History of Lorentz transformations


The history of Lorentz transformations comprises the development of linear transformations forming the Lorentz group or Poincaré group preserving the Lorentz interval and the Minkowski inner product.
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′.

Overview

Most general Lorentz transformations

The general quadratic form q with coefficients of a symmetric matrix A, the associated bilinear form b, and the linear transformations of q and b into q and b using the transformation matrix g, can be written as
The case n=1 is the binary quadratic form introduced by Lagrange and Gauss, n=2 is the ternary quadratic form introduced by Gauss, n=3 is the quaternary quadratic form etc.
The general Lorentz transformation follows from by setting A=A′=diag and det g=±1. It forms an indefinite orthogonal group called the Lorentz group O, while the case det g=+1 forms the restricted Lorentz group SO. The quadratic form q becomes the Lorentz interval in terms of an indefinite quadratic form of Minkowski space, and the associated bilinear form b becomes the Minkowski inner product:
Such general Lorentz transformations for various dimensions were used by Gauss, Jacobi, Lebesgue, Bour, Somov, Hill in order to simplify computations of elliptic functions and integrals. They were also used by Poincaré, Cox, Picard, Killing, Gérard, Hausdorff, Woods, Liebmann to describe hyperbolic motions, which were expressed in terms of Weierstrass coordinates of the hyperboloid model satisfying the relation or in terms of the Cayley–Klein metric of projective geometry using the "absolute" form. In addition, infinitesimal transformations related to the Lie algebra of the group of hyperbolic motions were given in terms of Weierstrass coordinates by Killing.
If in 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 ):
Such Lorentz transformations for various dimensions were used by Gauss, Jacobi, Lebesgue, Bour, Somov, Hill, Callandreau in order to simplify computations of elliptic functions and integrals, by Picard in relation to Hermitian quadratic forms, or by Woods in terms of the Beltrami–Klein model of hyperbolic geometry. In addition, infinitesimal transformations in terms of the Lie algebra of the group of hyperbolic motions leaving invariant the unit sphere were given by Lie and Werner and Killing.
Particular forms of Lorentz transformations or relativistic velocity additions, mostly restricted to 2, 3 or 4 dimensions, have been formulated by many authors using:
By using the imaginary quantities in x as well as ' in g, the Lorentz transformation assumes the form of an orthogonal transformation of Euclidean space forming the orthogonal group O if det g=±1 or the special orthogonal group SO if det g'=+1, the Lorentz interval becomes the Euclidean norm, and the Minkowski inner product becomes the dot product:
The cases n=1,2,3,4 of orthogonal transformations in terms of real coordinates were discussed by Euler and in n dimensions by Cauchy. The case in which one of these coordinates is imaginary and the other ones remain real was alluded to by Lie 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 and Sommerfeld.
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:
or in exponential form using Euler's formula :
Defining as real, spatial rotation in the form was introduced by Euler and in the form by Wessel. The interpretation of as Lorentz boost in which correspond to the imaginary quantities was given by Minkowski and Sommerfeld. As shown in the next section using hyperbolic functions, becomes while becomes.

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 :
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 is given by
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:
Finally, Lorentz boost assumes a simple form by using squeeze mappings in analogy to Euler's formula in :
Hyperbolic relations on the right of were given by Riccati, relations by Lambert. Lorentz transformations were given by Laisant, Cox, Lindemann, Gérard, Killing, Whitehead, Woods and Liebmann in terms of Weierstrass coordinates of the hyperboloid model. Hyperbolic angle sum laws equivalent to Lorentz boost were given by Riccati and Lambert, while the matrix representation was given by Glaisher and Günther. Lorentz transformations were given by Lindemann and Herglotz, while formulas equivalent to by Klein.
In line with equation one can use coordinates inside the unit circle, thus the corresponding Lorentz transformations obtain the form:
These Lorentz transformations were given by Escherich and Killing , as well as Beltrami and Schur in terms of Beltrami coordinates of hyperbolic geometry. By using the scalar product of, the resulting Lorentz transformation can be seen as equivalent to the hyperbolic law of cosines:
The hyperbolic law of cosines was given by Taurinus and Lobachevsky and others, while variant was given by Schur.

Lorentz transformation via velocity

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 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 becomes:
Or in four dimensions and by setting and adding an unchanged z the familiar form follows
Without relation to physics, similar transformations have been used by Lipschitz. In physics, analogous transformations have been introduced by Voigt and by Lorentz who analyzed Maxwell's equations, they were completed by Larmor and Lorentz, and brought into their modern form by Poincaré who gave the transformation the name of Lorentz. Eventually, Einstein 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 used them to argue that space and time are inseparably connected as spacetime. Minkowski and Varićak showed the relation to imaginary and hyperbolic functions. Important contributions to the mathematical understanding of the Lorentz transformation were also made by other authors such as Herglotz, Ignatowski, Noether and Klein, Borel.
Also Lorentz boosts for arbitrary directions in line with can be given as:
or in vector notation
Such transformations were formulated by Herglotz and Silberstein and others.
In line with equation, one can substitute in or, producing the Lorentz transformation of velocities in analogy to Beltrami coordinates of :
or using trigonometric and hyperbolic identities it becomes the hyperbolic law of cosines in terms of :
and by further setting u=u′=c the relativistic aberration of light follows:
The velocity addition formulas were given by Einstein and Poincaré, the aberration formula for cos by Einstein, while the relations to the spherical and hyperbolic law of cosines were given by Sommerfeld and Varićak. These formulas resemble the equations of an ellipse of eccentricity v/c, eccentric anomaly α' and true anomaly α, first geometrically formulated by Kepler and explicitly written down by Euler, Lagrange and many others in relation to planetary motions.

