This article is rated B-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects:

History of Science Low‑importance This article is part of the History of Science WikiProject, an attempt to improve and organize the history of science content on Wikipedia. If you would like to participate, you can edit the article attached to this page, or visit the project page, where you can join the project and/or contribute to the discussion. You can also help with the History of Science Collaboration of the Month.History of ScienceWikipedia:WikiProject History of ScienceTemplate:WikiProject History of Sciencehistory of science articles Low This article has been rated as Low-importance on the project's importance scale. Physics: History Mid‑importance This article is within the scope of WikiProject Physics, a collaborative effort to improve the coverage of Physics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.PhysicsWikipedia:WikiProject PhysicsTemplate:WikiProject Physicsphysics articles Mid This article has been rated as Mid-importance on the project's importance scale. This article is supported by History Taskforce.

The article says nothing about the symmetry of Lorentz transformations as it was known by English mathematicians. Relativity was rather quickly adopted due to familiarity with biquaternions and writings of William Kingdon Clifford and Alexander Macfarlane. Fundamental work by Gilbert N. Lewis and Edwin Bidwell Wilson set the Lorentz transformation into the context of synthetic geometry. Furthermore, Whittaker spelled out for everyone just how the Lorentz transformation works to express the Principle of Relativity. This article, like History of Special Relativity, neglects the developments in abstract algebra and transformation geometry that made possible relativity science. While I appreciate that numerous references and given and the viewpoint is orthodox, the article does not stand up the standards of due diligence in academic research. Rgdboer ( talk) 21:33, 25 March 2011 (UTC) reply

Well, then you should append a new section about this. -- D.H ( talk) 08:42, 22 April 2011 (UTC) reply

The basic idea behind the Lorentz transformation can be understood as Corner Flow from hydrodynamics. When you go to the essence of the matter, the planar mapping of a Lorentz boost is an old idea, older than the linear algebra which frames the subject today. Rgdboer ( talk) 20:07, 26 July 2011 (UTC) Corrected link to Corner Flow. Rgdboer ( talk) 21:33, 31 July 2011 (UTC) reply

Seven years have seen this article grow. Now there is considerable development of the prehistory. Yet the essence can be stated briefly, so a new section "Euler's gap" has been added to show appropriate study in the 17th and 18th centuries. — Rgdboer ( talk) 22:43, 24 March 2018 (UTC) reply

The assembled references and gloss are a treasure of spacetime thought. Now available extensively in Wikiversity, this encyclopedia article just opens the topic of repercussions of Lorentz mathematical physics. Still, classical geometry lies behind the symmetry. See hyperbolic orthogonality and conjugate diameters for related geometry invariant under the Lorentz motion. Good that the focus remains on the Boost and not on the Group! See hyperbolic quaternion for a 4-space that admits the squeeze mapping. The products of hyperbolic quaternions do not form a group. That finding in the 1890s stalled expression of the Lorentz symmetry until Minkowski space was set up in 1908. The idea of a Lorentz group became a trap that ensnared many. — Rgdboer ( talk) 02:06, 2 February 2024 (UTC) reply

Link to Corner Flow has been restored; "Euler's gap" remains in History of this article. Rgdboer ( talk) 02:11, 2 February 2024 (UTC) reply

This article has gone through impressive expansion, largely, it seems, thanks to a single editor (herewith a big Thank You!).

I noticed one sentence that requires improvement:

"He showed that Lorentz's application of the transformation on the equations of electrodynamics didn't fully satisfy the principle of relativity."

That suggests to the readers (at least, to me!) that this was something that was not shown by Lorentz; however that's not true. Moreover, it is too far from the way it is presented in the source:

"I was only led to modify and complete them in a few points of detail. [..] These formulas differ somewhat from those which had been found by Lorentz."

I would thus rephrase it as follows:

"He modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity."

I'm Ok with either "modified", as Poincare phrased it, or "corrected", which better characterises it.

