History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)

Lorentz transformation via Cayley–Hermite transformation
The E:general transformation (Q1) of any quadratic form into itself can also be given using arbitrary parameters based on the Cayley transform (I-T)−1·(I+T), where I is the identity matrix, T an arbitrary antisymmetric matrix, and by adding A as symmetric matrix defining the quadratic form (there is no primed A'  because the coefficients are assumed to be the same on both sides):

After Cayley (1846) introduced transformations related to sums of positive squares, Hermite (1853/54, 1854) derived transformations for arbitrary quadratic forms, whose result was reformulated in terms of matrices by Cayley (1855a, 1855b). For instance, the choice A=diag(1,1,1) gives an orthogonal transformation which can be used to describe spatial rotations corresponding to the w:Euler-Rodrigues parameters [a,b,c,d] discovered by Euler (1771) and Rodrigues (1840), which can be interpreted as the coefficients of quaternions. Setting d=1, the equations have the form:

Also the Lorentz interval and the general Lorentz transformation in any dimension can be produced by the Cayley–Hermite formalism. For instance, the E:most general Lorentz transformation (1a) with n=1 follows from ($$) with:

This becomes E:Lorentz boost (4a) by setting $$\tfrac{2a}{1+a^{2}}=\tfrac{v}{c}$$, which is equivalent to the relation $$\tfrac{2\beta_{0}}{1+\beta_{0}^{2}}=\tfrac{v}{c}$$ known from Loedel diagrams, thus ($$) can be interpreted as a Lorentz boost from the viewpoint of a "median frame" in which two other inertial frames are moving with equal speed $$\beta_0$$ in opposite directions.

Furthermore, Lorentz transformation E:(1a)) with n=2 is given by:

or using n=3:

The transformation of a binary quadratic form of which Lorentz transformation ($$) is a special case was given by Hermite (1854), equations containing Lorentz transformations ($$, $$, $$) as special cases were given by Cayley (1855), Lorentz transformation ($$) was given (up to a sign change) by Laguerre (1882), Darboux (1887), Smith (1900) in relation to Laguerre geometry, and Lorentz transformation ($$) was given by Bachmann (1869). In relativity, equations similar to ($$, $$) were first employed by Borel (1913) to represent Lorentz transformations.

Euler (1771) – Euler-Rodrigues parameter
Euler (1771) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as orthogonal transformation. The transformation in three dimensions was given as


 * $$\begin{matrix}X^{2}+Y^{2}+Z^{2}=x^{2}+y^{2}+z^{2}\\

\hline \begin{align}X & =Ax+By+Cz\\ Y & =Dx+Ey+Fz\\ Z & =Gx+Hy+Iz \end{align} \begin{matrix}\left|{\scriptstyle \begin{align}1 & =AA+DD+GG\\ 1 & =BB+EE+HH\\ 1 & =CC+FF+II\\ 0 & =AB+DE+GH\\ 0 & =AG+DF+GI\\ 0 & =BC+EF+HI \end{align} }\right.\end{matrix}\end{matrix}$$

in which the coefficiens A,B,C,D,E,F,G,H,I were related by Euler to four arbitrary parameter p,q,r,s, which where rediscovered by Olinde Rodrigues (1840) who related them to rotation angles :


 * $$\begin{matrix}\begin{align}A & =\frac{pp+qq-rr-ss}{u} & B & =\frac{2pq+2ps}{u} & C & =\frac{2qs-2pr}{u}\\

D & =\frac{2qr-2ps}{u} & E & =\frac{pp-qq+rr-ss}{u} & F & =\frac{2pq+2rs}{u}\\ G & =\frac{2qs+2pr}{u} & H & =\frac{2rs-2pq}{u} & I & =\frac{pp-qq-rr+ss}{u} \end{align} \\ (u=pp+qq+rr+ss) \end{matrix}$$

Cayley (1846–1855) – Cayley–Hermite transformation
The Euler–Rodrigues parameters discovered by Euler (1871) and Rodrigues (1840) leaving invariant $$x_{0}^{2}+x_{1}^{2}+x_{2}^{2}$$ were extended to $$x_{0}^{2}+\dots+x_{n}^{2}$$ by Arthur Cayley (1846) as a byproduct of what is now called the Cayley transform using the method of skew–symmetric coefficients. Following Cayley's methods, a general transformation for quadratic forms into themselves in three (1853) and arbitrary (1854) dimensions was provided by Hermite (1853, 1854). Hermite's formula was simplified and brought into matrix form equivalent to ($$) by Cayley (1855a)


