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′.
The general Lorentz transformation follows from (Q1) by setting A=A′=diag(-1,1,...,1) and det g=±1. It forms an indefinite orthogonal group called the Lorentz group O(1,n), while the case det g=+1 forms the restricted Lorentz group SO(1,n). The quadratic form q(x) becomes the Lorentz interval in terms of an indefinite quadratic form of Minkowski space (being a special case of pseudo-Euclidean space), and the associated bilinear form b(x) becomes the Minkowski inner product:[2][3]
If in (1a) are interpreted as homogeneous coordinates, then the corresponding inhomogenous coordinates follow by
so that the Lorentz transformation becomes a homography leaving invariant the equation of the unit sphere, which John Lighton Synge called "the most general formula for the composition of velocities" in terms of special relativity (the transformation matrix g stays the same as in (1a)):[8]
Lorentz transformation via imaginary orthogonal transformation[]
By using the imaginary quantities in x as well as (s=1,2...n) in g, the Lorentz transformation (1a) assumes the form of an orthogonal transformation of Euclidean space forming the orthogonal group O(n) if det g=±1 or the special orthogonal group SO(n) if det g=+1, the Lorentz interval becomes the Euclidean norm, and the Minkowski inner product becomes the dot product:[9]
(2a)
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The cases n=1,2,3,4 of orthogonal transformations in terms of real coordinates were discussed by Euler (1771) and in n dimensions by Cauchy (1829). The case in which one of these coordinates is imaginary and the other ones remain real was alluded to by Lie (1871) in terms of spheres with imaginary radius, while the interpretation of the imaginary coordinate as being related to the dimension of time as well as the explicit formulation of Lorentz transformations with n=3 was given by Minkowski (1907) and Sommerfeld (1909).
A well known example of this orthogonal transformation is spatial rotation in terms of trigonometric functions, which become Lorentz transformations by using an imaginary angle , so that trigonometric functions become equivalent to hyperbolic functions:
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Defining as real, spatial rotation in the form (2b-1) was introduced by Euler (1771) and in the form (2c-1) by Wessel (1799). The interpretation of (2b) as Lorentz boost (i.e. Lorentz transformation without spatial rotation) in which correspond to the imaginary quantities was given by Minkowski (1907) and Sommerfeld (1909). As shown in the next section using hyperbolic functions, (2b) becomes (3b) while (2c) becomes (3d).
Lorentz transformation via hyperbolic functions[]
The case of a Lorentz transformation without spatial rotation is called a Lorentz boost. The simplest case can be given, for instance, by setting n=1 in (1a):
or in matrix notation
(3a)
which resembles precisely the relations of hyperbolic functions in terms of hyperbolic angle. Thus by adding an unchanged -axis, a Lorentz boost or hyperbolic rotation for n=2 (being the same as a rotation around an imaginary angle in (2b) or a translation in the hyperbolic plane in terms of the hyperboloid model) is given by
or in matrix notation
(3b)
in which the rapidity can be composed of arbitrary many rapidities as per the angle sum laws of hyperbolic sines and cosines, so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between angle sum laws of circular trigonometry and spatial rotations. Alternatively, the hyperbolic angle sum laws themselves can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the unit hyperbola:
or in matrix notation
(3c)
Finally, Lorentz boost (3b) assumes a simple form by using squeeze mappings in analogy to Euler's formula in (2c):[10]
In the theory of relativity, Lorentz transformations exhibit the symmetry of Minkowski spacetime by using a constant c as the speed of light, and a parameter v as the relative velocity between two inertial reference frames. In particular, the hyperbolic angle in (3b) can be interpreted as the velocity related rapidity, so that is the Lorentz factor, the proper velocity, the velocity of another object, the velocity-addition formula, thus (3b) becomes:
(4a)
Or in four dimensions and by setting and adding an unchanged z the familiar form follows, using as Doppler factor:
In line with equation (1b), one can substitute in (3b) or (4a), producing the Lorentz transformation of velocities (or velocity addition formula) in analogy to Beltrami coordinates of (3e):
(4d)
or using trigonometric and hyperbolic identities it becomes the hyperbolic law of cosines in terms of (3f):[12][R 1][13]
(4e)
and by further setting u=u′=c the relativistic aberration of light follows:[17]
If one only requires the invariance of the light cone represented by the differential equation , which is the same as asking for the most general transformation that changes spheres into spheres, the Lorentz group can be extended by adding dilations represented by the factor λ. The result is the group Con(1,p) of spacetime conformal transformations in terms of special conformal transformations and inversions producing the relation
.
