# History of Topics in Special Relativity/Lorentz transformation (squeeze)

## Lorentz transformation via squeeze mappings

As already indicated in E:(3d) in exponential form or E:(6f) in terms of Cayley–Klein parameter, Lorentz boosts in terms of hyperbolic rotations can be expressed as w:squeeze mappings. Using w:asymptotic coordinates of a hyperbola (u,v), in relativity also known as w:light-cone coordinates, they have the general form (some authors alternatively add a factor of 2 or ${\displaystyle {\sqrt {2}}}$):[1]

${\displaystyle {\begin{matrix}&{\begin{array}{c}u=x_{0}-x_{1},\ v=x_{0}+x_{1}\\u'=x_{0}^{\prime }-x_{1}^{\prime },\ v'=x_{0}^{\prime }+x_{1}^{\prime }\end{array}}\\\hline (1)&(u',v')=\left(ku,\ {\frac {1}{k}}v\right)\\(2)&(u',v')=\left({\frac {1}{k}}v,\ ku\right)\\\hline &u'v'=uv\end{matrix}}}$

(9a)

with arbitrary k. This geometrically corresponds to the transformation of the area of one parallelogram to other ones of same area, whose sides touch a hyperbola and both asymptotes. While equation system (1) corresponds to proper Lorentz boosts, equation system (2) produces improper ones. For instance, solving (1) for ${\displaystyle x'_{0},x'_{1}}$ gives:

{\displaystyle {\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}\\\hline {\begin{aligned}x_{0}^{\prime }&={\frac {1}{2}}\left(k+{\frac {1}{k}}\right)x_{0}-{\frac {1}{2}}\left(k-{\frac {1}{k}}\right)x_{1}&&={\frac {x_{0}\left(k^{2}+1\right)-x_{1}\left(k^{2}-1\right)}{2k}}\\x_{1}^{\prime }&=-{\frac {1}{2}}\left(k-{\frac {1}{k}}\right)x_{0}+{\frac {1}{2}}\left(k+{\frac {1}{k}}\right)x_{1}&&={\frac {-x_{0}\left(k^{2}-1\right)+x_{1}\left(k^{2}+1\right)}{2k}}\\\\x_{0}&={\frac {1}{2}}\left(k+{\frac {1}{k}}\right)x_{0}^{\prime }+{\frac {1}{2}}\left(k-{\frac {1}{k}}\right)x_{1}^{\prime }&&={\frac {x_{0}^{\prime }\left(k^{2}+1\right)+x_{1}^{\prime }\left(k^{2}-1\right)}{2k}}\\x_{1}&={\frac {1}{2}}\left(k-{\frac {1}{k}}\right)x_{0}^{\prime }+{\frac {1}{2}}\left(k+{\frac {1}{k}}\right)x_{1}^{\prime }&&={\frac {x_{0}^{\prime }\left(k^{2}-1\right)+x_{1}^{\prime }\left(k^{2}+1\right)}{2k}}\end{aligned}}\end{matrix}}}

(9b)

The geometrical foundation of squeeze mapping (9a) was known for a long time since Apollonius (BC) and was used to generate hyperbolas by Speidell (1688) and Whiston (1710). The exact analytical form (9a-2) was given by Reynaud (1819) while (9a-1) was given by Laisant (1874) and Günther (1880/81) in relation to elliptic trigonometry, or by Lie (1879-81), Bianchi (1886, 1894), Darboux (1891/94), Eisenhart (1905) as Lie transform[1] of w:pseudospherical surfaces in terms of the w:Sine-Gordon equation, or by Lipschitz (1885/86) in transformation theory. From that, different forms of Lorentz transformation were derived: (9b) by Lipschitz (1885/86), Bianchi (1886, 1894), Eisenhart (1905), trigonometric Lorentz boost E:(8a) by Bianchi (1886, 1894) and Darboux (1891/94), and trigonometric Lorentz boost E:(8b) by Eisenhart (1905). Lorentz boost (9b) was rediscovered in the framework of special relativity by w:Hermann Bondi (1964)[2] in terms of w:Bondi k-calculus, by which k can be physically interpreted as Doppler factor. Since (9b) is equivalent to E:(6f) in terms of Cayley–Klein parameter by setting ${\displaystyle k=\alpha ^{2}}$, it can be interpreted as the 1+1 dimensional special case of Lorentz Transformation E:(6e) stated by Gauss around 1800 (posthumously published 1863), E:Selling (1873), E:Bianchi (1888), E:Fricke (1891) and E:Woods (1895).