Lorentz transformation via conformal, spherical wave, and Laguerre transformation

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 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. Both representations were studied by Lie and others. It was shown by Bateman & Cunningham, that the group Con is the most general one leaving invariant the equations of Maxwell's electrodynamics.
It turns out that Con is isomorphic to the special orthogonal group SO, and contains the Lorentz group SO as a subgroup by setting λ=1. More generally, Con is isomorphic to SO and contains SO as subgroup. This implies that Con is isomorphic to the Lorentz group of arbitrary dimensions SO. Consequently, the conformal group in the plane Con – known as the group of Möbius transformations – is isomorphic to the Lorentz group SO. This can be seen using tetracyclical coordinates satisfying the form, which were discussed by Pockels, Klein, Bôcher. The relation between Con and the Lorentz group was noted by Bateman & Cunningham and others.
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] introduced by Laguerre and discussed by Darboux and Smith leaving invariant with R as radius, thus the Laguerre group is isomorphic to the Lorentz group. A similar concept was studied by Scheffers in terms of contact transformations. Stephanos 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, provides a geometrical interpretation of Hamilton's biquaternions. The group isomorphism between the Laguerre group and Lorentz group was pointed out by Bateman, Cartan, Poincaré and others.

Lorentz transformation via Cayley–Hermite transformation

The general transformation of any quadratic form into itself can also be given using arbitrary parameters based on the Cayley transform −1·, where I is the identity matrix, T an arbitrary antisymmetric matrix, and by adding A as symmetric matrix defining the quadratic form :
After Cayley introduced transformations related to sums of positive squares, Hermite derived transformations for arbitrary quadratic forms, whose result was reformulated in terms of matrices by Cayley. For instance, the choice A=diag gives an orthogonal transformation which can be used to describe spatial rotations corresponding to the Euler-Rodrigues parameters discovered by Euler and Rodrigues, 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. For instance, Lorentz transformation with n=1 follows from with:
This becomes Lorentz boost by setting, which is equivalent to the relation known from Loedel diagrams, thus 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 with n=2 is given by:
or using n=3:
The transformation of a binary quadratic form of which Lorentz transformation is a special case was given by Hermite, equations containing Lorentz transformations as special cases were given by Cayley, Lorentz transformation was given by Laguerre, Darboux, Smith in relation to Laguerre geometry, and Lorentz transformation was given by Bachmann. In relativity, equations similar to were first employed by Borel to represent Lorentz transformations.
As described in equation, the Lorentz interval is closely connected to the alternative form, which in terms of the Cayley–Hermite parameters is invariant under the transformation:
This transformation was given by Cayley, even though he didn't relate it to the Lorentz interval but rather to. As shown in the next section in equation, many authors expressed the invariance of and its relation to the Lorentz interval by using the alternative [|Cayley–Klein parameters] and Möbius transformations.

Lorentz transformation via Cayley–Klein parameters, Möbius and spin transformations

The previously mentioned Euler-Rodrigues parameter a,b,c,d are closely related to Cayley–Klein parameter α,β,γ,δ introduced by Helmholtz, Cayley and Klein to connect Möbius transformations and rotations:
thus becomes:
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 , and the Möbius transformation of. When defined in terms of isometries of hyperblic space, the Hermitian matrix u 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". It also turns out that the related spin group Spin or special linear group SL acts as the double cover of the Lorentz group, while the Möbius group Con or projective special linear group PSL 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:
thus:
or in line with equation one can substitute so that the Möbius/Lorentz transformations become related to the unit sphere:
The general transformation u′ in was given by Cayley, while the general relation between Möbius transformations and transformation u′ leaving invariant the generalized circle was pointed out by Poincaré in relation to Kleinian groups. The adaptation to the Lorentz interval by which becomes a Lorentz transformation was given by Klein, Bianchi, Fricke. Its reformulation as Lorentz transformation was provided by Bianchi and Fricke. Lorentz transformation was given by Klein in relation to surfaces of second degree and the invariance of the unit sphere. In relativity, was first employed by Herglotz.
In the plane, the transformations can be written as:
thus
which includes the special case implying, reducing the transformation to a Lorentz boost in 1+1 dimensions:
Finally, by using the Lorentz interval related to a hyperboloid, the Möbius/Lorentz transformations can be written
The general transformation u′ and its invariant in was already used by Lagrange and Gauss in the theory of integer binary quadratic forms. The invariant was also studied by Klein in connection to hyperbolic plane geometry ), while the connection between u′ and with the Möbius transformation was analyzed by Poincaré in relation to Fuchsian groups. The adaptation to the Lorentz interval by which becomes a Lorentz transformation was given by Bianchi and Fricke. Lorentz Transformation was stated by [|Gauss around 1800], as well as Selling, Bianchi, Fricke, Woods in relation to integer indefinite ternary quadratic forms. Lorentz transformation was given by Bianchi and Eisenhart. Lorentz transformation of the hyperboloid was stated by Poincaré and Hausdorff.

Lorentz transformation via quaternions and hyperbolic numbers

The Lorentz transformations can also be expressed in terms of biquaternions: A Minkowskian quaternion 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 and the corresponding boost are as follows:
Hamilton and Cayley derived the quaternion transformation for spatial rotations, and Cayley gave the corresponding transformation leaving invariant the sum of four squares. Cox 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 . Stephanos 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 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 and Klein as well as Conway and Silberstein.
Often connected with quaternionic systems is the hyperbolic number, which also allows to formulate the Lorentz transformations:
After the trigonometric expression was given by Euler, and the hyperbolic analogue as well as hyperbolic numbers by Cockle in the framework of tessarines, it was shown by Cox 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. The hyperbolic versor was also discussed by Macfarlane in terms of hyperbolic quaternions. The expression for hyperbolic motions also appear in "biquaternions" defined by Vahlen.
More extended forms of complex and 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:
The Lorentz interval follows if the sign of one differs from all others. The general definite form as well as the general indefinite form and their invariance under transformation was discussed by Lipschitz, while hyperbolic motions were discussed by Vahlen by setting in transformation, 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, is equivalent to the Gudermannian function, and is equivalent to the Lobachevskian angle of parallelism :
This relation was first defined by Varićak.
a) Using one obtains the relations and, and the Lorentz boost takes the form:
This Lorentz transformation was derived by Bianchi and Darboux while transforming pseudospherical surfaces, and by Scheffers as a special case of contact transformation in the plane. In special relativity, it was used by Gruner 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:
This Lorentz transformation was derived by Eisenhart while transforming pseudospherical surfaces. In special relativity it was first used by Gruner while developing Loedel diagrams.