One can switch between two representations of this group by using an imaginary sphere radius coordinate x0=iR with the interval related to conformal transformations, or by using a real radius coordinate x0=R with the interval related to spherical wave transformations in terms of contact transformations preserving circles and spheres. It turns out that Con(1,3) is isomorphic to the special orthogonal group SO(2,4), and contains the Lorentz group SO(1,3) as a subgroup by setting λ=1. More generally, Con(q,p) is isomorphic to SO(q+1,p+1) and contains SO(q,p) as subgroup.[20] This implies that Con(0,p) is isomorphic to the Lorentz group of arbitrary dimensions SO(1,p+1). Consequently, the conformal group in the plane Con(0,2) – known as the group of Möbius transformations – is isomorphic to the Lorentz group SO(1,3).[21][22] This can be seen using tetracyclical coordinates satisfying the form .
A special case of Lie's geometry of oriented spheres is the Laguerre group, transforming oriented planes and lines into each other. It's generated by the Laguerre inversion leaving invariant with R as radius, thus the Laguerre group is isomorphic to the Lorentz group.[23][24]
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Both representations of Lie sphere geometry and conformal transformations were studied by Lie (1871) and others. It was shown by Bateman & Cunningham (1909–1910), that the group Con(1,3) is the most general one leaving invariant the equations of Maxwell's electrodynamics. Tetracyclical coordinates were discussed by Pockels (1891), Klein (1893), Bôcher (1894). The relation between Con(1,3) and the Lorentz group was noted by Bateman & Cunningham (1909–1910) and others.
The Laguerre inversion was introduced by Laguerre (1882) and discussed by Darboux (1887) and Smith (1900). A similar concept was studied by Scheffers (1899) in terms of contact transformations. Stephanos (1883) argued that Lie's geometry of oriented spheres in terms of contact transformations, as well as the special case of the transformations of oriented planes into each other (such as by Laguerre), provides a geometrical interpretation of Hamilton's biquaternions. The group isomorphism between the Laguerre group and Lorentz group was pointed out by Bateman (1910), Cartan (1912, 1915/55), Poincaré (1912/21) and others.
Lorentz transformation via Cayley–Hermite transformation[]
The general transformation (Q1) of any quadratic form into itself can also be given using arbitrary parameters based on the Cayley transform (I-T)−1·(I+T), where I is the identity matrix, T an arbitrary antisymmetric matrix, and by adding A as symmetric matrix defining the quadratic form (there is no primed A' because the coefficients are assumed to be the same on both sides):[25][26]
(Q2)
For instance, the choice A=diag(1,1,1) gives an orthogonal transformation which can be used to describe spatial rotations corresponding to the Euler-Rodrigues parameters[a,b,c,d] which can be interpreted as the coefficients of quaternions. Setting d=1, the equations have the form:
Also the Lorentz interval and the general Lorentz transformation in any dimension can be produced by the Cayley–Hermite formalism.[R 2][R 3][27][28] For instance, Lorentz transformation (1a) with n=1 follows from (Q2) with:
(5a)
This becomes Lorentz boost (4a or 4b) by setting , which is equivalent to the relation known from Loedel diagrams, thus (5a) can be interpreted as a Lorentz boost from the viewpoint of a "median frame" in which two other inertial frames are moving with equal speed in opposite directions.
