History of the Lorentz Transformation

The Lorentz transformation , like the Galilei transformation, links the coordinates of an event in a certain inertial system with the coordinates of the same event in another inertial system, which is moved in the positive x-direction with the speed v relative to the first system. However, in contrast to the Galileo transformation, it contains, in addition to the principle of relativity, the constancy of the speed of light in all inertial systems and thus forms the mathematical basis for the special theory of relativity . ${\displaystyle x,y,z,t}$ ${\displaystyle x',y',z',t'}$

The first formulations of this transformation were published by Woldemar Voigt (1887) and Hendrik Lorentz (1892, 1895), whereby these authors regarded the unpainted system as resting in the ether , and the “moving” painted system was identified with the earth. This transformation was completed by Joseph Larmor (1897, 1900) and Lorentz (1899, 1904) and brought into its modern form by Henri Poincaré (1905), who gave the transformation its name. Albert Einstein (1905) was finally able to derive the equations from a few basic assumptions and showed the connection between transformation and fundamental changes in the concepts of space and time.

In this article, the historical terms are replaced with modern ones, using the Lorentz transformation

${\displaystyle x^{\prime }=\gamma (x-vt),\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t^{\prime }=\gamma \left(t-x{\frac {v}{c^{2}}}\right)}$,

and the Lorentz factor :

${\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$,

v is the relative speed between bodies and c is the speed of light .

Spherical geometry in the 19th century

→ Main article : Ball wave transformation

One of the defining properties of the Lorentz transformation is its group structure, whereby the invariance of is fulfilled in all inertial systems. This means, for example, that a spherical wave in one inertial system is also a spherical wave in all other inertial systems, which is usually also used to derive the Lorentz transformation. However, long before experiments and physical theories made the introduction of the Lorentz transformation necessary, transformation groups such as the conformal transformation through reciprocal radii in Möbius geometry or the transformation through reciprocal directions in Laguerre geometry were discussed, especially in pure mathematics Transform balls into other balls. These can be seen as special cases of Lie's spherical geometry . However, the connection between these transformations and the Lorentz transformation in physics was only discovered after 1905. ${\displaystyle x^{2}+y^{2}+z^{2}-c^{2}t^{2}}$

In several works between 1847 and 1850, Joseph Liouville proved that the shape is invariant under the conformal group or the transformation by reciprocal radii maps the spheres into spheres. This proof was extended to all dimensions by Sophus Lie (1871) in the context of Lieschen spherical geometry . Harry Bateman and Ebenezer Cunningham showed in 1909 that not only the above quadratic relationship, but Maxwell's electrodynamics is also covariant under the conformal group of spherical wave transformations for any . This covariance is limited to sub-areas such as electrodynamics, whereas the entirety of the laws of nature in inertial systems is only covariant under the Lorentz group . ${\displaystyle \lambda \left(\delta x^{2}+\delta y^{2}+\delta z^{2}\right)}$ ${\displaystyle \lambda \left(\delta x_{1}^{2}+\dots +\delta x_{n}^{2}\right)}$ ${\displaystyle \lambda }$

A related transformation was given by Albert Ribaucour (1870) and especially Edmond Laguerre (1880–1885) - the transformation through reciprocal directions (also called "Laguerre inversion" or "Laguerre transformation"), which spheres in spheres and planes in Depicts levels. Laguerre wrote the corresponding formulas explicitly in 1882, and Gaston Darboux (1887) reformulated them for the coordinates (with R as the radius): ${\displaystyle x,y,z,R}$

${\displaystyle {\begin{aligned}x'&=x,\quad &z'&={\frac {1+k^{2}}{1-k^{2}}}z-{\frac {2kR}{1-k^{2}}},\\y'&=y,&R'&={\frac {2kz}{1-k^{2}}}-{\frac {1+k^{2}}{1-k^{2}}}R,\end{aligned}}}$

creates the following relationship:

${\displaystyle x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-R^{\prime 2}=x^{2}+y^{2}+z^{2}-R^{2}}$.