 * $${\scriptstyle (\mathrm{x,y,z}\dots)=\left(\left|\begin{matrix}a, & h, & g & \dots\\

h, & b, & f & \dots\\ g, & f, & c & \dots\\ \dots & \dots & \dots & \dots \end{matrix}\right|^{-1}\left|\begin{matrix}a, & h-\nu, & g+\mu & \dots\\ h+\nu, & b, & f-\lambda & \dots\\ g-\mu, & f+\lambda, & c & \dots\\ \dots & \dots & \dots & \dots \end{matrix}\right|\left|\begin{matrix}a, & h+\nu, & g-\mu & \dots\\ h-\nu, & b, & f+\lambda & \dots\\ g+\mu, & f-\lambda, & c & \dots\\ \dots & \dots & \dots & \dots \end{matrix}\right|^{-1}\left|\begin{matrix}a, & h, & g & \dots\\ h, & b, & f & \dots\\ g, & f, & c & \dots\\ \dots & \dots & \dots & \dots \end{matrix}\right|\right)\ ^{\frown}(x,y,z\dots)}$$

which he abbreviated in 1858, where $$\Upsilon$$ is any skew-symmetric matrix:


 * $$(\mathrm{x,y,z})=\left(\Omega^{-1}(\Omega-\Upsilon)(\Omega+\Upsilon)^{-1}\Omega\right)(x,y,z)$$

Using the parameters of (1855a), Cayley in a subsequent paper (1855b) particularly discussed several special cases, such as:


 * $$\begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}=ax^{2}+by^{2}\\

\hline (\mathrm{x,y})=\frac{1}{ab+\nu^{2}}\cdot\left[\begin{matrix}ab-\nu^{2}, & -2\nu b\\ 2\nu a, & ab-\nu^{2} \end{matrix}\right](x,y) \end{matrix}$$

or:


 * $${\scriptstyle \begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}+c\mathrm{z}^{2}=ax^{2}+by^{2}+cz^{2}\\

\hline (\mathrm{x,y,z})=\frac{1}{abc+a\lambda^{2}+b\mu^{2}+c\nu^{2}}\times\left[\begin{matrix}abc+a\lambda^{2}-b\mu^{2}-c\nu^{2}, & 2(\lambda\mu-c\nu)b & 2(\nu\lambda+b\mu)c\\ 2(\lambda\mu+c\nu)a, & abc-a\lambda^{2}+b\mu^{2}-c\nu^{2} & 2(\mu\nu-a\lambda)c\\ 2(\nu\lambda-b\mu)a & 2(\mu\nu+a\lambda)b & abc-a\lambda^{2}-b\mu^{2}-c\nu^{2} \end{matrix}\right](x,y,z) \end{matrix}}$$

or:


 * $${\scriptstyle \begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}+c\mathrm{z}^{2}+d\mathrm{w}^{2}=ax^{2}+by^{2}+cz^{2}+dw^{2}\\