Furthermore, Lorentz transformation (1a) with n=2 is given by:
(5b)
or using n=3:
(5c)
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The transformation of a binary quadratic form of which Lorentz transformation (5a) is a special case was given by Hermite (1854), equations containing Lorentz transformations (5a, 5b, 5c) as special cases were given by Cayley (1855), Lorentz transformation (5a) was given (up to a sign change) by Laguerre (1882), Darboux (1887), Smith (1900) in relation to Laguerre geometry, and Lorentz transformation (5b) was given by Bachmann (1869). In relativity, equations similar to (5b, 5c) were first employed by Borel (1913) to represent Lorentz transformations.
As described in equation (3d), the Lorentz interval is closely connected to the alternative form ,[29] which in terms of the Cayley–Hermite parameters is invariant under the transformation:
(5d)
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This transformation was given by Cayley (1884), even though he didn't relate it to the Lorentz interval but rather to .
Lorentz transformation via Cayley–Klein parameters, Möbius and spin transformations[]
The previously mentioned Euler-Rodrigues parameter a,b,c,d (i.e. Cayley-Hermite parameter in equation (Q3) with d=1) are closely related to Cayley–Klein parameter α,β,γ,δ in order to connect Möbius transformations and rotations:[30]
Also the Lorentz transformation can be expressed with variants of the Cayley–Klein parameters: One relates these parameters to a spin-matrix D, the spin transformations of variables (the overline denotes complex conjugate), and the Möbius transformation of . When defined in terms of isometries of hyperbolic space (hyperbolic motions), the Hermitian matrixu associated with these Möbius transformations produces an invariant determinant identical to the Lorentz interval. Therefore, these transformations were described by John Lighton Synge as being a "factory for the mass production of Lorentz transformations".[31] It also turns out that the related spin group Spin(3, 1) or special linear group SL(2, C) acts as the double cover of the Lorentz group (one Lorentz transformation corresponds to two spin transformations of different sign), while the Möbius group Con(0,2) or projective special linear group PSL(2, C) is isomorphic to both the Lorentz group and the group of isometries of hyperbolic space.
In space, the Möbius/Spin/Lorentz transformations can be written as:[32][31][33][34]
or in line with equation (1b) one can substitute so that the Möbius/Lorentz transformations become related to the unit sphere:
(6c)
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The general transformation u′ in (6a) was given by Cayley (1854), while the general relation between Möbius transformations and transformation u′ leaving invariant the generalized circle was pointed out by Poincaré (1883) in relation to Kleinian groups. The adaptation to the Lorentz interval by which (6a) becomes a Lorentz transformation was given by Klein (1889-1893, 1896/97), Bianchi (1893), Fricke (1893, 1897). Its reformulation as Lorentz transformation (6b) was provided by Bianchi (1893) and Fricke (1893, 1897). Lorentz transformation (6c) was given by Klein (1884) in relation to surfaces of second degree and the invariance of the unit sphere. In relativity, (6a) was first employed by Herglotz (1909/10).
In the plane, the transformations can be written as:[29][34]
(6d)
thus
(6e)
which includes the special case implying , reducing the transformation to a Lorentz boost in 1+1 dimensions:
(6f)
Finally, by using the Lorentz interval related to a hyperboloid, the Möbius/Lorentz transformations can be written
Lorentz transformation via quaternions and hyperbolic numbers[]
The Lorentz transformations can also be expressed in terms of biquaternions: A Minkowskian quaternion (or minquat) q having one real part and one purely imaginary part is multiplied by biquaternion a applied as pre- and postfactor. Using an overline to denote quaternion conjugation and * for complex conjugation, its general form (on the left) and the corresponding boost (on the right) are as follows:[36][37]
(7a)
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Hamilton (1844/45) and Cayley (1845) derived the quaternion transformation for spatial rotations, and Cayley (1854, 1855) gave the corresponding transformation leaving invariant the sum of four squares . Cox (1882/83) discussed the Lorentz interval in terms of Weierstrass coordinates in the course of adapting William Kingdon Clifford's biquaternions a+ωb to hyperbolic geometry by setting (alternatively, 1 gives elliptic and 0 parabolic geometry). Stephanos (1883) related the imaginary part of William Rowan Hamilton's biquaternions to the radius of spheres, and introduced a homography leaving invariant the equations of oriented spheres or oriented planes in terms of Lie sphere geometry. Buchheim (1884/85) discussed the Cayley absolute and adapted Clifford's biquaternions to hyperbolic geometry similar to Cox by using all three values of . Eventually, the modern Lorentz transformation using biquaternions with as in hyperbolic geometry was given by Noether (1910) and Klein (1910) as well as Conway (1911) and Silberstein (1911).