Some authors noted the extensive analogy to the Lorentz transformation (see Laguerre inversion and Lorentz transformation ) - is set , and it follows ${\displaystyle R=t}$ ${\displaystyle c=1}$ ${\displaystyle v=2k/\left(1+k^{2}\right)}$

${\displaystyle {\frac {1-k^{2}}{1+k^{2}}}={\sqrt {1-v^{2}}}={\frac {1}{\gamma }},\quad {\frac {2k}{1-k^{2}}}=v\gamma ,}$

which, if used in the above transformation, results in a very large analogy to a Lorentz transformation with the direction of movement, except that the sign of is reversed from to : ${\displaystyle z}$ ${\displaystyle t'}$ ${\displaystyle t-vz}$ ${\displaystyle vz-t}$

${\displaystyle x'=x,\quad y'=y,\quad z'=\gamma (z-vt),\quad t'=\gamma (vz-t).}$

In fact, the group isomorphism of the two groups was demonstrated by Élie Cartan , Henri Poincaré (1912) and others (see Laguerre group isomorphic to Lorentz group ).

Voigt (1887)

As part of a theoretical investigation of the Doppler effect of transverse waves in an incompressible elastic transmission medium, which served as a model for the light ether , Voigt (1887) developed the following transformation, which left the wave equation unchanged and in modern notation had the form:

${\displaystyle x^{\prime }=x-vt,\quad y^{\prime }={\frac {y}{\gamma }},\quad z^{\prime }={\frac {z}{\gamma }},\quad t^{\prime }=t-x{\frac {v}{c^{2}}}}$

When the right-hand sides of these equations are multiplied by a scale factor, the formulas of the Lorentz transformation result. The reason for this is that, as explained above, the electromagnetic equations are not only Lorentz invariant , but also scale invariant and even conformal invariant . The Lorentz transformation can be provided with the above scale factor , for example: ${\displaystyle \gamma }$ ${\displaystyle l={\sqrt {\lambda }}}$

${\displaystyle x^{\prime }=\gamma l\left(x-vt\right),\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }=\gamma l\left(t-x{\frac {v}{c^{2}}}\right)}$.

With you get the Voigt transformation, and with the Lorentz transformation. As Poincaré and Einstein in particular later showed, the transformations are only symmetrical and form a group, which is the prerequisite for compatibility with the principle of relativity . So the Voigt transformation is not symmetrical and violates the principle of relativity. The Lorentz transformation, on the other hand, is also valid for all natural laws outside of electrodynamics. However, with some problem solutions, such as the calculation of radiation phenomena in empty space, both transformations lead to the same end result. ${\displaystyle l=1/\gamma }$ ${\displaystyle l=1}$ ${\displaystyle l=1}$

Regarding the Doppler effect, Voigt's work from 1887 was referenced by Emil Kohl in 1903. Regarding the Lorentz transformation, Lorentz declared in 1909 and 1912 that Voigt's transformation was "equivalent" to the transformation with the above scale factor in his own work from 1904, and that if he had known these equations he could have used them in his electron theory. Hermann Minkowski praised Voigt's achievements in 1908 in space and time and in a discussion : ${\displaystyle l}$

" Minkowski : Historically, I may add that the transformations that play the main role in the principle of relativity, are first treated mathematically by Voigt in the 1887th With their help Voigt drew conclusions with regard to Doppler's principle.
Voigt : Mr. Minkowski reminds me of an old work of mine. These are applications of the Doppler principle, which occur in special parts, but not on the basis of the electromagnetic, but on the basis of the elastic theory of light. At that time, however, some of the same conclusions emerged that were later derived from electromagnetic theory. "

Heaviside, Thomson, Searle (1888, 1889, 1896)

In 1888 Oliver Heaviside studied the properties of moving charges according to Maxwell's electrodynamics. He calculates, among other things, that anisotropies in an electric field of moving charges must occur according to the following formula:

${\displaystyle \mathrm {E} =\left({\frac {q\mathrm {r} }{r^{2}}}\right)\left(1-{\frac {v^{2}\sin ^{2}\theta }{c^{2}}}\right)^{-3/2}}$.

Building on this, Joseph John Thomson (1889) discovered a method to substantially simplify calculations for moving charges by using the following mathematical transformation:

${\displaystyle x^{\prime }=\gamma x}$.

