Editing Squeeze mapping (section)

===Lie transform===
{{Further|History of Lorentz transformations#Lorentz transformation via squeeze mappings}}

Following [[Pierre Ossian Bonnet]]'s (1867) investigations on surfaces of constant curvatures, [[Sophus Lie]] (1879) found a way to derive new [[pseudospherical surface]]s from a known one. Such surfaces satisfy the [[Sine-Gordon equation]]:

:<math>\frac{d^{2}\Theta}{ds\ d\sigma}=K\sin\Theta ,</math>

where <math>(s,\sigma)</math> are asymptotic coordinates of two principal tangent curves and <math>\Theta</math> their respective angle. Lie showed that if <math>\Theta=f(s,\sigma)</math> is a solution to the Sine-Gordon equation, then the following squeeze mapping (now known as Lie transform<ref name=terng />) indicates other solutions of that equation:<ref>{{Cite journal|author=Lie, S.|year=1881|orig-year=1879|journal=Fortschritte der Mathematik|volume=11|title=Selbstanzeige: Über Flächen, deren Krümmungsradien durch eine Relation verknüpft sind|pages=529–531}} Reprinted in [https://archive.org/details/gesammabhand03lierich Lie's collected papers, Vol. 3, pp. 392–393].</ref>

:<math>\Theta=f\left(ms,\ \frac{\sigma}{m}\right) .</math>

Lie (1883) noticed its relation to two other transformations of pseudospherical surfaces:<ref>{{Cite journal|author=Lie, S.|year=1884|orig-year=1883|journal=Christ. Forh.|title=Untersuchungen über Differentialgleichungen IV}}. Reprinted in [https://archive.org/details/gesammabhand03lierich Lie's collected papers, Vol. 3, pp. 556–560].</ref> The [[Bäcklund transform]] (introduced by [[Albert Victor Bäcklund]] in 1883) can be seen as the combination of a Lie transform with a Bianchi transform (introduced by [[Luigi Bianchi]] in 1879.) Such transformations of pseudospherical surfaces were discussed in detail in the lectures on [[differential geometry]] by [[Gaston Darboux]] (1894),<ref>{{Cite book|author=Darboux, G.|year=1894|title=Leçons sur la théorie générale des surfaces. Troisième partie|publisher=Gauthier-Villars|location=Paris|url=https://archive.org/details/leonssurlathorie03darb|pages=[https://archive.org/details/leonssurlathorie03darb/page/381 381]–382}}</ref> [[Luigi Bianchi]] (1894),<ref>{{Cite book|author=Bianchi, L.|year=1894|title=Lezioni di geometria differenziale|publisher=Enrico Spoerri|location=Pisa|url=https://archive.org/details/lezionidigeomet00biangoog|pages=[https://archive.org/details/lezionidigeomet00biangoog/page/n443 433]–434}}</ref> or [[Luther Pfahler Eisenhart]] (1909).<ref>{{Cite book|author=Eisenhart, L. P.|year=1909|title=A treatise on the differential geometry of curves and surfaces|publisher=Ginn and Company|location=Boston|url=https://archive.org/details/treatonthediffer00eiserich|pages=[https://archive.org/details/treatonthediffer00eiserich/page/n306 289]–290}}</ref>

It is known that the Lie transforms (or squeeze mappings) correspond to Lorentz boosts in terms of [[light-cone coordinates]], as pointed out by Terng and Uhlenbeck (2000):<ref name=terng>{{Cite journal|author=Terng, C. L., & Uhlenbeck, K.|year=2000|journal=Notices of the AMS|volume=47|issue=1|title=Geometry of solitons|pages=17–25|url=https://www.ams.org/journals/notices/200001/fea-terng.pdf}}</ref>

:''Sophus Lie observed that the SGE [Sinus-Gordon equation] is invariant under Lorentz transformations. In asymptotic coordinates, which correspond to light cone coordinates, a Lorentz transformation is <math>(x,t)\mapsto\left(\tfrac{1}{\lambda}x,\lambda t\right)</math>.''

This can be represented as follows:

:<math>\begin{matrix}-c^{2}t^{2}+x^{2}=-c^{2}t^{\prime2}+x^{\prime2}\\
\hline \begin{align}ct' & =ct\gamma-x\beta\gamma &  & =ct\cosh\eta-x\sinh\eta\\
x' & =-ct\beta\gamma+x\gamma &  & =-ct\sinh\eta+x\cosh\eta
\end{align}
\\
\hline u=ct+x,\ v=ct-x,\ k=\sqrt{\tfrac{1+\beta}{1-\beta}}=e^{\eta}\\
u'=\frac{u}{k},\ v'=kv\\
\hline u'v'=uv
\end{matrix}</math>

where ''k'' corresponds to the Doppler factor in [[Bondi k-calculus|Bondi ''k''-calculus]], ''η'' is the [[rapidity]].