On a side note: I don't think that "The views of Lorentz and Einstein, together with Poincaré's four-dimensional approach, were further elaborated by [[Hermann Minkowski]" has been well sourced. The only factual and verifiable part is IMHO the second part, Poincaré's four-dimensional approach.

Regards, Harald88 ( talk) 18:34, 12 February 2012 (UTC) reply

Thanks for the suggestion, I've changed the Poincaré section (although the error in the formulas for charge density used by Lorentz are commonly interpreted (Keswani, Miller) as the consequence, that Lorentz didn't fully understood the relativistic velocity addition law). Second, Minkowski himself referred to Lorentz, Einstein, Poincaré, Planck in 1907 in his first paper; then to Lorentz, Einstein, Poincaré in his second 1907 paper; then to Voigt, Lorentz, Einstein in his 1908 paper. Strange, isn't it? I've given the primary sources now in the article, including a secondary source by S. Walter. -- D.H ( talk) 19:18, 12 February 2012 (UTC) reply

We have in the Poincare section, 'In July 1905 (published in January 1906)[A 19] Poincaré showed that the transformations are a consequence of the principle of least action' , but in the cited source Poincare himself seems to attribute this discovery to Lorentz, he says, 'We know how Lorentz deduced his equations from the principle of least action' . Martin Hogbin ( talk) 13:31, 25 March 2013 (UTC) reply

Good catch. Poincare also attributed other concepts to Lorentz, such as local clocks measuring local time and the Lorentz transformations forming a group. Most historians say that Lorentz was lacking these two concepts, and Lorentz himself seems to say that he got them from Einstein. Poincare certainly did not get them from Einstein, either directly or indirectly. We have no way to resolve these contradictions. Roger ( talk) 20:07, 25 March 2013 (UTC) reply

"Lorentz's equation" refer to his electrodynamic equations rather than to the Lorentz transformation. According to Schwartz ( doi: 10.1119/1.1976641), Poincaré was probably alluding to Lorentz's 1903 paper

" Contributions to the theory of electrons. Proc. Amst.; (see also Lorentz's extensive 1904 Encyclopedia article, pp. 145–288.)

There, Lorentz applied the principle of least action (Lagrangian variational principle) to his electron theory. -- D.H ( talk) 21:02, 25 March 2013 (UTC) reply

Dear D H

I must write in English but you may use German if you like since I can easily read scientific German (but not speak it). Many thanks for drawing my attention to these entries. Your work is excellent, here as elsewhere. A few comments:

(1) Weierstrass coordinates are those used by Killing and Lindemann with the k multiplier. Homogeneous coordinates without the k are not Weierstrass coordinates, e.g. Cox, Poincare, Haussdorf etc were not using Weierstrass coordinates. The importance of Weiertrass coordinates is that changing k to ik transforms from spherical to hyperbolic and vice versa.

2) Cayley: Only those papers on Cayley transform and bilinear substitutions are mentioned but I remember he did some work directly relevant to the Lorentz transformation though I cannot find it now.

3) The description of Larmor's work is very useful and should also be in the Larmor article in expanded form. His work is available online but is very difficult for anyone to understand.

4) No mention is made in the history article of the naming of the Lorentz transformation. The word 'boost' is used throughout. This name is comparatively recent being introduced only in the 2nd half of the 19th century. Before that it was called either a 'pure Lorentz transformation' (Poincare) or a 'special Lorentz transformation' (Minkowski). The word 'boost', recalling rocket propulsion, seems quite mathematically undescriptive to me, it is just a jargon term. The correct term would be based on Varicak's discovery that this transformation is a translation in hyperbolic space analogous to the Galilean transformation in Euclidean space.

