Editing Möbius transformation (section)

=== Lorentz transformation ===
{{main|Lorentz transformation}}
An isomorphism of the Möbius group with the [[Lorentz group]] was noted by several authors: Based on previous work of [[Felix Klein]] (1893, 1897)<ref>Felix Klein (1893), [https://archive.org/details/nichteuklidisch00liebgoog Nicht-Euklidische Geometrie], Autogr. Vorl., Göttingen;<br />[[Robert Fricke]] & Felix Klein (1897), [https://archive.org/details/vorlesungenber01fricuoft Autormorphe Funktionen I.], Teubner, Leipzig</ref> on [[automorphic function]]s related to hyperbolic geometry and Möbius geometry, [[Gustav Herglotz]] (1909)<ref>{{Citation|author=Herglotz, Gustav|year=1910|orig-year=1909|trans-title=On bodies that are to be designated as 'rigid' from the relativity principle standpoint|title=Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper|journal=Annalen der Physik|volume=336|issue=2|language=de |pages=393–415|doi=10.1002/andp.19103360208|bibcode = 1910AnP...336..393H |url=https://zenodo.org/record/1424161}}</ref> showed that [[hyperbolic motion]]s (i.e. [[isometry|isometric]] [[automorphism]]s of a [[hyperbolic space]]) transforming the [[unit sphere]] into itself correspond to Lorentz transformations, by which Herglotz was able to classify the one-parameter Lorentz transformations into loxodromic, elliptic, hyperbolic, and parabolic groups. Other authors include [[Emil Artin]] (1957),<ref>Emil Artin (1957) ''[[Geometric Algebra (book)|Geometric Algebra]]'', page 204</ref> [[H. S. M. Coxeter]] (1965),<ref>[[H. S. M. Coxeter]] (1967) "The Lorentz group and the group of homographies", in L. G. Kovacs & B. H. Neumann (editors) ''Proceedings of the International Conference on The Theory of Groups held at Australian National University, Canberra, 10—20 August 1965'', [[Gordon and Breach]] Science Publishers</ref> and [[Roger Penrose]], [[Wolfgang Rindler]] (1984),{{sfn|Penrose|Rindler|1984|pp=8–31}} [[Tristan Needham]] (1997)<ref>{{Cite book|last=Needham|first=Tristan|url=https://umv.science.upjs.sk/hutnik/NeedhamVCA.pdf|title=Visual Complex Analysis|publisher=Oxford University Press|year=1997|location=Oxford|pages=122–124}}</ref> and W. M. Olivia (2002).<ref>{{cite book|first=Waldyr Muniz|last=Olivia|date=2002|title=Geometric Mechanics|chapter=Appendix B: Möbius transformations and the Lorentz group|pages=195–221|publisher=Springer|mr=1990795 |isbn=3-540-44242-1}}</ref> 

[[Minkowski space]] consists of the four-dimensional real coordinate space '''R'''<sup>4</sup> consisting of the space of ordered quadruples {{nowrap|(''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}} of real numbers, together with a [[quadratic form]]
<math display="block">Q(x_0,x_1,x_2,x_3) = x_0^2-x_1^2-x_2^2-x_3^2.</math>

Borrowing terminology from [[special relativity]], points with {{math|''Q'' > 0}} are considered ''timelike''; in addition, if {{math|''x''<sub>0</sub> > 0}}, then the point is called ''future-pointing''. Points with {{math|''Q'' < 0}} are called ''spacelike''. The [[null cone]] ''S'' consists of those points where {{math|1=''Q'' = 0}}; the ''future null cone'' ''N''<sup>+</sup> are those points on the null cone with {{math|''x''<sub>0</sub> > 0}}. The [[celestial sphere]] is then identified with the collection of rays in ''N''<sup>+</sup> whose initial point is the origin of '''R'''<sup>4</sup>. The collection of [[linear transformation]]s on '''R'''<sup>4</sup> with positive [[determinant]] preserving the quadratic form ''Q'' and preserving the time direction form the [[restricted Lorentz group]] {{nowrap|SO<sup>+</sup>(1, 3)}}.

