Thehistory ofLorentz transformations comprises the development oflinear transformations forming theLorentz group orPoincaré group preserving theLorentz interval${\displaystyle -x_0^2+\cdots +x_n^2}$ and theMinkowski inner product${\displaystyle -x_0{y}_0+\cdots +x_n{y}_n}$.

Inmathematics, transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory ofquadratic forms,hyperbolic geometry,Möbius geometry, andsphere geometry, which is connected to the fact that the group ofmotions in hyperbolic space, theMöbius group orprojective special linear group, and theLaguerre group areisomorphic to theLorentz group.

Inphysics, Lorentz transformations became known at the beginning of the 20th century, when it was discovered that they exhibit the symmetry ofMaxwell's equations. Subsequently, they became fundamental to all of physics, because they formed the basis ofspecial relativity in which they exhibit the symmetry ofMinkowski spacetime, making thespeed of light invariant between different inertial frames. They relate the spacetime coordinates of two arbitraryinertial frames of reference with constant relative speedv. In one frame, the position of an event is given byx,y,z and timet, while in the other frame the same event has coordinatesx′,y′,z′ andt′.

Using the coefficients of asymmetric matrixA, the associatedbilinear form, and alinear transformations in terms oftransformation matrixg, the Lorentz transformation is given if the following conditions are satisfied:

It forms anindefinite orthogonal group called theLorentz group O(1,n), while the case detg=+1 forms the restrictedLorentz group SO(1,n). The quadratic form becomes theLorentz interval in terms of anindefinite quadratic form ofMinkowski space (being a special case ofpseudo-Euclidean space), and the associated bilinear form becomes theMinkowski inner product.^[1][2] Long before the advent of special relativity it was used in topics such as theCayley–Klein metric,hyperboloid model and other models ofhyperbolic geometry, computations ofelliptic functions and integrals, transformation ofindefinite quadratic forms,squeeze mappings of the hyperbola,group theory,Möbius transformations,spherical wave transformation, transformation of theSine-Gordon equation,Biquaternion algebra,split-complex numbers,Clifford algebra, and others.

Learning materials from Wikiversity:

• TheWikiversity: History of most general Lorentz transformations

includes contributions ofCarl Friedrich Gauss (1818),Carl Gustav Jacob Jacobi (1827, 1833/34),Michel Chasles (1829),Victor-Amédée Lebesgue (1837),Thomas Weddle (1847),Edmond Bour (1856),Osip Ivanovich Somov (1863),Wilhelm Killing (1878–1893),Henri Poincaré (1881),Homersham Cox (1881–1883),George William Hill (1882),Émile Picard (1882-1884),Octave Callandreau (1885),Sophus Lie (1885-1890),Louis Gérard (1892),Felix Hausdorff (1899),Frederick S. Woods (1901-05),Heinrich Liebmann (1904/05).

• TheWikiversity: History of Lorentz transformations via imaginary orthogonal transformation

includes contributions ofSophus Lie (1871),Hermann Minkowski (1907–1908),Arnold Sommerfeld (1909).

• TheWikiversity: History of Lorentz transformations via hyperbolic functions

includes contributions ofVincenzo Riccati (1757),Johann Heinrich Lambert (1768–1770),Franz Taurinus (1826),Eugenio Beltrami (1868),Charles-Ange Laisant (1874),Gustav von Escherich (1874),James Whitbread Lee Glaisher (1878),Siegmund Günther (1880/81),Homersham Cox (1881/82),Rudolf Lipschitz (1885/86),Friedrich Schur (1885-1902),Ferdinand von Lindemann (1890–91),Louis Gérard (1892),Wilhelm Killing (1893-97),Alfred North Whitehead (1897/98),Edwin Bailey Elliott (1903),Frederick S. Woods (1903),Heinrich Liebmann (1904/05),Philipp Frank (1909),Gustav Herglotz (1909/10),Vladimir Varićak (1910).