\hline (\mathrm{x,y,z,w})=\frac{1}{k}\cdot\left[\begin{align} & abcd-bc\rho^{2}+ca\sigma^{2}+ab\tau^{2}+ad\lambda^{2} & & 2b\left(-cd\nu-\tau\phi+d\lambda\mu-c\rho\sigma\right),\\ & \quad-bd\mu^{2}-cd\nu^{2}-\phi^{2}, & & abcd+bc\rho^{2}-ca\sigma^{2}+ab\tau^{2}-ad\lambda^{2}\\ & 2a\left(cd\nu+\tau\phi+d\lambda\mu-c\rho\sigma\right), & & \quad+bd\mu^{2}-cd\nu^{2}-\phi^{2},\\ & 2a\left(-bd\mu-\sigma\phi+d\lambda\nu-b\rho\tau\right), & & 2b\left(ad\lambda+\rho\phi+d\mu\nu-a\sigma\tau\right),\\ & 2a\left(bc\rho+\lambda\phi+c\nu\sigma-b\mu\tau\right), & & 2b\left(ac\sigma+\mu\phi-c\nu\rho+a\lambda\tau\right),\\ \\ & \quad2c\left(bd\mu+\sigma\phi+d\lambda\nu-b\rho\tau\right), & & \quad2d\left(-bc\rho-\lambda\phi+c\nu\sigma-b\mu\tau\right)\\ & \quad2c\left(-ad\lambda-\rho\phi+d\mu\nu-a\sigma\tau\right), & & \quad2d\left(-ac\sigma-\mu\phi-c\nu\rho+a\lambda\tau\right)\\ & \quad abcd+bc\rho^{2}+ca\sigma^{2}-ab\tau^{2}-ad\lambda^{2} & & \quad2d\left(-ab\tau-\nu\phi+b\mu\rho-a\lambda\sigma\right)\\ & \quad\quad-bd\mu^{2}+cd\nu^{2}-\phi^{2}, & & \quad abcd-bc\rho^{2}-ca\sigma^{2}-ab\tau^{2}+ad\lambda^{2}\\ & \quad2c\left(ab\tau+\nu\phi+b\mu\rho-a\lambda\sigma\right), & & \quad\quad+bd\mu^{2}+cd\nu^{2}-\phi^{2}, \end{align} \right]\cdot(x,y,z,w)\\ \left(\begin{align}k & =abcd+bc\rho^{2}+ca\sigma^{2}+ab\tau^{2}+ad\lambda^{2}+bd\mu^{2}+cd\nu^{2}+\phi^{2}\\ \phi & =\lambda\rho+\mu\sigma+\nu\tau \end{align} \right) \end{matrix}}$$

Hermite (1853, 1854) – Cayley–Hermite transformation
Charles Hermite (1853) extended the number theoretical work of E:Gauss (1801) and others (including himself) by additionally analyzing indefinite ternary quadratic forms that can be transformed into the Lorentz interval ±(x2+y2-z2), and by using Cayley's (1846) method of skew–symmetric coefficients he derived transformations leaving invariant almost all types of ternary quadratic forms. This was generalized by him in 1854 to n dimensions:


 * $$\begin{matrix}f\left(X_{1},X_{2},\dots\right)=f\left(x_{1},x_{2},\dots\right)\\

\hline X_{r}=2\xi_{r}-x_{r}=\xi_{r}-\frac{1}{2}\sum_{s=1}^{n}\lambda_{r,s}\frac{df}{d\xi_{s}}\\ \left(\lambda_{r,s}=-\lambda_{s,r},\ \lambda_{r,r}=0\right) \end{matrix}$$

This result was subsequently expressed in matrix form by Cayley (1855), while Ferdinand Georg Frobenius (1877) added some modifications in order to include some special cases of quadratic forms that cannot be dealt with by the Cayley–Hermite transformation.

For instance, the special case of the transformation of a binary quadratic form into itself was given by Hermite as follows:


 * $$\begin{matrix}f=ax^{2}+2bxy+cy^{2}\\

\hline \begin{align}X & =\frac{\left(1-2\lambda b+\lambda^{2}D\right)x-2\lambda cy}{1-\lambda^{2}D} & & =x(t-bu)-cuy\\ Y & =\frac{2\lambda ax+\left(1+2\lambda b+\lambda^{2}D\right)y}{1-\lambda^{2}D} & & =xau+(t+bu)y \end{align} \\ \left(b^{2}-ac=D,\ t=\frac{1+\lambda^{2}D}{1-\lambda^{2}D},\ u=\frac{2\lambda}{1-\lambda^{2}D},\ t^{2}-Du^{2}=1\right) \end{matrix}$$

Bachmann (1869) – Cayley–Hermite transformation
Paul Gustav Heinrich Bachmann (1869) adapted Hermite's (1853/54) transformation of ternary quadratic forms to the case of integer transformations. He particularly analyzed the Lorentz interval and its transformation, and also alluded to the analogue result of E:Gauss (1800) in terms of Cayley–Klein parameters, while Bachmann formulated his result in terms of the Cayley–Hermite transformation:


 * $$\begin{matrix}x^{2}+x^{\prime2}-x^{\prime\prime2}\\

\hline \begin{align}\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X & =\left(p^{2}-q^{2}+q^{\prime2}-q^{\prime\prime2}\right)x-2(pq+qq')x'-2(pq'+qq)x''\\ \left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X' & =2(pq-qq')x+\left(p^{2}+q^{2}-q^{\prime2}-q^{\prime\prime2}\right)x'+2(pq-q'q)x''\\ \left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X & =-2(pq'-qq')x+2(pq+q'q)x'+\left(p^{2}+q^{2}+q^{\prime2}+q^{\prime\prime2}\right)x'' \end{align} \end{matrix}$$

He described this transformation in 1898 in the first part of his "arithmetics of quadratic forms" as well.

