Celestial sphere and Möbius action

Statement

Under the identifications

SO^+(1,3)\cong PSL(2,\mathbb C)_{\mathbb R}, \qquad \mathscr C\cong\mathbb{CP}^1,

the action of the proper orthochronous Lorentz group on the celestial sphere is the standard Möbius action. If

A=\begin{pmatrix}a&b\\c&d\end{pmatrix},

then in the affine coordinate $\zeta=z_1/z_2$ the action is

\zeta\longmapsto\frac{a\zeta+b}{c\zeta+d},

with the usual interpretation at infinity.

Derivation from spinors

A null ray is represented by $zz^\dagger$ , and $A(zz^\dagger)A^\dagger=(Az)(Az)^\dagger$ . Projectivizing therefore sends $[z]$ to $[Az]$ . The central matrices $\pm I$ act trivially, so the action factors precisely through $PSL(2,\mathbb C)$ .

The resulting transformations preserve the conformal structure of the round $2$ -sphere, though not generally a chosen round metric. Orientation-reversing conformal maps require complex conjugation and lie outside $SO^+(1,3)$ .

References

Alan F. Beardon, The Geometry of Discrete Groups, Springer, 1983, Chapter 3. Publisher record.
Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Vol. 1, Cambridge University Press, 1984, §1.3. Publisher record.