Turbulent Instability of Anti-de Sitter Space

Transkrypt

1 Turbulent Instability of Anti-de Sitter Space Andrzej Rostworowski Uniwersytet Jagielloński joint work with Piotr Bizoń Phys.Rev.Lett (2011) (arxiv: ) now a part of joint project with Joanna Ja lmużna, Patryk Mach and Maciej Maliborski Leiden, 11th October 2012 Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

2 Maximally symmetric solutions of vacuum Einstein s equations and their stability R αβ 1 2 R g αβ + Λ g αβ = 0. Λ = 0: Minkowski (trivial, yet most important) asymptotically stable (Christodoulou&Klainerman 1993), Λ > 0: de Sitter (important in cosmology - Nobel Prize 2011) asymptotically stable (Friedrich 1986), Λ < 0: anti- de Sitter (most popular on arxiv due to AdS/CFT) Stability of AdS seems unexplored Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

3 Anti-de Sitter spacetime in d + 1 dimensions Geometrically, AdS d+1 is wrapped around hiperboloid ( X 0) 2 d ( + X k ) 2 ( X d+1 ) 2 = l 2 k=1 embedded in flat, SO(d, 2) invaraiant space Parametrization X 0 = l ds 2 = ( dx 0) 2 d ( + dx k ) 2 ( dx d+1 ) 2 k=1 1 + (tan x) 2 cos(t), X d+1 = l 1 + (tan x) 2 sin(t) induces X k = l tan x n k, ds 2 = 0 x < π/2, d ( n k ) 2 = 1, k=1 l 2 [ ] (cos x) 2 dt 2 + dx 2 + (sin x) 2 dω 2 S d 1 Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

4 Anti-de Sitter spacetime in d + 1 dimensions Induced metric ds 2 = l 2 [ ] (cos x) 2 dt 2 + dx 2 + (sin x) 2 dω 2 S, d 1 < t < +, 0 x < π/2. is the maximally symmetric solution of the vacuum Einstein equations R αβ 1 2 g αβr + Λg αβ = 0 with negative cosmological constant Λ < 0: Λ = d(d 1)/(2l 2 ) Conformal infinity x = π/2 is the timelike hypersurface I = R S d 1 with the boundary metric ds 2 I = dt2 + dω 2 S d 1 Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

5 Is AdS stable? A solution (of a dynamical system) is said to be stable if small perturbations of it at t = 0 remain small for all later times Minkowski (Λ = 0) and de Sitter spacetimes (Λ > 0) are known to be stable (actually asymptotically stable) (Christodoulou&Klainerman 1993 and Friedrich 1986). The key difference between these solutions and AdS: the main mechanism of stability - dissipation of energy (dispersion in Minkowski, expansion in de Sitter) - is absent in AdS because AdS is effectively bounded (for no flux boundary conditions at I it acts as a perfect cavity) Note that by positive energy theorems both Minkowski and AdS are the unique ground states among asymptotically flat/ads spacetimes Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

6 Model To deal with the problem of the stability of AdS we start with spherical symmetry (effectually dimensional problem) Spherically symmetric vacuum solutions are static (Birkhoff s theorem) we need matter to generate dynamics Simple matter model: massless scalar field φ in d+1 dimensions ( G αβ + Λ g αβ = 8πG α φ β φ 1 ) 2 g αβ µ φ µ φ, Λ = d(d 1)/(2l 2 ), g αβ α β φ = 0 In the corresponding asymptotically flat (Λ = 0) model Christodoulou proved dispersion for small data and collapse to a black hole for large data and Choptuik discovered critical phenomena at the threshold for black hole formation Remark: For even d 4 there is a way to bypass Birkhoff s theorem (cohomogeneity-two Bianchi IX ansatz, Bizoń, Chmaj, Schmidt 2005) Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

