Quasinormal modes of ads black holes

Quasinormal modes of AdS black holes Claude Warnick University of Warwick Cambridge, March 2014 Based on 1306.5760 Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 / 18 Goals Want to define quasinormal modes for general stationary AdS black hole spacetimes Avoid symmetry assumptions (in particular no separability) Avoid analyticity assumptions Want to understand the completeness (or otherwise) of the quasinormal mode spectrum To what extent is a perturbation captured by its QNM spectrum I will restrict attention to the well understood case of a scalar field on Schwarzschild-AdS, but full theorem is much more general Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 / 18 What are quasinormal modes? Quasinormal modes are characteristic oscillations of linear fields on black hole backgrounds They are time harmonic solutions of the field equations ψ ∝ est , They both oscillate and decay: the quasinormal frequencies, s, are complex Play a role in late time behaviour of the scalar field Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 / 18 The Schwarzschild-AdS Spacetime ι+ ι+ r = T H+ I I H− ι− Warnick (Univ Warwick) r = Quasinormal modes ι− Cambridge, March ‘14 / 18 The Schwarzschild-AdS Spacetime ι+ ι+ r = T H+ I I H− r = ι− ds Warnick (Univ Warwick) =− 1− 2M r + r2 l2 dτ + ι− dr 2 − 2M + r2 r l Quasinormal modes 2 + r dΩ Cambridge, March ‘14 / 18 The regular slicing ι+ ι+ r = T H+ I I H− ι− Warnick (Univ Warwick) r = Quasinormal modes ι− Cambridge, March ‘14 / 18 The regular slicing ι+ ι+ r = T H+ I I H− r = ι− ds =− 1− Warnick (Univ Warwick) 2M r + r2 l2 dt + 4M r 1+ r2 l2 ι− dtdr + Quasinormal modes + 2M + r2 r l 2 + r2 2 dr + r dΩ l Cambridge, March ‘14 / 18 Quasinormal modes as eigenvalues Consider the conformally coupled Klein-Gordon equation: gψ ψ|t=0 = ψ, − ψ l2 =0 ∂t ψ|t=0 = ψ , rψ → 0, as r → ∞ Solution exists for all t ≥ Want to understand late time behaviour of solutions to this equation In particular, we are interested in features characteristic of the spacetime (not of particular choices of initial data) Regularity at horizon is an important factor [Horowitz–Hubeny; Bizon et al.] Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 / 18 Wavepackets at the horizon Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 / 18 Wavepackets at the horizon Rate of decay determined by how sharply localised the wave packet is Measure localisation using Sobolev norms For a function u(x) defined on Rn , with Fourier transform u ˜(ξ), define ||u(x)||2H k = dn ξ(1 + |ξ|2 )k |˜ u(ξ)|2 Can extend definition to curved manifolds The larger k is, the smoother a function with ||u(x)||H k < ∞ is Crudely, an outgoing wavepacket localised at the horizon, with ||ψ(x, t)||H k < ∞ will decay like |ψ(x, t)| ∼ e−κ(k− ) where κ is the surface gravity Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 / 18 The main theorem Theorem (Discreteness of QNF [CMW, 2013]) The spectrum of A in the region (s) > 12 − k κ consists solely of isolated eigenvalues of finite multiplicity The eigenfunctions u are smooth at the horizon and if ψ = est u, we have gψ − ψ = l2 Related work: [Horowitz–Hubeny; Vasy; Bachelot; Gannot; Melrose–S´ a Baretto–Vasy; Dyatlov; S´ a Baretto–Zworski; Bony–H¨ afner; ] Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 11 / 18 The main theorem Corollary Let ψ(x, t) be a smooth solution of the Klein-Gordon equation on an asymptotically AdS black hole Then the Laplace transform ∞ ˆ s) = ψ(x, e−st ψ(x, t)dt extends meromorphically to C, and the location of its poles belong to a countable set ΛQN F which is independent of