History of Topics in Special Relativity/Lorentz transformation (velocity)

Boosts
In the theory of relativity, Lorentz transformations exhibit the symmetry of Minkowski spacetime by using a constant c as the speed of light, and a parameter v as the relative velocity between two inertial reference frames. The corresponding formulas are identical to E:Lorentz transformations via hyperbolic functions introduced long before relativity was developed. In particular, the hyperbolic angle $$\eta$$ can be interpreted as the velocity related rapidity $$\tanh\eta=\beta=v/c$$, so that $$\gamma=\cosh\eta$$ is the Lorentz factor, $$\beta\gamma=\sinh\eta$$ the proper velocity, $$u'=c\tanh q$$ the velocity of another object, $$u=c\tanh(q+\eta)$$ the velocity-addition formula, thus transformation E:(3b) becomes:

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

Transformations analogous to (A) have been introduced by Voigt (1887) in terms of an incompressible medium, and by Lorentz (1892, 1895) who analyzed Maxwell's equations, they were completed by Larmor (1897, 1900) and Lorentz (1899, 1904), and brought into their modern form by Poincaré (1905) who gave the transformation the name of Lorentz. Eventually, Einstein (1905) showed in his development of special relativity that the transformations follow from the principle of relativity and constant light speed alone by modifying the traditional concepts of space and time, without requiring a mechanical aether in contradistinction to Lorentz and Poincaré. Minkowski (1907–1908) used them to argue that space and time are inseparably connected as spacetime.

The matrix form (B) is a special case of the general boost matrix given by Hahn (1912) in terms of imaginary time, while variant (C) for arbitrary k was given by many authors (see E:Lorentz transformations via squeeze mappings) with the choice equivalent to $$k=\sqrt{\tfrac{c+v}{c-v}}$$ given by Born (1921).

Velocity addition and aberration
In exact analogy to Beltrami coordinates in equation E:(3e), one can substitute $$\left[\tfrac{u_{x}}{c},\ \tfrac{u_{y}}{c},\ \tfrac{u_{z}}{c}\right]=\left[\tfrac{x}{ct},\ \tfrac{y}{ct},\ \tfrac{z}{ct}\right]$$ in ($$-A), producing the Lorentz transformation of velocities (or velocity addition formula):

By restriction to velocities in the $$\left[x,y\right]$$ plane and using trigonometric and hyperbolic identities as in equation E:(3f), it becomes the hyperbolic law of cosines:

and by further setting u=u′=c one gets the well known E:Kepler formulas (3g), which express the relativistic aberration of light:

Formulas ($$, $$) were given by Einstein (1905) and Poincaré (1905/06), while the relations to the spherical and hyperbolic law of cosines were given by Sommerfeld (1909) and Varićak (1910). The aberration formula for cos(α) was given by Einstein (1905).

Lorentz transformation in arbitrary directions
Lorentz boosts for arbitrary directions in line with E:general Lorentz transformation (1a) are in vector notation

and the vectorial velocity addition formula in line with E:general Lorentz transformation (1b) follows by:

The special case of parallel and perpendicular directions in ($$) was given by Minkowski (1907/8) while the complete transformation was formulated by Ignatowski (1910), Herglotz (1911), Tamaki (1911). General velocity addition ($$) was given in equivalent form by Ignatowski (1910).

Rewritten in matrix notation, the general Lorentz boost has the form:

While Minkowski (1907/8) formulated the matrix form of Lorentz transformations in general terms, he didn't explicitly express the velocity related components of the general boost matrix. A complete representation of ($$) was given by Hahn (1912).

Other formulations
Important contributions to the mathematical understanding of the Lorentz transformation of space and time also include: Minkowski (1907–1908) as well as Frank (1909) and Varićak (1910) showed the relation to imaginary and hyperbolic functions, Herglotz (1909/10) used exponential squeeze mappings and Möbius transformations, Ignatowski (1910) didn't use the light speed postulate, Klein and Noether (1908-11) as well as Conway and Silberstein (1911) used Biquaternions, Plummer (1910) and Gruner (1921) used trigonometric Lorentz boosts, Borel (1913–14) used Cayley-Hermite parameter.

Voigt (1887)
Woldemar Voigt (1887) developed a transformation in connection with the Doppler effect and an incompressible medium, being in modern notation:


 * $$\begin{matrix}\text{original} & \text{modern}\\

\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 ($$). In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Optical phenomena in free space are scale, conformal (using the factor λ discussed above), and Lorentz invariant, so the combination is invariant too. For instance, Lorentz transformations can be extended by using $$l=\sqrt{\lambda}$$:


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

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 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, 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 $\Delta\Psi-\tfrac{1}{c^{2}}\tfrac{\partial^{2}\Psi}{\partial t^{2}}=0$] a transformation equivalent to the formulae (287) and (288) [namely $x^{\prime}=\gamma l\left(x-vt\right),\ y^{\prime}=ly,\ z^{\prime}=lz,\ t^{\prime}=\gamma l\left(t-\tfrac{v}{c^{2}}x\right)$]. 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 the free ether is contained in his paper. |undefined"

Also Hermann Minkowski said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in 1887. Voigt responded in the same paper by saying that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.