Often connected with quaternionic systems is the hyperbolic number, which also allows to formulate the Lorentz transformations:[38][39]
(7b)
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After the trigonometric expression (Euler's formula) was given by Euler (1748), and the hyperbolic analogue as well as hyperbolic numbers by Cockle (1848) in the framework of tessarines, it was shown by Cox (1882/83) that one can identify with associative quaternion multiplication. Here, is the hyperbolic versor with , while -1 denotes the elliptic or 0 denotes the parabolic counterpart (not to be confused with the expression in Clifford's biquaternions also used by Cox, in which -1 is hyperbolic). The hyperbolic versor was also discussed by Macfarlane (1892, 1894, 1900) in terms of hyperbolic quaternions. The expression for hyperbolic motions (and -1 for elliptic, 0 for parabolic motions) also appear in "biquaternions" defined by Vahlen (1901/02, 1905).
More extended forms of complex and (bi-)quaternionic systems in terms of Clifford algebra can also be used to express the Lorentz transformations. For instance, using a system a of Clifford numbers one can transform the following general quadratic form into itself, in which the individual values of can be set to +1 or -1 at will, while the Lorentz interval follows if the sign of one differs from all others.:[40][41]
(7c)
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The general definite form as well as the general indefinite form and their invariance under transformation (1) was discussed by Lipschitz (1885/86), while hyperbolic motions were discussed by Vahlen (1901/02, 1905) by setting in transformation (2), while elliptic motions follow with -1 and parabolic motions with 0, all of which he also related to biquaternions.
Lorentz transformation via trigonometric functions[]
The following general relation connects the speed of light and the relative velocity to hyperbolic and trigonometric functions, where is the rapidity in (3b), is equivalent to the Gudermannian function, and is equivalent to the Lobachevskian angle of parallelism:
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This relation was first defined by Varićak (1910).
a) Using one obtains the relations and , and the Lorentz boost takes the form:[42]
(8a)
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This Lorentz transformation was derived by Bianchi (1886) and Darboux (1891/94) while transforming pseudospherical surfaces, and by Scheffers (1899) as a special case of contact transformation in the plane (Laguerre geometry). In special relativity, it was used by Gruner (1921) while developing Loedel diagrams, and by Vladimir Karapetoff in the 1920s.
b) Using one obtains the relations and , and the Lorentz boost takes the form:[42]
(8b)
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This Lorentz transformation was derived by Eisenhart (1905) while transforming pseudospherical surfaces. In special relativity it was first used by Gruner (1921) while developing Loedel diagrams.
Lorentz transformation via squeeze mappings[]
As already indicated in equations (3d) in exponential form or (6f) in terms of Cayley–Klein parameter, Lorentz boosts in terms of hyperbolic rotations can be expressed as squeeze mappings. Using asymptotic coordinates of a hyperbola (u,v), they have the general form (some authors alternatively add a factor of 2 or ):[43]
(9a)
That this equation system indeed represents a Lorentz boost can be seen by plugging (1) into (2) and solving for the individual variables:
Variables u, v in (9a) can be rearranged to produce another form of squeeze mapping, resulting in Lorentz transformation (5b) in terms of Cayley-Hermite parameter:
(9c)
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These Lorentz transformations were given (up to a sign change) by Laguerre (1882), Darboux (1887), Smith (1900) in relation to Laguerre geometry.
