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

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

with arbitrary k. This geometrically corresponds to the transformation 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 $$x'_0, x'_1$$ gives:

The geometrical foundation of squeeze mapping ($$) was known for a long time since Apollonius (BC) and was used to generate hyperbolas by Speidell (1688) and Whiston (1710). Equation ($$-1) was implicitly used by Mercator (1668) and explicitly 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 of pseudospherical surfaces in terms of the Sine-Gordon equation, or by Lipschitz (1885/86) in transformation theory. Equation ($$-2) was given by Reynaud (1819).

From that, different forms of Lorentz transformation were derived: ($$) by Lipschitz (1885/86), Bianchi (1886, 1894), Eisenhart (1905), trigonometric Lorentz boost E:(8a) by Bianchi (1886, 1894) and Darboux (1891/94), and trigonometric Lorentz boost E:(8b) by Eisenhart (1905). Lorentz boost ($$) was rediscovered in the framework of special relativity by Hermann Bondi (1964) in terms of Bondi k-calculus, by which k can be physically interpreted as Doppler factor. Since ($$) is equivalent to E:(6f) in terms of Cayley–Klein parameter by setting $$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 ($$) in terms of homogeneous coordinates signifies squeeze mappings of the unit hyperbola in terms of a w:projective conic:

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

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

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:

Squeeze mappings in terms of $$\theta$$ were used by Darboux (1891/94) and Bianchi (1894), in terms of $$\eta$$ implicitly by Mercator (1668) and explicitly by Lindemann (1891), Elliott (1903), Herglotz (1909/10), in terms of $$\vartheta$$ by Eisenhart (1905), in terms of $$\beta$$ by Born (1921), w:Milne (1935) and Bondi (1964).

Apollonius (BC) – Hyperbola mapping
Apollonius of Perga (c. 240–190 BC, and maybe other Greek geometers such as Menaechmus even earlier) defined the following proposition Nr. XII in his second book on conic sections, which was translated into Latin several times by Giovanni Battista Memmo (1537), Federico Commandino (1566), Isaac Barrow (1675), and in particular by Edmond Halley (1710), with the Halley translation reading as follows:


 * Let there be a hyperbola whose asymptotes are AB, BΓ, and let some point Δ be taken in that section, from which ΔΕ, ΔΓ are drawn to ΑΒ, ΒΓ; and let another point H be taken in that section, through which HΘ, HK are drawn parallel to ΔΕ, ΔΖ: I say that the rectangle EΔZ is equal to the rectangle ΘHK.
 * Let ΔH be joined, and A is connected to Γ. Therefore, since the rectangle AΔΓ is equal to the rectangle AHΓ, it follows that AH is to AΔ as ΔΓ is to ΓΗ. But AH is to ΑΔ as ΗΘ is to ΕΔ, and ΔΓ is to ΓΗ as ΔZ is to ΗΚ; wherefore as ΘΗ is to ΔΕ, so ΔZ is to ΗK: therefore the rectangle EΔZ is equal to the rectangle ΘHK.

A modernized translation was given by w:Thomas Heath as follows:


 * 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. Let Qq meet the asymptotes in R,r. We have RQ.Qr=Rq.qr; therefore RQ:Rq=qr:Qr. But RQ:Rq=HQ:hq, and qr:Qr=qk:QK; therefore HQ:hq=qk:QK, or HQ.QK=hq.qk.

In the next proposition XIII, Apollonius showed that if a line is drawn parallel to the asymptotes, within the space between asymptotes and hyperbola, it must meet the hyperbola exactly once. In his demonstration, Apollonius used the previous proposition XII when comparing the area of several parallelograms whose sides are drawn parallel to the asymptotes.

Mercator (1668) – hyperbolic relations


While deriving the Mercator series, Nicholas Mercator (1668) demonstrated Apollonius' proposition on a rectangular hyperbola algebraically as follows:


 * $$\begin{matrix}AD=1+a,\ DF=\sqrt{2a+aa}\\ AH=\frac{1+a+\sqrt{2a+aa}}{\sqrt{2}},\ FH=\frac{1+a-\sqrt{2a+aa}}{\sqrt{2}}\\ AI=BI=\frac{1}{\sqrt{2}}\\ 1+a=c,\ \sqrt{2a+aa}=d,\ 1=cc-dd\\ AH*FH=\frac{cc-dd}{\sqrt{2}*\sqrt{2}}=\frac{1}{2}\\ AI*BI=\frac{1}{2}\\ \hline AH*FH=AI*BI\\ AH.AI::BI.FH \end{matrix}$$

Speidell (1688), Whiston (1710) – Hyperbola generation
The case of squeezing a given square or parallelogram as a means to generate hyperbolas was discussed by Euclid Speidell (1688):
 * [..] 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 $$\tfrac{100}{20},\tfrac{100}{30},\tfrac{100}{40},\tfrac{100}{50}$$ &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 [..].

In similar terms, William Whiston (1710/16) wrote:
 * 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;

Reynaud (1819) – Hyperbola mapping
Antoine André Louis Reynaud algebraically expressed squeeze mappings by writing:
 * "The system of equations $$(2)\ x=\frac{y'}{\alpha},\ y=\alpha x'$$ determines all points of the curve $$S$$, because $$x'$$ and $$y'$$ being given numbers, each arbitrary value of $$\alpha$$ gives a point $$x,y$$ of this curve. The elimination of the indeterminate $$\alpha$$ between equations (2) will therefore lead to the equation $$xy=x'y'$$ of the curve in question. This curve is therefore a hyperbola related to its asymptotes $$xX,yY$$."

