celestial sphere


Riemannian geometry


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Given a point in 4-dimensional Minkowski spacetime, its celestial (or heavenly) sphere is the space of lines in its light cone, hence the projective space of its light cone.

We can equivalently speak of rays in the past light cone (or rays in the future light cone); then your celestial sphere (the one around the point where your head is) is the sphere of which you directly perceive a portion when you look. (Since our eyes face forward, we actually see only a small portion of this sphere, but some birds see the entire sphere.) If you take the point to be Earth, then this celestial sphere is the sphere of the heavens as it appeared to the ancients.


Spinorial coordinates

By the exceptional spin isomorphism

Spin(3,1)SL(2,) Spin(3,1) \simeq SL(2,\mathbb{C})

one may identify points (x i)=(x 0,x 1,x 2,x 3)(x^i) = (x^0, x^1, x^2, x^3) in Minkowski spacetime with Hermitean matrices

(x αβ)(x iγ i αβ)=12(x 0+x 3 x 1+ix 2 x 1ix 2 x 0x 3) \left(x^{\alpha \beta}\right) \coloneqq (x^i \gamma_i^{\alpha \beta}) = \tfrac{1}{\sqrt{2}} \left( \array{ x^0 + x^3 & x^1 + i x^2 \\ x^1 - i x^2 & x^0 - x^3 } \right)

(where γ i\gamma_i denote the generators of the Clifford algebra given by the Pauli matrices). This is such that the Lorentz metric norm is just the determinant of this matrix

(x i)=2det((x αβ)). \Vert \left(x^i\right) \Vert = 2 det\left(\left(x^{\alpha \beta} \right)\right) \,.

From this one finds that (x i)\left(x^i\right) is lightlike precisely if there is a spinor κ\kappa, hence a pair of complex numbers ξ,η\xi, \eta \in \mathbb{C}

(κ a)=(ξ η), \left(\kappa^a\right) = \left( \array{ \xi \\ \eta } \right) \,,

such that

x αβ=κ ακ¯ β. x^{\alpha \beta} = \kappa^\alpha \overline{\kappa}^{\beta} \,.

Therefore the celestial sphere is equivalently the space of such pairs of complex numbers, modulo rescaling κcκ\kappa \mapsto c \kappa for 0c0 \neq c \in \mathbb{C}. This identifies the celestial sphere with the complex projective space

CelestialSphereP 1, CelestialSphere \simeq \mathbb{C}P^1 \,,

the Riemann sphere.


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