History of Lorentz transformations
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.
In this article the historical notations are replaced with modern notations, with the Lorentz transformation,
and the Lorentz factor,
v being the relative velocity of the two reference frames, and c the speed of light.
- 1 Sphere geometry in the 19th century
- 2 Voigt (1887)
- 3 Heaviside (1888), Thomson (1889), Searle (1896)
- 4 Lorentz (1892, 1895)
- 5 Larmor (1897, 1900)
- 6 Lorentz (1899, 1904)
- 7 Poincaré (1900, 1905)
- 8 Einstein (1905)
- 9 Minkowski (1907–1908)
- 10 Ignatowski (1910)
- 11 See also
- 12 References
- 13 Secondary sources
- 14 External links
Sphere geometry in the 19th century
One of the defining properties of the Lorentz transformation is its group structure which leaves the expression invariant. So a spherical wave in one frame remains spherical in another one, which is often used to derive the Lorentz transformation. However, long before experiments and physical theories made the introduction of the Lorentz transformation necessary, transformation groups and sphere geometries transforming sphere into spheres have been discussed, such as the Transformation by reciprocal radii within Möbius geometry, and the Transformation by reciprocal directions within Laguerre geometry. Both can be seen as special cases of Lie sphere geometry. The connections of these transformations to Maxwell's equations and the laws of physics were discovered, however, only after 1905 when the Lorentz transformation was already derived in a different way by physicists.
In several papers between 1847 and 1850 it was shown by Joseph Liouville[A 1] that the relation is invariant under the group of conformal transformations or the "Transformation by reciprocal radii" which transforms spheres into spheres. This theorem was extended to all dimensions by Sophus Lie (1871) so that is invariant too.[A 2] In 1909, Harry Bateman and Ebenezer Cunningham showed that not only the quadratic form but also Maxwells equations are covariant with respect to conformal transformation, irrespective of the choice of . This variant of conformal transformations were called spherical wave transformations by him.[A 3][A 4] 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.[A 5]
Albert Ribaucour (1870)[A 6] and in particular Edmond Laguerre (1880-1885)[A 7][A 8] employed another variant, namely the "transformation by reciprocal directions" or „Laguerre inversion/transformation“ which transforms spheres into spheres and planes into planes.[A 7] Laguerre explicitly wrote down the corresponding transformation formulas in 1882, with Gaston Darboux (1887) presenting them in respect to coordinates (R being the radius):[A 9]
producing the following relation:
thus the above transformation becomes similar to a Lorentz transformation with as direction of motion, except that the sign of is reversed from to :
If the right-hand sides of his equations are multiplied by 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 (using the factor discussed above), and Lorentz invariant, so the combination is invariant too. For instance, Lorentz transformations can be extended by using :[A 14][A 15]
gives the Voigt transformation, 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 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:
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 ] a transformation equivalent to the formulae (287) and (288) [namely ]. The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the free ether is contained in his paper.[A 15]
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.[A 16]
Heaviside (1888), Thomson (1889), Searle (1896)
In 1888, Oliver Heaviside[A 17] 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:
Thereby, inhomogeneous electromagnetic wave equations are transformed into a Poisson equation. Eventually, George Frederick Charles Searle[A 19] 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")[A 20] 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. 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 2 steps. At first the Galilean transformation - and later the expansion into the "fictitious" electromagnetic system with the aid of the Lorentz transformation. In order to explain the negative result of the Michelson–Morley experiment, he (1892b)[A 21] introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced length contraction in his theory (without proof as he admitted). The same hypothesis was already made by George FitzGerald in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.
In 1895,[A 22] Lorentz further elaborated on his theory and introduced the "theorem of corresponding states". This theorem states that a moving observer (relative to the ether) in his „fictitious“ field makes the same observations as a resting observers in his „real“ field for velocities to first order in v/c. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:
For solving optical problems Lorentz used the following transformation, whereby for the time variable he used the expression "local time" (Ortszeit):
Larmor (1897, 1900)
Larmor in 1897[A 23] and 1900[A 24] presented the transformations in two parts. Similar to Lorentz, he considered first the transformation from a rest system (x, y, z, t) to a moving system (x′, y′, z′, t′)
This transformation is just the Galilean transformation for the x, y, z coordinates but contains Lorentz’s "local time". Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor v²/c², and so he sought the transformations which were "accurate to second order" (as he put it). Thus he wrote the final transformations (where x* = x − vt) as:
Larmor showed that Maxwell's equations were invariant under this two-step transformation, "to second order in v/c", as he put it. 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 1/γ.