Rewriting (9a) in terms of w:homogeneous coordinates signifies squeeze mappings of the unit hyperbola in terms of a w:projective conic:

${\displaystyle {\begin{matrix}\left[u,v\right]=\left[{\frac {y_{1}}{y_{3}}},{\frac {y_{2}}{y_{3}}}\right]\quad \left(uv=1\quad \Rightarrow \quad y_{1}y_{2}-y_{3}^{2}=0\right)\\\left[k,{\frac {1}{k}}\right]=\left[{\frac {\alpha _{1}}{\alpha _{3}}},{\frac {\alpha _{2}}{\alpha _{3}}}\right]\quad \left(k{\frac {1}{k}}=1\quad \Rightarrow \quad \alpha _{1}\alpha _{2}-\alpha _{3}^{2}=0\right)\\\hline y'_{1}=\alpha _{1}y_{1}\\y'_{2}=\alpha _{2}y_{2}\\y'_{3}=\alpha _{3}y_{3}\\\hline y_{1}y_{2}-y_{3}^{2}=y'_{1}y'_{2}-y_{3}^{\prime 2}=0\\\hline uv={\frac {y_{1}y_{2}}{y_{3}^{2}}}=u'v'={\frac {y'_{1}y'_{2}}{y_{3}^{\prime 2}}}\end{matrix}}}$

(9c)

Such transformations were given by Klein (1871) to express motions in non-Euclidean space.

Furthermore, variables u, v in (9a) can be rearranged to produce another form of squeeze mapping, resulting in Lorentz transformation E:(5b) in terms of Cayley-Hermite parameter:

{\displaystyle {\begin{matrix}{\begin{matrix}u=x_{0}-x_{1}\\v=x_{0}+x_{1}\\u'=x_{0}^{\prime }-x_{1}^{\prime }\\v'=x_{0}^{\prime }+x_{1}^{\prime }\end{matrix}}\Rightarrow {\begin{matrix}u_{1}=x_{0}+x_{0}^{\prime }\\v_{1}=x_{0}-x_{0}^{\prime }\\u_{2}=x_{1}-x_{1}^{\prime }\\v_{2}=x_{1}+x_{1}^{\prime }\end{matrix}}\\\hline (u_{2},v_{2})=\left(au_{1},\ {\frac {1}{a}}v_{1}\right)\Rightarrow u_{2}v_{2}=u_{1}v_{1}\\(u',v')=\left({\frac {1+a}{1-a}}u,\ {\frac {1-a}{1+a}}v\right)\Rightarrow u'v'=uv\end{matrix}}\Rightarrow {\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}\\\hline {\begin{aligned}x_{0}^{\prime }&=x_{0}{\frac {1+a^{2}}{1-a^{2}}}-x_{1}{\frac {2a}{1-a^{2}}}&&={\frac {x_{0}\left(1+a^{2}\right)-x_{1}2a}{1-a^{2}}}\\x_{1}^{\prime }&=-x_{0}{\frac {2a}{1-a^{2}}}+x_{1}{\frac {1+a^{2}}{1-a^{2}}}&&={\frac {-x_{0}2a+x_{1}\left(1+a^{2}\right)}{1-a^{2}}}\\\\x_{0}&=x_{0}^{\prime }{\frac {1+a^{2}}{1-a^{2}}}+x_{1}^{\prime }{\frac {2a}{1-a^{2}}}&&={\frac {x_{0}^{\prime }\left(1+a^{2}\right)+x_{1}^{\prime }2a}{1-a^{2}}}\\x_{1}&=x_{0}^{\prime }{\frac {2a}{1-a^{2}}}+x_{1}^{\prime }{\frac {1+a^{2}}{1-a^{2}}}&&={\frac {x_{0}^{\prime }2a+x_{1}^{\prime }\left(1+a^{2}\right)}{1-a^{2}}}\end{aligned}}\end{matrix}}}

(9d)

These Lorentz transformations were given (up to a sign change) by Laguerre (1882), Darboux (1887), Smith (1900) in relation to Laguerre geometry.