On the basis of factors k or a, all previous Lorentz boosts (3b, 4a, 8a, 8b) can be expressed as squeeze mappings as well:
If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation (4b). 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.[46] For instance, Lorentz transformations can be extended by using :[R 5]
.
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,[47] 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 ] a transformation equivalent to the formulae (287) and (288) [namely ]. 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.[R 6]
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.[R 7]
Heaviside (1888), Thomson (1889), Searle (1896)[]
In 1888, Oliver Heaviside[R 8] 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:[48]
.
Consequently, Joseph John Thomson (1889)[R 9] 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 transformationz-vt in his equation[49]):
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)[R 11][50]
where x* is the Galilean transformationx-vt. Except the additional γ in the time transformation, this is the complete Lorentz transformation (4b).[50] 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)[R 12] 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:[R 13]
For solving optical problems Lorentz used the following transformation, in which the modified time variable was called "local time" (German: Ortszeit) by him:[R 14]
In 1897, Larmor extended the work of Lorentz and derived the following transformation[R 15]
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".[52][53] 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:[R 16]
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:[R 17]
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:[R 18]
by which he arrived at the complete Lorentz transformation (4b). 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 [..][R 19] 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 20]
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):[R 21]
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):[R 22]
This is equivalent to the complete Lorentz transformation (4b) when solved for x″ and t″ and with ε=1. Like Larmor, Lorentz noticed in 1899[R 23] 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.[54]
In 1904 he rewrote the equations in the following form by setting l=1/ε (again, x* must be replaced by x-vt):[R 24]
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.[R 25] However, he didn't achieve full covariance of the transformation equations for charge density and velocity.[55] When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:[56]
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:[R 26]
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.[57] 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 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.[R 30] In order to synchronise the clocks here on Earth (the x*, t* frame) a light signal from one clock (at the origin) is sent to another (at x*), and is sent back. It's supposed that the Earth is moving with speed v in the x-direction (= x*-direction) in some rest system (x, t) (i.e. the luminiferous aether system for Lorentz and Larmor). The time of flight outwards is
and the time of flight back is
.
The elapsed time on the clock when the signal is returned is δta+δtb and the time t*=(δta+δtb)/2 is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time t=δta is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus
identical to Lorentz (1892). By dropping the factor γ2 under the assumption that , Poincaré gave the result t*=t-vx*/c2, which is the form used by Lorentz in 1895.
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 (4b):[R 33]
.
Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name "Lorentz transformation".[58][59] 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.[60]
In July 1905 (published in January 1906)[R 34] 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 as a fourth imaginary coordinate, and he used an early form of four-vectors. He also formulated the velocity addition formula (4d), which he had already derived in unpublished letters to Lorentz from May 1905:[R 35]
.
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.[61][62][63]
The notation for this transformation is equivalent to Poincaré's of 1905 and (4b), except that Einstein didn't set the speed of light to unity:[R 36]
Einstein also defined the velocity addition formula (4d, 4e):[R 37]
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.[R 39][R 40] Minkowski particularly reformulated electrodynamics in a four-dimensional way (Minkowski spacetime).[64] For instance, he wrote x, y, z, it in the form x1, x2, x3, x4. By defining ψ as the angle of rotation around the z-axis, the Lorentz transformation assumes a form (with c=1) in agreement with (2b):[R 41]
Even though Minkowski used the imaginary number iψ, he for once[R 41] directly used the tangens hyperbolicus in the equation for velocity
with .
Minkowski's expression can also by written as ψ=atanh(q) and was later called rapidity. He also wrote the Lorentz transformation in matrix form equivalent to (2a) (n=3):[R 42]
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:[R 43]
Original spacetime diagram by Minkowski in 1908.