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.[A 25]
p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..][A 25]
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.[A 26]
Lorentz (1899, 1904)
Also Lorentz, by extending his theorem of corresponding states, derived in 1899 the complete transformations. However, he used the undetermined factor as an arbitrary function of . Like Larmor, Lorentz noticed in 1899[A 27] also some sort of time dilation effect in relation to the frequency of oscillating electrons "that in the time of vibrations be times as great as in ", where is the aether frame, Subsequently, in 1904 he wrote the equations in the following form (as mentioned above must be replaced by ):
Under the assumption that when , he demonstrated that must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor to unity, Lorentz's transformation now assumed the same form as Larmor's. 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 . 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.[A 28] 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:
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.
Poincaré (1900, 1905)
Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, Henri Poincaré[A 29][A 30] 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 c 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 (the x*, t* frame) a light signal from one clock (at the origin) is sent to another (at x*), and is sent back. It's supposed that the Earth is moving with speed v in the x-direction (= x*-direction) in some rest system (x,t) (i.e. the luminiferous aether system for Lorentz and Larmor). The time of flight outwards is
and the time of flight back is
The elapsed time on the clock when the signal is returned is δto + δtb and the time t* = (δto + δtb)/2 is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time t = δto is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus
Poincaré gave the result t* = t − vx*/c2, which is the form used by Lorentz in 1895. Poincaré dropped the factor ε ≅ 1 under the assumption that
On June 5, 1905 (published June 9)[A 14] 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 , 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 (published in January 1906)[A 33] 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 − c2t2 is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing ct√−1 as a fourth imaginary coordinate, and he used an early form of four-vectors.
On June 30, 1905 (published September 1905) Einstein[A 34] 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:
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.[A 35][A 36][A 37] Minkowski particularly reformulated electrodynamics in a four-dimensional way (Minkowski spacetime). For instance, he wrote in the form . By defining as the angle of rotation around the -axis, the Lorentz transformation assumes the form[A 36]
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:[A 38][A 39][A 40]
where . The variable 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 in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when , resulting in and the Lorentz transformation. With , no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved by Philipp Frank and Hermann Rothe (1911, 1912),[A 41][A 42] with various authors developing similar methods in subsequent years.
- Primary sources
- Liouville, Joseph (1850). "Théorème sur l'équation dx²+dy²+dz²=λ(dα²+dβ²+dγ²)". Journal de Mathématiques pures et Appliquées. 15: 103.
- Lie, Sophus (1871). "Ueber diejenige Theorie eines Raumes mit beliebig vielen Dimensionen, die der Krümmungs-Theorie des gewöhnlichen Raumes entspricht". Göttinger Nachrichten: 191–209.
- Bateman, Harry (1910). "The Transformation of the Electrodynamical Equations". Proceedings of the London Mathematical Society. 8: 223–264. doi:10.1112/plms/s2-8.1.223.
- Cunningham, Ebenezer (1909/10). "The principle of Relativity in Electrodynamics and an Extension Thereof". Proceedings of the London Mathematical Society. 8: 77–98. doi:10.1112/plms/s2-8.1.77. Check date values in:
- Klein, Felix (1910/21). "Über die geometrischen Grundlagen der Lorentzgruppe". Gesammelte mathematische Abhandlungen. 1: 533–552. doi:10.1007/978-3-642-51960-4_31. Check date values in:
- Ribaucour, Albert (1870). "Sur la déformation des surfaces". Comptes rendus. 70: 330–333.
- Laguerre, Edmond (1881). "Sur la transformation par directions réciproques". Comptes rendus. 92: 71–73.
- Laguerre, Edmond (1882). "Transformations par semi-droites réciproques". Nouvelles annales de mathématiques. 1: 542–556.
- Darboux, Gaston (1887). Leçons sur la théorie générale des surfaces. Première partie. Paris: Gauthier-Villars. pp. 254–256.
- Bateman, Harry (1912). "Some geometrical theorems connected with Laplace's equation and the equation of wave motion". American Journal of Mathematics. 34: 325–360. doi:10.2307/2370223. (submitted 1910, published 1912)
- Müller, Hans Robert (1948). "Zyklographische Betrachtung der Kinematik der speziellen Relativitätstheorie". Monatshefte für Mathematik und Physik. 52: 337–353. doi:10.1007/bf01525338.
- Poincaré, Henri (1912). "Rapport sur les travaux de M. Cartan (fait à la Faculté des sciences de l'Université de Paris)". Acta Mathematica. 38 (1): 137–145. doi:10.1007/bf02392064. Written by Poincaré in 1912, printed in Acta Mathematica in 1914 though belatedly published in 1921.