ψ Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 12 / 18 The main theorem Corollary Suppose there exist QNM with | (sn )| → ∞ as n → ∞ and such that for some C −C ( (sn ))− α < (sn ) ≤ Then for any such that > 0, there exists a solution ψ with initial data in D1 (A) ||ψ||H (Σt ) + ||T ψ||L2 (Σt ) ≥ Warnick (Univ Warwick) tα+ Quasinormal modes , as t → ∞ Cambridge, March ‘14 13 / 18 The main theorem No separability of the equations is assumed Regularity as a boundary condition is very natural Can extend to any other of the usual linear fields (Dirac, Maxwell, etc.) Can extend to arbitrary locally stationary black holes Unlike the usual definition using ‘ingoing’ boundary conditions, QNM are honest eigenfunctions of an operator on a Hilbert space Do not need to restrict to perturbations supported away from the horizon Can show that ‘ingoing’ QNF are a subset of these QNF, and they typically agree Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 14 / 18 Introduction Example: Schwarzschild-AdS Completeness of the quasinormal mode spectrum Conclusions Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 15 / 18 Completeness of the spectrum Since QNF spectrum is countable, is it true by analogy with Fourier series that if ψ is a solution of KGE, then ∞ s t e i ui (x)? ψ(x, t) = i=0 Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 16 / 18 Completeness of the spectrum Since QNF spectrum is countable, is it true by analogy with Fourier series that if ψ is a solution of KGE, then ∞ s t e i ui (x)? ψ(x, t) = i=0 ∞ si t i=0 e ui (x) In fact, can arrange that converges, ∞ s t e i ui (x) ψ(x, t) ∼ as t → ∞ i=0 but nevertheless ∞ s t e i ui (x) ψ(x, t) = i=0 for any finite t Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 16 / 18 Incompleteness for AdS Schwarzschild ι+ ι+ r = I I Σ ι− Warnick (Univ Warwick) r = Quasinormal modes ι− Cambridge, March ‘14 17 / 18 Incompleteness for AdS Schwarzschild ι+ ι+ r = I I Σ ι− Warnick (Univ Warwick) r = Quasinormal modes ι− Cambridge, March ‘14 17 / 18 Incompleteness for AdS Schwarzschild ι+ ι+ r = I I Σ Σ ι− Warnick (Univ Warwick) r = Quasinormal modes ι− Cambridge, March ‘14 17 / 18 Incompleteness for AdS Schwarzschild ι+ ι+ r = I I Σ Σ ι− Warnick (Univ Warwick) r = Quasinormal modes ι− Cambridge, March ‘14 17 / 18 Incompleteness for AdS Schwarzschild ι+ ι+ r = Σ I ι− Warnick (Univ Warwick) Σ r = Quasinormal modes I ι− Cambridge, March ‘14 17 / 18 Incompleteness for AdS Schwarzschild ι+ I ι− Warnick (Univ Warwick) ι+ r = Σ r = Quasinormal modes I ι− Cambridge, March ‘14 17 / 18 Incompleteness for AdS Schwarzschild ι+ ι+ r = I ι− r = ψ ∼ as t → ∞, Warnick (Univ Warwick) I Σ ι− but ψ ≡ Quasinormal modes Cambridge, March ‘14 17 / 18 Conclusions QNM should be thought of as eigenvalues of the infinitesimal generator of the solution operator on H k × H k−1 for a regular slicing The QNF are a discrete, countable set of points in the complex plane The QNM not form a complete basis for H k × H k−1 Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 18 / 18 ... general Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 / 18 What are quasinormal modes? Quasinormal modes are characteristic oscillations of linear fields on black hole backgrounds They... Warwick) Quasinormal modes Cambridge, March ‘14 14 / 18 Introduction Example: Schwarzschild -AdS Completeness of the quasinormal mode spectrum Conclusions Warnick (Univ Warwick) Quasinormal modes. .. (Univ Warwick) Quasinormal modes Cambridge, March ‘14 / 18 The spectrum of (Dk (A), A) (s) (s) k=1 Warnick (Univ Warwick) Quasinormal modes Cambridge, March ‘14 10 / 18 The spectrum of (Dk (A),