Sommerfeld (1909) – Spherical trigonometry[]
Using an imaginary rapidity such as Minkowski, Arnold Sommerfeld (1909) formulated a transformation equivalent to Lorentz boost (3b), and the relativistic velocity addition (4d) in terms of trigonometric functions and the spherical law of cosines:[R 44]
Bateman and Cunningham (1909–1910) – Spherical wave transformation[]
In line with Lie's (1871) research on the relation between sphere transformations with an imaginary radius coordinate and 4D conformal transformations, it was pointed out by Bateman and Cunningham (1909–1910), that by setting u=ict as the imaginary fourth coordinates one can produce spacetime conformal transformations. Not only the quadratic form , 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.[R 45][R 46] 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.[R 47] In particular, by setting λ=1 the Lorentz group SO(1,3) can be seen as a 10-parameter subgroup of the 15-parameter spacetime conformal group Con(1,3).
Bateman (1910/12)[65] 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),[24][R 48]Henri Poincaré (1912/21)[R 49] and others.
Herglotz (1909/10) – Möbius transformation[]
Following Klein (1889–1897) and Fricke & Klein (1897) concerning the Cayley absolute, hyperbolic motion and its transformation, Gustav Herglotz (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic. The general case (on the left) equivalent to Lorentz transformation (6a) and the hyperbolic case (on the right) equivalent to Lorentz transformation (3d) or squeeze mapping (9d) are as follows:[R 50]
Varićak (1910) – Hyperbolic functions[]
Following Sommerfeld (1909), hyperbolic functions were used by Vladimir Varićak in several papers starting from 1910, who represented the equations of special relativity on the basis of hyperbolic geometry in terms of Weierstrass coordinates. For instance, by setting l=ct and v/c=tanh(u) with u as rapidity he wrote the Lorentz transformation in agreement with (3b):[R 51]
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.[10]
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:[R 53][R 54]
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 (4b). 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),[R 55] with various authors developing similar methods in subsequent years.[66]
Noether (1910), Klein (1910) – Quaternions[]
Felix Klein (1908) described Cayley's (1854) 4D quaternion multiplications as "Drehstreckungen" (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor), and pointed out that the modern principle of relativity as provided by Minkowski is essentially only the consequent application of such Drehstreckungen, even though he didn't provide details.[R 56]
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 , 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 equivalent to (7a):[R 57]
Besides citing quaternion related standard works such as Cayley (1854), Noether referred to the entries in Klein's encyclopedia by Eduard Study (1899) and the French version by Élie Cartan (1908).[67] Cartan's version contains a description of Study's dual numbers, Clifford's biquaternions (including the choice 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 58]
Arthur W. Conway in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:[R 60]
Also Ludwik Silberstein in November 1911[R 61] as well as in 1914,[68] formulated the Lorentz transformation in terms of velocity v:
Silberstein cites Cayley (1854, 1855) and Study's encyclopedia entry (in the extended French version of Cartan in 1908), as well as the appendix of Klein's and Sommerfeld's book.
Further information: Lorentz transformation § Vector transformations
Gustav Herglotz (1911)[R 62] showed how to formulate the transformation equivalent to (4c) in order to allow for arbitrary velocities and coordinates v=(vx, vy, vz) and r=(x, y, z):
This was simplified using vector notation by Ludwik Silberstein (1911 on the left, 1914 on the right):[R 63]
These formulas were called "general Lorentz transformation without rotation" by Christian Møller (1952),[71] who in addition gave an even more general Lorentz transformation in which the Cartesian axes have different orientations, using a rotation operator. In this case, v′=(v′x, v′y, v′z) is not equal to -v=(-vx, -vy, -vz), but the relation holds instead, with the result
Borel (1913–14) – Cayley–Hermite parameter[]
Borel (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and Cayley's (1846) parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions equivalent to (5b):[R 64]
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:[R 66]
This is equivalent to Lorentz transformation (8a) by the identity
In another paper Gruner used the alternative relations:[R 67]
This is equivalent to Lorentz Lorentz boost (8b) by the identity .
