2016 2017 2019
Innocenti V. Maresin
Steklov Mathematical Institute, Moscow

# Celestial geometry of Lorentzian manifolds

An attempt on a physical spacetime theory

## Introduction: spacetime and its structure

• 4-manifold, and its additional structure (Lorentzian metric and time orientation).[slide] Now we introduce a “reference frame” and may remove the manifold.[slide]
•  Physical spacetime
↙   ↘
4-manifold
(generally, 1 + d)
Lorentzian metric;
time orientation
Physical spacetime
¦ ↘
4-manifold  ¦
¦
Lorentzian metric;
time orientation
↓  ↓  ↓
Reference frame
• The light cone and its projectivization (the sky).(not ready)
• Symmetries of Tx X:
• Lorentz transformations and the sky;[slide]
•  x ↦ const⁢ ( x0 + x3 x1 + i⁢x2 ) x1 − i⁢x2 x0 − x3
C ∈ SL(2, ℂ)
xAA′CALxLL′L′A′
that is a tensor product of two 2-dimensional reps.

Remind that SL(2, ℂ) 2:1 covers SO+(1,3).

S1,0 = Γhol(O(1,0)), S0,1 = Γantihol(O(0,1)) where O(?,?) are ℂ-line bundles.

• rise of SL(2, ℂ) and the spinors;
• “T” symmetry.[slide]
• C+ is the boundary of the chronological future cone.
S is the base of C+.

## Celestial geometry proper

• The bundle of skies:[slide]
SX :=
x ∈ X
Sx
dim SX = (1 + d) + (d − 1) = 2⁢d.
• the geodesic flow and respective foliation;[slide]
• Geodesic flow FSX:  dxv, ∇v = 0
(v ∈ Cx+ ⊂ Tx X represents an element of Sx).
[x,v] – an equiv. class in SX, maximally extended geodesic curve.

Always smooth SX2⁢d
Foliation N2⁢d − 1(d − 1)-sphere
X1 + d
(not necessarily a manifold)
• achronal directions: sky, foliation, and wave;[slide][slide]
• TvSX := ker dx
[FSX, TvSX] = ?
Linear span of [FSX, TvSX] is ker ϑ, where
ϑ : T(SX) → LX ; ϑv⋅dx .
So, ker ϑ has:
• 1 foliation (geodesic flow) dimension;
• d − 1 sky dimensions;
• d − 1 wave dimensions; all three sums to dim ker ϑ = 2⁢d − 1.
The rest in SX is the time (chronal) dimension (obtained as [TvSX, wave]).
• contact structure.[slide]
•  2⁢d     1       2⁢d − 1
( T(SX) / FSX )|wT[w]N

The direct image [ ](oriented hyperplanes specified by ϑ) is a contact structure, where N is smooth.

• Conformal reference frame:
• definition;[slide]
• Conformal reference frame (Ω, M, j) for X Such open Ω ⊂ SX that x ∈ X: Ωx ≠ ∅.

A smooth manifold Md.

Such smooth map j : Ω → M that:

(f) w1, w2 ∈ Ω:
[w1] = [w2]j(w1) = j(w2)
;
(c) ϑ|Ωj(T* M);
(d) j : Tv Ω → TM has the maximal rank (d − 1) everywhere.

• the basic example;[slide]
• 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 and the celestial transform;[slide]
• sM : MS1,0S0,1
 x ↦ const⁢ ( x0 + x3 x1 + i⁢x2 ) x1 − i⁢x2 x0 − x3
Recall that S1,0 = Γhol(O(1,0)),
S0,1 = Γantihol(O(0,1)).
sM : M → Γ(L), where LL = O(1,1).

Note: sM(null 4-vector ≠ 0) vanishes at exactly one point of S.

We can prove that (SM, L, sM) is a conformal reference frame for M.

• standard c.r.f. for the Minkowski space;
• LX and explicit definition of ϑ;[slide]
•  Tv SX ⌠↓ T SX │dx↡ ϑ ↘ N SX ─→ LℝX ↓0 s
NSX = T(SX) / TvSX = x(TX).
NSx = Sx × Tx X.
sTx X : Tx X → Γ(Lx).
sTx X : NSxLx .
LX :=
x ∈ X
Lx
• differential geometry – differentiating the sky image;[slide]
•  Tv Ω ─→ dj(Tv Ω) ⌠↓ dj ⌠↓ T Ω ─→ j∗(T M) → T M │dx↡ p ↘ │↡ N Ω ─→ NM Ω ↓0 d.sky.im. ↓0
The differential dj is contrasted to the pushforward
j : T Ω → TM .

Theorem: ker p = ker ϑ|Ω .
Proof: from ker ϑ|Ω = ker djTv Ω , that follows from dim ker dj = d and ker djTv Ω = 0.

• differential geometry – the flow of time;[slide]
• a : LXNM Ω solves the equation aϑ = p.
dAA′sky.im. = a(HAA′)
where H and are complex conjugate bases in S1,0x and S0,1x respectivily.

Follows from the definition of ϑ.

• 2⁢d-, (2⁢d − 1)-, and d-dimensional pictures. (not ready)

## Application to the real-world spacetime

• Importance of conformal boundaries:
• A conformal boundary as reference frame;[slide]
• Now M is a conformal boundary of X or some smooth piece of it.
Again, j – projection along geodesics. Ω – from (d) cond.
• The FLRW cosmology implies that some special (absolute) conformal reference frame exists.
• No more freedom of coordinates.
• Theory on X ⇔ theory on M.

## CRF-based causality

[slide]
(Conformal reference frame)-based causality

Some drawings on the blackboard will possibly help.

• Introduction.
• A singularity of the sky image.