Knowlpedia

Construction

Celestial sphere

The sphere of future null directions, naturally identified with CP¹.

Core idea

In , let

N+={v≠0:q(v)=0, t>0}\mathcal N^+=\{v\ne0:q(v)=0,\ t>0\}

be the future null cone. The celestial sphere is its space of positive rays,

C=N+/R>0.\mathscr C=\mathcal N^+/\mathbb R_{>0}.

Every ray has a unique representative with t=1t=1, whose spatial part lies on S2S^2, so C≅S2\mathscr C\cong S^2.

Complex projective description

Under the , a future null ray is represented by zz†zz^\dagger with z∈C2∖{0}z\in\mathbb C^2\setminus\{0\}. Two spinors yield the same ray exactly when their complex lines agree. Therefore

C≅P(C2)=CP1≅C^.\mathscr C\cong\mathbb P(\mathbb C^2)=\mathbb{CP}^1 \cong\widehat{\mathbb C}.

This identification equips the celestial sphere with its standard complex structure.

Geometric meaning

At an event in Minkowski space, future null directions are possible directions of incoming or outgoing light. A choice of inertial observer identifies this abstract sphere with ordinary viewing directions; a different observer acts by a Lorentz transformation, which becomes a in the projective coordinate.

References
  1. Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Vol. 1, Cambridge University Press, 1984, §§1.2–1.3. Publisher record.
  2. Barrett O'Neill, Semi-Riemannian Geometry With Applications to Relativity, Academic Press, 1983, Chapter 5. Publisher record.