7 Convenient parametrization of asymptotically AdS spacetimes ds 2 = l 2 [ ] (cos x) 2 Ae 2δ dt 2 + A 1 dx 2 + (sin x) 2 dω 2 S, d 1 where A and δ are functions of (t, x). Auxiliary variables Φ = φ and Π = A 1 e δ φ ( = x, = t ) Field equations (using units where 8πG = d 1) A d (sin x)2 = (1 A) (cos x) (sin x) A ( Φ 2 + Π 2), (cos x) (sin x) δ = (cos x) (sin x) ( Φ 2 + Π 2), ( Φ = Ae Π) δ 1 [, Π = (tan x) d 1 (tan x) d 1 Ae Φ] δ. AdS space: φ 0, A 1, δ 0; now we want to perturb AdS solving the initial-boundary value problem for this system starting with some small, smooth initial data (φ, φ) t=0 Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

8 Boundary conditions We assume that initial data (φ, φ) t=0 are smooth Smoothness at the center implies that near x = 0 (Λ irrelevant) φ(t, x) = f 0 (t) + O(x 2 ), δ(t, x) = O(x 2 ), A(t, x) = 1 + O(x 2 ) Smoothness at spatial infinity and conservation of the total mass M imply that near x = π/2 (using z = π/2 x) ( φ(t, x) = f (t) z d + O z d+2) (, δ(t, x) = δ (t) + O z 2d), A(t, x) = 1 (M/l d 2 )z d + O (z d+2) Remark: There is no freedom in prescribing boundary data Local well-posedness (Friedrich 1995, Holzegel&Smulevici 2011) mass function and asymptotic mass: m(t, x) = (1 A(t, ρ)) (l tan x) d 2 ( 1 + tan 2 x ) π/2 ( AΦ 2 + AΠ 2) (tan x) d 1 dx M = lim m(t, x) = ld 2 x π/2 Andrzej Rostworowski (UJ) Turbulent Instability of0anti-de Sitter Space Leiden, 11th October / 22

9 Reminder: asymptotically flat (Λ = 0) self-gravitating scalar field Christodoulou ( ): dispersion for small data and collapse to a black hole for large data (proof of the weak cosmic censorship) Consider a family of initial data Φ(p) which interpolates between dispersion and collapse (Choptuik 1993) There exists a critical value of the parameter p such that p < p dispersion p > p black hole Universal behavior in the near-critical region p p 1 m BH (p p) γ with universal exponent γ discretely self-similar attractor with universal period Critical solution (p = p ) is a non-generic naked singularity Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

10 Movie from now d = 3 qualitatively the same behaviour in any d 3 d = 2 case is very special (Pretorius&Choptuik 2000) Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

11 Critical behavior [ Initial data: Φ(0, x) = 0, Π(0, x) = ε exp ( tan x ) ] 2 σ We fix σ = 1/16 and vary ε. There is a decreasing sequence of critical amplitudes ε n for which the evolution, after 0.02 making n reflections from the AdS boundary, locally asymptotes Choptuik s 0.01 solution. In each small right neighborhood of ε n x H ε m BH (ε) (ε ε n ) γ with γ It seems that BH size vs. amplitude lim n ε n = 0 Remark: The generic endstate of evolution is the Schwarzschild-AdS BH of mass M (in accord with Holzegel&Smulevici 2011) Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

12 Key evidence for instability (a) ε = 6/π * 2 3/2 ε = 6/π * 2 ε = 6/π * 2 1/2 ε = 6/π Π 2 (t,0) Ricci scalar R = 2 ( Φ 2 Π 2) /l 2 12/l 2 t Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

13 Key evidence for instability ε -2 Π 2 (ε 2 t,0) (b) ε = 6/π * 2 3/2 ε = 6/π * 2 ε = 6/π * 2 1/2 ε = 6/π ε 2 t Onset of instability at time t = O(ε 2 ) Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