Lorentz transformation via squeeze mappings

As already indicated in equations in exponential form or 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, they have the general form :
That this equation system indeed represents a Lorentz boost can be seen by plugging into and solving for the individual variables:
Lorentz transformation of asymptotic coordinates have been used Laisant and Günther in relation to elliptic trigonometry, or by Lie, Bianchi, Darboux, Eisenhart of pseudospherical surfaces in terms of the Sine-Gordon equation, or by Lipschitz in transformation theory. From that, different forms of Lorentz transformation were derived: by Lipschitz, Bianchi, Eisenhart, trigonometric Lorentz boost by Bianchi and Darboux, and trigonometric Lorentz boost by Eisenhart. Lorentz boost was rediscovered in the framework of special relativity by Hermann Bondi in terms of Bondi k-calculus, by which k can be physically interpreted as Doppler factor. Since is equivalent to in terms of Cayley–Klein parameter by setting, it can be interpreted as the 1+1 dimensional special case of Lorentz Transformation stated by Gauss around 1800, Selling, Bianchi, Fricke and Woods.
Variables u, v in can be rearranged to produce another form of squeeze mapping, resulting in Lorentz transformation in terms of Cayley-Hermite parameter:
These Lorentz transformations were given by Laguerre, Darboux, Smith in relation to Laguerre geometry.
On the basis of factors k or a, all previous Lorentz boosts can be expressed as squeeze mappings as well:
Squeeze mappings in terms of were used by Darboux and Bianchi, in terms of by Lindemann and Herglotz, in terms of by Eisenhart, in terms of by Bondi.

Lorentz transformations in pure mathematics before 1905

Historical formulas for Lorentz boosts and velocity additions

Euler (1735-1771)

True and eccentric anomaly

geometrically formulated Kepler's equation and the relations between the mean, true, and eccentric anomaly. The relation between the true anomaly z and the eccentric anomaly P was algebraically expressed by Leonhard Euler as follows:
and in 1748:
while Joseph-Louis Lagrange expressed them as follows

Orthogonal transformation

Euler demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as orthogonal transformation, as well as under rotations using Euler angles. The case of two dimensions is given by
or three dimensions
These coefficiens A,B,C,D,E,F,G,H,I were related by Euler to four arbitrary parameter p,q,r,s, which where rediscovered by Olinde Rodrigues who related them to rotation angles now called Euler–Rodrigues parameters in line with equation :
The orthogonal transformation in four dimensions was given by him as

Euler's formula and rotation

The [|above] orthogonal transformations representing Euclidean rotations can also be expressed by using Euler's formula. After this formula was derived by Euler in 1748
it was used by Caspar Wessel to describe Euclidean rotations in the complex plane:

Riccati (1757) – hyperbolic functions

introduced hyperbolic functions in 1757, in particular he formulated the angle sum laws for hyperbolic sine and cosine:
He furthermore showed that and follow by setting and in the above formulas.

Lambert (1768–1770) – hyperbolic functions

While Riccati discussed the hyperbolic sine and cosine, Johann Heinrich Lambert introduced the expression tang φ or abbreviated tφ as the tangens hyperbolicus of a variable u, or in modern notation tφ=tanh:
In he rewrote the addition law for the hyperbolic tangens or as:
Lambert also formulated the addition laws for the hyperbolic cosine and sine :

Gauss (1798–1818)

Binary quadratic forms

After the invariance of the sum of squares under linear substitutions was discussed by Euler, the general expressions of a binary quadratic form and its transformation was formulated by Lagrange as follows
which is equivalent to '. The theory of binary quadratic forms was considerably expanded by Carl Friedrich Gauss in his Disquisitiones Arithmeticae. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:
which is equivalent to '. As pointed out by Gauss, F and F′ are called "proper equivalent" if αδ-βγ=1, so that F is contained in F′ as well as F′ is contained in F. In addition, if another form F″ is contained by the same procedure in F′ it is also contained in F and so forth.

Ternary quadratic forms

Gauss also discussed ternary quadratic forms with the general expression
which is equivalent to . Gauss called these forms definite when they have the same sign such as x2+y2+z2, or indefinite in the case of different signs such as x2+y2-z2. While discussing the classification of ternary quadratic forms, Gauss presented twenty special cases, among them these six variants:

Cayley–Klein parameter

The determination of all transformations of the Lorentz interval into itself was explicitly worked out by Gauss around 1800, for which he provided a coefficient system α,β,γ,δ:
Gauss' result was cited by Bachmann, Selling, Bianchi, Leonard Eugene Dickson. The parameters α,β,γ,δ, when applied to spatial rotations, were later called Cayley–Klein parameters.

