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

Lorentz transformation via conformal, spherical wave, and Laguerre transformation

If one only requires the invariance of the light cone represented by the differential equation ${\displaystyle -dx_{0}^{2}+\dots +dx_{n}^{2}=0}$, which is the same as asking for the most general transformation that changes spheres into spheres, the Lorentz group can be extended by adding dilations represented by the factor λ. The result is the group Con(1,p) of spacetime w:conformal transformations in terms of w:special conformal transformations and inversions producing the relation

${\displaystyle -dx_{0}^{2}+\dots +dx_{n}^{2}=\lambda \left(-dx_{0}^{\prime 2}+\dots +dx_{n}^{\prime 2}\right)}$.

One can switch between two representations of this group by using an imaginary sphere radius coordinate x0=iR with the interval ${\displaystyle dx_{0}^{2}+\dots +dx_{n}^{2}}$ related to conformal transformations, or by using a real radius coordinate x0=R with the interval ${\displaystyle -dx_{0}^{2}+\dots +dx_{n}^{2}}$ related to w:spherical wave transformations in terms of w:contact transformations preserving circles and spheres. Both representations were studied by Lie (1871) and others. It was shown by Bateman & Cunningham (1909–1910), that the group Con(1,3) is the most general one leaving invariant the equations of Maxwell's electrodynamics.

It turns out that Con(1,3) is isomorphic to the w:special orthogonal group SO(2,4), and contains the Lorentz group SO(1,3) as a subgroup by setting λ=1. More generally, Con(q,p) is isomorphic to SO(q+1,p+1) and contains SO(q,p) as subgroup.[1] This implies that Con(0,p) is isomorphic to the Lorentz group of arbitrary dimensions SO(1,p+1). Consequently, the conformal group in the plane Con(0,2) – known as the group of w:Möbius transformations – is isomorphic to the Lorentz group SO(1,3).[2][3] This can be seen using tetracyclical coordinates satisfying the form ${\displaystyle -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0}$, which were discussed by Pockels (1891), Klein (1893), Bôcher (1894). The relation between Con(1,3) and the Lorentz group was noted by Bateman & Cunningham (1909–1910) and others.

A special case of Lie's geometry of oriented spheres is the Laguerre group, transforming oriented planes and lines into each other. It's generated by the Laguerre inversion introduced by Laguerre (1882) and discussed by Darboux (1887) and Smith (1900) leaving invariant ${\displaystyle x^{2}+y^{2}+z^{2}-R^{2}}$ with R as radius, thus the Laguerre group is isomorphic to the Lorentz group. A similar concept was studied by Scheffers (1899) in terms of contact transformations. Stephanos (1883) argued that Lie's geometry of oriented spheres in terms of contact transformations, as well as the special case of the transformations of oriented planes into each other (such as by Laguerre), provides a geometrical interpretation of Hamilton's w:biquaternions. The w:group isomorphism between the Laguerre group and Lorentz group was pointed out by Bateman (1910), Cartan (1912, 1915/55), Poincaré (1912/21) and others.[4][5]

Historical notation

Lie (1871) – Conformal, spherical, and orthogonal transformations

In several papers between 1847 and 1850 it was shown by w:Joseph Liouville[M 1] that the relation λ(δx2+δy2+δz2) is invariant under the group of w:conformal transformations generated by inversions transforming spheres into spheres, which can be related w:special conformal transformations or w:Möbius transformations. (The conformal nature of the linear fractional transformation ${\displaystyle {\tfrac {a+bz}{c+dz}}}$ of a complex variable ${\displaystyle z}$ was already discussed by Euler (1777)).[M 2][6]

Liouville's theorem was extended to all dimensions by w:Sophus Lie (1871a).[M 3][7] In addition, Lie described a manifold whose elements can be represented by spheres, where the last coordinate yn+1 can be related to an imaginary radius by iyn+1:[M 3]

${\displaystyle {\begin{matrix}\sum _{i=1}^{i=n}(x_{i}-y_{i})^{2}+y_{n+1}^{2}=0\\\downarrow \\\sum _{i=1}^{i=n+1}(y_{i}^{\prime }-y_{i}^{\prime \prime })^{2}=0\end{matrix}}}$

If the second equation is satisfied, two spheres y′ and y″ are in contact. Lie then defined the correspondence between contact transformations in Rn and conformal point transformations in Rn+1: The sphere of space Rn consists of n+1 parameter (coordinates plus imaginary radius), so if this sphere is taken as the element of space Rn, it follows that Rn now corresponds to Rn+1. Therefore, any transformation (to which he counted orthogonal transformations and inversions) leaving invariant the condition of contact between spheres in Rn, corresponds to the conformal transformation of points in Rn+1.

Eventually, Lie (1871/72) pointed out that conformal point transformations consist of motions (such as w:rigid transformations and orthogonal transformations), similarity transformations, and inversions.[M 4]

As shown by Bateman and Cunningham (1909), the spacetime conformal group Con(1,3) of "w:spherical wave transformations" corresponds to the transformations of Lie's sphere geometry in which the radius indicates the fourth coordinate, while the Lorentz group SO(1,3) is a subgroup of Con(1,3).

Klein, Pockels, Bôcher (1871-91) – Conformal transformation and polyspherical coordinates

In relation to line geometry, w:Felix Klein (1871/72)[M 5] used coordinates satisfying the condition ${\displaystyle s_{1}^{2}+s_{2}^{2}+s_{2}^{2}+s_{2}^{2}+s_{5}^{2}=0}$. They were introduced in 1868 (belatedly published in 1872/73) by w:Gaston Darboux[M 6] as a system of five coordinates in R3 (later called "pentaspherical" coordinates) in which the last coordinate is imaginary. w:Sophus Lie (1871)[M 7] more generally used n+2 coordinates in Rn (later called "polyspherical" coordinates) satisfying ${\displaystyle \scriptstyle \sum _{i=1}^{i=n+2}x_{i}^{2}=0}$ in which the last coordinate is imaginary, as a means to discuss conformal transformations generated by inversions. These simultaneous publications can be explained by the fact that Darboux, Lie, and Klein corresponded with each other by letter.

When the last coordinate is defined as real, the corresponding polyspherical coordinates satisfy the form of a sphere. Initiated by lectures of Klein between 1889–1890, his student Friedrich Pockels (1891) used such real coordinates, emphasizing that all of these coordinate systems remain invariant under conformal transformations generated by inversions:[M 8]

${\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n+1}^{2}-x_{n+2}^{2}=0{\text{ or }}\sum _{1}^{n+1}x_{h}^{2}-x_{n+2}^{2}=0}$

Special cases were described by Klein (1893):[M 9]

${\displaystyle y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}-y_{5}^{2}=0}$ (pentaspherical).
${\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0}$ (tetracyclical).

Both systems were also described by w:Maxime Bôcher (1894) in an expanded version of a thesis supervised by Klein.[M 10]

Polyspherical coordinates indicate that the conformal group Con(0,p) is isomorphic to the Lorentz group SO(1,p+1).[8] For instance, Con(0,2) – known as Möbius group – is related to tetracyclical coordinates satisfying ${\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0}$, which is nothing other than the Lorentz interval invariant under the Lorentz group SO(1,3).

Laguerre (1882) – Laguerre inversion

After previous work by w:Albert Ribaucour (1870),[M 11] a transformation which transforms oriented spheres into oriented spheres, oriented planes into oriented planes, and oriented lines into oriented lines, was explicitly formulated by w: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 Lie's transformations of oriented spheres. In two dimensions the transformation or oriented lines has the form (R being the radius):[M 12]

{\displaystyle \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^{2}-D^{\prime 2}&=R^{2}-R^{\prime 2}\\D-D'&=\alpha (R-R')\\D+D'&={\frac {1}{\alpha }}(R+R')\end{aligned}}}

This is equivalent (up to a sign change) to a Lorentz boost with ${\displaystyle {\tfrac {2\alpha }{1+\alpha ^{2}}}={\tfrac {v}{c}}}$.

Stephanos (1883) – Biquaternions

w:Cyparissos Stephanos (1883)[M 13] showed that Hamilton's biquaternion a0+a1ι1+a2ι2+a3ι3 can be interpreted as an oriented sphere in terms of Lie's sphere geometry (1871), having the vector a1ι1+a2ι2+a3ι3 as its center and the scalar ${\displaystyle a_{0}{\sqrt {-1}}}$ as its radius. Its norm ${\displaystyle a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}}$ is thus equal to the power of a point of the corresponding sphere. In particular, the norm of two quaternions N(Q1-Q2) (the corresponding spheres are in contact with N(Q1-Q2)=0) is equal to the tangential distance between two spheres. The general contact transformation between two spheres then can be given by a w:homography using 4 arbitrary quaternions A,B,C,D and two variable quaternions X,Y:[M 14][9][10]

${\displaystyle XAY+XB+CY+D=0}$ (or ${\displaystyle X=-{\frac {CY+D}{AY+B}}}$).

Stephanos pointed out that the special case A=0 denotes transformations of oriented planes (see Laguerre (1882)).

The Lorentz group SO(1,3) is a subgroup of the conformal group Con(1,3) in terms of Lie's transformations of orientied spheres in which the radius indicates the fourth coordinate. The Lorentz group is isomorphic to the group of Laguerre's transformation of oriented planes.

Darboux (1887) – Laguerre inversion

Following Laguerre (1882), w:Gaston Darboux (1887) presented the Laguerre inversions in four dimensions using coordinates x,y,z,R:[M 15]

{\displaystyle {\begin{matrix}x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-R^{\prime 2}=x^{2}+y^{2}+z^{2}-R^{2}\\\hline {\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}}\end{matrix}}}

This is equivalent (up to a sign change for R) to a Lorentz boost with ${\displaystyle {\tfrac {2k}{1+k^{2}}}={\tfrac {v}{c}}}$.

Darboux rewrote these equations as follows:

{\displaystyle {\begin{matrix}x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-R^{\prime 2}=x^{2}+y^{2}+z^{2}-R^{2}\\\hline {\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.

