Conformal reference frame
(Ω, *M*, *j*)
for *X*

Such open Ω ⊂ S*X*
that ∀*x* ∈ *X*:
Ω_{x} ≠ ∅.

A smooth manifold *M*^{d}.

Such smooth map *j* : Ω → *M* that:

(f)
∀*w*_{1}, *w*_{2} ∈ Ω:

[*w*_{1}] = [*w*_{2}]
⇒ *j*(*w*_{1}) = *j*(*w*_{2});

(c) *ϑ*|_{Ω} ⊂
*j*^{∗}(*T** *M*);

(d) *j*_{∗} :
*T*_{v} Ω → *T* *M*
has the maximal rank (*d* − 1) everywhere.

*X* := the portion of
the original spacetime after

*M*.

*M* – a Cauchy surface in a globally hyperbolic manifold.

*j* – projection along geodesics.
Ω – from (d) cond.

L^{ℝ} and the celestial transform;[slide]
s_{M} : **M** → **S**^{1,0} ⊗ **S**^{0,1}

*x* ↦ |
const |
( |
*x*^{0} + *x*^{3} |
*x*^{1} + *ix*^{2} |
) |

*x*^{1} − *ix*^{2} |
*x*^{0} − *x*^{3} |

Recall that

**S**^{1,0} =
Γ_{hol}(O(1,0)),

**S**^{0,1} =
Γ_{antihol}(O(0,1)).

s_{M} : **M** →
Γ(L^{ℝ}),
where

L^{ℝ} ⊂
L^{ℂ} = O(1,1).

Note: s_{M}(null 4-vector ≠ 0) vanishes at exactly
one point of S.

We can prove that (S**M**, L^{ℝ}, s_{M})
is a conformal reference frame for **M**.

standard c.r.f. for the Minkowski space;
L^{ℝ}*X* and
explicit definition of *ϑ*;[slide]
*T*_{v} S*X* |

⌠ ↓ |

*T* S*X* |

│ dx ↡ |
*ϑ*
↘ |

*N* S*X* |
─→ |
L^{ℝ}*X* |

↓ 0 |
s |

*N* S*X*
= *T*(S*X*) / *T*_{v} S*X*
= x^{∗}(*T* *X*).

*N* S_{x}
= S_{x} × *T*_{x} X.

s_{Tx X} :
*T*_{x} X →
Γ(L^{ℝ}_{x}).

s_{Tx X} :
*N* S_{x} →
L^{ℝ}_{x} .

L^{ℝ}*X* :=

∐
*x* ∈ *X*

L^{ℝ}_{x}
differential geometry – differentiating the sky image;[slide]
*T*_{v} Ω |
─→ |
d*j*(*T*_{v} Ω) |

⌠ ↓ |
d*j* |
⌠ ↓ |

*T* Ω |
─→ |
*j*^{∗}(*T* *M*) |
→ |
*T* *M* |

│ dx ↡ |
*p*
↘ |
│ ↡ |

*N* Ω |
─→ |
*N*_{M} Ω |

↓ 0 |
d.sky.im. |
↓ 0 |

The

*differential* d*j*
is contrasted to the

*pushforward*
*j*_{∗} :
*T* Ω → *T* *M* .

Theorem: ker *p*
= ker *ϑ*|_{Ω} .

Proof: from ker *ϑ*|_{Ω}
= ker d*j* ⊕ *T*_{v} Ω ,
that follows from dim ker d*j* = *d*
and ker d*j* ∩ *T*_{v} Ω
= 0.

differential geometry – the flow of time;[slide]
*a* : L^{ℝ}*X*
→ *N*_{M} Ω
solves the equation
*a*∘*ϑ* = *p*.

d_{AA′} sky.im.
= *a*(*H*_{A}⋅*H̅*_{A′})

where

*H* and *H̅*
are complex conjugate bases in
**S**^{1,0}_{x} and
**S**^{0,1}_{x} respectivily.
Follows from the definition of *ϑ*.