Laguerre (1882) – Laguerre inversion
After previous work by Albert Ribaucour (1870), a transformation which transforms oriented spheres into oriented spheres, oriented planes into oriented planes, and oriented lines into oriented lines, was explicitly formulated by Edmond Laguerre (1882) as "transformation by reciprocal directions" which was later called "Laguerre inversion/transformation". It can be seen as a special case of the conformal group in terms of E:Lie's transformations of oriented spheres. In two dimensions the transformation or oriented lines has the form (R being the radius):


 * $$\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^{2}-D^{\prime2} & =R^{2}-R^{\prime2}\\ D-D' & =\alpha(R-R')\\ D+D' & =\frac{1}{\alpha}(R+R') \end{align} $$

Darboux (1887) – Laguerre inversion
Following Laguerre (1882), Gaston Darboux (1887) presented 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 \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} \end{matrix}$$

Smith (1900) – Laguerre inversion
Percey F. Smith (1900) followed Laguerre (1882) and Darboux (1887) and defined the Laguerre inversion as follows:
 * $$\begin{matrix}p^{\prime2}-p^{2}=R^{\prime2}-R^{2}\\

\hline 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}$$

Borel (1913–14) – Cayley–Hermite parameter
Émile Borel (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and Cayley's (1846) parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions equivalent to ($$):


 * $$\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\

\hline {\scriptstyle \begin{align}\delta a & =\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, & \delta b & =2(\lambda\mu+\nu\rho), & \delta c & =-2(\lambda\nu+\mu\rho),\\ \delta a' & =2(\lambda\mu-\nu\rho), & \delta b' & =-\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, & \delta c' & =2(\lambda\rho-\mu\nu),\\ \delta a & =2(\lambda\nu-\mu\rho), & \delta b & =2(\lambda\rho+\mu\nu), & \delta c'' & =-\left(\lambda^{2}+\mu^{2}+\nu^{2}+\rho^{2}\right), \end{align} }\\ \left(\delta=\lambda^{2}+\mu^{2}-\rho^{2}-\nu^{2}\right)\\ \lambda=\nu=0\rightarrow\text{Hyperbolic rotation} \end{matrix}$$

In four dimensions equivalent to ($$):


 * $$\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\

\hline {\scriptstyle \begin{align} & \left(\mu^{2}+\nu^{2}-\alpha^{2}\right)\cos\varphi+\left(\lambda^{2}-\beta^{2}-\gamma^{2}\right)\operatorname{ch}{\theta} & & -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi-\gamma\operatorname{sh}{\theta}\\ & -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi+\gamma\operatorname{sh}{\theta} & & \left(\mu^{2}+\nu^{2}-\beta^{2}\right)\cos\varphi+\left(\mu^{2}-\alpha^{2}-\gamma^{2}\right)\operatorname{ch}{\theta}\\ & -(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi-\beta\operatorname{sh}{\theta} & & -(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\lambda\sin\varphi+\alpha\operatorname{sh}{\theta}\\ & (\gamma\mu-\beta\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta} & & -(\alpha\nu-\lambda\gamma)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\ \\ & \quad-(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi+\beta\operatorname{sh}{\theta} & & \quad(\beta\nu-\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta}\\ & \quad-(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})-\lambda\sin\varphi-\alpha\operatorname{sh}{\theta} & & \quad(\lambda\gamma-\alpha\nu)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\ & \quad\left(\lambda^{2}+\mu^{2}-\gamma^{2}\right)\cos\varphi+\left(\nu^{2}-\alpha^{2}-\beta^{2}\right)\operatorname{ch}{\theta} & & \quad(\alpha\mu-\beta\lambda)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta}\\ & \quad(\beta\gamma-\alpha\mu)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta} & & \quad-\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\cos\varphi+\left(\lambda^{2}+\mu^{2}+\nu^{2}\right)\operatorname{ch}{\theta} \end{align} }\\ \left(\alpha^{2}+\beta^{2}+\gamma^{2}-\lambda^{2}-\mu^{2}-\nu^{2}=-1\right) \end{matrix}$$