SD@Convergence Workshop Gravitational Collapse in SD Andrea Napoletano Sapienza Universit` a di Roma In collaboration with: Flavio Mercati Henrique Gomes Tim Koslowski
Isotropic Wormhole Solution � 2 � 1 − α 1 + α � 4 � ds 2 = dt 2 + � dr 2 + r 2 d Ω 2 � 4 r 1 + α 4 r 4 r ◮ No Singularity ◮ Symmetry under inversion r → α 2 16 r ◮ Two Schwarzschild exteriors glued together at the throat THE ISOTROPIC SOLUTION IS ETERNAL “A Birkhoff theorem for Shape Dynamics” H. Gomes arXiv:1305.0310
The Thin Shell Model WHAT? Dynamical creation of the isotropic solution through gravitational collapse HOW? Spherically symmetric infinitely thin shell of dust
The Thin Shell Model: Starting Point Metric tensor and conjugate momentum f µ 2 0 0 0 0 µ p ij = sin ( θ ) 1 g ij = 0 σ 0 0 2 s 0 σ sin ( θ ) 2 0 0 1 2 s sin ( θ ) − 2 0 0 Hamiltonian, diffeo and conformal constraints � P 2 1 2 f σµ 2 s − f 2 µ 3 + µ ( σ ′ ) 2 + 4 σµ 3 + 4 σµ ′ σ ′ − 4 σµσ ′′ � � µ 2 + M 2 = δ ( r − R ) 2 σµ 2 P µ f ′ − 1 2 s σ ′ = − δ ( r − R ) 2 µ f + s σ = 0
Solution to Constraints Solutions Integration Constants ◮ f = p rr µ = A ◮ A in , A out √ σ ◮ α in , α out ( σ ′ ) 2 ◮ µ 2 = g rr = 4 α √ σ + 4 σ + A 2 σ 4 spatial integration constants 2 inside the shell, 2 outside the shell
Metric EoM and LFE Metric Tensor � σ sN 2 f µ 2 N � � � � + 2 µξµ ′ + 2 µ 2 ξ ′ j sin 2 θ � δ r i δ r + ξσ ′ δ θ i δ θ j + δ φ i δ φ ˙ g ij = j + σ µ Conjugate Momentum sin θ p ij = − 5 f 2 N µ 2 + 4 σ ( N ′ σ ′ − µ 2 ξ f ′ ) + N ( − 4 σµ 2 + ( σ ′ ) 2 ) + 4 f σµ ( ξµ ′ + µξ ′ ) � � δ i r δ j ˙ r 4 σµ 3 sin θ � ( f 2 N µ 3 + N µ ( σ ′ ) 2 + 2 σ 2 ( 2 N ′ µ ′ + µ 2 ( ξ s ′ + s ξ ′ ) − 2 µ N ′′ ) + 4 σ 2 µ 2 − 2 σ ( − N σ ′ µ ′ + µ ( N ′ σ ′ + N σ ′′ ))) φ sin − 2 θ � � δ i θ δ j θ + δ i φ δ j � P 2 /µ 4 + sin θδ i r δ j N ( R ) δ ( r − R ) r � P 2 /µ 2 + M 2 2 Lapse Fixing Equation 1 6 f 2 N µ 3 + 4 σ 2 N ′ σ ′ + N σ ′′ � + N σ ′ µ ′ + N µ 3 � � � � + N µ ( σ ′ ) 2 − µ 4 σµ 2 P 2 /µ 2 3 Ns 2 − 8 N ′′ � + σ 2 � � + 8 N ′ µ ′ �� µ + N δ ( r − R ) = 0 � P 2 /µ 2 + M 2 2
Solutions to EoM Lapse function � r σ ′ dy µ 3 ( y ) � � N = c 1 + c 2 2 µ √ σ ( σ ′ ( y )) 2 r a Shift vector ξ = ˙ σ A σ ′ + 2 σ ′ N ( r ) 1 σ Metric tensor and conjugate momentum c 2 = − 2 ˙ α = 0 ˙ A
Assumptions and jump conditions Integration Constants A in A out α in α out c 1 in c 1 out Assumptions Jump Conditions ◮ Continuity of the metric tensor ◮ Diffeo constraint ◮ No mass inside the shell ◮ Hamiltonian constraint ◮ Asymptotic flatness at infinity ◮ LFE ◮ Continuity of the Lapse ◮ EoM for p ij ◮ N = 1 at infinity A in A out α out c 1 out α in = 0