This allows inhomogeneous electromagnetic wave equations to be transformed into a Poisson equation . Finally, George Frederick Charles Searle (1896) noted that Heavisides' expression for moving charges leads to a deformation of the electric field, which he called the "Heaviside ellipsoid" with an axis ratio of . ${\displaystyle 1/\gamma :1:1\!}$

Lorentz (1892, 1895)

In 1892 Lorentz developed the basic features of a model, later referred to as Lorentz's theory of ether , in which the ether is completely at rest, whereby the speed of light has the same value in all directions. In order to be able to calculate the optics of moving bodies, Lorentz introduced the following auxiliary variables for the transformation from the ether system into a system that is moving relative to it:

${\displaystyle x^{\prime }=\gamma x^{*},\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t^{\prime }=t-\gamma ^{2}x^{*}{\frac {v}{c^{2}}}}$

where the Galileo transformation is. While the “true” time is now for systems resting in the ether, the time is a mathematical auxiliary variable which is used for calculations of systems moving in the ether. A similar “local time” was already used by Voigt, but Lorentz later stated that he had no knowledge of his work at the time. It is also unknown whether he was familiar with Thomson's work. ${\displaystyle x^{\ast }}$ ${\displaystyle x-vt}$ ${\displaystyle t}$ ${\displaystyle t'}$

In 1895 he further developed Lorentz's electrodynamics much more systematically, whereby a fundamental concept was the "theorem of the corresponding states" for quantities . It follows from this that an observer moving in the ether makes approximately the same observations in his "fictitious" (electromagnetic) field as an observer resting in the ether makes approximately the same observations in his "real" field. That means, as long as the velocities are comparatively low relative to the ether, the Maxwell equations have the same form for all observers. For the electrostatics of moving bodies, he used the transformations that changed the dimensions of the body as follows: ${\displaystyle v/c}$

${\displaystyle x^{\prime }=\gamma x^{*},\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t^{\prime }=t}$

As an additional and independent hypothesis, Lorentz (1892b, 1895) asserted (without proof as he admitted) that intermolecular forces and thus also material bodies are deformed in a similar way, and led to the explanation of the Michelson-Morley experiment, the length contraction a. The same hypothesis had already been put forward in 1889 by George FitzGerald , whose considerations were based on the work of Heaviside. But while for Lorentz the contraction in length was a real, physical effect, for him the local time initially only meant an agreement or a useful calculation method. For the optics of moving bodies, however, he used the transformations:

${\displaystyle x^{\prime }=x^{*},\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t^{\prime }=t-x^{*}{\frac {v}{c^{2}}}}$

With the help of local time, Lorentz was able to explain the aberration of light , the Doppler effect and the dependence of the speed of light in moving liquids measured in the Fizeau experiment . It is important that Lorentz and later Larmor always formulated the transformations in two steps. First the Galileo transformation, and then, separately, the extension to the "fictitious" electromagnetic system with the help of the Lorentz transformation. The equations only got their symmetrical shape through Poincaré.

Larmor (1897, 1900)

At that time Larmor knew that the Michelson – Morley experiment was precise enough to show movement-related effects of size , and so he was looking for a transformation that is also valid for these sizes. Although he followed a very similar scheme to Lorentz's, he went beyond his work of 1895 and modified the equations so that he was the first to set up the complete Lorentz transformation in 1897 and a little more clearly in 1900: ${\displaystyle v^{2}/c^{2}\,}$

${\displaystyle x^{\prime }=\gamma x^{*},\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t^{\prime }={\frac {t}{\gamma }}-\gamma x^{*}{\frac {v}{c^{2}}}}$

He showed that the Maxwell equations were invariant under this 2-step transformation (however, he only performed the proof for quantities of the second order, not for all orders). Larmor also noted that if an electrical constitution of the molecules is assumed, the contraction in length is a consequence of the transformation. He was also the first to notice a kind of time dilation as a consequence of the equations, because periodic processes of moving objects are relatively slower than those of stationary objects. ${\displaystyle 1/\gamma }$

Larmor paid tribute to Lorentz in two papers published in 1904, in which he used the term "Lorentz transformation" for the transformation (for greater than first order) 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)

