Recent analytical and numerical studies of asymptotically AdS - PowerPoint PPT Presentation

Recent analytical and numerical studies of asymptotically AdS spacetimes in spherical symmetry Maciej Maliborski and Andrzej Rostworowski Institute of Physics, Jagiellonian University, Krak´ ow New frontiers in dynamical gravity, Cambridge, 24th March, 2014

Outline and motivation ◮ The phase space of solutions to the Einstein equations with Λ < 0 has a complicated structure. Close to the pure anti-de Sitter (AdS) space there exists a variety of coherent structures: geons [ Dias,Horowitz&Santos, 2011 ] , [ Dias,Horowitz,Marolf&Santos, 2012 ] , boson stars (standing waves) [ Buchel,Liebling&Lehner, 2013 ] and time-periodic solutions [ M&Rostworowski, 2013 ] . ◮ The construction of non-generic configurations which stay close to the AdS solution does not imply their stability. ◮ To understand the mechanism of (in)stability of asymptotically AdS solutions we study a spherically symmetric complex self-gravitating massless scalar field (the simplest model possessing standing wave solutions). ◮ We conjecture that the dispersive spectrum of linear perturbations of standing waves makes them immune to the instability. 1/19

Complex (real) self-gravitating massless scalar field � φ − 1 � 2 g αβ ∇ µ φ ∇ µ ¯ ∇ α φ ∇ β ¯ , Λ = − d ( d − 1) / (2 ℓ 2 ) , G αβ +Λ g αβ = 8 πG φ g αβ ∇ α ∇ β φ = 0 . ◮ Spherically symmetric parametrization of asymptotically AdS spacetimes ℓ 2 ds 2 = − Ae − 2 δ dt 2 + A − 1 dx 2 + sin 2 x d Ω 2 � � , cos 2 x S d − 1 where ( t, x ) ∈ R × [0 , π/ 2) ◮ Field equations with auxiliary variables: Φ = φ ′ and Π = A − 1 e δ ˙ φ A ′ = d − 2 + 2 sin 2 x (1 − A ) + Aδ ′ , δ ′ = − sin x cos x | Φ | 2 + | Π | 2 � � , sin x cos x � ′ , 1 � ′ . ˙ ˙ Ae − δ Π tan d − 1 x Ae − δ Φ � � Φ = Π = tan d − 1 x ◮ Units 8 πG = d − 1 and notation ′ = ∂ x , ˙= ∂ t 2/19

Boundary conditions ◮ We require smooth evolution and finiteness of the total mass m ( t, x ) = sin d − 2 x � � 1 − A ( t, x ) , cos d x π/ 2 � | Φ | 2 + | Π | 2 � tan d − 1 x dx . � M = x → π/ 2 m ( t, x ) = lim A 0 Then, there is no freedom in prescribing boundary data at x = π/ 2 : reflecting boundary conditions ◮ Conserved charge for the complex field � π/ 2 Π tan d − 1 x dx . φ ¯ Q = −ℑ 0 3/19

Linear perturbations of AdS ◮ Linear equation on an AdS background [ Ishibashi&Wald, 2004 ] 1 ¨ tan d − 1 x ∂ x � � φ + Lφ = 0 , L = − tan d − 1 x ∂ x , ◮ Eigenvalues and eigenvectors of L are ( j = 0 , 1 , . . . ) � j !( j + d − 1)! cos d x P ( d/ 2 − 1 ,d/ 2) ω 2 j = ( d +2 j ) 2 , e j ( x ) = 2 (cos 2 x ) , j Γ( j + d/ 2) ◮ AdS is linearly stable, linear solution � φ ( t, x ) = a j cos( ω j t + b j ) e j ( x ) , j ≥ 0 with a j , b j determined by the initial data φ (0 , x ) and ˙ φ (0 , x ) 4/19

Real scalar field — time-periodic solutions ◮ We search for solutions of the form ( | ε | ≪ 1 ) φ ( t, x ) = ε cos( ω γ t ) e γ ( x ) + O ( ε 3 ) , solution bifurcating from single eigenmode ◮ We make an ansatz for the ε -expansion ε λ φ λ ( τ, x ) , � φ = ε cos( τ ) e γ ( x ) + odd λ ≥ 3 ε λ δ λ ( τ, x ) , ε λ A λ ( τ, x ) . � � δ = 1 − A = even λ ≥ 2 even λ ≥ 2 where we rescaled time variable ε λ ω γ,λ , � τ = Ω t, Ω = ω γ + even λ ≥ 2 5/19

