A view of spacetime near spatial infinity Juan A. Valiente Kroon, School of Mathematical Sciences, Queen Mary, University of London, United Kingdom. November 24th, 2006. 1
✆ ☎ � ✟ ✞ ✝ ✆ ☎ ✄ � ✂ ✁ � ✁ ✂ ✄ The i 0 problem There is a lack of general results about the evolution of data near spatial infinity. x 0 ρ 0 i S One of the difficulties of the analysis lies in the fact that on an initial hypersurface S , the rescaled conformal Weyl tensor behaves like: Ω C 3 ) as r d O ( r 0 In order to overcome this difficulty, one has to resolve the structure contained in the point i 0 . 2
✍ ✠ ✡ ✌ ✎ ✏ ✑ ✡ ✡ ✆ ☛ ✡ ✌ ✆ ✠ ✡ ✠ Blow-up of i 0 into the cylinder at spatial infinity a + I x 0 I τ ρ 0 i 0 I S S ρ − I The conformal factor is given by: Ω 2 f ( ) 1 ✡☞☛ 2 ) f ( ) O ( where ✠✓✒ is given in terms of initial data on S . a H. Friedrich. Gravitational fields near spacelike and null infinity . J. Geom. Phys. 24 , 83-163 (1998). 3
✘ ✖ ✗ ✕ ✕ ✗ ✘ ✘ ✖ ✔ ✕ ✗ ✔ ✖ ✕ ✘ ✘ For suitable classes of initial data ( S h ) —e.g. time symmetric data ( 0) with smooth conformal metric, ✖✚✙ time asymmetric ( 0), conformally flat data, ✛✜✙ stationary data, and ... the standard Cauchy problem can be reformulated as a regular finite initial value problem for the conformal field equations. Features: the data and equations are regular on a manifold with boundary; spacelike and null infinity have a finite representation with their structure and location known a priori. 4
✖ ✦ ✕ ✔ ✘ ✕ ✦ ✔ ✕ ✣ ✕ ✗ ✙ ✔ ✖ ✕ ✣ ✕ ✗ ✩ ✤ ✕ ✖ ✔ ✣ ✕ ✕ ✖ ✖ ✕ ✦ ✖ ✕ ✦ About the initial data: Construct maximal initial data (˜ h ˜ ) by means of the conformal Ansatz: ˜ 4 h 2 h ˜ ✣✥✤ ✖✢✙ ✖✢✙ so that the constraint equations reduce to: D 0 ✖✚✙ 1 1 7 D D 8 r ✕★✧ 8 5
✘ ✫ ✖ ✕ ✦ ✔ ✲ ✵ ✴ ✴ ✖ ✵ ✴ ✰ ✵ ✵ ✕ ✱ ✰ ✰ ✙ ✖ ✕ ✦ ✙ ✰ ✲ ✖ ✩ ✰ ✰ ✸ ✙ ✖ ✕ ✲ ✴ ✴ ✕ ✱ ✧ ✖ ✕ ✪ ✧ ✕ ✖ ✫ ✖ ✕ ✔ ✴ ✖ ✦ ✦ ✔ ✖ ✕ ✫ ✖ ✕ ✦ ✕ ✕ ✖ ✦ ✕ ✦ ✘ ✩ ✖ ✕ ✪ ✣ ✕ ✕ ✕ ✰ ✙ ✱ ✧ ✕ ✰ ✪ ✖ Consider conformally flat initial data: 4 h ✖✚✙ To solve the momentum constraint write: J Q A ✦✯✮ ✖✚✙ ✖✬✫ ✖✭✫ where A A 3 n n 3 x 3 J n J n n J n 3 x ✖✬✳ ✕✶✳ 3 Q Q n Q n ( n n ) Q n 2 x 2 O (1 x ) (higher multipoles) ✦✷✮ 6
✘ ✸ ✼ ✸ ✙ ✫ ✼ ✙ ✣ ✣ ✽ ✫ ✼ ✹ ✹ ✹ ✦ ✮ ✕ ✖ ✘ ✙ The term is calculated out of a smooth complex function . If ✹✻✺ ✹✾✽ with , smooth, then the conformal factor admits the ✹✿✺ parametrisation 1 W solely a . with W ( i ) m 2 and expandible in powers of a S Dain & H Friedrich, Asymptotically flat initial data with prescribed regularity at infin- ity Comm. Math. Phys. 222 , 569 (2001) 7