Lorentz also derived the complete transformation in 1899 by extending the theorem of the corresponding states. However, he used the indefinite factor as a function of . Like Larmor, Lorentz noticed a kind of time dilation, as he recognized that the vibrations of an oscillating electron, which moves relative to the ether, run more slowly. Due to further negative ether wind experiments ( Trouton Noble experiment , experiments by Rayleigh and Brace ) Lorentz was forced to formulate his theory in such a way that ether wind effects of all sizes remain undetectable. To this end, he wrote the Lorentz transformation in the same form as Larmor, with an initially undefined factor : ${\displaystyle \epsilon }$ ${\displaystyle v}$ ${\displaystyle v/c}$ ${\displaystyle l}$

${\displaystyle x^{\prime }=\gamma lx^{*},\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }={\frac {l}{\gamma }}t-\gamma lx^{*}{\frac {v}{c^{2}}}}$

In this context he derived the correct equations for the velocity dependence of the electromagnetic mass as early as 1899 and in 1904 he concluded that this transformation must be applied to all forces of nature, not just electrical ones, and therefore the length contraction is a consequence of the transformation.

Likewise, Lorentz stipulated that with must also be, and later showed that this is only the case if it is generally given - from this he concluded that the Lorentz contraction can only occur in the direction of movement. With this he formulated the actual Lorentz transformation, but did not achieve the covariance of the transformation equations of charge density and velocity. He therefore wrote in 1912 about his work from 1904: ${\displaystyle v=0}$ ${\displaystyle l=1}$ ${\displaystyle l=1}$

“You will notice that in this paper I have not quite reached the transformation equations of Einstein's theory of relativity. [...] The awkwardness of some of the further considerations in this work is related to this fact. "

In the spring of 1905 Richard Gans had published a summary of the 1904 article by Lorentz in No. 4 of the bi-weekly Beiblätter zu den Annalen der Physik , about which Albert Einstein also published summaries of important international articles in his special field of thermodynamics and at the same time used to contribute statistical mechanics. It is noteworthy that Einstein claims not to have known Lorentz's work from 1904, although 14 days later he himself published a whole series of summaries in the same journal, in issue no. 5, signed with the abbreviation "AE" are. Einstein's biographer Abraham Pais came to the conclusion in his biography of Einstein, after carefully examining the documents, that Einstein was not yet familiar with the Lorentz transformation when he was preparing his 1905 essay.

Poincaré (1900, 1905)

Neither Lorentz nor Larmor gave a clear interpretation of the origin of local time. In 1900, however, Poincaré interpreted local time as the result of a synchronization carried out with light signals. He assumed that two observers A and B moving in the ether synchronize their clocks with optical signals. Since they believe they are at rest, they assume a constant speed of light in all directions, so that all they have to do now is take into account the times of flight and cross their signals to check the synchronicity of the clocks. On the other hand, from the point of view of an observer resting in the ether, one clock runs towards the signal and the other runs away from it. The clocks are therefore not synchronous ( relativity of simultaneity ), but only show the local time for quantities of the first order in . But since the moving observer has no means to decide whether they are moving or not, they will not notice the error. In contrast to Lorentz, Poincaré understood local time as well as length contraction as a real physical effect. Similar explanations were given later by Emil Cohn (1904) and Max Abraham (1905). ${\displaystyle v/c}$ ${\displaystyle t^{\prime }=t-vx/c^{2}}$

On June 5, 1905 (published June 9), Poincaré simplified the equations (which are equivalent to those of Larmor and Lorentz) and gave them their modern symmetrical form, in which, in contrast to Larmor and Lorentz, he converted the Galileo transformation into the new one Transformation directly integrated. Obviously Poincaré was not aware of Larmor's work, because he only referred to Lorentz and was therefore the first to use the expression "Lorentz transformation" (whereby the expression "Lorentz transformation" was used by Emil Cohn in 1900 for the 1895 equations by Lorentz has been):

${\displaystyle x^{\prime }=\gamma l(x-vt),\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }=\gamma l\left(t-vx\right)}$

and vice versa:

${\displaystyle x=\gamma l(x^{\prime }+vt^{\prime }),\quad y=ly^{\prime },\quad z=lz^{\prime },\quad t=\gamma l\left(t^{\prime }+vx^{\prime }\right)}$