In connection with the geometry of the celestial sphere, the group of transformations {{nowrap|SO<sup>+</sup>(1, 3)}} is identified with the group {{nowrap|PSL(2, '''C''')}} of Möbius transformations of the sphere. To each {{math|(''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>) ∈ '''R'''<sup>4</sup>}}, associate the [[hermitian matrix]]
<math display="block">X=\begin{bmatrix}
x_0+x_1 & x_2+ix_3\\
x_2-ix_3 & x_0-x_1
\end{bmatrix}.</math>

The [[determinant]] of the matrix ''X'' is equal to {{math|''Q''(''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}}. The [[special linear group]] acts on the space of such matrices via

{{NumBlk2|:|<math>X \mapsto AXA^*</math>|1}}

for each {{nowrap|''A'' ∈ SL(2, '''C''')}}, and this action of {{nowrap|SL(2, '''C''')}} preserves the determinant of ''X'' because {{math|1=det ''A'' = 1}}. Since the determinant of ''X'' is identified with the quadratic form ''Q'', {{nowrap|SL(2, '''C''')}} acts by Lorentz transformations. On dimensional grounds, {{nowrap|SL(2, '''C''')}} covers a neighborhood of the identity of {{nowrap|SO(1, 3)}}. Since {{nowrap|SL(2, '''C''')}} is connected, it covers the entire restricted Lorentz group {{nowrap|SO<sup>+</sup>(1, 3)}}. Furthermore, since the [[kernel (group theory)|kernel]] of the action ({{EquationNote|1}}) is the subgroup {{mset|±''I''}}, then passing to the [[quotient group]] gives the [[group isomorphism]]

{{NumBlk2|:|<math>\operatorname{PSL}(2,\Complex)\cong \operatorname{SO}^+(1,3).</math>|2}}

Focusing now attention on the case when {{nowrap|(''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}} is null, the matrix ''X'' has zero determinant, and therefore splits as the [[outer product]] of a complex two-vector ''ξ'' with its complex conjugate:

{{NumBlk2|:|<math>X = \xi\bar{\xi}^\text{T}=\xi\xi^*.</math>|3}}

The two-component vector ''ξ'' is acted upon by {{nowrap|SL(2, '''C''')}} in a manner compatible with ({{EquationNote|1}}). It is now clear that the kernel of the representation of {{nowrap|SL(2, '''C''')}} on hermitian matrices is {{mset|±''I''}}.

The action of {{nowrap|PSL(2, '''C''')}} on the celestial sphere may also be described geometrically using [[stereographic projection]]. Consider first the hyperplane in '''R'''<sup>4</sup> given by ''x''<sub>0</sub>&nbsp;=&nbsp;1. The celestial sphere may be identified with the sphere ''S''<sup>+</sup> of intersection of the hyperplane with the future null cone ''N''<sup>+</sup>. The stereographic projection from the north pole {{nowrap|(1, 0, 0, 1)}} of this sphere onto the plane {{nowrap|1=''x''<sub>3</sub> = 0}} takes a point with coordinates {{nowrap|(1, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}} with
<math display="block">x_1^2+x_2^2+x_3^2=1</math>
to the point
<math display="block">\left(1, \frac{x_1}{1-x_3}, \frac{x_2}{1-x_3},0\right).</math>

Introducing the [[complex number|complex]] coordinate
<math display="block">\zeta = \frac{x_1+ix_2}{1-x_3},</math>
the inverse stereographic projection gives the following formula for a point {{nowrap|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}} on ''S''<sup>+</sup>:

{{NumBlk2|:|<math>
\begin{align}
x_1 &= \frac{\zeta+\bar{\zeta}}{\zeta\bar{\zeta}+1}\\
x_2 &= \frac{\zeta-\bar{\zeta}}{i(\zeta\bar{\zeta}+1)}\\
x_3 &= \frac{\zeta\bar{\zeta}-1}{\zeta\bar{\zeta}+1}.
\end{align}
</math>|4}}

The action of {{nowrap|SO<sup>+</sup>(1, 3)}} on the points of ''N''<sup>+</sup> does not preserve the hyperplane ''S''<sup>+</sup>, but acting on points in ''S''<sup>+</sup> and then rescaling so that the result is again in ''S''<sup>+</sup> gives an action of {{nowrap|SO<sup>+</sup>(1, 3)}} on the sphere which goes over to an action on the complex variable ''ζ''. In fact, this action is by fractional linear transformations, although this is not easily seen from this representation of the celestial sphere. Conversely, for any fractional linear transformation of ''ζ'' variable goes over to a unique Lorentz transformation on ''N''<sup>+</sup>, possibly after a suitable (uniquely determined) rescaling.