Time-periodic solution — perturbative construction ◮ We decompose functions φ λ , δ λ , A λ in the eigenbasis � φ λ = f λ,j ( τ ) e j ( x ) , j ≥ 0 � � δ λ = d λ,j ( τ ) ( e j ( x ) − e j (0)) , A λ = a λ,j ( τ ) e j ( x ) , j ≥ 0 j ≥ 0 with expansion coefficients being periodic functions ◮ This reduces the constraint equations to algebraic system and the wave equation to a set of forced harmonic oscillator equations � π/ 2 S λ e k ( x ) tan d − 1 x dx , ω 2 γ ∂ ττ + ω 2 � � f λ,k = k 0 with initial conditions ˙ f λ,k (0) = c λ,k , f λ,k (0) = ˜ c λ,k , ◮ We use the integration constants { c λ,k , ˜ c λ,k } and frequency expansion coefficients ω γ,λ to remove all of the resonant terms cos( ω k /ω γ ) τ or sin( ω k /ω γ ) τ 6/19

Time-periodic solution — numerical construction We make an ansatz ( τ = Ω t ) � � φ = f i,j cos((2 i + 1) τ ) e j ( x ) , 0 ≤ i<N 0 ≤ j<K � � Π= p i,j sin((2 i + 1) τ ) e j ( x ) . 0 ≤ i<N 0 ≤ j<K ◮ Solution given by the set of Π � 2 2 × K × N + 1 numbers ◮ Two equations on each of K × N collocation points Τ ◮ One extra equation for the � x k , Τ n � dominant mode condition � 0 Π � 2 f i,γ = ε . x 0 ≤ i<N 7/19

Time-periodic solution — perturbative and numerical results For d = 4 , γ = 0 1. � 10 � 10 Ε � j 5 10 15 0 � 50 � 100 log 10 � f i,j � Ε �� � 150 0 10 20 i 30 40 � � φ = f i,j cos((2 i + 1) τ ) e j ( x ) , 0 ≤ i<N 0 ≤ j<K 8/19

Time-periodic solution — non-linear stability Closed curves on the 4.7 � 10 � 13 4.7 � 10 � 13 slices of phase space – strong evidence for the non-linear stability . p 10 p 10 0 0 Sections of the phase space spanned by the set � 4.7 � 10 � 13 � 4.7 � 10 � 13 of Fourier coefficients � 4.9 � 10 � 11 4.9 � 10 � 11 � 2. � 10 � 14 2. � 10 � 14 0 0 f 6 f 10 { f j ( t ) , p k ( t ) } , 4.9 � 10 � 11 6.7 � 10 � 12 � φ = f j ( t ) e j ( x ) , 0 ≤ j<K f 6 f 7 0 0 � Π = p j ( t ) e j ( x ) . 0 ≤ j<K � 4.9 � 10 � 11 � 6.7 � 10 � 12 � 1.7 � 10 � 6 1.7 � 10 � 6 � 1.6 � 10 � 8 1.6 � 10 � 8 0 0 [Animation] f 2 f 4 The d = 4 , γ = 0 , ε = 0 . 01 case. 9/19

Complex scalar field — standing waves ◮ The standing wave ansatz φ ( t, x ) = e i Ω t f ( x ) , Ω > 0 , δ ( t, x ) = d ( x ) , A ( t, x ) = A ( x ) , with f ( x ) a real function. The field equations reduce to − Ω 2 e d 1 tan d − 1 x A e − d f ′ � ′ , � A f = tan d − 1 x A ′ = d − 2 + 2 sin 2 x (1 − A ) + Ad ′ , sin x cos x � 2 � � � Ω e d d ′ = − sin x cos x f ′ 2 + A f , ◮ Boundary conditions d ′ ( π/ 2) = 0 , f ( π/ 2) = 0 , A ( π/ 2) = 1 , f ′ (0) = 0 , A (0) = 1 , d (0) = 0 . 10/19