Homogeneous coordinates

Gauss discussed planetary motions together with formulating elliptic functions. In order to simplify the integration, he transformed the expression
into
in which the eccentric anomaly E is connected to the new variable T by the following transformation including an arbitrary constant k, which Gauss then rewrote by setting k=1:
Subsequently, he showed that these relations can be reformulated using three variables x,y,z and u,u′,u″, so that
can be transformed into
in which x,y,z and u,u′,u″ are related by the transformation:

Taurinus (1826) – Hyperbolic law of cosines

After the addition theorem for the tangens hyperbolicus was given by Lambert, hyperbolic geometry was used by Franz Taurinus, and later by Nikolai Lobachevsky and others, to formulate the hyperbolic law of cosines:

Jacobi (1827, 1833/34) – Homogeneous coordinates

Following Gauss, Carl Gustav Jacob Jacobi extended Gauss' transformation to 3 dimensions in 1827:
Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation in the case of 2 dimensions:
Extending his previous result, Jacobi started with Cauchy's orthogonal transformation for n dimensions, and by using an imaginary substitution he formulated Gauss' transformation in the case of n dimensions:
He also stated the following transformation leaving invariant the Lorentz interval:

Cauchy (1829) – Orthogonal transformation

extended the orthogonal transformation of Euler to arbitrary dimensions

Lebesgue (1837) – Homogeneous coordinates

summarized the previous work of Gauss, Jacobi, Cauchy. He started with the orthogonal transformation
In order to achieve the invariance of the Lorentz interval
he gave the following instructions as to how the previous equations shall be modified: In equation change the sign of the last term of each member. In the first n-1 equations of change the sign of the last term of the left-hand side, and in the one which satisfies α=n change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations the last term will change sign. In equations the last terms of the right-hand side will change sign, and so will the left-hand side of the n-th equation. In equations the signs of the last terms of the left-hand side will change, moreover in the n-th equation change the sign of the right-hand side. In equations the last terms will change sign.
He went on to redefine the variables of the Lorentz interval and its transformation:

Hamilton (1844/45) – Quaternions

's, in an abstract of a lecture held on November 1844 and published 1845/47, showed that spatial rotations can be formulated using his theory of quaternions by employing versors as pre- and postfactor, with α as unit vector and a as real angle:
In a footnote added before printing, he showed that this is equivalent to Cayley's rotation formula by setting
Hamilton acknowledged Cayley's independent discovery and priority for first printed publication, but noted that he himself communicated formula already in October 1844 to Charles Graves.

Cayley (1846–1884)

Euler–Rodrigues parameter and Cayley–Hermite transformation

The Euler–Rodrigues parameters discovered by Euler and Rodrigues leaving invariant were extended to by Arthur Cayley as a byproduct of what is now called the Cayley transform using the method of skew–symmetric coefficients. Following Cayley's methods, a general transformation for quadratic forms into themselves in three and arbitrary dimensions was provided by Hermite. Hermite's formula was simplified and brought into matrix form equivalent to by Cayley
which he abbreviated in 1858, where is any skew-symmetric matrix:
Using the parameters of, Cayley in a subsequent paper particularly discussed several special cases, such as:
or:
or:

Cayley–Klein parameter

Already in 1854, Cayley published an alternative method of transforming quadratic forms by using certain parameters α,β,γ,δ in relation to an improper homographic transformation of a surface of second order into itself:
By setting and rewriting M and M' in terms of four different parameters he demonstrated the invariance of, and subsequently showed the relation to 4D quaternion transformations. Fricke & Klein credited Cayley by calling the above transformation the most general space collineation of first kind of an absolute surface of second kind into itself. Parameters α,β,γ,δ are similar to what was later called Cayley–Klein parameters in relation to spatial rotations.

Quaternions

In 1845, Cayley showed that the Euler-Rodrigues parameters in equation representing rotations can be related to quaternion multiplication by pre- and postfactors :
and in 1848 he used the abbreviated form
In 1854 he showed how to transform the sum of four squares into itself:
or in 1855:

Cayley absolute and hyperbolic geometry

In 1859, Cayley found out that a quadratic form or projective quadric can be used as an "absolute", serving as the basis of a projective metric. For instance, using the absolute x2+y2+z2=0, he defined the distance of two points as follows
and he also alluded to the case of the unit sphere x2+y2+z2=1. In the hands of Klein, all of this became essential for the discussion of non-Euclidean geometry and associated quadratic forms and transformations, including the Lorentz interval and Lorentz transformation.
Cayley himself also discussed some properties of the Beltrami–Klein model and the pseudosphere, and formulated coordinate transformations using the Cayley-Hermite formalism:

Cockle (1848) - Tessarines

introduced the tessarine algebra as follows:
While is the ordinary imaginary unit, the new unit led him to formulate the following relation:

Hermite (1853, 1854) – Cayley–Hermite transformation

extended the number theoretical work of Gauss and others by additionally analyzing indefinite ternary quadratic forms that can be transformed into the Lorentz interval ±, and by using Cayley's method of skew–symmetric coefficients he derived transformations leaving invariant almost all types of ternary quadratic forms. This was generalized by him in 1854 to n dimensions:
This result was subsequently expressed in matrix form by Cayley, while Ferdinand Georg Frobenius added some modifications in order to include some special cases of quadratic forms that cannot be dealt with by the Cayley–Hermite transformation.
For instance, the special case of the transformation of a binary quadratic form into itself was given by Hermite as follows:

Bour (1856) – Homogeneous coordinates

Following Gauss, Edmond Bour wrote the transformations:

Somov (1863) – Homogeneous coordinates

Following Gauss, Jacobi, and Bour, Osip Ivanovich Somov wrote the transformation systems:

Beltrami (1868) – Beltrami coordinates

introduced coordinates of the Beltrami–Klein model of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:
, and for arbitrary dimensions in

Bachmann (1869) – Cayley–Hermite transformation

adapted Hermite's transformation of ternary quadratic forms to the case of integer transformations. He particularly analyzed the Lorentz interval and its transformation, and also alluded to the analogue result of Gauss in terms of Cayley–Klein parameters, while Bachmann formulated his result in terms of the Cayley–Hermite transformation:
He described this transformation in 1898 in the first part of his "arithmetics of quadratic forms" as well.

