The history of Lorentz transformations comprises the development of linear transformations forming the Lorentz group or Poincaré group preserving the Lorentz interval − x 0 2 + ⋯ + x n 2 {\displaystyle -x_{0}^{2}+\cdots +x_{n}^{2}} and the Minkowski inner product − x 0 y 0 + ⋯ + x n y n {\displaystyle -x_{0}y_{0}+\cdots +x_{n}y_{n}} . In mathematics, transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory of quadratic forms, hyperbolic geometry, Möbius geometry, and sphere geometry, which is connected to the fact that the group of motions in hyperbolic space, the Möbius group or projective special linear group, and the Laguerre group are isomorphic to the Lorentz group. In physics, Lorentz transformations became known at the beginning of the 20th century, when it was discovered that they exhibit the symmetry of Maxwell's equations. Subsequently, they became fundamental to all of physics, because they formed the basis of special relativity in which they exhibit the symmetry of Minkowski spacetime, making the speed of light invariant between different inertial frames. They relate the spacetime coordinates of two arbitrary inertial frames of reference with constant relative speed v. In one frame, the position of an event is given by x,y,z and time t, while in the other frame the same event has coordinates x′,y′,z′ and t′.
Mathematical prehistory Using the coefficients of a symmetric matrix g, the associated bilinear form, and a linear transformations in terms of transformation matrix A, the Lorentz transformation is given if the following conditions are satisfied:
− x 0 2 + ⋯ + x n 2 = − x 0 ′ 2 + ⋯ + x n ′ 2 − x 0 y 0 + ⋯ + x n y n = − x 0 ′ y 0 ′ + ⋯ + x n ′ y n ′ x ′ = A ⋅ x x = A − 1 ⋅ x ′ g ⋅ A T ⋅ g = A − 1 A T ⋅ g ⋅ A = g A ⋅ g ⋅ A T = g g = d i a g ( − 1 , 1 , … , 1 ) det A = ± 1 {\displaystyle {\begin{matrix}{\begin{aligned}-x_{0}^{2}+\cdots +x_{n}^{2}&=-x_{0}^{\prime 2}+\dots +x_{n}^{\prime 2}\\-x_{0}y_{0}+\cdots +x_{n}y_{n}&=-x_{0}^{\prime }y_{0}^{\prime }+\cdots +x_{n}^{\prime }y_{n}^{\prime }\end{aligned}}\\\hline {\begin{matrix}\mathbf {x} '=\mathbf {A} \cdot \mathbf {x} \\\mathbf {x} =\mathbf {A} ^{-1}\cdot \mathbf {x} '\end{matrix}}\\\hline {\begin{matrix}{\begin{aligned}\mathbf {g} \cdot \mathbf {A} ^{\mathrm {T} }\cdot \mathbf {g} &=\mathbf {A} ^{-1}\\\mathbf {A} ^{\rm {T}}\cdot \mathbf {g} \cdot \mathbf {A} &=\mathbf {g} \\\mathbf {A} \cdot \mathbf {g} \cdot \mathbf {A} ^{\mathrm {T} }&=\mathbf {g} \end{aligned}}\end{matrix}}\\\hline \mathbf {g} ={\rm {diag}}(-1,1,\dots ,1)\\\det \mathbf {A} =\pm 1\end{matrix}}}
It forms an indefinite orthogonal group called the Lorentz group O(1,n), while the case det A=+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. 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.
Electrodynamics and special relativity
Overview In 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:
− c 2 t 2 + x 2 + y 2 + z 2 = − c 2 t ′ 2 + x ′ 2 + y ′ 2 + z ′ 2 t ′ = γ ( t − x v c 2 ) x ′ = γ ( x − v t ) y ′ = y z ′ = z | t = γ ( t ′ + x v c 2 ) x = γ ( x ′ + v t ′ ) y = y ′ z = z ′ ⇒ ( c t ′ + x ′ ) = ( c t + x ) c + v c − v ( c t ′ − x ′ ) = ( c t − x ) c − v c + v {\displaystyle {\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}+z^{2}=-c^{2}t^{\prime 2}+x^{\prime 2}+y^{\prime 2}+z^{\prime 2}\\\hline \left.{\begin{aligned}t'&=\gamma \left(t-x{\frac {v}{c^{2}}}\right)\\x'&=\gamma (x-vt)\\y'&=y\\z'&=z\end{aligned}}\right|{\begin{aligned}t&=\gamma \left(t'+x{\frac {v}{c^{2}}}\right)\\x&=\gamma (x'+vt')\\y&=y'\\z&=z'\end{aligned}}\end{matrix}}\Rightarrow {\begin{aligned}(ct'+x')&=(ct+x){\sqrt {\frac {c+v}{c-v}}}\\(ct'-x')&=(ct-x){\sqrt {\frac {c-v}{c+v}}}\end{aligned}}}
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. 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. 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,
Voigt (1887) Woldemar Voigt (1887) developed a transformation in connection with the Doppler effect and an incompressible medium, being in modern notation:
original modern ξ 1 = x 1 − ϰ t η 1 = y 1 q ζ 1 = z 1 q τ = t − ϰ x 1 ω 2 q = 1 − ϰ 2 ω 2 | x ′ = x − v t y ′ = y γ z ′ = z γ t ′ = t − v x c 2 1 γ = 1 − v 2 c 2 {\displaystyle {\begin{matrix}{\text{original}}&{\text{modern}}\\\hline \left.{\begin{aligned}\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{aligned}}\right|&{\begin{aligned}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{aligned}}\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 {\displaystyle l} :
x ′ = γ l ( x − v t ) , y ′ = l y , z ′ = l z , t ′ = γ l ( t − x v c 2 ) {\displaystyle 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 Δ Ψ − 1 c 2 ∂ 2 Ψ ∂ t 2 = 0 {\displaystyle \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 ′ = γ l ( x − v t ) , y ′ = l y , z ′ = l z , t ′ = γ l ( t − v c 2 x ) {\displaystyle 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. 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:
E = ( q r r 2 ) ( 1 − v 2 sin 2 θ c 2 ) − 3 / 2 {\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) 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):
original modern z = { 1 − ω 2 v 2 } 1 2 z ′ |