Standing waves — perturbative construction ◮ We look for solutions of the system of the form ( | ε | ≪ 1 ) f 1 ( x ) = e γ ( x ) ε λ f λ ( x ) , � f ( x )= e γ (0) , odd λ ≥ 1 ε λ d λ ( x ) , ε λ A λ ( x ) , � � d ( x )= 1 − A ( x ) = even λ ≥ 2 even λ ≥ 2 ε λ ω γ,λ , � Ω = ω γ + even λ ≥ 2 where e γ ( x ) is a dominant mode in the solution in the limit ε → 0 ◮ Decomposition into the eigenbasis e j ( x ) ˆ � f λ ( x )= f λ,j e j ( x ) , j � ˆ � ˆ d λ ( x )= d λ,j ( e j ( x ) − e j (0)) , A λ ( x ) = A λ,j e j ( x ) . j j ◮ System of differential equations → set of algebraic equations for the Fourier coefficients 11/19

Standing waves — numerical construction ◮ Ansatz N − 1 ˆ � f ( x )= f j e j ( x ) , j =0 N − 1 N − 1 � ˆ � ˆ d ( x )= d j ( e j ( x ) − e j (0)) , A ( x ) = 1 − A j e j ( x ) . j =0 j =0 ◮ Solution given by the set of 3 N + 1 numbers ◮ Three equations on each of N collocation points { x j ∈ (0 , π/ 2) : e N ( x j ) = 0 , j = 0 , . . . , N − 1 } . ◮ One extra equation for the value of central density f (0) = ε . ◮ Non-linear system solved with the Newton-Raphson algorithm 12/19

Standing waves — perturbative and numerical results 13 7 7 13 � � 0 10 10 10 10 6 4 0 2 � 2 � 4 � 6 10 1 8 5 6 Λ � 10 4 15 2 19 0 10 20 30 36 0 0.0 0.1 0.2 0.3 0.4 0.5 j � The Fourier coefficients ˆ Frequency Ω of a fundamental f λ,j of a standing wave versus f (0) = ε for ground state standing wave in d = 4 . d = 2 , 3 , 4 , 5 , 6 . 13/19

Standing waves — linear perturbation [ Gleiser&Watkins, 1989 ] , [ Choptuik&Hawley, 2000 ] , [ Buchel,Liebling&Lehner, 2013 ] ◮ Perturbative ansatz ( | µ | ≪ 1) φ ( t, x ) = e i Ω t � � f ( x ) + µ ψ ( t, x ) + · · · , A ( t, x ) = A ( x ) (1 + µ α ( t, x ) + · · · ) , δ ( t, x ) = d ( x ) + µ ( α ( t, x ) − β ( t, x )) + · · · , with a harmonic time dependence ψ ( t, x ) = ψ 1 ( x ) e i X t + ψ 2 ( x ) e − i X t , α ( t, x ) = α ( x ) cos X t , β ( t, x ) = β ( x ) cos X t . ◮ Set of linear algebraic-differential equations for α ( x ) , β ( x ) and ψ 1 ( x ) , ψ 2 ( x ) 14/19

Standing waves — linear perturbation ◮ Numerical approach or perturbative ε -expansion ◮ Linear problem – the condition for the χ 0 j − ( χ 0 − ω γ ) 2 = 0 , � Lψ 1 , 0 − ( χ 0 − ω γ ) 2 ψ 1 , 0 = 0 , � ω 2 ⇒ k − ( χ 0 + ω γ ) 2 = 0 . Lψ 2 , 0 − ( χ 0 + ω γ ) 2 ψ 2 , 0 = 0 , ω 2 ◮ Thus ◮ ψ 1 , 0 = e j ( x ) , ψ 2 , 0 ≡ 0 , and χ 0 = ω γ ± ω j , ◮ ψ 1 , 0 ≡ 0 , ψ 2 , 0 = e k ( x ) , and χ 0 = − ω γ ± ω k . ◮ From a form of perturbative ansatz we take χ ± ψ 1 , 0 = e ζ ( x ) , ψ 2 , 0 = 0 , 0 = ω γ ± ω ζ . 15/19