- Voigt, Woldemar (1887), "Ueber das Doppler'sche Princip" [On the Principle of Doppler], Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen (2): 41–51
- Poincaré, Henri (1905b), "Sur la dynamique de l'électron" [On the Dynamics of the Electron], Comptes Rendus, 140: 1504–1508
- Lorentz, Hendrik Antoon (1909/16), The theory of electrons and its applications to the phenomena of light and radiant heat on the Internet Archive, Leipzig & Berlin: B.G. Teubner Check date values in:
|date=(help); External link in
- Bucherer, A. H. (1908), "Messungen an Becquerelstrahlen. Die experimentelle Bestätigung der Lorentz-Einsteinschen Theorie. (Measurements of Becquerel rays. The Experimental Confirmation of the Lorentz-Einstein Theory)", Physikalische Zeitschrift, 9 (22): 758–762. For Minkowski's and Voigt's statements see p. 762.
- Heaviside, Oliver (1889), "On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric", Philosophical Magazine, 5, 27 (167): 324–339, doi:10.1080/14786448908628362
- Thomson, Joseph John (1889), "On the Magnetic Effects produced by Motion in the Electric Field", Philosophical Magazine, 5, 28 (170): 1–14, doi:10.1080/14786448908619821
- Searle, George Frederick Charles (1897), "On the Steady Motion of an Electrified Ellipsoid", Philosophical Magazine, 5, 44 (269): 329–341, doi:10.1080/14786449708621072
- Lorentz, Hendrik Antoon (1892a), "La Théorie electromagnétique de Maxwell et son application aux corps mouvants", Archives néerlandaises des sciences exactes et naturelles, 25: 363–552
- Lorentz, Hendrik Antoon (1892b), "De relatieve beweging van de aarde en den aether" [The Relative Motion of the Earth and the Aether], Zittingsverlag Akad. V. Wet., 1: 74–79
- Lorentz, Hendrik Antoon (1895), Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern [Attempt of a Theory of Electrical and Optical Phenomena in Moving Bodies], Leiden: E.J. Brill
- Larmor, Joseph (1897), "On a Dynamical Theory of the Electric and Luminiferous Medium, Part 3, Relations with material media", Philosophical Transactions of the Royal Society, 190: 205–300, Bibcode:1897RSPTA.190..205L, doi:10.1098/rsta.1897.0020
- Larmor, Joseph (1900), Aether and Matter, Cambridge University Press
- Larmor, Joseph (1904). "On the intensity of the natural radiation from moving bodies and its mechanical reaction". Philosophical Magazine. 7 (41): 578–586. doi:10.1080/14786440409463149.
- Larmor, Joseph (1904). "On the ascertained Absence of Effects of Motion through the Aether, in relation to the Constitution of Matter, and on the FitzGerald-Lorentz Hypothesis". Philosophical Magazine. 7 (42): 621–625. doi:10.1080/14786440409463156.
- Lorentz, Hendrik Antoon (1899), "Simplified Theory of Electrical and Optical Phenomena in Moving Systems", Proceedings of the Royal Netherlands Academy of Arts and Sciences, 1: 427–442
- Lorentz, Hendrik Antoon (1904), "Electromagnetic phenomena in a system moving with any velocity smaller than that of light", Proceedings of the Royal Netherlands Academy of Arts and Sciences, 6: 809–831
- Poincaré, Henri (1900), "La théorie de Lorentz et le principe de réaction", Archives néerlandaises des sciences exactes et naturelles, 5: 252–278. See also the English translation.
- Poincaré, Henri (1904/6), "The Principles of Mathematical Physics", Congress of arts and science, universal exposition, St. Louis, 1904, 1, Boston and New York: Houghton, Mifflin and Company, pp. 604–622 Check date values in:
- Cohn, Emil (1904), "Zur Elektrodynamik bewegter Systeme II" [On the Electrodynamics of Moving Systems II], Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 1904/2 (43): 1404–1416
- Abraham, M. (1905). "§ 42. Die Lichtzeit in einem gleichförmig bewegten System". Theorie der Elektrizität: Elektromagnetische Theorie der Strahlung. Leipzig: Teubner.
- Poincaré, Henri (1906), "Sur la dynamique de l'électron" [On the Dynamics of the Electron], Rendiconti del Circolo matematico di Palermo, 21: 129–176, doi:10.1007/BF03013466
- Einstein, Albert (1905), "Zur Elektrodynamik bewegter Körper" (PDF), Annalen der Physik, 322 (10): 891–921, Bibcode:1905AnP...322..891E, doi:10.1002/andp.19053221004. See also: English translation.
- Minkowski, Hermann (1915) , "Das Relativitätsprinzip", Annalen der Physik, 352 (15): 927–938, Bibcode:1915AnP...352..927M, doi:10.1002/andp.19153521505
- Minkowski, Hermann (1908), "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern" [The Fundamental Equations for Electromagnetic Processes in Moving Bodies], Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111
- Minkowski, Hermann (1908/9), "Space and Time", Physikalische Zeitschrift, 10: 75–88 Check date values in:
- Ignatowsky, W. v. (1910). "Einige allgemeine Bemerkungen über das Relativitätsprinzip". Physikalische Zeitschrift. 11: 972–976.