Scheffers (1899) – Contact transformation

w:Georg Scheffers (1899) synthetically determined all finite w:contact transformations preserving circles in the plane, consisting of dilatations, inversions, and the following one preserving circles and lines (compare with Laguerre inversion by Laguerre (1882) and Darboux (1887)):[M 16]

${\displaystyle {\begin{matrix}\sigma ^{\prime 2}-\rho ^{\prime 2}=\sigma ^{2}-\rho ^{2}\\\hline \rho '={\frac {\rho }{\cos \omega }}+\sigma \tan \omega ,\quad \sigma '=\rho \tan \omega +{\frac {\sigma }{\cos \omega }}\end{matrix}}}$

This is equivalent to Lorentz transformation by the identity ${\displaystyle \sin \omega =v/c}$.

Smith (1900) – Laguerre inversion

w:Percey F. Smith (1900) followed Laguerre (1882) and Darboux (1887) and defined the Laguerre inversion as follows:[M 17]

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

This is equivalent (up to a sign change) to a Lorentz boost with ${\displaystyle {\tfrac {2\kappa }{1+\kappa ^{2}}}={\tfrac {v}{c}}}$.

Bateman and Cunningham (1909–1910) – Spherical wave transformation

In line with Lie's (1871) research on the relation between sphere transformations with an imaginary radius coordinate and 4D conformal transformations, it was pointed out by Bateman and Cunningham (1909–1910), that by setting u=ict as the imaginary fourth coordinates one can produce spacetime conformal transformations. Not only the quadratic form ${\displaystyle \lambda \left(dx^{2}+dy^{2}+dz^{2}+du^{2}\right)}$, but also w:Maxwells equations are covariant with respect to these transformations, irrespective of the choice of λ. These variants of conformal or Lie sphere transformations were called w:spherical wave transformations by Bateman.[R 1][R 2] However, this covariance is restricted to certain areas such as electrodynamics, whereas the totality of natural laws in inertial frames is covariant under the w:Lorentz group.[R 3] In particular, by setting λ=1 the Lorentz group SO(1,3) can be seen as a 10-parameter subgroup of the 15-parameter spacetime conformal group Con(1,3).

Bateman (1910/12)[11] also alluded to the identity between the Laguerre inversion and the Lorentz transformations. In general, the isomorphism between the Laguerre group and the Lorentz group was pointed out by w:Élie Cartan (1912, 1915/55),[5][R 4] w:Henri Poincaré (1912/21)[R 5] and others.

References

Historical mathematical sources

1. Liouville (1847)
2. Euler (1777), p. 140
3. Lie (1871), pp. 199–209
4. Lie (1871/72), p. 186
5. Klein (1871/72), p. 268
6. Darboux (1872/73), p. 137
7. Lie (1871), p. 208
8. Pockels (1891), pp. 197–206
9. Klein (1893c), pp. 200ff (pentaspherical), pp. 373ff (tetracyclical)
10. Bôcher (1894), pp. 30–34, 40–43
11. Ribaucour (1870)
12. Laguerre (1882), pp. 550–551.
13. Stephanos (1883), p. 590ff
14. Stephanos (1883), p. 592
15. Darboux (1887)
16. Scheffers (1899), p. 158
17. Smith (1900), p. 159

Historical relativity sources

1. Bateman (1909/10), pp. 223ff
2. Cunningham (1909/10), pp. 77ff
3. Klein (1910)
4. Cartan (1912), p. 23
5. Poincaré (1912/21), p. 145

Secondary sources

1. Schottenloher (2008), section 2.2
2. Kastrup (2008), section 2.4.1
3. Schottenloher (2008), section 2.3
4. Coolidge (1916), p. 370
5. Cartan & Fano (1915/55), sections 14–15
6. Kastrup (2008), section 2.1
7. Kastrup (2008), section 2.3
8. Kastrup (2008), p. 22
9. Cartan & Study (1908), p. 460
10. Rothe (1916), p. 1399
11. Bateman (1910/12), pp. 358–359
• Cartan, É.; Study, E. (1908). "Nombres complexes". Encyclopédie des Sciences Mathématiques Pures et Appliquées 1.1: 328–468.
• Cartan, É.; Fano, G. (1955). "La théorie des groupes continus et la géométrie". Encyclopédie des Sciences Mathématiques Pures et Appliquées 3.1: 39–43.  (Only pages 1–21 were published in 1915, the entire article including pp. 39–43 concerning the groups of Laguerre and Lorentz was posthumously published in 1955 in Cartan's collected papers, and was reprinted in the Encyclopédie in 1991.)
• Coolidge, Julian (1916). A treatise on the circle and the sphere. Oxford: Clarendon Press.
• Kastrup, H. A. (2008). "On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics". Annalen der Physik 520 (9–10): 631–690. doi:10.1002/andp.200810324.
• Rothe, H. (1916). "Systeme geometrischer Analyse". Encyclopädie der Mathematischen Wissenschaften 3.1.1: 1282–1425.
• Schottenloher, M. (2008). A Mathematical Introduction to Conformal Field Theory. Springer. ISBN 978-3540706908.