A more invariant description of the stereographic projection which allows the action to be more clearly seen is to consider the variable {{nowrap|1=''ζ'' = ''z'':''w''}} as a ratio of a pair of homogeneous coordinates for the complex projective line '''CP'''<sup>1</sup>. The stereographic projection goes over to a transformation from {{nowrap|'''C'''<sup>2</sup> − {{mset|0}}}} to ''N''<sup>+</sup> which is homogeneous of degree two with respect to real scalings

{{NumBlk2|:|<math>(z,w)\mapsto (x_0,x_1,x_2,x_3)=(z\bar{z}+w\bar{w}, z\bar{z}-w\bar{w}, z\bar{w}+w\bar{z}, i^{-1}(z\bar{w}-w\bar{z}))</math>|5}}

which agrees with ({{EquationNote|4}}) upon restriction to scales in which <math>z\bar{z}+w\bar{w}=1.</math> The components of ({{EquationNote|5}}) are precisely those obtained from the outer product
<math display="block">
\begin{bmatrix}
x_0+x_1 & x_2+ix_3 \\
x_2-ix_3 & x_0-x_1
\end{bmatrix} =
2\begin{bmatrix} z \\ w \end{bmatrix}
\begin{bmatrix} \bar{z} & \bar{w} \end{bmatrix}.
</math>

In summary, the action of the restricted Lorentz group SO<sup>+</sup>(1,3) agrees with that of the Möbius group {{nowrap|PSL(2, '''C''')}}. This motivates the following definition. In dimension {{nowrap|''n'' ≥ 2}}, the '''Möbius group''' Möb(''n'') is the group of all orientation-preserving [[conformal geometry|conformal]] [[isometry|isometries]] of the round sphere ''S''<sup>''n''</sup> to itself. By realizing the conformal sphere as the space of future-pointing rays of the null cone in the Minkowski space '''R'''<sup>1,n+1</sup>, there is an isomorphism of Möb(''n'') with the restricted Lorentz group SO<sup>+</sup>(1,''n''+1) of Lorentz transformations with positive determinant, preserving the direction of time.

Coxeter began instead with the equivalent quadratic form {{tmath|1= Q(x_1,\ x_2,\ x_3 \ x_4) = x_1^2 + x_2^2 + x_3^2 - x_4^2 }}.

He identified the Lorentz group with transformations for which {{mset|''x'' | Q(''x'') {{=}} −1}} is [[invariant (mathematics)#Invariant set|stable]]. Then he interpreted the ''x''{{null}}'s as [[homogeneous coordinates]]  and {{mset|''x'' | Q(''x'') {{=}} 0}}, the [[null cone]], as the [[Cayley absolute]] for a hyperbolic space of points {{mset|''x'' | Q(''x'') < 0}}. Next, Coxeter introduced the variables
<math display="block">\xi = \frac {x_1}{x_4} , \ \eta = \frac {x_2}{x_4}, \ \zeta = \frac {x_3}{x_4} </math>
so that the Lorentz-invariant quadric corresponds to the sphere {{tmath|1= \xi^2 + \eta^2 + \zeta^2 = 1 }}. Coxeter notes that [[Felix Klein]] also wrote of this correspondence, applying stereographic projection from {{nowrap|(0, 0, 1)}} to the complex plane <math display="inline">z = \frac{\xi + i \eta}{1 - \zeta}.</math> Coxeter used the fact that circles of the inversive plane represent planes of hyperbolic space, and the general homography is the product of inversions in two or four circles, corresponding to the general hyperbolic displacement which is the product of inversions in two or four planes.