Klein (1871–1897)

Cayley absolute and non-Euclidean geometry

Elaborating on Cayley's definition of an "absolute", Felix Klein defined a "fundamental conic section" in order to discuss motions such as rotation and translation in the non-Euclidean plane, and another fundamental form by using homogeneous coordinates x,y related to a circle with radius 2c with measure of curvature. When c is positive, the measure of curvature is negative and the fundamental conic section is real, thus the geometry becomes hyperbolic :
In he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to an ellipsoid or two-sheet hyperboloid.

Möbius transformation, spin transformation, Cayley–Klein parameter

In while devising the Erlangen program, Klein discussed the general relation between projective metrics, binary forms and conformal geometry transforming a sphere into itself in terms of linear transformations of the complex variable x+iy. Following Klein, these relations were discussed by Ludwig Wedekind using. Klein then showed that all finite groups of motions follow by determining all finite groups of such linear transformations of x+iy into itself. In, Klein classified the substitutions of with αδ-βγ=1 into hyperbolic, elliptic, parabolic, and in he added the loxodromic substitution as the combination of elliptic and hyperbolic ones.
In Klein related the linear fractional transformations to Cayley–Klein parameters , to Euler–Rodrigues parameters , and to the unit sphere by means of stereographic projection, and also discussed transformations preserving surfaces of second degree equivalent to the transformation given by Cayley :
In his lecture in the winter semester of 1889/90, he discussed the hyperbolic plane by using the Lorentz interval in terms of a circle with radius 2k as the basis of hyperbolic geometry, and another quadratic form to discuss the "kinematics of hyperbolic geometry" consisting of motions and congruent displacements of the hyperbolic plane into itself:
In his lecture in the summer semester of 1890, he discussed general surfaces of second degree, including an "oval" surface corresponding to hyperbolic space and its motions:
In, Klein again defined hyperbolic motions and explicitly used t as time coordinate:
Klein's work was summarized and extended by Bianchi and Fricke, obtaining equivalent Lorentz transformations.

Conformal transformation and polyspherical coordinates

In relation to line geometry, Klein used coordinates satisfying the condition. They were introduced in 1868 by Gaston Darboux as a system of five coordinates in R3 in which the last coordinate is imaginary. Sophus Lie more generally used n+2 coordinates in Rn satisfying in which the last coordinate is imaginary, as a means to discuss conformal transformations generated by inversions. These simultaneous publications can be explained by the fact that Darboux, Lie, and Klein corresponded with each other by letter.
When the last coordinate is defined as real, the corresponding polyspherical coordinates satisfy the form of a sphere. Initiated by lectures of Klein between 1889–1890, his student Friedrich Pockels used such real coordinates, emphasizing that all of these coordinate systems remain invariant under conformal transformations generated by inversions:
Special cases were described by Klein :
Both systems were also described by Maxime Bôcher in an expanded version of a thesis supervised by Klein.

Lie (1871–1893)

Conformal, spherical, and orthogonal transformations

In several papers between 1847 and 1850 it was shown by Joseph Liouville that the relation λ is invariant under the group of conformal transformations generated by inversions transforming spheres into spheres, which can be related special conformal transformations or Möbius transformations..
Liouville's theorem was extended to all dimensions by Sophus Lie. In addition, Lie described a manifold whose elements can be represented by spheres, where the last coordinate yn+1 can be related to an imaginary radius by iyn+1:
If the second equation is satisfied, two spheres y′ and y″ are in contact. Lie then defined the correspondence between contact transformations in Rn and conformal point transformations in Rn+1: The sphere of space Rn consists of n+1 parameter, so if this sphere is taken as the element of space Rn, it follows that Rn now corresponds to Rn+1. Therefore, any transformation leaving invariant the condition of contact between spheres in Rn, corresponds to the conformal transformation of points in Rn+1.
Eventually, Lie pointed out that conformal point transformations consist of motions, similarity transformations, and inversions.

Transforming pseudospherical surfaces

Lie derived an operation from Pierre Ossian Bonnet's investigations on surfaces of constant curvatures, by which pseudospherical surfaces can be transformed into each other. Lie gave explicit formulas for this operation in two papers published in 1881: If are asymptotic coordinates of two principal tangent curves and their respective angle, and is a solution of the Sine-Gordon equation, then the following operation is also a solution from which infinitely many new surfaces of same curvature can be derived:
In he wrote these relations as follows:
In he showed that the combination of Lie transform O with Bianchi transform I produces Bäcklund transform B of pseudospherical surfaces: B=OIO−1.

Lie group, hyperbolic motions, and infinitesimal transformations

In, Lie identified the projective group of a general surface of second degree with the group of non-Euclidean motions. In a thesis guided by Lie, Hermann Werner discussed this projective group by using the equation of a unit hypersphere as the surface of second degree, and also gave the corresponding infinitesimal projective transformations :
More generally, Lie defined non-Euclidean motions in terms of two forms in which the imaginary form with denotes the group of elliptic motions, the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:
Summarizing, Lie discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form:

Selling (1873–74) – Quadratic forms

Continuing the work of Gauss on definite ternary quadratic forms and Hermite on indefinite ternary quadratic forms, Eduard Selling used the auxiliary coefficients ξ,η,ζ by which a definite form and an indefinite form f can be rewritten in terms of three squares:
In addition, Selling showed that auxiliary coefficients ξ,η,ζ can be geometrically interpreted as point coordinates which are in motion upon one sheet of a two-sheet hyperboloid, which is related to Selling's formalism for the reduction of indefinite forms by using definite forms.
Selling also reproduced the Lorentz transformation given by Gauss, to whom he gave full credit, and called it the only example of a particular indefinite ternary form known to him that has ever been discussed:

Laisant (1874)

Elliptic polar coordinates

extended circular trigonometry to elliptic trigonometry. In his model, polar coordinates x, y of circular trigonometry are related to polar coordinates x', y' of elliptic trigonometry by the relation
He noticed the geometrical implication that any elliptic polar system of coordinates obtained by this formula is located on the same equilateral hyperbola having its asymptotes as axes.