He set the speed of light equal to 1 and, like Lorentz, showed that a bet must be made. Poincaré was able to derive this more generally from the fact that the totality of the transformations form a symmetrical group only under this condition, which is necessary for the validity of the principle of relativity. He also showed that Lorentz's application of the transformations does not fully satisfy the relativity principle. Poincaré, on the other hand, was able to fully demonstrate the Lorentz covariance of the Maxwell-Lorentz equations in addition to showing the group property of the transformation. ${\displaystyle l=1}$

A significantly expanded version of this document from July 1905 (published January 1906) contained the knowledge that the combination is invariant; he introduced the term as the fourth coordinate of a four-dimensional space ; he was already using four-vectors before Minkowski; he showed that the transformations are a consequence of the principle of least effect ; and he demonstrated in more detail than before their group characteristics, where he coined the name Lorentz group ("Le groupe de Lorentz"). Like Lorentz, Poincaré continued to distinguish between “true” coordinates in the ether and “apparent” coordinates for moving observers. ${\displaystyle x^{2}+y^{2}+z^{2}-c^{2}t^{2}}$ ${\displaystyle ct{\sqrt {-1}}}$

Einstein (1905)

On June 30, 1905 (published September 1905) Einstein presented a radically new interpretation and derivation of the transformation within the framework of the special theory of relativity , which was based on two postulates, namely the principle of relativity and the principle of the constancy of the speed of light. While Poincaré had only derived the original Lorentz local time from 1895 by optical synchronization, Einstein was able to derive the entire transformation using a similar synchronization method, and thereby show that operational considerations in terms of space and time were sufficient and that no ether was required for this (whether Einstein was influenced by Poincaré's synchronization method is not known). In contrast to Lorentz, who saw local time only as a mathematical trick, Einstein showed that the "effective" coordinates of the Lorentz transformation are indeed coordinates of inertial systems with equal rights. In a certain way, Poincaré has already portrayed this in this way, but the latter still differentiated between "true" and "apparent" time. Formally, Einstein's version of the transformation was identical to that of Poincaré, although Einstein did not set the speed of light equal to 1. Einstein was also able to show that the transformations form a group:

${\displaystyle x^{\prime }=\gamma (x-vt),\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t^{\prime }=\gamma \left(t-x{\frac {v}{c^{2}}}\right)}$

From the transformations, Einstein was able to derive effects such as time dilation , length contraction , Doppler effect , aberration of light , or the relativistic addition of speed as a consequence of this new understanding of space and time, without having to make any assumptions about the structure of matter or a substantial ether .

Minkowski (1907-1908)

The work on the principle of relativity by Lorentz, Einstein, Planck , together with Poincaré's four-dimensional approach, was continued by Hermann Minkowski in the years 1907 to 1908, especially with the inclusion of group theoretical arguments. His main achievement was the four-dimensional reformulation of electrodynamics. For example, he wrote in the form , and if the angle of rotation is about the z axis, then the Lorentz transformations take the form: ${\displaystyle x,y,z,it}$ ${\displaystyle x_{1},x_{2},x_{3},x_{4}}$ ${\displaystyle \psi }$

${\displaystyle x'_{1}=x_{1},\quad x'_{2}=x_{2},\quad x'_{3}=x_{3}\cos i\psi +x_{4}\sin i\psi ,\quad x'_{4}=-x_{3}\sin i\psi +x_{4}\cos i\psi ,}$

where and . He also introduced the graphical representation of the Lorentz transformation using Minkowski diagrams : ${\displaystyle \cos i\psi ={1}/{\sqrt {1-v^{2}}}}$ ${\displaystyle c=1}$

Original space-time diagram by Minkowski from 1908.