14 Weakly nonlinear perturbations We seek an approximate solution starting from small initial data (φ, φ) t=0 = (εf (x), εg(x)) Perturbation series φ = εφ 1 + ε 3 φ δ = ε 2 δ 2 + ε 4 δ A = ε 2 A 2 + ε 4 A where (φ 1, φ 1 ) t=0 = (f (x), g(x)) and (φ j, φ j ) t=0 = (0, 0) for j > 1. Inserting this expansion into the field equations and collecting terms of the same order in ε, we obtain a hierarchy of linear equations which can be solved order-by-order. Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

15 First order Linearized equation (Ishibashi&Wald 2004) φ 1 + Lφ 1 = 0, L = 1 tan 2 x ( x tan 2 ) x x The operator L is essentially self-adjoint on L 2 ([0, π/2), tan 2 x dx). Eigenvalues and eigenvectors of L are (j = 0, 1,... ) ( ) ωj 2 = (3 + 2j) 2, e j (x) = N j (cos x) 3 j, 3 + j F 3/2 (sin x)2 AdS is linearly stable Linearized solution φ 1 (t, x) = a j cos(ω j t + β j ) e j (x) j=0 where amplitudes a j and phases β j are determined by the initial data. Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

16 Second order (back-reaction on the metric) so It follows that M = ε2 2 A sin2 x ( sin x cos x A 2 = sin x cos x φ φ 2) 1 ( δ 2 = sin x cos x φ φ 2) 1 A 2 (t, x) = cos3 x sin x x δ 2 (t, x) = 0 x 0 quadratic in φ 1! π/2 0 ( φ1 (t, y) 2 + φ 1(t, y) 2) tan 2 y dy ( φ 1 (t, y) 2 + φ 1(t, y) 2) sin y cos y dy ( φ 1 (t, y) 2 + φ 1(t, y) 2) tan 2 y dy + O(ε 4 ) Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22 (1)

17 Third order φ 3 + Lφ 3 = S(φ 1, A 2, δ 2 ), where S := 2(A 2 + δ 2 ) φ 1 + (Ȧ 2 + δ 2 ) φ 1 + (A 2 + δ 2 )φ 1. ( ) Let φ 3 (t, x) = j c j(t) e j (x). Projecting Eq.( ) on the basis {e j } we obtain an infinite set of decoupled forced harmonic oscillators for the generalized Fourier coefficients c j (t) := (e j, φ 3 ) c j + ω 2 j c j = S j := (e j, S) and (c j, ċ j ) t=0 = 0 If S j has a part oscilating with a resonant frequency ω j it will give rise to a secular term in c j, that is c j t sin ω j t or c j t cos ω j t. S cubic in φ 1 contains all frequencies ± ω 1 ± ω 2 ± ω 3, where ω k Ω 1 and φ 1 (t, x) = k [ω k Ω 1 ] a k cos(ω k t + β k ) e k (x), ω k - odd integers all frequencies in S potentially resonant! Not all resonances survive the projection (e j, S). Some of those, which do survive can be compensated with frequency shifts in φ 1 and are harmless for stability, but the others put stability in question! Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

18 Example 1: single-mode data φ(0, x) = ε e 0 (x) First order φ 1 (t, x) = cos(ω 0 t)e 0 (x), ω 0 = 3 (ω j = 3 + 2j) Third order φ 3 (t, x) = j=0 c j(t)e j (x), (c j, ċ j ) t=0 = 0 and c j + ω 2 j c j = b j,0 cos(ω 0 t) + b j,3 cos(ω 3 t). But b 3,3 = 0 (!) and only j = 0 is resonant. log[π 2 (t,0)] ε=0.25 ε= ε= t The j = 0 resonance can be easily removed by the two-scale method (slow-time phase modulation) which gives φ 1 = cos((ω π ε2 ) t) e 0 (x). This suggests that there are non-generic initial data which may stay close to AdS solution Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