Equipollences

In his French translation of Giusto Bellavitis' principal work on equipollences, Laisant added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as
Here, OA and OB are conjugate semi-diameters of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:
producing the invariant relation
Substituting into, he showed that OM retains its form
He also defined velocity and acceleration by differentiation of.

Escherich (1874) – Beltrami coordinates

discussed the plane of constant negative curvature based on the Beltrami–Klein model of hyperbolic geometry by Beltrami. Similar to Christoph Gudermann who introduced axial coordinates x=tan and y=tan in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions x=tanh and y=tanh in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:

Glaisher (1878)

It was shown by James Whitbread Lee Glaisher that the hyperbolic addition laws can be written as matrix multiplication

Killing (1878–1893)

Weierstrass coordinates

described non-Euclidean geometry by using Weierstrass coordinates obeying the form
or
where k is the reciprocal measure of curvature, denotes Euclidean geometry, elliptic geometry, and hyperbolic geometry. In he pointed out the possibility and some characteristics of a transformation preserving the above form. In he wrote the corresponding transformations as a general rotation matrix
In he wrote the Weierstrass coordinates and their transformation as follows:
In he also gave the transformation for n dimensions:
In he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force. Regarding the geometrical interpretation of his transformations, Killing argued in that by setting and using p,x,y as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid , by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space. Finally, in he wrote:
and for n dimensions

Translation in the hyperbolic plane

The case of translation was given by Killing in the form
In 1898, Killing wrote that relation in a form similar to Escherich, and derived the corresponding Lorentz transformation for the two cases were v is unchanged or u is unchanged:

Infinitesimal transformations and Lie group

After Lie identified the projective group of a general surface of second degree with the group of non-Euclidean motions, Killing defined the infinitesimal projective transformations in relation to the unit hypersphere:
and in he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:
In he pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to form and transitive when related to form, and he also showed the relation of Weierstrass coordinates to the notation of Killing and Werner, Lie :

Günther (1880/81)

Elliptic polar coordinates

Following Laisant, Siegmund Günther demonstrated the relation betwenn circular polar coordinates and elliptic polar coordinates as
showing that any elliptic polar system of coordinates obtained by this formula is located on the same equilateral hyperbola having its asymptotes as axes.

Matrix multiplication

Following Glaisher, he formulated the hyperbolic addition laws in matrix form as

Poincaré (1881 – 1887)

Weierstrass coordinates

connected the work of Hermite and Selling on indefinite quadratic forms with non-Euclidean geometry. He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates connected by a transformation with integer coefficients:
He went on to describe the properties of "hyperbolic coordinates". Poincaré mentioned the hyperboloid model also in.

Möbius transformation

Poincaré also demonstrated the connection of his above formulas to Möbius transformations:
Poincaré also used the Möbius transformation in relation to Fuchsian functions and the discontinuous Fuchsian group, being a special case of the hyperbolic group leaving invariant the "fundamental circle". He then extended Klein's study on the relation between Möbius transformations and hyperbolic, elliptic, parabolic, and loxodromic substitutions, and while formulating Kleinian groups he used the following transformation leaving invariant the generalized circle:
In 1886, Poincaré investigated the relation between indefinite ternary quadratic forms and Fuchsian functions and groups:

Cox (1881–1883)

Weierstrass coordinates

– referring to similar rectangular coordinates used by Gudermann and George Salmon on a sphere, and to Escherich as reported by Johannes Frischauf in the hyperbolic plane – defined the Weierstrass coordinates and their transformation:
Cox also gave the Weierstrass coordinates and their transformation in hyperbolic space:
The case of translation was also given by him, where the y-axis remains unchanged:

Quaternions

Subsequently, Cox also described hyperbolic geometry in terms of an analogue to quaternions and Hermann Grassmann's exterior algebra. To that end, he used hyperbolic numbers as a means to transfer point P to point Q in the hyperbolic plane, which he wrote in the form:
In he showed the equivalence of PQ=-cosh+ι·sinh with "quaternion multiplication", and in he described QP−1=cosh+ι·sinh as being "associative quaternion multiplication". He also showed that the position of point P in the hyperbolic plane may be determined by three quantities in terms of Weierstrass coordinates obeying the relation z2-x2-y2=1.
Cox went on to develop an algebra for hyperbolic space analogous to Clifford's biquaternions. While Clifford used biquaternions of the form a+ωb in which ω2=0 denotes parabolic space and ω2=1 elliptic space, Cox discussed hyperbolic space using the imaginary quantity and therefore ω2=-1. He also obtained relations of quaternion multiplication in terms of Weierstrass coordinates:

Hill (1882) – Homogeneous coordinates

Following Gauss, George William Hill formulated the equations

Laguerre (1882) – Laguerre inversion

After previous work by Albert Ribaucour, a transformation which transforms oriented spheres into oriented spheres, oriented planes into oriented planes, and oriented lines into oriented lines, was explicitly formulated by Edmond Laguerre as "transformation by reciprocal directions" which was later called "Laguerre inversion/transformation". It can be seen as a special case of the conformal group in terms of [|Lie's transformations of oriented spheres]. In two dimensions the transformation or oriented lines has the form :

Picard (1882-1884) – Quadratic forms

analyzed the invariance of indefinite ternary Hermitian quadratic forms with integer coefficients and their relation to discontinuous groups, extending Poincaré's Fuchsian functions of one complex variable related to a circle, to "hyperfuchsian" functions of two complex variables related to a hypersphere. He formulated the following special case of an Hermitian form:
Or in in relation to indefinite binary Hermitian quadratic forms:
Or in :
Or in :

Stephanos (1883) – Biquaternions

showed that Hamilton's biquaternion a0+a1ι1+a2ι2+a3ι3 can be interpreted as an oriented sphere in terms of Lie's sphere geometry, having the vector a1ι1+a2ι2+a3ι3 as its center and the scalar as its radius. Its norm is thus equal to the power of a point of the corresponding sphere. In particular, the norm of two quaternions N is equal to the tangential distance between two spheres. The general contact transformation between two spheres then can be given by a homography using 4 arbitrary quaternions A,B,C,D and two variable quaternions X,Y:
Stephanos pointed out that the special case A=0 denotes transformations of oriented planes.