Euler's gap[]
In pursuing the history in years before Lorentz enunciated his expressions, one looks to the essence of the concept. In mathematical terms, Lorentz transformations are squeeze mappings, the linear transformations that turn a square into a rectangles of the same area. Before Euler, the squeezing was studied as quadrature of the hyperbola and led to the hyperbolic logarithm. In 1748 Euler issued his precalculustextbook where the number e is exploited for trigonometry in the unit circle. The first volume of Introduction to the Analysis of the Infinite had no diagrams, allowing teachers and students to draw their own illustrations.
There is a gap in Euler's text where Lorentz transformations arise. A feature of natural logarithm is its interpretation as area in hyperbolic sectors. In relativity the classical concept of velocity is replaced with rapidity, a hyperbolic angle concept built on hyperbolic sectors. A Lorentz transformation is a hyperbolic rotation which preserves differences of rapidity, just as the circular sector area is preserved with a circular rotation. Euler's gap is the lack of hyperbolic angle and hyperbolic functions, later developed by Johann H. Lambert. Euler described some transcendental functions: exponentiation and circular functions. He used the exponential series With the imaginary unit i2 = – 1, and splitting the series into even and odd terms, he obtained
For physics, one space dimension is insufficient. But to extend split-complex arithmetic to four dimensions leads to hyperbolic quaternions, and opens the door to abstract algebra's
hypercomplex numbers. Reviewing the expressions of Lorentz and Einstein, one observes that the Lorentz factor is an algebraic function of velocity. For readers uncomfortable with transcendental functions cosh and sinh, algebraic functions may be more to their liking.
Bucherer, A. H. (1908), "Messungen an Becquerelstrahlen. Die experimentelle Bestätigung der Lorentz-Einsteinschen Theorie. (Measurements of Becquerel rays. The Experimental Confirmation of the Lorentz-Einstein Theory)", Physikalische Zeitschrift, 9 (22): 758–762. For Minkowski's and Voigt's statements see p. 762.
Klein, F. (1911). Hellinger, E. (ed.). Elementarmethematik vom höheren Standpunkte aus. Teil I (Second Edition). Vorlesung gehalten während des Wintersemesters 1907-08. Leipzig: Teubner. hdl:2027/mdp.39015068187817.
Larmor, Joseph (1929) [1897], "On a Dynamical Theory of the Electric and Luminiferous Medium. Part 3: Relations with material media", Mathematical and Physical Papers: Volume II, Cambridge University Press, pp. 2–132, ISBN978-1-107-53640-1 (Reprint of Larmor (1897) with new annotations by Larmor.)
Larmor, Joseph (1900), Aether and Matter, Cambridge University Press
Poincaré, Henri (1906) [1904], "The Principles of Mathematical Physics" , Congress of arts and science, universal exposition, St. Louis, 1904, 1, Boston and New York: Houghton, Mifflin and Company, pp. 604–622
^Plummer (1910), pp. 258-259: After deriving the relativistic expressions for the aberration angles φ' and φ, Plummer remarked on p. 259: Another geometrical representation is obtained by assimilating φ' to the eccentric and φ to the true anomaly in an ellipse whose eccentricity is v/U = sin β.
^Robinson (1990), chapter 3-4, analyzed the relation between "Kepler's formula" and the "physical velocity addition formula" in special relativity.
Baccetti, Valentina; Tate, Kyle; Visser, Matt (2012). "Inertial frames without the relativity principle". Journal of High Energy Physics. 2012 (5): 119. arXiv:1112.1466. Bibcode:2012JHEP...05..119B. doi:10.1007/JHEP05(2012)119.
Cartan, É.; Study, E. (1908). "Nombres complexes". Encyclopédie des Sciences Mathématiques Pures et Appliquées. 1.1: 328–468.
