Mathematical Introduction to Conformal Field Theory|publisher=Springer|isbn=978-3540706908}}
• {{Cite book|author=Silberstein, L.|year=1914|title=The Theory of Relativity|publisher=Macmillan|location=London|url=https://archive.org/details/theoryofrelativi00silbrich}}
• {{Cite journal |author=Sobczyk, G.|year=1995 |journal=The College Mathematics Journal |title= The Hyperbolic Number Plane|pages=268–280|volume=26 |issue=4|doi=10.2307/2687027|jstor=2687027 }}
• {{Cite book|author=Sommerville, D. M. L. Y. |year=1911|title=Bibliography of non-Euclidean geometry|location=London|url=https://archive.org/details/bibliographyofno00sommuoft|publisher=London Pub. by Harrison for the University of St. Andrews}}
• {{Citation |author=Synge, J. L. |year=1956 |title=Relativity: The Special Theory |publisher=North Holland}}
• {{Cite journal |author=Synge, J.L.|year=1972 |journal=Communications of the Dublin Institute for Advanced Studies |title= Quaternions, Lorentz transformations, and the Conway–Dirac–Eddington matrices|volume=21|url=http://repository.dias.ie/id/eprint/128}}
• {{Cite journal|author=Terng, C. L.|author2=Uhlenbeck, K.|name-list-style=amp|year=2000|journal=Notices of the AMS|volume=47|issue=1|title=Geometry of solitons|pages=17–25|url=https://www.ams.org/journals/notices/200001/fea-terng.pdf}}
• {{Cite journal |author=Touma, J. R. |author2=Tremaine, S. |author3=Kazandjian, M. V. |name-list-style=amp |year=2009|title=Gauss's method for secular dynamics, softened|journal=Monthly Notices of the Royal Astronomical Society|volume=394|issue=2 |pages=1085–1108|arxiv=0811.2812|doi=10.1111/j.1365-2966.2009.14409.x|doi-access=free |bibcode=2009MNRAS.394.1085T |s2cid=14531003 }}
• {{Cite journal |author=Volk, O.|year=1976 |journal=Celestial Mechanics |title= Miscellanea from the history of celestial mechanics|volume=14|issue=3|pages=365–382|url=http://adsabs.harvard.edu/full/1976CeMec..14..365V |doi=10.1007/bf01228523 |bibcode=1976CeMec..14..365V|s2cid=122955645 }}
• {{Cite book |author=Walter, Scott A.|year=1999 |editor1=H. Goenner |editor2=J. Renn |editor3=J. Ritter |editor4=T. Sauer |chapter= Minkowski, mathematicians, and the mathematical theory of relativity |title=The Expanding Worlds of General Relativity|series=Einstein Studies |volume=7 |pages=45–86 |location=Boston|publisher=Birkhäuser |chapter-url=http://scottwalter.free.fr/papers/1999-mmm-walter.html|isbn=978-0-8176-4060-6}}
• {{Cite book|author=Walter, Scott A.|year=1999b|editor=J. Gray|chapter=The non-Euclidean style of Minkowskian relativity|title=The Symbolic Universe: Geometry and Physics|pages=91–127|location=Oxford|publisher=Oxford University Press|chapter-url=http://scottwalter.free.fr/papers/1999-symbuniv-walter.html}}
• {{Cite book|author=Walter, Scott A.|chapter=Figures of Light in the Early History of Relativity (1905–1914) |series=Einstein Studies |title=Beyond Einstein|volume=14|pages=3–50|editor=Rowe D. |editor2=Sauer T. |editor3=Walter S.|location=New York|publisher=Birkhäuser|year=2018|chapter-url=https://philpapers.org/rec/WALFOL-2|doi=10.1007/978-1-4939-7708-6_1|isbn=978-1-4939-7708-6|s2cid=31840179 }}

• Mathpages: [http://mathpages.com/rr/s1-04/1-04.htm 1.4 The Relativity of Light]

{{History of physics}}

{{DEFAULTSORT:Lorentz transformations}}

Category:Equations

Category:History of physics

Category:Hendrik Lorentz

Category:Historical treatment of quaternions