5) I was also glad to get the link to the Clebsch-Lindemann reference. But their contribution to the hyperboloid model is unclear. And I suspect this remark applies to other references given in the Wiki article though I am unable to checkup on the originals. (Note 'hyperboloid' should actually be 'cylindrical hyperboloid')

Kind regards, JFB80 — Preceding unsigned comment added by JFB80 ( talk • contribs) 22:28, 26 March 2018 (UTC) reply

Thanks for your appreciation. While it's true that Killing's equation including k is more general by including several types of non-Euclidean Geometry, the authors themselves call their coordinates in hyperbolic plane as "Weierstrass coordinates". For instance:

• Hausdorff (1899, pp 164-165) called ${\displaystyle x=\sinh BP,\ y=\sinh AP,\ p=\cosh OP}$ with ${\displaystyle p^{2}-x^{2}-y^{2}=1}$ in hyperbolic geometry as "die Weierstrass'schen Coordinaten x,y,p des Punktes P (im Grunde homogene Coordinaten, die durch eine Bedingungsgleichung verknüpft sind)".
• Liebmann (1905, pp. 167-168), called ${\displaystyle x'=\sinh \xi ,\ y'=\sinh y,\ p'=\cosh r}$ with ${\displaystyle p'^{2}-x'^{2}-y'^{2}=1}$ in hyperbolic geometry as "die Weierstraßschen Koordinaten".
• Variĉak (1912, pp. 112-114), described ${\displaystyle x=\sinh \xi ,\ y=\sinh \eta ,\ l=\cosh r}$ and z=0, as well as the more general case satisfying ${\displaystyle l^{2}-x^{2}-y^{2}-z^{2}=1}$ to which he writes: "Diese Relation besteht zwischen den Weierstraßschen Koordinaten eines jeden Punktes."
• As as secondary source see Müller (1910, Klein's encyclopedia, vol.3.1.1, p. 661). He attributed the Weierstrass coordinates mainly to Killing (1885), and applications of such coordinates to Story (1882), Gérard (1892), Hausdorff (1899), Liebmann (1905). Note that Story (1882) describes the case of elliptic geometry, therefore I didn't include him in the article. -- D.H ( talk) 08:50, 27 March 2018 (UTC) reply

D H: I have not been able to check on all the sources but I think the answer is fairly clear from Varićak (1912, 1924). The Weierstrass coordinates can be defined purely geometrically in hyperbolic space as lengths of limiting arcs from a point to the axes together with the radial distance. If you assume the radius of curvature is unity (c=1 in relativity) then you get the equations of Varićak. Here it is possible to express the lengths of these limiting arcs as sinh ξ, sinh η and cosh ρ using lengths of perpendiculars ξ, η to the axes and radial distance ρ. If the same definition is used without assuming unit radius of curvature (c not 1 in relativity) then there result the equations with the k as used by Killing and Lindeman. JFB80 ( talk) 16:57, 30 March 2018 (UTC) reply

I should add the remark that it is of course possible to use the Weierstrass parameterization without understanding its geometrical meaning. Few people (myself included) understand what are limiting arcs in hyperbolic space. The parameterization with the k multiplier is particularly convenient as it allows the switch from hyperbolic to spherical and vice versa and can be used as 'Weierstrass coordinates' without Varićak's geometrical interpretation (cf. Sommerville's 'Non-Euclidean Geometry') JFB80 ( talk) 05:36, 31 March 2018 (UTC) reply

Well, all these authors use essentially the same construction. Some with k=k, or k=ik, or k=2ik, but most of them use implicitly or explicitly the most simple choice k=i from the outset. Some describe only hyperbolic geometry, some all kinds of geometry. Some call them homogeneous coordinates, or homogeneous Weierstrass coordinates, or only Weierstrass coordinates. Let's compare some expressions:

Cox 1881, k=i (p. 186 and 187, “homogeneous coordinates”):	${\displaystyle {\begin{matrix}x=\sinh r\cos \theta ,\ y=\sinh r\sin \theta ,\ z=\cosh r,\ (r=OP);\\or:\ x=\sinh PN,\ y=\sinh PM,\ z=\cosh OP\\z^{2}-x^{2}-y^{2}=1\end{matrix}}}$
Killing 1885 k=k (p. 18, “Weierstrass coordinates”):	${\displaystyle {\begin{matrix}p=\cos {\frac {r}{k}},\ x=k\sin {\frac {r}{k}}\sin \varphi ,\ y=k\sin {\frac {r}{k}}\cos \varphi ,\ (r=OP)\\or:\ x=k\sin {\frac {a}{k}},\ y=k\sin {\frac {b}{k}},\ p=\cos {\frac {r}{k}}\\k^{2}p^{2}+x^{2}+y^{2}=k^{2}\end{matrix}}}$
Lindemann 1891, k=2ik: (p. 481, “homogenen Coordinaten des Punktes P” with reference to Killing and Weierstrass):	${\displaystyle {\begin{matrix}\xi =2ik\sin {\frac {p}{2ki}},\ \eta =2ik\sin {\frac {q}{2ki}},\ \zeta =\cos {\frac {r}{2ki}}\\\xi ^{2}+\eta ^{2}-4k^{2}\zeta ^{2}=-4k^{2}\end{matrix}}}$
Gérard 1892 k=i (p. 164ff. “coordonnèes homogènes du point M”):	${\displaystyle {\begin{matrix}X=\sinh OM\cos \omega ,\ Y=\sinh OM\sin \omega ,\ Z=\cosh OM\\or:\ X=\sinh QM,\ Y=\sinh PM,\ Z=\cosh OM\\X^{2}+Y^{2}-Z^{2}=-1\end{matrix}}}$
Hausdorff 1899, k=i (p. 164ff., “die Weierstrasschen Coordinaten x,y,p des Punktes P (im Grunde homogene Koordinaten...):	${\displaystyle {\begin{matrix}x=r\cos \varphi ,\ y=r\sin \varphi ,\ y=\sinh y,\ (r=\sinh OP)\\or:\ x=\sinh BP,\ y=\sinh AP,\ p=\cosh OP\\p^{2}-x{}^{2}-y^{2}=1\end{matrix}}}$
Woods 1903 k=k (p. 45ff., “called by Killing the Weierstrass coordinates”.):	${\displaystyle {\begin{matrix}x_{0}=\cos kr,\ x_{i}=a_{i}{\frac {\sin kr}{k}}\\x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\end{matrix}}}$
Liebmann 1905 k=i (p. 167, “Weierstraßschen Koordinaten”):	${\displaystyle {\begin{matrix}p'=\cosh x\cosh y,\ x'=\cosh y\sinh x,\ y'=\sinh y\\or:\ x'=\sinh \xi ,\ p'=\cosh r,\ y'=\sinh y\\p'^{2}-x'^{2}-y'^{2}=1\end{matrix}}}$
Coolidge 1909 k=k (p. 64):	${\displaystyle {\begin{matrix}\xi =k\sin {\frac {\rho }{k}}\cos \alpha ,\ \eta =k\sin {\frac {\rho }{k}}\cos \beta ,\ \omega =\cos {\frac {\rho }{k}}\\\xi ^{2}+\eta ^{2}+k^{2}\omega ^{2}=k^{2}\end{matrix}}}$
Müller 1910 k=k (p. 660f, cites Killing (1885), Story (1882), Gérard (1892), Hausdorff (1899), Liebmann (1905):	${\displaystyle {\begin{matrix}x=k\sin {\frac {r}{k}}\sin \lambda ,\ y=k\sin {\frac {r}{k}}\sin \mu ,\ z=k\sin {\frac {r}{k}}\sin \nu ,\ p=\cos {\frac {r}{k}}\\x^{2}+y^{2}+z^{2}=k^{2}\left(1-p^{2}\right)\end{matrix}}}$
Varicak 1912 k=i (p. 112-113 “Weierstraßschen Koordinaten”, mentions Liebmann (1905) before on p. 111):	${\displaystyle {\begin{matrix}x=\sinh X\cosh Y,\ y=\sinh Y,\ l=\cosh X\cosh Y,\ (z=0){\text{ }}\\or:\ x=\sinh \xi ,\ y=\sinh \eta ,\ l=\cosh r\\{\text{General case:}}\ l^{2}-x^{2}-y^{2}-z^{2}=1\end{matrix}}}$
Sommerville 1919 k=k (p. 127, “Weierstrass point-coordinates. The three homogeneous coordinates...”):	${\displaystyle {\begin{matrix}x=k\sin {\frac {r}{k}}\cos \theta ,\ y=k\sin {\frac {r}{k}}\cos \theta ,\ z=\cos {\frac {r}{k}}\\or:\ x=k\sin {\frac {u}{k}},\ y=k\sin {\frac {v}{k}}\\x^{2}+y^{2}+k^{2}z^{2}=k^{2}\end{matrix}}}$
Liebmann 1923 k=i ( p. 77ff, “Weierstraßsche Koordinaten”):	${\displaystyle {\begin{matrix}p=\cosh r,\ x=\sinh r\cos \varphi ,\ y=\sinh r\sin \varphi \\p^{2}-x{}^{2}-y^{2}=1\end{matrix}}}$