• TheWikiversity: History of Lorentz transformations via sphere transformation

includes contributions ofPierre Ossian Bonnet (1856),Albert Ribaucour (1870),Sophus Lie (1871a),Gaston Darboux (1873-87),Edmond Laguerre (1880),Cyparissos Stephanos (1883),Georg Scheffers (1899),Percey F. Smith (1900),Harry Bateman andEbenezer Cunningham (1909–1910).

• TheWikiversity: History of Lorentz transformations via Cayley–Hermite transformation

was used byArthur Cayley (1846–1855),Charles Hermite (1853, 1854),Paul Gustav Heinrich Bachmann (1869),Edmond Laguerre (1882),Gaston Darboux (1887),Percey F. Smith (1900),Émile Borel (1913).

• TheWikiversity: History of Lorentz transformations via Cayley–Klein parameters, Möbius and spin transformations

includes contributions ofCarl Friedrich Gauss (1801/63),Felix Klein (1871–97),Eduard Selling (1873–74),Henri Poincaré (1881),Luigi Bianchi (1888-93),Robert Fricke (1891–97),Frederick S. Woods (1895),Gustav Herglotz (1909/10).

• TheWikiversity: History of Lorentz transformations via quaternions and hyperbolic numbers

includes contributions ofJames Cockle (1848),Homersham Cox (1882/83),Cyparissos Stephanos (1883),Arthur Buchheim (1884),Rudolf Lipschitz (1885/86),Theodor Vahlen (1901/02),Fritz Noether (1910),Felix Klein (1910),Arthur W. Conway (1911),Ludwik Silberstein (1911).

• TheWikiversity: Lorentz transformation via trigonometric functions

includes contributions ofLuigi Bianchi (1886),Gaston Darboux (1891/94),Georg Scheffers (1899),Luther Pfahler Eisenhart (1905),Vladimir Varićak (1910),Henry Crozier Keating Plummer (1910),Paul Gruner (1921).

• TheWikiversity: History of Lorentz transformations via squeeze mappings

includes contributions ofAntoine André Louis Reynaud (1819),Felix Klein (1871),Charles-Ange Laisant (1874),Sophus Lie (1879-84),Siegmund Günther (1880/81),Edmond Laguerre (1882),Gaston Darboux (1883–1891),Rudolf Lipschitz (1885/86),Luigi Bianchi (1886–1894),Ferdinand von Lindemann (1890/91),Mellen W. Haskell (1895),Percey F. Smith (1900),Edwin Bailey Elliott (1903),Luther Pfahler Eisenhart (1905).

In thespecial relativity, Lorentz transformations exhibit the symmetry ofMinkowski spacetime by using a constantc as thespeed of light, and a parameterv as the relativevelocity between twoinertial reference frames. Using the above conditions, the Lorentz transformation in 3+1 dimensions assume the form:

In physics, analogous transformations have been introduced byVoigt (1887) related to an incompressible medium, and byHeaviside (1888), Thomson (1889), Searle (1896) andLorentz (1892, 1895) who analyzedMaxwell's equations. They were completed byLarmor (1897, 1900) andLorentz (1899, 1904), and brought into their modern form byPoincaré (1905) who gave the transformation the name of Lorentz.^[3] Eventually,Einstein (1905) showed in his development ofspecial relativity that the transformations follow from theprinciple of relativity and constant light speed alone by modifying the traditional concepts of space and time, without requiring amechanical aether in contradistinction to Lorentz and Poincaré.^[4]Minkowski (1907–1908) used them to argue that space and time are inseparably connected asspacetime.