19 Example 2: two-mode data φ(0, x) = ε (e 0 (x) + e 1 (x)) First order φ 1 (t, x) = cos(ω 0 t)e 0 (x) + cos(ω 1 t)e 1 (x), ω 0 = 3, ω 1 = 5 Third order φ 3 (t, x) = j=0 c j(t)e j (x), (c j, ċ j ) t=0 = 0 and c j +ω 2 j c j = k [ω k Ω 3 ]b j,k cos(ω k t), where Ω 3 = { ±ω 0,1 ±ω 0,1 ±ω 0,1 Π 2 (t,0) Here Ω 3 = {1, 3, 5, 7, 9, 11, 13, 15}, but the resonance (b j,j 0) only if ω j {3, 5, 7}. ω 0 ω 0 + (87/π)ɛ 2, ε = 1/8 ω 1 ω 1 + (413/π)ɛ 2 shifts ε = 2 1/2 /16 ε = 1/16 remove the resonances ω j = 3, 5, but the resonance ω j = 7 cannot be removed. Thus we get the secular term c 2 (t) t sin(7t). We expect this term to be a progenitor of the onset of exponential instability t Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

20 Conjectures Our numerical and formal perturbative computations lead us to: Conjecture 1 Anti-de Sitter space is unstable against the formation of a black hole under arbitrarily small generic perturbations Proof (and a precise formulation) is left as a challenge. Note that we do not claim that all perturbed solutions end up as black holes. Conjecture 2 There are non-generic initial data which may stay close to AdS solution; Einstein-scalar-AdS equations may admit time-quasiperiodic solutions Proof: KAM theory for PDEs? Analogous conjecture for vacuum Einstein s equations by Dias, Horowitz and Santos (2011) (existence of geons) Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

21 Turbulence: transfer of energy from low to high frequencies Let Π j := ( A Π, e j ) and Φ j := ( A Φ, e j ). Then M = l π/2 0 ( AΦ 2 + AΠ 2) (tan x) 2 dx = E j (t), where E j := Π 2 j + ω 2 j Φ 2 j can be interpreted as the j-mode energy. j= ε = 3 t=0 t=232*π/2 t=464*π/2 t=696*π/2 t=920*π/2 fit E j = C / j ε = 3 t=0 t=232*π/2 t=464*π/2 t=696*π/2 t=920*π/2 fit E j = C / j E j 10-7 E j j j Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22

22 Power-law scaling E j ingoing t=0 ingoing t=t H /2 ingoing t=t H centered t=0 centered t=t H /2 centered t=t H fit E j = C / j E j ingoing t=0 ingoing t=t H /2 ingoing t=t H centered t=0 centered t=t H /2 centered t=t H fit E j = C / j j j E j ingoing t=0 ingoing t=t H /2 ingoing t=t H centered t=0 centered t=t H /2 centered t=t H fit E j = C / j E j BCS 4+1 ingoing t=0 ingoing t=t H /2 ingoing t=t H centered t=0 centered t=t H /2 centered t=t H fit E j = C / j Andrzej Rostworowski (UJ) j Turbulent Instability of Anti-de Sitter Space Leiden, 11th October j / 22

23 Final remarks Weakly turbulent behavior seems to be common for (non-integrable) nonlinear wave equations on bounded domains (e.g. NLS on torus, Colliander,Keel, Staffilani,Takaoka,Tao 2008, Carles,Faou 2010) and our work shows that Einstein s equations are not an exception. For Einstein s equations the transfer of energy to high frequencies cannot proceed forever because concentration of energy on smaller and smaller scales inevitably leads to the formation of a black hole. The role of negative cosmological constant is purely kinematical, that is the only role of Λ is to confine the evolution in an effectively bounded domain. Similar turbulent dynamics has been observed for small perturbations of Minkowski in a box (Maliborski 2012) Generalizations: different matter models (Buchel,Lehner&Liebling 2012), relaxing symmetry (Dias,Horowitz&Santos 2011), multi-scale analysis, (in)stability of AdS black holes (Holzegel&Smulevici 2011, Horowitz 2012), instability of AdS 2+1 (work in progress by Ja lmużna) Andrzej Rostworowski (UJ) Turbulent Instability of Anti-de Sitter Space Leiden, 11th October / 22