Regards, -- D.H ( talk) 07:32, 31 March 2018 (UTC) reply

It's also interesting that Killing derived Weierstrass coordinates in his earlier papers only by "simplification": In his (1877/78) paper, pp. 73-74, he started with the Riemann metric, changed the variables to obtain Beltrami coordinates, and "in order to avoid the denominators" in Beltrami's expressions he again changed the variables by a simple substitution to obtain coordinates obeying ${\displaystyle k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}}$ which he attributed to a seminar by Weierstrass in 1872. In his (1879/80) paper, p. 273, he started with coordinates ${\displaystyle u^{2}+v^{2}+w^{2}=1}$ for positive ${\displaystyle k^{2}}$ and ${\displaystyle u^{2}-v^{2}-w^{2}=1}$ for negative ${\displaystyle k^{2}}$, and “in order to summarize both cases, let us change somewhat the meaning of v and w and write by following Weierstrass the equation in the form ${\displaystyle k^{2}u^{2}+v^{2}+w^{2}=k^{2}}$." -- D.H ( talk) 08:47, 31 March 2018 (UTC) reply

I made a rather large number of the following corrections in this article, and probably missed some:

wrong:

Cayley-Dirac

right:

Cayley–Dirac

wrong:

pp. 236-318

right:

pp. 236–318

Michael Hardy ( talk) 18:16, 2 April 2018 (UTC) reply

At the beginning of the article the Minkowski inner product was defined as ${\displaystyle -x_{0}y_{0}+\cdots +x_{n}y_{n}}$ with the corresponding expression for the quadratic form when x = y. I suggest using the form which Minkowski himself used in "Space and Time" i.e. ${\displaystyle x_{0}y_{0}-\cdots -x_{n}y_{n}}$ which is used in e.g. Landau & Lifschitz. This sign convention + - - - also occurs for k=i with Weierstrass coordinates. The alternative sign convention - + + + is used by those concerned with General Relativity. But since this is a historical article, not at all concerned with General Relativity, I think it is more appropriate to call the Minkowski inner product the one Minkowski used. JFB80 ( talk) 20:54, 25 September 2018 (UTC) reply

Killing used formulas such as

${\displaystyle k^{2}p^{2}+x^{2}+y^{2}=k^{2}}$

${\displaystyle k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}}$

${\displaystyle k^{2}x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}=k^{2}}$

${\displaystyle k^{2}y_{0}y_{0}^{\prime }+y_{1}y_{1}^{\prime }+\cdots +y_{n}y_{n}^{\prime }=k^{2}x_{0}x_{0}^{\prime }+x_{1}x_{1}^{\prime }+\cdots +x_{n}x_{n}^{\prime }}$

which implies the "- + +" sign convention. It's a matter of taste anyway. -- D.H ( talk) 14:24, 27 September 2018 (UTC) reply