✖ ✳ ✗ ✙ ✔ ✹ ✖ ❁ ✴ ✕ ✕ ❂ ❁ ✤ ✗ ✴ ✦ ✕ ✕ ✖ For later use, we define the tensor C R R D ( ) ✖❀✙ R 4 R where is the part of the second fundamental form arising from the real part of . 8
✖ ❁ ✙ ✦ ✕ ✕ ✗ ✖ ✔ ✹ ✖ ✴ ✤ ✳ ✕ ✘ ❁ ✕ ✗ ✴ ✖ ✹ ✕ ❂ For later use, we define the tensor C R R D ( ) ✖❀✙ R 4 R where is the part of the second fundamental form arising from the real part of . C R can be thought of as the magnetic part of the Weyl tensor arising from Re( ). 8-a
✔ ❆ ✔ ❄ ❄ ✔ ❄ ✩ ✘ ✕ ❅ ❅ ❄ ✙ ✕ ✫ ✫ ✔ ❄ ✫ ❆ ❄ ✙ ✔ ❅ ✔ ❄ ✔ ✔ ❄ ✘ ✙ ✙ ❄ ❃ ✔ ❄ ✔ The conformal propagation equations near spatial infinity: The unknowns are given by the components of the frame, connection, and Ricci tensor Γ ABCD Φ ABCD ) v ( c AB and the components of the Weyl spinor ( 4 ) 0 1 2 3 The evolution equations are given by: v Kv Q ( v v ) L B ( Γ ABCD ) A 0 A 9
✘ ❈ ✙ ✙ ✼ ❊ ✙ ❉ ❋ ● ❉ ✩ ❈ ✫ ✔ ✔ ✔ ✔ ❅ ❆ ✔ ❈ ✧ ❇ ✙ The matrix associated to the term in the Bianchi propagation equations is given by: A 0 2diag(1 1 1 1 1 ) – Thus, the equations degenerate at the sets where null infinity touches spatial infinity: I 0 1 – Standard methods of symmetric hyperbolic systems cannot be used to analyse the equations near I . 10
❈ ■ ✔ ❂ ✔ ❄ ✔ ✼ ✘ ❏ ✼ ❂ ✔ ❍ ✩ ❄ ✔ ✘ ❍ ❈ ■ ❏ Transport equations on I The procedure by which i 0 is replaced by I leads to an unfolding of the evolution process near spatial infinity which permits an analysis to arbitrary order and in all detail. Consistent with our choice of initial data assume that the field quantities admit the following Taylor like expansions: 1 1 ∑ ∑ p ! v ( p ) ( p ) p p v j j ( ) j ( ) j p ! p 0 p 0 11
❄ ❈ ■ ❈ ✘ ❍ ✔ ❄ ✔ ❂ ✼ ✔ ❏ ✔ ❂ ✔ ❏ ✼ ✩ ■ ✘ ❍ ❄ ✘ Transport equations on I The procedure by which i 0 is replaced by I leads to an unfolding of the evolution process near spatial infinity which permits an analysis to arbitrary order and in all detail. Consistent with our choice of initial data assume that the field quantities admit the following Taylor like expansions: 1 1 ∑ ∑ p ! v ( p ) ( p ) p p v j j ( ) j ( ) j p ! p 0 p 0 In order to determine the coefficients v ( p ) ( p ) and exploit the fact j j that the cylinder I is a total characteristic of the propagation equations: – The equations reduce to an interior system on I . 11-a