Heaviside (1888), Thomson (1889), Searle (1896)
In 1888, Oliver Heaviside investigated the properties of charges in motion according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:


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

Consequently, Joseph John Thomson (1889) 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 ):


 * $$\begin{matrix}\text{original} & \text{modern}\\

\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. Eventually, George Frederick Charles Searle noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called "Heaviside-Ellipsoid" of axial ratio


 * $$\begin{matrix}\text{original} & \text{modern}\\

\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}$$

Lorentz (1892, 1895)
In order to explain the aberration of light and the result of the Fizeau experiment in accordance with Maxwell's equations, Lorentz in 1892 developed a model ("Lorentz ether theory") in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson)


 * $$\begin{matrix}\text{original} & \text{modern}\\

\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) introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced length contraction in his theory (without proof as he admitted). The same hypothesis was already made by George FitzGerald in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.

In 1895, Lorentz further elaborated on his theory and introduced the "theorem of corresponding states". This theorem states that a moving observer (relative to the ether) in his "fictitious" field makes the same observations as a resting observers in his "real" field for velocities to first order in v/c. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:


 * $$\begin{matrix}\text{original} & \text{modern}\\

\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" (Ortszeit) by him:


 * $$\begin{matrix}\text{original} & \text{modern}\\

\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.

Larmor (1897, 1900)
In 1897, Larmor extended the work of Lorentz and derived the following transformation


 * $$\begin{matrix}\text{original} & \text{modern}\\

\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". 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:

"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:


 * $$\begin{matrix}\text{original} & \text{modern}\\

\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:


 * $$\begin{matrix}\text{original} & \text{modern}\\

\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:

"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 [..] 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."

Lorentz (1899, 1904)
Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 (again, x* must be replaced by x-vt):


 * $$\begin{matrix}\text{original} & \text{modern}\\

\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):


 * $$\begin{matrix}\text{original} & \text{modern}\\

\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 1899 also some sort of time dilation effect in relation to the frequency of oscillating electrons "that in S the time of vibrations be kε times as great as in S0", where S0 is the aether frame.

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


 * $$\begin{matrix}\text{original} & \text{modern}\\

\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. However, he didn't achieve full covariance of the transformation equations for charge density and velocity. When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:

"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:


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

or by Wilhelm Wien in July 1904:


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

or by Emil Cohn in November 1904 (setting the speed of light to unity):


 * $$x=\frac{x_{0}}{k},\quad y=y_{0},\quad z=z_{0},\quad t=kt_{0},\quad t_{1}=t_{0}-w\cdot r_{0},\quad k^{2}=\frac{1}{1-w^{2}}$$

or by Richard Gans in February 1905:


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

Local time
Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, Henri Poincaré in 1900 commented on the origin of Lorentz's "wonderful invention" of local time. He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed $$c$$ 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. 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


 * $$\delta t_{a}=\frac{x^{\ast}}{\left(c-v\right)}$$

and the time of flight back is


 * $$\delta t_{b}=\frac{x^{\ast}}{\left(c+v\right)}$$.

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


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

identical to Lorentz (1892). By dropping the factor γ2 under the assumption that $$\tfrac{v^{2}}{c^{2}}\ll1$$, 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) and Max Abraham (1905).

Lorentz transformation
On June 5, 1905 (published June 9) Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form ($$):


 * $$\begin{align}x^{\prime} & =kl(x+\varepsilon t)\\

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". Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by setting l=1, and modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity, i.e. making them fully Lorentz covariant.

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


 * $$\xi'=\frac{\xi+\varepsilon}{1+\xi\varepsilon},\ \eta'=\frac{\eta}{k(1+\xi\varepsilon)}$$.

Einstein (1905) – Special relativity
On June 30, 1905 (published September 1905) Einstein published what is now called special relativity and gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. While Lorentz considered "local time" to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in v/c this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations concern the nature of space and time.