On the basis of factors k or a, all previous Lorentz boosts E:(3b), E:(4a), E:(8a), E:(8b), can be expressed as squeeze mappings as well:

${\displaystyle {\begin{array}{r|c|c|c|c|c|c}&(9a)&(9d)&(3b)&(4a)&(8a)&(8b)\\\hline {\frac {u'}{u}}={\frac {x_{0}^{\prime }-x_{1}^{\prime }}{x_{0}-x_{1}}}=&k&{\frac {1+a}{1-a}}&e^{\eta }&{\sqrt {\tfrac {1+\beta }{1-\beta }}}&{\frac {1+\sin \theta }{\cos \theta }}&{\frac {1+\cos \vartheta }{\sin \vartheta }}=\cot {\frac {\vartheta }{2}}\\\hline {\frac {u_{2}}{u_{1}}}={\frac {x_{1}-x_{1}^{\prime }}{x_{0}+x_{0}^{\prime }}}=&{\frac {k-1}{k+1}}&a&\tanh {\frac {\eta }{2}}&{\frac {\gamma -1}{\beta \gamma }}&{\frac {1-\cos \theta }{\sin \theta }}=\tan {\frac {\theta }{2}}&{\frac {1-\sin \vartheta }{\cos \vartheta }}\\\hline &{\frac {k^{2}-1}{k^{2}+1}}&{\frac {2a}{1+a^{2}}}&\tanh \eta &\beta &\sin \theta &\cos \vartheta \\\hline &{\frac {k^{2}+1}{2k}}&{\frac {1+a^{2}}{1-a^{2}}}&\cosh \eta &\gamma &\sec \theta &\csc \vartheta \\\hline &{\frac {k^{2}-1}{2k}}&{\frac {2a}{1-a^{2}}}&\sinh \eta &\beta \gamma &\tan \theta &\cot \vartheta \end{array}}}$

(9e)

Squeeze mappings in terms of ${\displaystyle \theta }$ were used by Darboux (1891/94) and Bianchi (1894), in terms of ${\displaystyle \eta }$ by Lindemann (1891) and Elliott (1903), in terms of ${\displaystyle \vartheta }$ by Eisenhart (1905), in terms of ${\displaystyle \beta }$ by Bondi (1964).

## Historical notation

### Apollonius (BC), Speidell (1688), Whiston (1710) – Hyperbola mapping

w:Apollonius of Perga (c. 240–190 BC, and maybe other Greek geometers such as w:Menaechmus even earlier) defined a proposition, which was translated and adapted to the modern reader by w:Thomas Heath as follows:[M 1]

If Q, q be any two points on a hyperbola, and parallel straight lines QH, qh be drawn to meet one asymptote at any angle, and QK, qk (also parallel to one another) meet the other asymptote at any angle, then
HQ·QK = hq·qk.

In the next proposition, Apollonius (adapted to the modern reader by Heath) applied this result to the case where the lines or sides of parallelograms are drawn parallel to the asymptotes:[M 2]

Let E be a point on one asymptote, and let EF be drawn parallel to the other. Then EF produced shall meet the curve in one point only. For, if possible, let it not meet the curve. Take Q, any point on the curve, and draw QH, QK each parallel to one asymptote and meeting the other; let a point F be taken on EF such that HQ·QK=CE·EF. Join CF and produce it to meet the curve in q; and draw qh, qk respectively parallel to QH, QK. Then
hq·qk = HQ·QK [previous proposition]
and HQ·QK=CE·EF, by hypothesis, hq·qk=GE·EF: which is impossible, because hq > EF, and qk > CE. Therefore EF will meet the hyperbola in one point, as R. Again, EF will not meet the hyperbola in any other point. For, if possible, let EF meet it in R as well as R, and let RM, RM be drawn parallel to QK. Then
ER·RM = ER·RM [previous proposition]
which is impossible, because ER > ER. Therefore EF does not meet the hyperbola in a second point R.

The identity HQ·QK = hq·qk demonstrates the invariance of the area of all parallelograms that are constructed in line with the first proposition, thereby representing all points of a hyperbola defined by HQ·QK = const. In case HQ, QK, hq, qk are all drawn parallel to the respective asymptotes (as in the second proposition), HQ·QK = hq·qk becomes equivalent to u'v' = uv in (9a), signifying squeezed parallelograms located between the asymptotes and the hyperbola, thus Apollonius' propositions provide the foundation of squeeze mapping. That is, the invariant area HQ·QK = const. together with const=1 gives HQ=1/QK, which implies that QK is inverse proportional to HQ. Thus when HQ is increased into k·HQ using some factor k, it follows that QK must be proportionally diminished into QK/k in order to preserve invariance of area.