Ignatowski (1910)

While earlier derivations of the Lorentz transformation were based on optics, electrodynamics or the invariance of the speed of light from the outset, Vladimir Sergejewitsch Ignatowski (1910) showed that it is possible to derive the following transformation between two inertial systems from the principle of relativity (and related group-theoretical arguments) :

${\displaystyle x^{\prime }=p(x-vt),\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t^{\prime }=p(t-nvx)}$

where . The variable can be seen as a spacetime constant, the value of which is determined from experiment or a known physical law. Ignatowski used the Heaviside ellipsoid mentioned above, which represents a contraction of electrostatic fields in the direction of movement. This is in agreement with Ignatowski's transformation if one sets what, and thus, the Lorentz transformation follows. results in no length changes and consequently the Galileo transformation. Ignatowski's method was improved and expanded upon by Philipp Frank and Hermann Rothe (1911, 1912), and many authors followed who developed similar methods. ${\displaystyle p=1/{\sqrt {1-nv^{2}}}}$ ${\displaystyle n}$ ${\displaystyle x/\gamma }$ ${\displaystyle n=1/c^{2}}$ ${\displaystyle p=\gamma }$ ${\displaystyle n=0}$

See also

• History of the special theory of relativity

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Primary sources

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Individual evidence

1. ↑ Walter (2012)
2. ↑ Kastrup (2008), Section 2.3
3. ↑ Klein & Blaschke (1926)
4. ↑ Klein & Blaschke (1926), p. 259
5. ↑ ^{a b} Pais (1982), chap. 6b
6. ↑ A derivation method possibly used by Voigt is contained in section 1.4 The Relativity of Light of the treatise Reflections on Relativity on the MathPages website (where the scale factor was used instead ): "In order to make the transformation formula for agree with the Galilean transformation , Voigt chose , so he did not actually arrive at the Lorentz transformation, but nevertheless he had shown roughly how the wave equation could actually be relativistic - just like the dynamic behavior of inertial particles - provided we are willing to consider a transformation of the space and time coordinates that differs from the Galilean transformation. " ${\displaystyle A=\gamma l}$ ${\displaystyle l}$ ${\displaystyle x}$ ${\displaystyle A=1}$
7. ↑ ^{a b} Miller (1981), 114-115
8. ↑ Lorentz (1916), writes in the footnote on p. 198: "1) In a paper" About the Doppler principle ", 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 ] a transformation equivalent to the formulas (287) and (288) [namely the above transformation with the scale factor ]. 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. " ${\displaystyle \Delta \Psi -{\tfrac {1}{c^{2}}}{\tfrac {\partial ^{2}\Psi }{\partial t^{2}}}=0}$ ${\displaystyle l}$
9. ↑ I add the remark that Voigt already in 1887 [...] in a work “On Doppler Principle” on equations of the form ${\displaystyle \Delta \psi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\psi }{\partial t^{2}}}=0}$has applied a transformation which is equivalent to that contained in equations (4) and (5) [namely the above transformation with the scale factor ] of my work. ${\displaystyle l}$
10. ↑ Walter (1999a), p. 59
11. ↑ Brown (2003)
12. ↑ ^{a b} Miller (1981), 98-99
13. ↑ ^{a b} Miller (1982), chap. 1.4 & 1.5
14. ↑ Janssen (1995), chap. 3.1
15. ^ Macrossan (1986)
16. ↑ Darrigol (2000), chap. 8.5
17. ↑ Jannsen (1995), chap. 3.3
18. ↑ ^{a b} Miller (1981), chap. 1.12.2
19. ↑ Richard Gans : HA Lorentz, Electromagnetic processes in a system that moves with an arbitrary speed (less than that of light) (Versl. K. Ak. Van Wet. 12 , pp. 986-1009, 1904) . In: Supplements to the Annals of Physics , Volume 29, 1905, No. 4, pp. 168–170.
20. ↑ In Issue No. 5 of the supplements to the Annals of Physics , Volume 29, 1905, the abbreviation “AE” appears on pages 235 (twice), 236, 237 (three times), 238, 240, 242 and 247. In the Issues No. 6 to No. 11 from 1905 do not contain any summaries written by Einstein, only again in issue No. 12 on pages 624, 629, 635 (twice) and 636.
21. ^ Pais, Subtle is the Lord, Oxford UP 1982, p. 133
22. ↑ ^{a b c} Darrigol (2005), chap. 4th
23. ↑ ^{a b} Darrigol (2005), chap. 6th
24. ↑ ^{a b} Pais (1982), chap. 6c
25. ↑ ^{a b} Katzir (2005), 280–288
26. ↑ ^{a b c} Miller (1981), chap. 6th
27. ↑ ^{a b c} Pais (1982), chap. 7th
28. ↑ Walter (1999a)
29. ↑ Baccetti (2011), see references 1–25 there.