- Ignatowsky, W. v. (1911). "Das Relativitätsprinzip". Archiv der Mathematik und Physik. 18: 17–40.
- Ignatowsky, W. v. (1911). "Eine Bemerkung zu meiner Arbeit: "Einige allgemeine Bemerkungen zum Relativitätsprinzip"". Physikalische Zeitschrift. 12: 779.
- Frank, Philipp; Rothe, Hermann (1910). "Über die Transformation der Raum-Zeitkoordinaten von ruhenden auf bewegte Systeme". Annalen der Physik. 339 (5): 825–855. Bibcode:1911AnP...339..825F. doi:10.1002/andp.19113390502.
- Frank, Philipp; Rothe, Hermann (1912). "Zur Herleitung der Lorentztransformation". Physikalische Zeitschrift. 13: 750–753.
- Brown, Harvey R. (2001), "The origins of length contraction: I. The FitzGerald-Lorentz deformation hypothesis", American Journal of Physics, 69 (10): 1044–1054, arXiv:, Bibcode:2001AmJPh..69.1044B, doi:10.1119/1.1379733 See also "Michelson, FitzGerald and Lorentz: the origins of relativity revisited", Online.
- Darrigol, Olivier (2000), Electrodynamics from Ampère to Einstein, Oxford: Oxford Univ. Press, ISBN 0-19-850594-9
- Darrigol, Olivier (2005), "The Genesis of the theory of relativity" (PDF), Séminaire Poincaré, 1: 1–22, doi:10.1007/3-7643-7436-5_1
- Janssen, Michel (1995), A Comparison between Lorentz's Ether Theory and Special Relativity in the Light of the Experiments of Trouton and Noble (Thesis)
- Katzir, Shaul (2005), "Poincaré's Relativistic Physics: Its Origins and Nature", Physics in perspective, 7 (3): 268–292, Bibcode:2005PhP.....7..268K, doi:10.1007/s00016-004-0234-y
- Macrossan, M. N. (1986), "A Note on Relativity Before Einstein", The British Journal for the Philosophy of Science, 37: 232–234, doi:10.1093/bjps/37.2.232
- Miller, Arthur I. (1981), Albert Einstein’s special theory of relativity. Emergence (1905) and early interpretation (1905–1911), Reading: Addison–Wesley, ISBN 0-201-04679-2
- Pais, Abraham (1982), Subtle is the Lord: The Science and the Life of Albert Einstein, New York: Oxford University Press, ISBN 0-19-520438-7
- Walter, Scott (1999a), "Minkowski, mathematicians, and the mathematical theory of relativity", in H. Goenner; J. Renn; J. Ritter; T. Sauer, Einstein Studies, 7, Birkhäuser, pp. 45–86
- Baccetti, Valentina; Tate, Kyle; Visser, Matt (2012). "Inertial frames without the relativity principle". Journal of High Energy Physics: 119. arXiv:. Bibcode:2012JHEP...05..119B. doi:10.1007/JHEP05(2012)119.
- Walter, Scott (1999b). "The non-Euclidean style of Minkowskian relativity". In J. Gray. The Symbolic Universe: Geometry and Physics. Oxford: University Press. pp. 91–127.
- Walter, Scott (2012). "Figures of light in the early history of relativity". To appear in Einstein Studies, D. Rowe, ed., Basel: Birkhäuser.
- Klein, Felix; Blaschke, Wilhelm (1926). Vorlesungen über höhere Geometrie. Berlin: Springer.
- Coolidge, Julian (1916). A treatise on the circle and the sphere. Oxford: Clarendon Press.
- Walter (2012)
- Kastrup (2008), section 2.3
- Coolidge (1916), p. 370
- Klein & Blaschke (1926), p. 259
- Miller (1981), 114–115
- Pais (1982), Kap. 6b
- Voigt's transformations and the beginning of the relativistic revolution, Ricardo Heras, arXiv:1411.2559 
- Brown (2003)
- Miller (1981), 98-99
- Miller (1982), 1.4 & 1.5
- Janssen (1995), 3.1
- Darrigol (2000), Chap. 8.5
- Macrossan (1986)
- Jannsen (1995), Kap. 3.3
- Miller (1981), Chap. 1.12.2
- Jannsen (1995), Chap. 3.5.6
- Darrigol (2005), Kap. 4
- Pais (1982), Chap. 6c
- Katzir (2005), 280–288
- Miller (1981), Chap. 1.14
- Miller (1981), Chap. 6
- Pais (1982), Kap. 7
- Darrigol (2005), Chap. 6
- Walter (1999a)
- Baccetti (2011), see references 1-25 therein.
- Mathpages: 1.4 The Relativity of Light