Cartan, É.; Fano, G. (1955) [1915]. "La théorie des groupes continus et la géométrie". Encyclopédie des Sciences Mathématiques Pures et Appliquées. 3.1: 39–43. (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.)
Fjelstad, P. (1986). "Extending special relativity via the perplex numbers". American Journal of Physics. 54 (5): 416–422. Bibcode:1986AmJPh..54..416g. doi:10.1119/1.14605.
Hawkins, Thomas (2013). "The Cayley–Hermite problem and matrix algebra". The Mathematics of Frobenius in Context: A Journey Through 18th to 20th Century Mathematics. Springer. ISBN978-1461463337.
Kastrup, H. A. (2008). "On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics". Annalen der Physik. 520 (9–10): 631–690. arXiv:0808.2730. Bibcode:2008AnP...520..631K. doi:10.1002/andp.200810324.
Katzir, Shaul (2005), "Poincaré's Relativistic Physics: Its Origins and Nature", Physics in Perspective, 7 (3): 268–292, Bibcode:2005PhP.....7..268K, doi:10.1007/s00016-004-0234-y
von Laue, M. (1921). Die Relativitätstheorie, Band 1 (fourth edition of "Das Relativitätsprinzip" ed.). Vieweg.; First edition 1911, second expanded edition 1913, third expanded edition 1919.
Lorente, M. (2003). "Representations of classical groups on the lattice and its application to the field theory on discrete space-time". Symmetries in Science. VI: 437–454. arXiv:hep-lat/0312042. Bibcode:2003hep.lat..12042L.
Majerník, V. (1986). "Representation of relativistic quantities by trigonometric functions". American Journal of Physics. 54 (6): 536–538. doi:10.1119/1.14557.
Meyer, W.F. (1899). "Invariantentheorie". Encyclopädie der Mathematischen Wissenschaften. 1.1: 322–455.
Naimark,M. A. (2014) [1964]. Linear Representations of the Lorentz Group. Oxford. ISBN978-1483184982.
Pacheco, R. (2008). "Bianchi–Bäcklund transforms and dressing actions, revisited". Geometriae Dedicata. 146 (1): 85–99. arXiv:0808.4138. doi:10.1007/s10711-009-9427-5.
Pauli, Wolfgang (1921), "Die Relativitätstheorie", Encyclopädie der Mathematischen Wissenschaften, 5 (2): 539–776 In English: Pauli, W. (1981) [1921]. Theory of Relativity. Fundamental Theories of Physics. 165. Dover Publications. ISBN978-0-486-64152-2.
Pais, Abraham (1982), Subtle is the Lord: The Science and the Life of Albert Einstein, New York: Oxford University Press, ISBN978-0-19-520438-4
Penrose, R.; Rindler W. (1984), Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields, Cambridge University Press, ISBN978-0521337076
Reynolds, W. F. (1993). "Hyperbolic geometry on a hyperboloid". The American Mathematical Monthly. 100 (5): 442–455. doi:10.1080/00029890.1993.11990430. JSTOR2324297.
Rindler, W. (2013) [1969]. Essential Relativity: Special, General, and Cosmological. Springer. ISBN978-1475711356.
Robinson, E.A. (1990). Einstein's relativity in metaphor and mathematics. Prentice Hall. ISBN9780132464970.
Rosenfeld, B.A. (1988). A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. New York: Springer. ISBN978-1441986801.
Terng, C. L. & Uhlenbeck, K. (2000). "Geometry of solitons"(PDF). Notices of AMS. 47 (1): 17–25.
Touma, J. R.; Tremaine, S. & Kazandjian, M. V. (2009). "Gauss's method for secular dynamics, softened". Monthly Notices of the Royal Astronomical Society. 394 (2): 1085–1108. arXiv:0811.2812. doi:10.1111/j.1365-2966.2009.14409.x.