Recent analytical and numerical studies

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 < 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 Gαβ +Λ gαβ = 8πG ∇α φ ∇β φ¯ − gαβ ∇µ φ∇µ φ¯ , Λ = −d(d−1)/(2 ) , g αβ ∇α ∇β φ = Spherically symmetric parametrization of asymptotically AdS spacetimes −Ae−2δ dt2 + A−1 dx2 + sin2 x dΩ2S d−1 , cos2 x where (t, x) ∈ R × [0, π/2) ds2 = Field equations with auxiliary variables: Φ = φ and Π = A−1 eδ φ˙ d − + sin2 x (1 − A) + Aδ , δ = − sin x cos x |Φ|2 + |Π|2 , sin x cos x ˙ = Ae−δ Π , Π ˙ = Φ tand−1 x Ae−δ Φ tand−1 x A = Units 8πG = d − and notation = ∂x , ˙ = ∂t 2/19 Boundary conditions We require smooth evolution and finiteness of the total mass m(t, x) = sind−2 x − A(t, x) , cosd x π/2 M = lim m(t, x) = x→π/2 A |Φ|2 + |Π|2 tand−1 x dx Then, there is no freedom in prescribing boundary data at x = π/2: reflecting boundary conditions Conserved charge for the complex field π/2 Q=− ¯ tand−1 x dx φΠ 3/19 Linear perturbations of AdS Linear equation on an AdS background [Ishibashi&Wald, 2004] φ¨ + Lφ = , L=− ∂x tand−1 x ∂x , tand−1 x Eigenvalues and eigenvectors of L are (j = 0, 1, ) ωj2 = (d+2j)2 , ej (x) = j!(j + d − 1)! (d/2−1,d/2) cosd x Pj (cos 2x) , Γ(j + d/2) AdS is linearly stable, linear solution φ(t, x) = aj cos(ωj t + bj ) ej (x) , j≥0 ˙ x) with aj , bj determined by the initial data φ(0, x) and φ(0, 4/19 Real scalar field — time-periodic solutions 1) We search for solutions of the form (|ε| φ(t, x) = ε cos(ωγ t)eγ (x) + O(ε3 ) , solution bifurcating from single eigenmode We make an ansatz for the ε-expansion ελ φλ (τ, x) , φ = ε cos(τ )eγ (x) + odd λ≥3 λ δ= ε δλ (τ, x) , even λ≥2 1−A= ελ Aλ (τ, x) even λ≥2 where we rescaled time variable τ = Ωt, ελ ωγ,λ , Ω = ωγ + even λ≥2 5/19 Time-periodic solution — perturbative construction We decompose functions φλ , δλ , Aλ in the eigenbasis φλ = fλ,j (τ )ej (x), j≥0 δλ = j≥0 dλ,j (τ ) (ej (x) − ej (0)) , Aλ = aλ,j (τ )ej (x) , 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 ωγ2 ∂τ τ + ωk2 fλ,k = Sλ ek (x) tand−1 x dx , with initial conditions fλ,k (0) = cλ,k , f˙λ,k (0) = c˜λ,k , We use the integration constants {cλ,k , c˜λ,k } and frequency expansion coefficients ωγ,λ to remove all of the resonant terms cos(ωk /ωγ )τ or sin(ωk /ωγ )τ 6/19 Time-periodic solution — numerical construction We make an ansatz (τ = Ωt) φ= fi,j cos((2i + 1)τ )ej (x) , 0≤i