Reduced Phase Space On-Shell Relation � M 2 � M 2 ρ 2 � M 2 � 2 + ρ A out 2 + A in A out − ρ 3 ( α out ) 2 − ( α out + ρ ) A in − α out − 2 ρ − α out − 2 ρ = 0 16 ρ 8 32 ρ Independent Variables ◮ ρ = σ ( R ) = g θθ ( R ) Area of the Shell ◮ A in A out Related to the Momentum P ◮ α out Related to the ADM mass UNDERDETERMINED SYSTEM The on-shell relation and the condition α out = const . define a 2-dimensional manifold - not a one dimensional curve
What is the under determination? ◮ Spherically symmetric thin shell ◮ Conformal constraint g ij p ij = 0 ◮ General result: Birkhoff Theorem Schwarzschild space-time in maximal foliation MAXIMAL FOLIATION OF SCHWARZCHILD IS NOT UNIQUE “3+1 Formalism in General Relativity” E.Gourgoulhon
GR point of view Isotropic static foliation A out = 0 � 2 � 1 − α α � 4 � � ds 2 = dt 2 + dr 2 + r 2 d Ω 2 4 r � 1 + 1 + α 4 r 4 r C foliation C ↔ A out 1 ds 2 = f ( C ( τ )) ˙ C ( τ ) 2 d τ 2 + g ( C ( τ )) ˙ dy 2 + y 2 d Ω 2 C ( τ ) d τ dr + 2 . y + C ( τ ) 2 1 + α 4 y 2 A 4-dimensional diffeomorphism connects the two solutions THERE IS NO AMBIGUITY IN GR
Why then the ambiguity in SD? ◮ SD rests on a preferred notion of simultaneity ◮ SD symmetries are 3d diffeomorphisms and 3d conformal transformations ◮ The 2 maximal foliations of Schwarzschild are not connected by a 3d conformodiffeo What fixes A out ? The rest of the universe
Twin Shell Model SD is naturally defined in a compact universe A spherically symmetric compact universe with S 3 topology is the simplest model ds 2 = d ψ 2 + sin 2 ψ d θ 2 + sin 2 θ d φ 2 � � B N S It requires at least two thin shell to not have any mass at the origin
The twin shell model: Starting point Ansatz � f � µ 2 , σ, σ sin 2 θ p ij = diag 2 s sin − 2 θ ξ i = { ξ, 0 , 0 } � � , 1 2 s , 1 g ij = diag sin θ µ Hamiltonian Constraint p ij T p T � √ g − √ g R � ij � p � 2 − 12 Λ � � g rr P 2 − 1 + 4 π � a + M 2 d θ d φ δ ( ψ − Ψ a ) a = 0 √ g 6 a ∈{ S , N } Diffeo constraint � d θ d φ ∇ j p j i − 4 π δ ψ � − 2 δ ( ψ − Ψ a ) P a = 0 i a ∈{ S , N } Conformal constraint: CMC Foliation g ij p ij − √ g � p � = 0