Buchheim (1884–85) – Biquaternions

applied Clifford's biquaternions and their operator ω to different forms of geometries. He defined the scalar ω2=e2 which in the case -1 denotes hyperbolic space, 1 elliptic space, and 0 parabolic space. He derived the following relations consistent with the Cayley–Klein absolute:

Darboux (1883–1891)

Transformations of pseudospherical surfaces

represented Lie's transformation of pseudospheres into each other as follows:
Similar to Bianchi, Darboux showed that the Lie transform gives rise to the following relations:

Laguerre inversion

Following Laguerre, Gaston Darboux presented the Laguerre inversions in four dimensions using coordinates x,y,z,R:
Darboux rewrote these equations as follows:

Callandreau (1885) – Homography

Following Gauss and Hill, Octave Callandreau formulated the equations

Lipschitz (1885–86)

Boosts

formulated transformations leaving invariant the sum of squares, which he rewrote as. This led to the problem of finding transformations leaving invariant the pairs for which he gave the following solution:

Clifford algebra

More generally, Lipschitz used Clifford algebra in order to formulate the orthogonal transformation of a sum or squares into itself, for which he used real variables and constants, thus Λ becomes a real quaternion for n=3. He went further and discussed transformations in which both variables x,y... and constants are complex, thus Λ becomes a complex quaternion for n=3. The transformation system for both real and complex quantities has the form:
Lipschitz noticed that this corresponds to the transformations of quadratic forms given by Hermite and Cayley. He then modified his equations to discuss the general indefinite quadratic form, by defining some variables and constants as real and some of them as purely imaginary:
resulting into

Schur (1885/86, 1900/02) – Beltrami coordinates

discussed spaces of constant Riemann curvature, and by following Beltrami he used the transformation
In he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:
where can have values >0, <0 or ∞.
He also defined the triangle

Bianchi (1886–1893)

Transformation of pseudospherical surfaces

investigated Lie's transformation of pseudospheres into each other, obtaining the result:
In 1894, Bianchi redefined the variables u,v as asymptotic coordinates, by which the transformation obtains the form:

Möbius and spin transformations

Related to Klein's and Poincaré's work on non-Euclidean geometry and indefinite quadratic forms, Bianchi analyzed the differential Lorentz interval in term of conic sections and hyperboloids, alluded to the linear fractional transformation of and its conjugate with parameters α,β,γ,δ in order to preserve the Lorentz interval, and gave credit to Gauss who obtained the same coefficient system:
In 1893, Bianchi gave the coefficients in the case of four dimensions:
Solving for Bianchi obtained:

Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute

discussed hyperbolic geometry in his edition of the lectures on geometry of Alfred Clebsch. Citing Killing and Poincaré in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set
In addition, following Klein he employed the Cayley absolute related to surfaces of second degree, by using the following quadratic form and its transformation
into which he put
From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations or translations :

Fricke (1891–1897) – Möbius and spin transformations

– following the work of his teacher Klein as well as Poincaré on automorphic functions and group theory – obtained the following transformation for an integer ternary quadratic form
And the general case of four dimensions in 1893:
Supported by Felix Klein, Fricke summarized his and Klein's work in a treatise concerning automorphic functions. Using a sphere as the absolute, in which the interior of the sphere is denoted as hyperbolic space, they defined hyperbolic motions, and stressed that any hyperbolic motion corresponds to "circle relations" :

Gérard (1892) – Weierstrass coordinates

– in a thesis examined by Poincaré – discussed Weierstrass coordinates in the plane using the following invariant and its Lorentz transformation equivalent to :
He gave the case of translation as follows:

Macfarlane (1892–1900) – Hyperbolic quaternions

– analogous to Cockle and Cox – defined the hyperbolic versor in terms of hyperbolic numbers
and in 1894 he defined the "exspherical" versor
and used them to analyze hyperboloids of one- or two sheets. This was further extended by him in in order to express trigonometry in terms of hyperbolic quaternions rebβ, with β2=+1 and, the hyperbolic number x+yβ, and the hyperbolic versor ebβ.

Woods (1895–1905)

Spin transformation

In a thesis supervised by Felix Klein, Frederick S. Woods further developed Bianchi's treatment of surfaces satisfying the Lorentz interval, and used the transformation of Gauss and Bianchi while discussing automorphisms of that surface:

Beltrami and Weierstrass coordinates

In he defined the following invariant quadratic form and its projective transformation in terms of Beltrami coordinates :
Alternatively, Woods – citing Killing – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation :
and the case of translation:
and the loxodromic substitution for hyperbolic space:

Whitehead (1897/98) – Universal algebra

discussed the kinematics of hyperbolic space as part of his study of universal algebra, and obtained the following transformation:

Scheffers (1899) – Contact transformation

synthetically determined all finite contact transformations preserving circles in the plane, consisting of dilatations, inversions, and the following one preserving circles and lines and Darboux ):

Hausdorff (1899)

Weierstrass coordinates

– citing Killing – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:

Möbius transformation

Hausdorff also discussed the relation of the above coordinates to conformal Möbius transformations:

Smith (1900) – Laguerre inversion

followed Laguerre and Darboux and defined the Laguerre inversion as follows:

Vahlen (1901/02) – Clifford algebra and Möbius transformation

Modifying Lipschitz's variant of Clifford numbers, Theodor Vahlen formulated Möbius transformations and biquaternions in order to discuss motions in n-dimensional non-Euclidean space, leaving the following quadratic form invariant :