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


 * $$\begin{align}\tau & =\beta\left(t-\frac{v}{V^{2}}x\right)\\

\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 ($$, $$):


 * $$\begin{matrix}x=\frac{w_{\xi}+v}{1+\frac{vw_{\xi}}{V^{2}}}t,\ y=\frac{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}{1+\frac{vw_{\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=\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 ($$):


 * $$\cos\varphi'=\frac{\cos\varphi-\frac{v}{V}}{1-\frac{v}{V}\cos\varphi}$$

Imaginary Lorentz transformation
The work on the principle of relativity by Lorentz, Einstein, Planck, together with Poincaré's four-dimensional approach, were further elaborated and combined with the hyperboloid model by Hermann Minkowski in 1907 and 1908. Minkowski particularly reformulated electrodynamics in a four-dimensional way (Minkowski spacetime). For instance, he wrote x, y, z, it in the form x1, x2, x3, x4. By defining ψ as the angle of rotation around the z-axis, the Lorentz transformation ($$-A) assumes the form (with c=1):


 * $$\begin{align}x'_{1} & =x_{1}\\

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


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

Minkowski's expression can also by written as ψ=atanh(q) and was later called rapidity.

Vector representation
Minkowski wrote the Lorentz transformation ($$) in vectorial form for the special case of directions being only parallel ($$\mathfrak{r_{v}}$$) or perpendicular ($$\mathfrak{r_{\bar{v}}}$$) to the velocity:


 * $$\begin{matrix}\mathfrak{r'_{v}}=\frac{\mathfrak{r_{v}}-qt}{\sqrt{1-q^{2}}},\quad\mathfrak{r'_{\bar{v}}}=\mathfrak{r_{\bar{v}}},\quad t'=\frac{-q\mathfrak{r_{v}}+t}{\sqrt{1-q^{2}}}\\

\mathfrak{r_{v}}=\frac{\mathfrak{r'_{v}}+qt'}{\sqrt{1-q^{2}}},\quad\mathfrak{r_{\bar{v}}}=\mathfrak{r'_{\bar{v}}},\quad t=\frac{q\mathfrak{r'_{v}}+t'}{\sqrt{1-q^{2}}}\\ \left[\mathfrak{r}=\left(x,y,z\right)=\left(\mathfrak{r_{v}},\mathfrak{r_{\bar{v}}}\right),\ |\mathfrak{v}|=q\right] \end{matrix}$$

Matrix representation
Minkowski used matrices in order to write the E:general Lorentz transformation (1a), of which boost matrix ($$) is a special case:


 * $$\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}+x_{4}^{\prime2}\\

\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}$$

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



Ignatowski (1910)
While earlier derivations and formulations of the Lorentz transformation relied from the outset on optics, electrodynamics, or the invariance of the speed of light, Vladimir Ignatowski (1910) showed that it is possible to use the principle of relativity (and related group theoretical principles) alone, in order to derive the following transformation between two inertial frames:


 * $$\begin{align}dx' & =p\ dx-pq\ dt\\

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), with various authors developing similar methods in subsequent years.

Ignatowski, Herglotz, Tamaki (1910-11) – Vector transformation
Vladimir Ignatowski (1910, published 1911) defined the vectorial velocity addition ($$) as well as general Lorentz boost ($$) as


 * $$\begin{matrix}\begin{matrix}\mathfrak{v} =\frac{\mathfrak{v}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{v}'+pq\mathfrak{c}_{0}}{p\left(1+nq\mathfrak{c}_{0}\mathfrak{v}'\right)} & \left|\begin{align}\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}\\ \\ \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}$$

An equivalent transformation was given by Gustav Herglotz (1911) using v=(vx, vy, vz) and r=(x, y, z):


 * $$\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} $$

Kajuro Tamaki (1911) represented ($$) as follows (as his paper was based on a 4-vector calculus, Tamaki's schematic is not representing a matrix despite looking very similar to the boost matrix in ($$)):


 * $$\begin{matrix}\begin{array}{c|c|c|c|c}

& x'_{1} & x'_{2} & x'_{3} & x'_{4}\\ \hline x_{1} & 1+l^{2}\left(\cos\psi-1\right) & lm\left(\cos\psi-1\right) & ln\left(\cos\psi-1\right) & l\sin\psi\\ \hline x_{2} & lm\left(\cos\psi-1\right) & 1+m^{2}\left(\cos\psi-1\right) & mn\left(\cos\psi-1\right) & m\sin\psi\\ \hline x_{3} & ln\left(\cos\psi-1\right) & mn\left(\cos\psi-1\right) & 1+n^{2}\left(\cos\psi-1\right) & n\sin\psi\\ \hline x_{4} & -l\sin\psi & -m\sin\psi & -n\sin\psi & \cos\psi \end{array}\\ \hline \psi=i\varphi,\ -i\tan i\varphi=-\frac{v}{c},\ \cos i\varphi=\frac{1}{\sqrt{1-(v/c)^{2}}}=\beta,\ -\sin i\varphi=\frac{i(v/c)}{\sqrt{1-(v/c)^{2}}}=i\beta(v/c)\\ \hline \mathbf{r}=\mathbf{r}'+(\beta-1)\mathbf{v}_{1}(\mathbf{v}_{1}\mathbf{r}')+\beta\mathbf{v}t'\\ \mathbf{r}'=\mathbf{r}+(\beta-1)\mathbf{v}_{1}(\mathbf{v}_{1}\mathbf{r})-\beta\mathbf{v}t \end{matrix}$$