As his works became accessible in several Latin translations, Apollonius' propositions became well known and were applied in textbooks in the 17th and 18th century by geometers such as w:Grégoire de Saint-Vincent, w:John Wallis, w:Philippe de La Hire. Furthermore, the inverse case of squeezing a given square or parallelogram as a means to generate hyperbolas was discussed by w:Euclid Speidell (1688):[M 3]

[..] from a Square and an infinite company of Oblongs on a Superficies, each Equal to that Square, how a Curve is begotten which shall have the same properties and affections of an Hyperbola inscribed within a Right Angled Cone
[..] There is a Square ABCD, whose Side or Root is 10, let DB be prolonged in infinitum, and continually divided equally by the Root, or DB, and those Equal Divisions numbered by 10, 20, 30, 40, 50, 60, 70, &c. in infinitum: Upon these Numbers let Perpendiculars be erected, which call Ordinates, and each of those Perpendiculars of that length, that Perpendiculars let fall from the aforesaid Perpendiculars to the Side or Base CD (which call Complement Ordinates) the Oblongs made of the Ordinate Perpendiculars, and Complement Ordinate Perpendiculars may be ever Equal to the Square AD, which is easily done thus, for it is ${\displaystyle {\tfrac {100}{20}},{\tfrac {100}{30}},{\tfrac {100}{40}},{\tfrac {100}{50}}}$ &c. produces the Length of the Ordinate Perpendiculars
[..] all the Oblongs made of the Ordinates, and Complement Ordinates are each of them equal to the Square AD, which is here 100
[..] the like Demonstration serves for all the Oblongs or Parallelograms standing upon the Base CD, by the Tips or Angular Points of those Parallelograms, or from the Ends of all the Ordinates standing upon 20, 30, 40, 50, 60, 70, in infinitum, draw the Curve Line from A towards E, so shall you describe the Curve AEFGS [..].

This corresponds to squeeze mappings (9a) with u=v=10 and k=1,2,3,4,5,6,7,..., thus u'v'=uv=100.

In similar terms, w:William Whiston (1710/16) wrote:[M 4]

But it is to be acknowledg'd, that many Properties of an Hyperbola are better known from another manner of generating the Figure; which Way is this: Let LL and MM be infinite Right Lines intersecting each other in any Angle whatever in the Point C: From any Point whatever, as D or e, let Dc, Dd, be drawn parallel to the first Lines, or (ec, ed), which with the Lines first drawn make the Parallelograms as DcCd, or ecCd; Now conceive two sides of the Parallelogram as Dc, Dd, or ec, ed, to be so mov'd this way and that way, that they always keep the same Parallelism, and that at the same time the Area's always remain equal: That is to say, that Dc and ec remain always Parallel to MM, and Dd or ed always Parallel to LL; and that the Area of every Parallelogram be equal to every other, one Side being increas'd in the same Proportion wherein the other is diminish'd. By this means the Point D or e will describe a Curve-Line within the Angle comprehended by the first Lines;

This corresponds to squeeze mappings (9a).

### Reynaud (1819) – Hyperbola mapping

w:Antoine André Louis Reynaud algebraically expressed squeeze mappings by writing:[M 5]

"The system of equations ${\displaystyle (2)\ x={\frac {y'}{\alpha }},\ y=\alpha x'}$ determines all points of the curve ${\displaystyle S}$, because ${\displaystyle x'}$ and ${\displaystyle y'}$ being given numbers, each arbitrary value of ${\displaystyle \alpha }$ gives a point ${\displaystyle x,y}$ of this curve. The elimination of the indeterminate ${\displaystyle \alpha }$ between equations (2) will therefore lead to the equation ${\displaystyle xy=x'y'}$ of the curve in question. This curve is therefore a hyperbola related to its asymptotes ${\displaystyle xX,yY}$."

This is equivalent to (improper) Lorentz transformation (9a-2).