Liebmann (1904–05) – Weierstrass coordinates

– citing Killing, Gérard, Hausdorff – used the invariant quadratic form and its Lorentz transformation equivalent to
and the case of translation:

Eisenhart (1905) – Pseudospherical surfaces

followed Bianchi and Darboux by writing the Lie's transformation of pseudospherical surfaces:

Electrodynamics and special relativity

Voigt (1887)

developed a transformation in connection with the Doppler effect and an incompressible medium, being in modern notation:
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 :
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:
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:
Consequently, Joseph John Thomson found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation :
Thereby, inhomogeneous electromagnetic wave equations are transformed into a Poisson equation. Eventually, George Frederick Charles Searle noted in that Heaviside's expression leads to a deformation of electric fields which he called "Heaviside-Ellipsoid" of axial ratio

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 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
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 introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced length contraction in his theory. 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 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:
For solving optical problems Lorentz used the following transformation, in which the modified time variable was called "local time" by him:
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
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 system in the ratio 1/γ". Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than '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:
In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time t″=t′-εvx′/c2 instead of the 1897 expression t′=t-vx/c2 by replacing v/c2 with εv/c2, so that t″ is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the x′, y′, z′, t′ coordinates:
Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor '2, and so he sought the transformations which were "accurate to second order". Thus he wrote the final transformations as:
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 and Poincaré 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:

Lorentz (1899, 1904)

Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 :
Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows :
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 :
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:
Lorentz's 1904 transformation was cited and used by Alfred Bucherer in July 1904:
or by Wilhelm Wien in July 1904:
or by Emil Cohn in November 1904 :
or by Richard Gans in February 1905:

Poincaré (1900, 1905)

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 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 a light signal from one clock is sent to another, and is sent back. It's supposed that the Earth is moving with speed v in the x-direction in some rest system . 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*=/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. 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.
Similar physical interpretations of local time were later given by Emil Cohn and Max Abraham.

Lorentz transformation

On June 5, 1905 Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form :
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 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, which he had already derived in unpublished letters to Lorentz from May 1905:

Einstein (1905) – Special relativity

On June 30, 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:
Einstein also defined the velocity addition formula :
and the light aberration formula :

Minkowski (1907–1908) – Spacetime

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. 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 in agreement with :
Even though Minkowski used the imaginary number iψ, he for once directly used the tangens hyperbolicus in the equation for velocity
Minkowski's expression can also by written as ψ=atanh and was later called rapidity. He also wrote the Lorentz transformation in matrix form equivalent to :
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:

Sommerfeld (1909) – Spherical trigonometry

Using an imaginary rapidity such as Minkowski, Arnold Sommerfeld formulated a transformation equivalent to Lorentz boost, and the relativistc velocity addition in terms of trigonometric functions and the spherical law of cosines:

Bateman and Cunningham (1909–1910) – Spherical wave transformation

In line with Lie's research on the relation between sphere transformations with an imaginary radius coordinate and 4D conformal transformations, it was pointed out by Bateman and Cunningham, 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. 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. In particular, by setting λ=1 the Lorentz group can be seen as a 10-parameter subgroup of the 15-parameter spacetime conformal group.
Bateman 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, Henri Poincaré and others.

Herglotz (1909/10) – Möbius transformation

Following Klein and Fricke & Klein concerning the Cayley absolute, hyperbolic motion and its transformation, Gustav Herglotz classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic. The general case equivalent to Lorentz transformation and the hyperbolic case equivalent to Lorentz transformation or squeeze mapping are as follows:

Varićak (1910) – Hyperbolic functions

Following Sommerfeld, 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 with u as rapidity he wrote the Lorentz transformation in agreement with :
and showed the relation of rapidity to the Gudermannian function and the angle of parallelism:
He also related the velocity addition to the hyperbolic law of cosines:
Subsequently, other authors such as E. T. Whittaker or Alfred Robb used similar expressions, which are still used in modern textbooks.

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 showed that it is possible to use the principle of relativity alone, in order to derive the following transformation between two inertial frames:
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, with various authors developing similar methods in subsequent years.

Noether (1910), Klein (1910) – Quaternions

described Cayley's 4D quaternion multiplications as "Drehstreckungen", 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.
In an appendix to Klein's and Sommerfeld's "Theory of the top", 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 :
Besides citing quaternion related standard works such as Cayley, Noether referred to the entries in Klein's encyclopedia by Eduard Study and the French version by Élie Cartan. Cartan's version contains a description of Study's dual numbers, Clifford's biquaternions, and Clifford algebra, with references to Stephanos, Buchheim, Vahlen and others.
Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:
or in March 1911

Conway (1911), Silberstein (1911) – Quaternions

in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:
Also Ludwik Silberstein in November 1911 as well as in 1914, formulated the Lorentz transformation in terms of velocity v:
Silberstein cites Cayley and Study's encyclopedia entry, as well as the appendix of Klein's and Sommerfeld's book.

Herglotz (1911), Silberstein (1911) – Vector transformation

showed how to formulate the transformation equivalent to in order to allow for arbitrary velocities and coordinates v=' and r=':
This was simplified using vector notation by Ludwik Silberstein :
Equivalent formulas were also given by Wolfgang Pauli, with Erwin Madelung providing the matrix form
These formulas were called "general Lorentz transformation without rotation" by Christian Møller, 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′=' is not equal to -v=', but the relation holds instead, with the result

Borel (1913–14) – Cayley–Hermite parameter

Borel started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and Cayley's parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions equivalent to :
In four dimensions equivalent to :

Gruner (1921) – Trigonometric Lorentz boosts

In order to simplify the graphical representation of Minkowski space, Paul Gruner developed what is now called Loedel diagrams, using the following relations:
In another paper Gruner used the alternative relations:

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 precalculus textbook 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
This development misses the alternative:
Here Euler could have noted split-complex numbers along with complex numbers.
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
. 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.

Historical mathematical sources

In English:
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