Regarding special representations of the Lorentz transformations:Minkowski (1907–1908) andSommerfeld (1909) used imaginary trigonometric functions,Frank (1909) andVarićak (1910) usedhyperbolic functions,Bateman and Cunningham (1909–1910) usedspherical wave transformations,Herglotz (1909–10) used Möbius transformations,Plummer (1910) andGruner (1921) used trigonometric Lorentz boosts,Ignatowski (1910) derived the transformations without light speed postulate,Noether (1910) and Klein (1910) as wellConway (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)^{[R 1]} developed a transformation in connection with theDoppler effect and an incompressible medium, being in modern notation:^[5][6]

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 arescale,conformal, andLorentz invariant, so the combination is invariant too.^[6] For instance, Lorentz transformations can be extended by using factor${\displaystyle l}$:^{[R 2]}

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

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 aprinciple of relativity in general. It was demonstrated by Poincaré and Einstein that one has to setl=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,^[7] and that was acknowledged in 1909:

In a paper "Über das Doppler'sche Princip", published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (7) (§ 3 of this book) [namely${\displaystyle \mathrm{\Delta }\mathrm{\Psi }-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathrm{\Psi }}{\mathrm{\partial }{t}^2}=0}$] a transformation equivalent to the formulae (287) and (288) [namely${\displaystyle x^{\mathrm{\prime }}=\gamma l\left(x-vt\right),y^{\mathrm{\prime }}=ly,z^{\mathrm{\prime }}=lz,t^{\mathrm{\prime }}=\gamma l\left(t-\frac{v}{c^2}x\right)}$]. The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for thefree ether is contained in his paper.^{[R 3]}

AlsoHermann 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.^{[R 4]}

In 1888,Oliver Heaviside^{[R 5]} investigated the properties ofcharges in motion according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:^[8]

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

Consequently,Joseph John Thomson (1889)^{[R 6]} 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 theGalilean transformationz-vt in his equation^[9]):

Thereby,inhomogeneous electromagnetic wave equations are transformed into aPoisson equation.^[9] Eventually,George Frederick Charles Searle^{[R 7]} noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called "Heaviside-Ellipsoid" ofaxial ratio

^[9]

In order to explain theaberration of light and the result of theFizeau experiment in accordance withMaxwell'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)^{[R 8][10]}

wherex^* is theGalilean transformationx-vt. Except the additional γ in the time transformation, this is the complete Lorentz transformation.^[10] Whilet 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 theMichelson–Morley experiment, he (1892b)^{[R 9]} introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introducedlength contraction in his theory (without proof as he admitted). The same hypothesis had been made previously byGeorge 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 inv/c. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:^{[R 10]}

For solving optical problems Lorentz used the following transformation, in which the modified time variable was called "local time" (German:Ortszeit) by him:^{[R 11]}

With this concept Lorentz could explain theDoppler effect, theaberration of light, and theFizeau experiment.^[11]

In 1897, Larmor extended the work of Lorentz and derived the following transformation^{[R 12]}

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 theMichelson–Morley experiment. It's notable that Larmor was the first who recognized that some sort oftime 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".^[12][13] 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 ofv/c:^{[R 13]}

Nothing need be neglected: the transformation isexact ifv/c² is replaced byεv/c² in the equations and also in the change following fromt tot′, as is worked out inAether 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 timet″=t′-εvx′/c² instead of the 1897 expressiont′=t-vx/c² by replacingv/c² withεv/c², so thatt″ is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for thex′, y′, z′, t′ coordinates:^{[R 14]}

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 (wherex′=x-vt andt″ as given above) as:^{[R 15]}

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 inv/c" – it was later shown by Lorentz (1904) and Poincaré (1905) that they are indeed invariant under this transformation to all orders inv/c.

Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term "Lorentz transformation" for Lorentz's first order transformations of coordinates and field configurations:

p. 583: [..] Lorentz's transformation for passing from the field of activity of a stationary electrodynamic material system to that of one moving with uniform velocity of translation through the aether.
p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..]^{[R 16]}
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.^{[R 17]}

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 byx-vt):^{[R 18]}

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 oft′ has to be inserted):^{[R 19]}

This is equivalent to the complete Lorentz transformation when solved forx″ andt″ and with ε=1. Like Larmor, Lorentz noticed in 1899^{[R 20]} also some sort of time dilation effect in relation to the frequency of oscillating electrons"that inS the time of vibrations bekε times as great as inS₀", whereS₀ is the aether frame.^[14]

In 1904 he rewrote the equations in the following form by settingl=1/ε (again,x* must be replaced byx-vt):^{[R 21]}

Under the assumption thatl=1 whenv=0, he demonstrated thatl=1 must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factorl 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 inv/c. He also derived the correct formulas for the velocity dependence ofelectromagnetic mass, and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones.^{[R 22]} However, he didn't achieve full covariance of the transformation equations for charge density and velocity.^[15] When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:^[16]

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 byAlfred Bucherer in July 1904:^{[R 23]}

${\displaystyle x^{\mathrm{\prime }}=\sqrt{s}x,y^{\mathrm{\prime }}=y,z^{\mathrm{\prime }}=z,t^{\prime }=\frac{t}{\sqrt{s}}-\sqrt{s}\frac{u}{v^2}x,s=1-\frac{u^2}{v^2}}$

or byWilhelm Wien in July 1904:^{[R 24]}

${\displaystyle x=k{x}^{\prime },y=y^{\prime },z=z^{\prime },t^{\prime }=kt-\frac{v}{k{c}^2}x}$

or byEmil Cohn in November 1904 (setting the speed of light to unity):^{[R 25]}

${\displaystyle x=\frac{x_0}{k},y=y_0,z=z_0,t=k{t}_0,t_1=t_0-w\cdot r_0,k^2=\frac{1}{1-w^2}}$

or byRichard Gans in February 1905:^{[R 26]}

${\displaystyle x^{\mathrm{\prime }}=kx,y^{\mathrm{\prime }}=y,z^{\mathrm{\prime }}=z,t^{\prime }=\frac{t}{k}-\frac{kwx}{c^2},k^2=\frac{c^2}{c^2-w^2}}$

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.^[17] 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${\displaystyle c}$ in both directions, which lead to what is nowadays calledrelativity of simultaneity, although Poincaré's calculation does not involve length contraction or time dilation.^{[R 27]} In order to synchronise the clocks here on Earth (thex*, t* frame) a light signal from one clock (at the origin) is sent to another (atx*), and is sent back. It's supposed that the Earth is moving with speedv in thex-direction (=x*-direction) in some rest system (x, t) (i.e. theluminiferous aether system for Lorentz and Larmor). The time of flight outwards is

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

and the time of flight back is

${\displaystyle \delta t_b=\frac{x^{\ast }}{\left(c+v\right)}}$.

The elapsed time on the clock when the signal is returned isδt_a+δt_b and the timet*=(δt_a+δt_b)/2 is ascribed to the moment when the light signal reached the distant clock. In the rest frame the timet=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

${\displaystyle t^{\ast }=t-\frac{{\gamma }^{2}v{x}^{\ast }}{c^2}}$

identical to Lorentz (1892). By dropping the factor γ² under the assumption that${\displaystyle \frac{v^2}{c^2}\ll 1}$, Poincaré gave the resultt*=t-vx*/c², which is the form used by Lorentz in 1895.

Similar physical interpretations of local time were later given byEmil Cohn (1904)^{[R 28]} andMax Abraham (1905).^{[R 29]}

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:^{[R 30]}

.

Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name "Lorentz transformation".^[18][19] Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by settingl=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.^[20]

In July 1905 (published in January 1906)^{[R 31]} Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of theprinciple of least action; he demonstrated in more detail the group characteristics of the transformation, which he calledLorentz group, and he showed that the combinationx²+y²+z²-t² is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing${\displaystyle ct\sqrt{-1}}$ as a fourth imaginary coordinate, and he used an early form offour-vectors. He also formulated the velocity addition formula, which he had already derived in unpublished letters to Lorentz from May 1905:^{[R 32]}

${\displaystyle {\xi }^{\prime }=\frac{\xi +\epsilon }{1+\xi \epsilon },{\eta }^{\prime }=\frac{\eta }{k(1+\xi \epsilon )}}$.