### Klein (1871) – Projective conic section

Elaborating on the w:Cayley–Klein metric, w:Felix Klein (1871) defined a w:projective conic in order to discuss motions such as rotation and translation in the non-Euclidean plane:[M 6]

{\displaystyle {\begin{matrix}x_{1}x_{2}-x_{3}^{2}=0\\\hline {\begin{aligned}x_{1}&=\alpha _{1}y_{1}\\x_{2}&=\alpha _{2}y_{2}\\x_{3}&=\alpha _{3}y_{3}\end{aligned}}\\\left(\alpha _{1}\alpha _{2}-\alpha _{3}^{2}=0\right)\\\hline {\frac {x_{1}x_{2}}{x_{3}^{2}}}={\text{invariant}}\end{matrix}}}

When the conic section is a hyperbola this is equivalent to squeeze mapping (9c). This becomes (9a) using ${\displaystyle \left[u,v\right]=\left[{\tfrac {x_{1}}{x_{3}}},{\tfrac {x_{2}}{x_{3}}}\right],\ \left[k,{\tfrac {1}{k}}\right]=\left[{\tfrac {\alpha _{1}}{\alpha _{3}}},{\tfrac {\alpha _{2}}{a_{3}}}\right]}$.

### Laisant (1874) – Elliptic polar coordinates

w:Charles-Ange Laisant extended circular trigonometry to elliptic trigonometry. In his model, polar coordinates x, y of circular trigonometry are related to polar coordinates x', y' of elliptic trigonometry by the relation[M 7]

${\displaystyle {\begin{matrix}x'=ax,\ y'={\frac {y}{a}}\\x'y'=xy\end{matrix}}}$

He noticed the geometrical implication that any elliptic polar system of coordinates obtained by this formula is located on the same equilateral hyperbola having its asymptotes as axes.

This is equivalent to Lorentz transformation (9a).

### Lie (1879-84) – Transforming pseudospherical surfaces

w:Sophus Lie (1879/80) derived an operation from w:Pierre Ossian Bonnet's (1867) investigations on surfaces of constant curvatures, by which pseudospherical surfaces can be transformed into each other.[M 8] Lie gave explicit formulas for this operation in two papers published in 1881: If ${\displaystyle (s,\sigma )}$ are asymptotic coordinates of two principal tangent curves and ${\displaystyle \Theta }$ their respective angle, and ${\displaystyle \Theta =f(s,\sigma )}$ is a solution of the Sine-Gordon equation ${\displaystyle {\tfrac {d^{2}\Theta }{ds\ d\sigma }}=K\sin \Theta }$, then the following operation (now called Lie transform) is also a solution from which infinitely many new surfaces of same curvature can be derived:[M 9]

${\displaystyle \Theta =f(s,\sigma )\Rightarrow \Theta =f\left(ms,\ {\frac {\sigma }{m}}\right)}$

In (1880/81) he wrote these relations as follows:[M 10]

${\displaystyle \vartheta =\Phi (s,S)\Rightarrow \vartheta =\Phi \left(ms,\ {\frac {S}{m}}\right)}$

In (1883/84) he showed that the combination of Lie transform O with Bianchi transform I produces w:Bäcklund transform B of pseudospherical surfaces:[M 11]

${\displaystyle B=OIO^{-1}}$

As shown by Bianchi (1886) and Darboux (1891/94), the Lie transform is equivalent to Lorentz transformations (9a) and (9b) in terms of light-cone coordinates 2s=u+v and 2σ=u-v. In general, it can be shown that the Sine-Gordon equation is Lorentz invariant.

### Günther (1880/81) – Elliptic polar coordinates

Following Laisant (1874), w:Siegmund Günther demonstrated the relation between circular polar coordinates and elliptic polar coordinates as[M 12]

${\displaystyle {\begin{matrix}x'=ax,\ y'={\frac {1}{a}}y\\x'y'=xy\end{matrix}}}$

showing that any elliptic polar system of coordinates obtained by this formula is located on the same equilateral hyperbola having its asymptotes as axes.

This is equivalent to Lorentz transformation (9a).

### Laguerre (1882) – Laguerre inversion

A transformation (later known as "Laguerre inversion") of E:oriented lines and spheres was given by w:Edmond Laguerre with R being the radius and D the distance of its center to the axis:[M 13]

{\displaystyle {\begin{matrix}D^{2}-D^{\prime 2}=R^{2}-R^{\prime 2}\\\hline \left.{\begin{aligned}D'&={\frac {D\left(1+\alpha ^{2}\right)-2\alpha R}{1-\alpha ^{2}}}\\R'&={\frac {2\alpha D-R\left(1+\alpha ^{2}\right)}{1-\alpha ^{2}}}\end{aligned}}\right|{\begin{aligned}D-D'&=\alpha (R-R')\\D+D'&={\frac {1}{\alpha }}(R+R')\end{aligned}}\end{matrix}}}

This is equivalent (up to a sign change for R) to a squeeze mapping in terms of Lorentz boost (9d).