Solutions to EoM and OS Relation Diffeo and Hamiltonian constraints ( σ ′ ) 2 A µ 2 = f = 1 3 � p � σ + 1 1 2 + 4 σ + A 2 ( 2 σ + 1 � p � 2 − 12 Λ σ 2 σ 2 3 A � p � + 4 α ) σ � � 9 We get two on shell relations M 4 � � � � A B A S S 2 − M 2 2 � 2 16 � + 4 − T ρ S − 4 A B − A S 2 − 4 T S 4 ρ S 16 ρ S ρ S M 2 � � X + X 2 = 0 S 4 � � − 3 A S A S − A B + 2 ρ S ρ S M 4 � � � � A B A N 16 ρ N 2 − M 2 N 2 � 2 16 � + 4 − T ρ N − 4 A B − A N 2 − 4 T N ρ N 4 ρ N � M 2 � X + X 2 = 0 N 4 − 3 A N � A N − A B � + 2 ρ N ρ N � 2 � 1 ρ 2 = σ (Ψ) � 12 Λ − � p � 2 � X = A B � p � + 4 α B T = 3 9
Twin Shell DoF ◮ ρ S ρ N ◮ 2 on shell relations ◮ A S A N A B ◮ α S = 0 α N = 0 ; α B = const ◮ α S α N α B ◮ � p � V 4 matter DoF (2 for each shell) 2 scale DoF (volume and momentum) THE SYSTEM IS FULLY DETERMINED
Towards Asymptotic Flatness ◮ Take the limit A N → ∞ ◮ Set to 0 each of the coefficients of A N ◮ 3 equations that can be solved in terms of ρ n α B A B 2 possible solutions � � � � � 16 − M 2 � 16 − M 2 � 1 � 4 − N T � 4 − N T 1 � � � � 16 − M 2 ρ N = , α B = − N T + , A B = 0 2 T 2 T 8 2 � � � � 1 � � 16 − M 2 � 16 − M 2 N T + 4 N T + 4 � 1 � � � 16 − M 2 � � ρ N = , α B = N T − A B = 0 2 T 8 2 2 T Asymptotic flatness becomes a good approximation for T → 0 V ∝ 1 T ⇔ lim T → 0 V = ∞
Deriving A out = 0 Late time limit T → 0 M N M N ρ N → α B → − A B → 0 4 4 M 2 √ 2 N T → 0 − ρ N ∼ → ∞ α B ∼ − A B → 0 √ T 32 ρ N → M N α out = α B = − M N A out = A B = 0 4 4
Reduced phase space A out = 0 On Shell Relation for the Thin Shell M 2 ρ 2 M 2 � � 2 − ρ 3 ( α out ) 2 − ( α out + ρ ) A in − α out − 2 ρ = 0 8 32 ρ A out diverges when the dynamics freezes or the shell escapes
Thin Shell Symplectic Reduction Symplectic potential � drd θ d φ p ij δ g ij + 4 π P δ R θ = � R � ∞ � � dr µ dr µ − 8 π √ σ + δ A out + 4 π P δ R δ A in √ σ 0 R Symplectic form � � µ ( R ) µ ( R ) ω = δθ = − 8 π δ A in − δ A out ∧ δ R , � � σ ( R ) σ ( R ) ω = 4 π δ P ∧ δ R P and R are the conjugate dynamical variables
ADM Mass Definition 1 � � � r 2 sin θ ∇ j g rj − ¯ ¯ g ij g ij ∇ r ¯ M ADM = lim d θ d φ 16 π r →∞ � � M ADM = 1 r µ 2 − σ ′ + 1 2 lim r σ r →∞ Assumption: Asymptotic flatness lim r →∞ σ = r 2 M ADM = − α out 2 α out plays the role of the ADM Hamiltonian
Thin Shell A out = 0 : Isotropic Gauge Isotropic gauge condition & A out = 0 µ 2 r σ ( r ) = 1 1 − α out � 4 σ ( r ) = r 2 � ⇒ r 2 4 r EoM in terms of R and P : α out ( R , P ) ◮ On shell relation M 2 ρ 2 M 2 � � 2 − ρ 3 ( α out ) 2 − ( α out + ρ ) A in − α out − 2 ρ = 0 8 32 ρ ◮ Isotropic relation � 4 � α out ρ 2 = R 2 1 − 4 R ◮ Diffeo jump condition RP A in − = 0 2 The solution of this system α out = α out ( R , P )
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