On June 30, 1905 (published September 1905) Einstein published what is now calledspecial 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 inv/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.^[21][22][23]

The notation for this transformation is equivalent to Poincaré's of 1905, except that Einstein didn't set the speed of light to unity:^{[R 33]}

Einstein also defined the velocity addition formula:^{[R 34]}

${\displaystyle \begin{array}{c}x=\frac{w_{\xi }+v}{1+\frac{v{w}_{\xi }}{V^2}}t,y=\frac{\sqrt{1-{\left(\frac{v}{V}\right)}^2}}{1+\frac{v{w}_{\xi }}{V^2}}{w}_{\eta }t\\ U^2={\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2,w^2=w_{\xi }^2+w_{\eta }^2,\alpha =\mathrm{arctg}\frac{w_y}{w_x}\\ U=\frac{\sqrt{\left(v^2+w^2+2vw\cos \alpha \right)-{\left(\frac{vw\sin \alpha }{V}\right)}^2}}{1+\frac{vw\cos \alpha }{V^2}}\end{array}|\begin{array}{c}\frac{u_x-v}{1-\frac{u_{x}v}{V^2}}=u_{\xi }\\ \frac{u_y}{\beta \left(1-\frac{u_{x}v}{V^2}\right)}=u_{\eta }\\ \frac{u_z}{\beta \left(1-\frac{u_{x}v}{V^2}\right)}=u_{\zeta }\end{array}}$

and the light aberration formula:^{[R 35]}

${\displaystyle \cos {\phi }^{\prime }=\frac{\cos \phi -\frac{v}{V}}{1-\frac{v}{V}\cos \phi }}$

The work on the principle of relativity by Lorentz, Einstein,Planck, together with Poincaré's four-dimensional approach, were further elaborated and combined with thehyperboloid model byHermann Minkowski in 1907 and 1908.^{[R 36][R 37]} Minkowski particularly reformulated electrodynamics in a four-dimensional way (Minkowski spacetime).^[24] For instance, he wrotex, y, z, it in the formx₁, x₂, x₃, x₄. By defining ψ as the angle of rotation around thez-axis, the Lorentz transformation assumes the form (withc=1):^{[R 38]}

Even though Minkowski used the imaginary number iψ, he for once^{[R 38]} directly used thetangens hyperbolicus in the equation for velocity

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

Minkowski's expression can also by written as ψ=atanh(q) and was later calledrapidity. He also wrote the Lorentz transformation in matrix form:^{[R 39]}

As a graphical representation of the Lorentz transformation he introduced theMinkowski diagram, which became a standard tool in textbooks and research articles on relativity:^{[R 40]}

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 thespherical law of cosines:^{[R 41]}

Hyperbolic functions were used byPhilipp Frank (1909), who derived the Lorentz transformation usingψ asrapidity:^{[R 42]}

${\displaystyle \begin{array}{c}{x}^{\prime }=x\phi (a)\mathrm{c}\mathrm{h}\psi +t\phi (a)\mathrm{s}\mathrm{h}\psi \\ t^{\prime }=-x\phi (a)\mathrm{s}\mathrm{h}\psi +t\phi (a)\mathrm{c}\mathrm{h}\psi \\ \mathrm{t}\mathrm{h}\psi =-a,\mathrm{s}\mathrm{h}\psi =\frac{a}{\sqrt{1-a^2}},\mathrm{c}\mathrm{h}\psi =\frac{1}{\sqrt{1-a^2}},\phi (a)=1\\ x^{\prime }=\frac{x-at}{\sqrt{1-a^2}},y^{\prime }=y,z^{\prime }=z,t^{\prime }=\frac{-ax+t}{\sqrt{1-a^2}}\end{array}}$

In line withSophus Lie's (1871) research on the relation between sphere transformations with an imaginary radius coordinate and 4D conformal transformations, it was pointed out byBateman andCunningham (1909–1910), that by settingu=ict as the imaginary fourth coordinates one can produce spacetime conformal transformations. Not only the quadratic form${\displaystyle \lambda \left(d{x}^2+d{y}^2+d{z}^2+d{u}^2\right)}$, but alsoMaxwells equations are covariant with respect to these transformations, irrespective of the choice of λ. These variants of conformal or Lie sphere transformations were calledspherical wave transformations by Bateman.^{[R 43][R 44]} However, this covariance is restricted to certain areas such as electrodynamics, whereas the totality of natural laws in inertial frames is covariant under theLorentz group.^{[R 45]} In particular, by setting λ=1 the Lorentz groupSO(1,3) can be seen as a 10-parameter subgroup of the 15-parameter spacetime conformal groupCon(1,3).

Bateman (1910–12)^[25] also alluded to the identity between theLaguerre inversion and the Lorentz transformations. In general, the isomorphism between the Laguerre group and the Lorentz group was pointed out byÉlie Cartan (1912, 1915–55),^{[R 46]}Henri Poincaré (1912–21)^{[R 47]} and others.

FollowingFelix 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:^{[R 48]}

FollowingSommerfeld (1909), hyperbolic functions were used byVladimir Varićak in several papers starting from 1910, who represented the equations of special relativity on the basis ofhyperbolic geometry in terms of Weierstrass coordinates. For instance, by settingl=ct andv/c=tanh(u) withu as rapidity he wrote the Lorentz transformation:^{[R 49]}

and showed the relation of rapidity to theGudermannian function and theangle of parallelism:^{[R 49]}

${\displaystyle \frac{v}{c}=\mathrm{th}u=\mathrm{tg}\psi =\sin \mathrm{gd}(u)=\cos \mathrm{\Pi }(u)}$

He also related the velocity addition to thehyperbolic law of cosines:^{[R 50]}

${\displaystyle \begin{array}{c}\mathrm{ch}u=\mathrm{ch}{u}_1\mathrm{c}h{u}_2+\mathrm{sh}{u}_1\mathrm{sh}{u}_2\cos \alpha \\ \mathrm{ch}{u}_i=\frac{1}{\sqrt{1-{\left(\frac{v_i}{c}\right)}^2}},\mathrm{sh}{u}_i=\frac{v_i}{\sqrt{1-{\left(\frac{v_i}{c}\right)}^2}}\\ v=\sqrt{v_1^2+v_2^2-{\left(\frac{v_1{v}_2}{c}\right)}^2}\left(a=\frac{\pi }{2}\right)\end{array}}$

Subsequently, other authors such asE. T. Whittaker (1910) orAlfred 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 functions^{[R 51]}

${\displaystyle \begin{array}{c}\tau =t\sec \beta -x\tan \beta /U\\ \xi =x\sec \beta -Ut\tan \beta \\ \eta =y,\zeta =z,\\ \sin \beta =v/U\end{array}}$

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 relatedgroup theoretical principles) alone, in order to derive the following transformation between two inertial frames:^{[R 52][R 53]}

The variablen 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 byx/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation whenn=1/c², resulting inp=γ and the Lorentz transformation. Withn=0, no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved byPhilipp Frank andHermann Rothe (1911, 1912),^{[R 54]} with various authors developing similar methods in subsequent years.^[26]

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.^{[R 55]}

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${\displaystyle \omega =\sqrt{-1}}$, 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:^{[R 56]}

Besides citing quaternion related standard works byArthur Cayley (1854), Noether referred to the entries in Klein's encyclopedia byEduard Study (1899) and the French version byÉlie Cartan (1908).^[27] Cartan's version contains a description of Study'sdual numbers, Clifford's biquaternions (including the choice${\displaystyle \omega =\sqrt{-1}}$ 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:^{[R 57]}

or in March 1911^{[R 58]}