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


 * $$\begin{matrix}x_{1}x_{2}-x_{3}^{2}=0\\

\hline \begin{align}x_{1} & =\alpha_{1}y_{1}\\ x_{2} & =\alpha_{2}y_{2}\\ x_{3} & =\alpha_{3}y_{3} \end{align} \\ \left(\alpha_{1}\alpha_{2}-\alpha_{3}^{2}=0\right)\\ \hline \frac{x_{1}x_{2}}{x_{3}^{2}}=\text{invariant} \end{matrix}$$

Laisant (1874) – Elliptic polar coordinates
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


 * $$\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.

Lie (1879-84) – Transforming pseudospherical surfaces
Sophus Lie (1879/80) derived an operation from Pierre Ossian Bonnet's (1867) investigations on surfaces of constant curvatures, by which pseudospherical surfaces can be transformed into each other. Lie gave explicit formulas for this operation in two papers published in 1881: If $$(s,\sigma)$$ are asymptotic coordinates of two principal tangent curves and $$\Theta$$ their respective angle, and $$\Theta=f(s,\sigma)$$ is a solution of the Sine-Gordon equation $$\tfrac{d^{2}\Theta}{ds\ d\sigma}=K\sin\Theta$$, then the following operation (now called Lie transform) is also a solution from which infinitely many new surfaces of same curvature can be derived:


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

In (1880/81) he wrote these relations as follows:


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

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


 * $$B=OIO^{-1}$$

Günther (1880/81) – Elliptic polar coordinates
Following Laisant (1874), Siegmund Günther demonstrated the relation between circular polar coordinates and elliptic polar coordinates as


 * $$\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.

Laguerre (1882) – Laguerre inversion
A transformation (later known as "Laguerre inversion") of E:oriented lines and spheres was given by Edmond Laguerre with R being the radius and D the distance of its center to the axis:


 * $$\begin{matrix}D^{2}-D^{\prime2}=R^{2}-R^{\prime2}\\

\hline \left.\begin{align}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{align} \right|\begin{align}D-D' & =\alpha(R-R')\\ D+D' & =\frac{1}{\alpha}(R+R') \end{align} \end{matrix}$$

Transforming pseudospherical surfaces
Gaston Darboux (1883) followed Lie (1879/81) by transforming pseudospheres into each other as follows:


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

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


 * $$\begin{align}(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{align} $$.

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


 * $$\begin{matrix}x^{\prime2}+y^{\prime2}+z^{\prime2}-R^{\prime2}=x^{2}+y^{2}+z^{2}-R^{2}\\

\hline \left.\begin{align}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{align} \right|\begin{align}z'+R' & =\frac{1+k}{1-k}(z-R)\\ z'-R' & =\frac{1-k}{1+k}(z+R) \end{align} \end{matrix}$$

Lipschitz (1885/86) - Quadratic forms
Rudolf Lipschitz (1885/86) discussed transformations leaving invariant the sum of squares
 * $$x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}$$

which he rewrote as
 * $$x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0$$.

This led to the problem of finding transformations leaving invariant the pairs $$x_{a}^{2}-y_{a}^{2}$$ (where a=1...n) for which he gave the following solution:


 * $$\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\

\hline \begin{align}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{align} \quad(1)\\ \hline\begin{align}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{align} \quad(2) \end{matrix}$$

Bianchi (1886–1894) – Transforming pseudospherical surfaces
Luigi Bianchi (1886) followed Lie (1879/80) by writing the transformation of pseudospheres into each other, obtaining the result:


 * $$\begin{align}(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{align} $$.

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


 * $$\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}$$.

Lindemann (1890/91) – Weierstrass coordinates and Cayley absolute
Ferdinand von Lindemann employed the Cayley absolute related to surfaces of second degree and its transformation


 * $$\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{align}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{align} \end{matrix}$$

into which he put


 * $$\begin{matrix}\begin{align}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{align} \\ \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}$$

Haskell (1895) – Hyperbola mapping
Mellen W. Haskell applied the linear transformation
 * $$\alpha'=k\alpha,\ \beta'=k^{-1}\beta$$

in order to transform a hyperbola into itself.

Smith (1900) – Laguerre inversion
Percey F. Smith followed Laguerre (1882) and Darboux (1887) and defined the Laguerre inversion as follows:


 * $$\begin{matrix}p^{\prime2}-p^{2}=R^{\prime2}-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}$$

Elliott (1903) – Invariant theory
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:


 * $$\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}$$

Eisenhart (1905) – Transforming pseudospherical surfaces
Luther Pfahler Eisenhart followed Lie (1879/81), Bianchi (1886, 1894) and Darboux (1891/94) in transforming pseudospherical surfaces:


 * $$\begin{align}(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{align}$$.

Herglotz (1909/10) – Special relativity
In relation to special relativity, Gustav Herglotz (1909/10) defined the Lorentz boost as follows:


 * $$\begin{align}x & =x', & t-z & =(t'-z')e^{\vartheta}\\

y & =y', & t+z & =(t'+z')e^{-\vartheta} \end{align}$$

Born (1921) – Special relativity
In the second edition of “Einstein's theory of relativity”, Max Born (1921) discussed the relation of the Lorentz transformation and the hyperbola:


 * $$\begin{matrix}x'-ct'=\frac{1+\beta}{\alpha}\left(x-ct\right)\\

x'+ct'=\frac{1-\beta}{\alpha}\left(x+ct\right)\\ \left[\alpha=\sqrt{1-\beta^{2}}\right]\\ \hline \eta=x-ct,\ \xi=x+ct\\ \xi\eta=(x-ct)(x+ct)=x^{2}-c^{2}t^{2}\\ \eta=\frac{1}{\xi} \end{matrix}$$