### Darboux (1883–1891)

#### Transforming pseudospherical surfaces

w:Gaston Darboux (1883) followed Lie (1879/81) by transforming pseudospheres into each other as follows:[M 14]

${\displaystyle f(x,y)\Rightarrow f\left({\frac {x}{m}},\ ym\right)}$

This becomes Lorentz boost (9a) by interpreting x, y as light-cone coordinates.

Similar to Bianchi (1886), Darboux (1891/94) showed that the Lie transform gives rise to the following relations:[M 15]

{\displaystyle {\begin{aligned}(1)\quad &u+v=2\alpha ,\ u-v=2\beta ;\\(2)\quad &\omega =\varphi \left(\alpha ,\beta \right)\Rightarrow \omega =\varphi \left(\alpha m,\ {\frac {\beta }{m}}\right)\\(3)\quad &\omega =\psi (u,v)\Rightarrow \omega =\psi \left({\frac {u+v\sin h}{\cos h}},\ {\frac {v+u\sin h}{\cos h}}\right)\end{aligned}}}.

Equations (1) together with transformation (2) gives Lorentz boost (9a) in terms of light-cone coordinates.

#### Laguerre inversion

Following Laguerre (1882), Darboux (1887) formulated the Laguerre inversions in four dimensions using coordinates x,y,z,R:[M 16]

{\displaystyle {\begin{matrix}x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-R^{\prime 2}=x^{2}+y^{2}+z^{2}-R^{2}\\\hline \left.{\begin{aligned}x'&=x,&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}}\right|{\begin{aligned}z'+R'&={\frac {1+k}{1-k}}(z-R)\\z'-R'&={\frac {1-k}{1+k}}(z+R)\end{aligned}}\end{matrix}}}

This is equivalent (up to a sign change for R) to a squeeze mapping in terms of Lorentz boost (9d) where Darboux's k corresponds to a.

### Lipschitz (1885/86) - Quadratic forms

w:Rudolf Lipschitz (1885/86) formulated transformations leaving invariant the sum of squares ${\displaystyle x_{1}^{2}+x_{2}^{2}\dots +x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n}^{2}}$, which he rewrote as ${\displaystyle x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots +x_{n}^{2}-y_{n}^{2}=0}$. This led to the problem of finding transformations leaving invariant the pairs ${\displaystyle x_{a}^{2}-y_{a}^{2}}$ (a=1...n) for which he gave the following solution:[M 17]

{\displaystyle {\begin{matrix}x_{a}^{2}-y_{a}^{2}={\mathfrak {x}}_{a}^{2}-{\mathfrak {y}}_{a}^{2}\\\hline {\begin{aligned}x_{a}-y_{a}&=\left({\mathfrak {x}}_{a}-{\mathfrak {y}}_{a}\right)r_{a}\\x_{a}+y_{a}&=\left({\mathfrak {x}}_{a}+{\mathfrak {y}}_{a}\right){\frac {1}{r_{a}}}\end{aligned}}\quad (1)\\\hline \Rightarrow {\begin{aligned}2{\mathfrak {x}}_{a}&=\left(r_{a}+{\frac {1}{r_{a}}}\right)x_{a}+\left(r_{a}-{\frac {1}{r_{a}}}\right)y_{a}\\2{\mathfrak {y}}_{a}&=\left(r_{a}-{\frac {1}{r_{a}}}\right)x_{a}+\left(r_{a}+{\frac {1}{r_{a}}}\right)y_{a}\end{aligned}}\quad (2)\end{matrix}}}

Equation system (1) represents Lorentz boost or squeeze mapping (9a), and (2) represents Lorentz boost (9b).