I disagree. The whole point of using Weierstrass coordinates is that it covers both the Euclidean and hyperbolic cases. So that by using the form

${\displaystyle k^{2}x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}=k^{2}}$

we have for k=1

${\displaystyle x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}=1}$

and for k=i an equation which becomes

${\displaystyle x_{0}^{2}-x_{1}^{2}-\dots -x_{n}^{2}=1}$

where in both cases the norm squared is set equal to 1. Here you can only use the Minkowski form to give a norm squared. The other sign convention cannot define a norm so it is not a matter of taste. JFB80 ( talk) 15:25, 27 September 2018 (UTC) reply

People use these coordinates as a convenient way to express non-euclidean geometries such as elliptic or hyperbolic - but not everyone uses the requirement that the right-hand side must be positive in the hyperbolic case. See for instance Sommerville (1919), p. 127. Starting from ${\displaystyle x^{2}+y^{2}+k^{2}z^{2}=k^{2}}$ he set k=ik for hyperbolic geometry, arriving (at the same page) at ${\displaystyle x^{2}+y^{2}-k^{2}z^{2}=-k^{2}}$. I still see no point in discussing the validity of "-+++" versus "+---". -- D.H ( talk) 06:26, 28 September 2018 (UTC) reply

1. This article is far too long and my computer struggles to edit it.
2. This article is not a History of Lorentz transformations, it is a repository of mathematical details of various author's work on the topic. Rename this article.
3. The amount of math is grossly excessive. Cut it.
4. Many of the older math cited uses old or outdated notation (eg. Ch instead of cosh).
5. The arrangement of the math into sections of a rectangle with lines is confusing and nonstandard.
6. Many results are boxed and ugly.
7. Notation between math highlighted from different sources is inconsistent, rendering the whole article indecipherable.
8. How many of the works cited in this article are actually relevant to a history of Lorentz transformations and its development?

This article needs severe help. It is way too large due to (in my opinion) excessive information (mathematical and sources) and has atrocious math readability. QueensanditsCrazy ( talk) 17:52, 31 August 2020 (UTC) reply

Second—Even if we were to split the article, it would consist of 2 article each on the verge of needing to be split again. There is no good way to split it, though, this is one historical topic and should not be a category. The solution to the problem is to follow the points by @ QueensanditsCrazy: and prune the article. In addition, many details can go on other pages where they are more relevant. Footlessmouse ( talk) 18:57, 6 October 2020 (UTC) reply

As the main author I have to admit that it became too long indeed. So I moved the historical math sections into the Wikiversity project v:History of Topics in Special Relativity, and added links in the relevant sections. This reduced the article size by about 200KB. -- D.H ( talk) 06:04, 7 October 2020 (UTC) reply

@ D.H: Wikiversity actually sounds like a great place for a lot of this material - thanks! Let me know if u want more help with this article here QueensanditsCrazy ( talk) 15:29, 8 October 2020 (UTC) reply

Actually, I found this a very interesting and useful summary of the confusing history of the Lorentz transformation.

My feeling is that the author really should write a book or perhaps a review article and publish it, so it won't be lost.

I've mixed feelings about it as a W. article. Maybe Wikiversity or some other part of Wikimedia? But, I'd vote to keep it here. Kbk ( talk) 20:45, 22 July 2022 (UTC) reply

Since it is already on Wikiversity for years, where it is regularly updated by me, I've replaced the long mathematical overview sections with short summaries and links. D.H ( talk) 08:16, 6 August 2022 (UTC) reply

after "In 1904 he rewrote the equations in the following form by ....." the formula in modern stile for time is wrong in last part after =

You did put the correct version. 79.202.46.238 ( talk) 21:42, 9 March 2023 (UTC) reply