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The Lorentz transformations relate the space-time coordinates, which specify the position x, y, z and time t of an event, relative to a particular inertial frame of reference (the "rest system"), and the coordinates of the same event relative to another coordinate system moving in the positive x-direction at a constant speed v, relative to the rest system. It was devised as a theoretical transformation which makes the velocity of light invariant between different inertial frames. The coordinates of the event in this "moving system" are denoted x′, y′, z′ and t′. The rest system was sometimes identified with the luminiferous aether, the postulated medium for the propagation of light, and the moving system was commonly identified with the earth as it moved through this medium. Early approximations of the transformation were published by Voigt (1887) and Lorentz (1895). They were completed by Larmor (1897, 1900) and Lorentz (1899, 1904) and were brought into their modern form by Poincaré (1905), who gave the transformation the name of Lorentz. Eventually, Einstein (1905) showed in his development of special relativity that the transformations follow from the principle of relativity and the constant light speed alone, without requiring a mechanical aether, and are changing the traditional concepts of space and time. Subsequently, Minkowski used them to argue that space and time are inseparably connected as spacetime. Important contributions to the mathematical understanding of the Lorentz transformation were also made by other authors such as Vladimir Varićak (1910) and Vladimir Ignatowski (1910).
Contents
- Sphere geometry in the 19th century
- Voigt 1887
- Heaviside 1888 Thomson 1889 Searle 1896
- Lorentz 1892 1895
- Larmor 1897 1900
- Lorentz 1899 1904
- Local time
- Lorentz transformation
- Einstein 1905
- Minkowski 19071908
- Variak 1910
- Ignatowski 1910
- References
The Lorentz transformation has the form
v being the relative velocity of the two reference frames, and c the speed of light, and the Lorentz factor,
In this article the historical notations are placed on the left, and modern notations on the right.
Sphere geometry in the 19th century
One of the defining properties of the Lorentz transformation is its group structure which leaves the expression
In several papers between 1847 and 1850 it was shown by Joseph Liouville that the relation
Albert Ribaucour (1870) and in particular Edmond Laguerre (1880-1885) employed another variant, namely the "transformation by reciprocal directions" or „Laguerre inversion/transformation“ which transforms spheres into spheres and planes into planes. Laguerre explicitly wrote down the corresponding transformation formulas in 1882, with Gaston Darboux (1887) presenting them in respect to coordinates
producing the following relation:
Several authors showed the close relation to the Lorentz transformation (see Laguerre inversion and Lorentz transformation) – by setting
thus the above transformation becomes similar to a Lorentz transformation with
Furthermore, the group isomorphism between the Laguerre group and Lorentz group was pointed out by Élie Cartan, Henri Poincaré and others (see Laguerre group isomorphic to Lorentz group).
Voigt (1887)
Woldemar 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
Voigt sent his 1887 paper to Lorentz in 1908, and that was acknowledged in 1909:
In a paper „Über das Doppler'sche Princip“, published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (6) (§ 3 of this book) [namely
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 (1889) found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the Galilean transformation
Thereby, inhomogeneous electromagnetic wave equations are transformed into a Poisson equation. Eventually, George Frederick Charles Searle noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called "Heaviside-Ellipsoid" of axial ratio
Lorentz (1892, 1895)
In order to explain the aberration of light and the result of the Fizeau experiment in accordance with Maxwell's equations, Lorentz in 1892 developed a model ("Lorentz ether theory") in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson).
where x* is the Galilean transformation x-vt. Except the additional
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
For solving optical problems Lorentz used the following transformation, in which the modified time variable was called "local time" (German: Ortszeit) 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. 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
In 1900 he modified the above local time
This transformation is just the Galilean transformation for the
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
Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term "Lorentz transformation" for Lorentz's first order transformations of coordinates and field configurations:
p. 583: [..] Lorentz's transformation for passing from the field of activity of a stationary electrodynamic material system to that of one moving with uniform velocity of translation through the aether.
p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..]
p. 622: [..] the transformation first developed by Lorentz: namely, each point in space is to have its own origin from which time is measured, its "local time" in Lorentz's phraseology, and then the values of the electric and magnetic vectors [..] at all points in the aether between the molecules in the system at rest, are the same as those of the vectors [..] at the corresponding points in the convected system at the same local times.
Lorentz (1899, 1904)
Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a tranformation equivalent to the one from 1892 (again,
Then he introduced a factor
This is identical to the complete Lorentz transformation when solved for
In 1904 he rewrote the equations in the following form by setting
Under the assumption that
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.
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
and the time of flight back is
The elapsed time on the clock when the signal is returned is
identical to Lorentz (1892). Poincaré gave the result
Similar physical interpretations of local time were later given by Emil Cohn (1904) and Max Abraham (1905).
Lorentz transformation
On June 5, 1905 (published June 9) Poincaré simplified the 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
In July 1905 (published in January 1906) Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the principle of least action; he demonstrated in more detail the group characteristics of the transformation, which he called Lorentz group, and he showed that the combination
Einstein (1905)
On June 30, 1905 (published September 1905) Einstein published what is now called special relativity and gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. While Lorentz considered "local time" to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in v/c this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations concern the nature of space and time.
The notation for this transformation is identical to Poincaré's of 1905, except that Einstein didn't set the speed of light to unity:
Minkowski (1907–1908)
The work on the principle of relativity by Lorentz, Einstein, Planck, together with Poincaré's four-dimensional approach, were further elaborated by Hermann Minkowski in 1907 and 1908. Minkowski particularly reformulated electrodynamics in a four-dimensional way (Minkowski spacetime). For instance, he wrote
Even though Minkowski ordinarily used the imaginary number
Minkowski's expression can also by written as
Varićak (1910)
Minkowski's rapidity in terms of real hyperbolic functions was systematically employed by Vladimir Varićak in several papers starting from 1910, who represented the equations of special relativity on the basis of hyperbolic geometry. For instance, by setting
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.
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:
The variable