### Bianchi (1886–1894) – Transforming pseudospherical surfaces

w:Luigi Bianchi (1886) followed Lie (1879/80) by writing the transformation of pseudospheres into each other, obtaining the result:[M 18]

{\displaystyle {\begin{aligned}(1)\quad &u+v=2\alpha ,\ u-v=2\beta ;\\(2)\quad &\Omega \left(\alpha ,\beta \right)\Rightarrow \Omega \left(k\alpha ,\ {\frac {\beta }{k}}\right);\\(3)\quad &\theta (u,v)\Rightarrow \theta \left({\frac {u+v\sin \sigma }{\cos \sigma }},\ {\frac {u\sin \sigma +v}{\cos \sigma }}\right)=\Theta _{\sigma }(u,v);\\&{\text{Inverse:}}\left({\frac {u-v\sin \sigma }{\cos \sigma }},\ {\frac {-u\sin \sigma +v}{\cos \sigma }}\right)\\(4)\quad &{\frac {1}{2}}\left(k+{\frac {1}{k}}\right)={\frac {1}{\cos \sigma }},\ {\frac {1}{2}}\left(k-{\frac {1}{k}}\right)={\frac {\sin \sigma }{\cos \sigma }}\end{aligned}}}.

Equations (1) together with transformation (2) gives Lorentz boost (9a) in terms of light-cone coordinates. Plugging equations (4) into (3) gives Lorentz boost (9b) in terms of Bondi's k factor.

In 1894, Bianchi redefined the variables u,v as asymptotic coordinates, by which the transformation obtains the form:[M 19]

${\displaystyle {\begin{matrix}\Omega \left(u,v\right)\Rightarrow \omega (u,v);\quad \Omega \left(u,v\right)=\omega \left(ku,\ {\frac {v}{k}}\right);\\k={\frac {1+\sin \sigma }{\cos \sigma }}\Rightarrow \Omega \left(u,v\right)=\omega \left({\frac {1+\sin \sigma }{\cos \sigma }}u,\ {\frac {1-\sin \sigma }{\cos \sigma }}v\right)\end{matrix}}}$.

This is consistent with one of the choices in (9e) where Bianchi's angle σ corresponds to θ.

### Lindemann (1890/91) – Weierstrass coordinates and Cayley absolute

w:Ferdinand von Lindemann employed the Cayley absolute related to surfaces of second degree and its transformation[M 20]

{\displaystyle {\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\X_{1}X_{4}+X_{2}X_{3}=\Xi _{1}\Xi _{4}+\Xi _{2}\Xi _{3}\\\hline {\begin{aligned}X_{1}&=\left(\lambda +\lambda _{1}\right)U_{4}&\Xi _{1}&=\left(\lambda -\lambda _{1}\right)U_{4}&X_{1}&={\frac {\lambda +\lambda _{1}}{\lambda -\lambda _{1}}}\Xi _{1}\\X_{2}&=\left(\lambda +\lambda _{3}\right)U_{4}&\Xi _{2}&=\left(\lambda -\lambda _{3}\right)U_{4}&X_{2}&={\frac {\lambda +\lambda _{3}}{\lambda -\lambda _{3}}}\Xi _{2}\\X_{3}&=\left(\lambda -\lambda _{3}\right)U_{2}&\Xi _{3}&=\left(\lambda +\lambda _{3}\right)U_{2}&X_{3}&={\frac {\lambda -\lambda _{3}}{\lambda +\lambda _{3}}}\Xi _{3}\\X_{4}&=\left(\lambda -\lambda _{1}\right)U_{1}&\Xi _{4}&=\left(\lambda +\lambda _{1}\right)U_{1}&X_{4}&={\frac {\lambda -\lambda _{1}}{\lambda +\lambda _{1}}}\Xi _{4}\end{aligned}}\end{matrix}}}

into which he put[M 21]

{\displaystyle {\begin{matrix}{\begin{aligned}X_{1}&=x_{1}+2kx_{4},&X_{2}&=x_{2}+ix_{3},&\lambda +\lambda _{1}&=\left(\lambda -\lambda _{1}\right)e^{a},\\X_{4}&=x_{1}-2kx_{4},&X_{3}&=x_{2}-ix_{3},&\lambda +\lambda _{3}&=\left(\lambda -\lambda _{3}\right)e^{\alpha i},\end{aligned}}\\\hline \Omega _{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\\ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}\end{matrix}}}

This is equivalent to squeeze mapping (9a, as well as 9e in terms of η) with ${\displaystyle e^{\alpha i}=1}$ and 2k=1 .

### Haskell (1895) – Hyperbola mapping

w:Mellen W. Haskell applied the linear transformation

${\displaystyle \alpha '=k\alpha ,\ \beta '=k^{-1}\beta }$

in order to transform a hyperbola into itself.[M 22]

This is equivalent to Lorentz transformation (9a).

### Smith (1900) – Laguerre inversion

w:Percey F. Smith followed Laguerre (1882) and Darboux (1887) and defined the Laguerre inversion as follows:[M 23]

${\displaystyle {\begin{matrix}p^{\prime 2}-p^{2}=R^{\prime 2}-R^{2}\\\hline \kappa ={\frac {R'-R}{p'-p}}\\p'={\frac {\kappa ^{2}+1}{\kappa ^{2}-1}}p-{\frac {2\kappa }{\kappa ^{2}-1}}R,\quad R'={\frac {2\kappa }{\kappa ^{2}-1}}p-{\frac {\kappa ^{2}+1}{\kappa ^{2}-1}}R\end{matrix}}}$

This is equivalent (up to a sign change) to Lorentz transformation (9d).

### Elliott (1903) – Invariant theory

w:Edwin Bailey Elliott (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:[M 24]

${\displaystyle {\begin{matrix}x=X\cosh \phi +Y\sinh \phi ,\quad y=X\sinh \phi +Y\cosh \phi \\\hline X+Y=e^{-\phi }(x+y),\quad X-Y=e^{\phi }(x-y)\end{matrix}}}$

The second line is equivalent to squeeze mapping or Lorentz boost (9a, as well as 9e in terms of η).

### Eisenhart (1905) – Transforming pseudospherical surfaces

w:Luther Pfahler Eisenhart followed Lie (1879/81), Bianchi (1886, 1894) and Darboux (1891/94) in transforming pseudospherical surfaces:[M 25]

{\displaystyle {\begin{aligned}(1)\quad &\alpha ={\frac {u+v}{2}},\ \beta ={\frac {u-v}{2}}\\(2)\quad &\omega \left(\alpha ,\beta \right)\Rightarrow \omega \left(m\alpha ,\ {\frac {\beta }{m}}\right)\\(3)\quad &\omega (u,v)\Rightarrow \omega (\alpha +\beta ,\ \alpha -\beta )\Rightarrow \omega \left(\alpha m+{\frac {\beta }{m}},\ \alpha m-{\frac {\beta }{m}}\right)\\&\Rightarrow \omega \left[{\frac {\left(m^{2}+1\right)u+\left(m^{2}-1\right)v}{2m}},\ {\frac {\left(m^{2}-1\right)u+\left(m^{2}+1\right)v}{2m}}\right]\\(4)\quad &m={\frac {1-\cos \sigma }{\sin \sigma }}\Rightarrow \omega \left({\frac {u-v\cos \sigma }{\sin \sigma }},\ {\frac {v-u\cos \sigma }{\sin \sigma }}\right)\end{aligned}}}.

Equations (1) together with transformation (2) gives Lorentz boost (9a) in terms of light-cone coordinates. Transformation (3) is equivalent to Lorentz boost (9b) in terms of Bondi's k factor. Eisenhart's angle σ corresponds to ϑ in (9e).

## References

### Historical mathematical sources

1. Apollonius/Heath (1896), Proposition 34 (Apollonius, Book II, Prop. 12).
2. Apollonius/Heath (1896), Proposition 35 (Apollonius, Book II, Prop. 13).
3. Speidell (1688), pp. 4-5
4. Whiston (1710/16). In the English version (1716) see pp. 16-17. In the original Latin version (1710) see pp. 16-18
5. Reynaud (1819), p. 247
6. Klein (1871), pp. 601–602
7. Laisant (1874a), pp. 73–76
8. Lie (1879/80), Collected papers, vol. 3, p. 389
9. Lie (1879/81), Collected papers, vol. 3, p. 393
10. Lie (1880/81), Collected papers, vol. 3, pp. 477–478
11. Lie (1883/84), Collected papers, vol. 3, p. 556
12. Günther (1880/81), pp. 383–385
13. Laguerre (1882), pp. 550–551.
14. Darboux (1883), p. 849
15. Darboux (1891/94), pp. 381–382
16. Darboux (1887)
17. Lipschitz (1886), pp. 90–92
18. Bianchi (1886), eq. 1 can be found on p. 226, eq. (2) on p. 240, eq. (3) on pp. 240–241, and for eq. (4) see the footnote on p. 240.
19. Bianchi (1894), pp. 433–434
20. Lindemann & Clebsch (1890/91), pp. 361–362
21. Lindemann & Clebsch (1890/91), p. 496