Hahn (1912) – Matrix transformation
Elaborating on Minkowski's (1907/8) matrix representation of the Lorentz transformations, Emil Hahn (1912) used matrix calculus in order to define the Lorentz boost for arbitrary directions (including the exponential form of the boost matrix) in line with ($$), using imaginary rapidity $$i\psi$$ and imaginary time $$x_4 =i\omega t$$:


 * $$\begin{matrix}\boldsymbol{x}'-\boldsymbol{x}'_{0}=\mathbb{I}_{-\mathbf{c}}(u)\boldsymbol{x}\\

\hline \begin{align}\mathbb{I}_{-\mathbf{c}}(u) & =\mathbb{J}\mathbb{G}_{\mathbf{c}}(u)\mathbb{J}^{-1} & (7,p.30)\\ & =\left(\begin{matrix}\mathbf{E}+(r-1)\mathbf{c}\overset{\perp}{\mathbf{c}}; & \frac{iur(u)}{\omega}\mathbf{c}\\ -\frac{iur(u)}{\omega}\overset{\perp}{\mathbf{c}}; & r(u) \end{matrix}\right) & (7,p.30)\\ & =\left(\begin{matrix}\mathbf{E}+(\cos i\psi-1)\mathbf{c}\overset{\perp}{\mathbf{c}}; & \sin i\psi\mathbf{c}\\ -\sin i\psi\mathbf{c}; & \cos i\psi \end{matrix}\right) & (8,p.30)\\ & =\left(\begin{matrix}1+(\cos i\psi-1)c_{1}c_{1} & (\cos i\psi-1)c_{1}c_{2} & (\cos i\psi-1)c_{1}c_{3} & \sin i\psi\,c_{1}\\ (\cos i\psi-1)c_{2}c_{1} & 1+(\cos i\psi-1)c_{2}c_{2} & (\cos i\psi-1)c_{2}c_{3} & \sin i\psi\,c_{2}\\ (\cos i\psi-1)c_{3}c_{1} & (\cos i\psi-1)c_{3}c_{2} & 1+(\cos i\psi-1)c_{3}c_{3} & \sin i\psi\,c_{3}\\ -\sin i\psi\,c_{1} & -\sin i\psi\,c_{2} & -\sin i\psi\,c_{3} & 1+(\cos i\psi-1) \end{matrix}\right) & (1,p.36)\\ \hline \mathbb{I}_{-\mathbf{c}}(u) & =\mathbb{E}-\sin i\psi\left\langle \mathbf{0},\mathbf{c}\right\rangle +(1-\cos i\psi)\left\langle \mathbf{0},\mathbf{c}\right\rangle ^{2} & (10,p.30)\\ \mathbb{I}_{\mathbf{c}}(u) & =e^{\left\langle \mathbf{0},\mathbf{c}\right\rangle i\psi} & (p.31) \end{align} \\ \hline \boldsymbol{x}=\left(\begin{matrix}x_{1} & 0 & 0 & 0\\ x_{2} & 0 & 0 & 0\\ x_{3} & 0 & 0 & 0\\ x_{4} & 0 & 0 & 0 \end{matrix}\right),\ \boldsymbol{x}'=\left(\begin{matrix}x_{1}^{\prime} & 0 & 0 & 0\\ x_{2}^{\prime} & 0 & 0 & 0\\ x_{3}^{\prime} & 0 & 0 & 0\\ x_{4}^{\prime} & 0 & 0 & 0 \end{matrix}\right),\ \mathbb{J}=\left(\begin{matrix}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & i\omega \end{matrix}\right),\ \mathbb{G}=\left(\begin{matrix}r_{11} & r_{12} & r_{13} & p_{1}\\ r_{21} & r_{22} & r_{23} & p_{2}\\ r_{31} & r_{32} & r_{33} & p_{3}\\ q_{1} & q_{2} & q_{3} & r \end{matrix}\right)\\ r(u)=\frac{1}{\sqrt{1-\frac{u^{2}}{\omega^{2}}}}=\cos i\psi,\ |\mathbb{I}|=1,\ \left\langle \mathbf{0},\mathbf{c}\right\rangle =\left(\begin{matrix}0 & 0 & 0 & c_{1}\\ 0 & 0 & 0 & c_{2}\\ 0 & 0 & 0 & c_{3}\\ c_{1} & c_{2} & c_{3} & 0 \end{matrix}\right) \end{matrix}$$