✘ ◗ ❘ ▼ ✒ ❖ ✆ ❑ ✒ ❚ ❖ ▲ ❑ ◗ ✡ ❖ ❙ ❯ P ❖ ✘ ❄ ✘ ✡ ✑ ✝ ✁ ✍ ❑ ◗ ✁ ✎ ✝ ❖ ✒ ❖ ✒ ▼ ◆ ✡ ✒ ✡ ✆ ✝ ✒ ✒ ✝ ▲ ❖ ❑ ✝ ✡ Exploiting the total characteristic one can obtain a hierarchy of interior equations for the coefficients in the expansions: p 1 ∑ v ( p ) Kv ( p ) Q ( v (0) v ( p ) ) Q ( v ( p ) v (0) ) Q ( v ( j ) v ( p j ) ) L ( j ) ( p j ) L ( p ) (0) j 1 p p ∑ B ( Γ (0) B ( Γ ( j ) A 0 (0) ( p ) A C ( p ) ( p ) ( p ) ( p j ) ( j ) ( p j ) ABCD ) ABCD ) A C j 1 j which can be solved recursively —the equations are linear and decoupled. v ( p ) ( p ) and are completely determined by the expansions of the j j initial data on S near spatial infinity. Thus, one can relate properties of the initial data with the asymptotic behaviour of the spacetime near null and spatial infinities. 12
❄ ❉ Obstructions to the smoothness of null infinity: Due to the degeneracy of the Bianchi propagation equations at the critical sets I , any hint of non-smoothness is bound to arise first in ( p ) . the coefficients 13
❱ ✤ ❈ ❉ ❳ ❳ ❄ ✤ ✘ ❱ ❄ ❲ ❄ ✤ ❲ ✙ Obstructions to the smoothness of null infinity: Due to the degeneracy of the Bianchi propagation equations at the critical sets I , any hint of non-smoothness is bound to arise first in ( p ) . the coefficients ( p ) in spherical harmonics: Decompose p l ∑ ∑ ( p ) a j ; p m ( ) 2 Y lm l j j m l l j 2 13-a
❈ ✙ ✤ ✘ ✤ ❱ ❳ ❳ ❈ ❲ ❨❩ ✤ ✤ ❲ ❱ ✔ ✧ ❳ ✙ ✔ ✩ ❄ ✩ ❄ ✩ ✘ ✔ ❄ ❈ ✙ ❉ ❋ ❳ Obstructions to the smoothness of null infinity: Due to the degeneracy of the Bianchi propagation equations at the critical sets I , any hint of non-smoothness is bound to arise first in ( p ) . the coefficients ( p ) in spherical harmonics: Decompose p l ∑ ∑ ( p ) a j ; p m ( ) 2 Y lm l j j m l l j 2 A first analysis of the equations at the level of the linearised Bianchi equations —spin 2 zero-rest-mass field— reveals that the coefficients a j ; p m ( ) 2 Y pm m p p p j develop a certain type of logarithmic singularities at 1. 13-b
✘ ✩ ❈ ✫ ✹ ✧ ❈ ✤ ✩ ❬ ✫ ❈ ❈ ✩ ❬ ✔ ✤ ✫ ❈ ✔ ✫ ✧ ✙ ✤ ✤ ❳ ❳ ❈ ✙ ✧ ❈ ❬ ✫ ❬ ❈ More precisely, j ln(1 ) p 2 j (1 ) p 2 a j ; p m ( ) A p (1 ) p j ln(1 ) p j (1 ) p 2 2 B p (1 ) (polynom in ) for p 2 3 . – A p and B p depend on Re( ) only. 14
✘ ❭ ❈ ❬ ✤ ✫ ❈ ❈ ✙ ✔ ✔ ✩ ✩ ✩ ✹ ✘ ✴ ❬ ✙ ❭ ✔ ✩ ✩ ✩ ✔ ✔ ❪ ✙ ✖ ✕ ✴ ❪ ❪ ✫ ✫ ❈ ❳ ❬ ✙ ✤ ✫ ✧ ❳ ❈ ❬ ✤ ✫ ✧ ❈ ❈ ✧ ❈ ✤ More precisely, j ln(1 ) p 2 j (1 ) p 2 a j ; p m ( ) A p (1 ) p j ln(1 ) p j (1 ) p 2 2 B p (1 ) (polynom in ) for p 2 3 . – A p and B p depend on Re( ) only. These singularities can be precluded by imposing a certain regularity condition at the initial hypersurface: 1 C R ( D D )( i ) 0 p for p 0 5, where denotes the symmetric tracefree part. 14-a
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