History of Lorentz transformations

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.

[3] Eventually, Einstein (1905) showed in his development of special relativity that the transformations follow from the principle of relativity and constant light speed alone by modifying the traditional concepts of space and time, without requiring a mechanical aether in contradistinction to Lorentz and Poincaré.

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.

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

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 3]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.

He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:[8] Consequently, Joseph John Thomson (1889)[R 6] found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the Galilean transformation z-vt in his equation[9]): Thereby, inhomogeneous electromagnetic wave equations are transformed into a Poisson equation.

[9] Eventually, George Frederick Charles Searle[R 7] noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called "Heaviside-Ellipsoid" of axial ratio 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 8][10] where x* is the Galilean transformation x-vt.

[10] While t is the "true" time for observers resting in the aether, t′ is an auxiliary variable only for calculating processes for moving systems.

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.

[12][13] Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than (v/c)2 – when his 1897 paper was reprinted in 1929, Larmor added the following comment in which he described how they can be made valid to all orders of v/c:[R 13] 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 14] Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor (v/c)2, and so he sought the transformations which were "accurate to second order" (as he put it).

[14] In 1904 he rewrote the equations in the following form by setting l=1/ε (again, x* must be replaced by x-vt):[R 21] 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.

[15] When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:[16] One will notice that in this work the transformation equations of Einstein’s Relativity Theory have not quite been attained.

[..] On this circumstance depends the clumsiness of many of the further considerations in this work.Lorentz's 1904 transformation was cited and used by Alfred Bucherer in July 1904:[R 23] or by Wilhelm Wien in July 1904:[R 24] or by Emil Cohn in November 1904 (setting the speed of light to unity):[R 25] or by Richard Gans in February 1905:[R 26] Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time.

[17] He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed

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.

[18][19] 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.

[20] In July 1905 (published in January 1906)[R 31] 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 also formulated the velocity addition formula, which he had already derived in unpublished letters to Lorentz from May 1905:[R 32] 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 of relativity and the principle of the constancy of the speed of light.

w:Henry Crozier Keating Plummer (1910) defined the Lorentz boost in terms of trigonometric functions[R 51] 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 52][R 53] 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.

[26] Felix Klein (1908) described Cayley's (1854) 4D quaternion multiplications as "Drehstreckungen" (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor), and pointed out that the modern principle of relativity as provided by Minkowski is essentially only the consequent application of such Drehstreckungen, even though he didn't provide details.

[R 55] In an appendix to Klein's and Sommerfeld's "Theory of the top" (1910), Fritz Noether showed how to formulate hyperbolic rotations using biquaternions with

He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations:[R 56] Besides citing quaternion related standard works by Arthur Cayley (1854), Noether referred to the entries in Klein's encyclopedia by Eduard Study (1899) and the French version by Élie Cartan (1908).

[27] Cartan's version contains a description of Study's dual numbers, Clifford's biquaternions (including the choice

Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:[R 57] or in March 1911[R 58] Arthur W. Conway in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:[R 59] Also Ludwik Silberstein in November 1911[R 60] as well as in 1914,[28] 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.

Vladimir Ignatowski (1910, published 1911) showed how to reformulate the Lorentz transformation in order to allow for arbitrary velocities and coordinates:[R 61] Gustav Herglotz (1911)[R 62] also showed how to formulate the transformation 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] Equivalent formulas were also given by Wolfgang Pauli (1921),[29] with Erwin Madelung (1922) providing the matrix form[30] These formulas were called "general Lorentz transformation without rotation" by Christian Møller (1952),[31] who in addition gave an even more general Lorentz transformation in which the Cartesian axes have different orientations, using a rotation operator

Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations.