Quasinormal modes of ads black holes

Quasinormal modes of AdS black holesClaude Warnick University of Warwick Cambridge, March 2014 Based on 1306.5760... Want to define quasinormal modes for general stationary AdS blackhole

Quasinormal modes of AdS black holes

Claude Warnick

University of Warwick

Cambridge, March 2014

Based on 1306.5760

Want to define quasinormal modes for general stationary AdS blackhole 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 onSchwarzschild-AdS, but full theorem is much more general

What are quasinormal modes?

Quasinormal modes are characteristic oscillations of linear fields onblack hole backgrounds

They are time harmonic solutions of the field equations

The Schwarzschild-AdS Spacetime

I I

The Schwarzschild-AdS Spacetime

H +

I I

The regular slicing

I I

The regular slicing

I I

 dtdr +

1 + 2M

r + r2 l2



1 + r2 l2

 2 dr2+ r2dΩ2

Quasinormal modes as eigenvalues

Consider the conformally coupled Klein-Gordon equation:

gψ −l22ψ = 0ψ|t=0= ψ, ∂tψ|t=0= ψ0,

rψ → 0, as r → ∞

Solution exists for all t ≥ 0

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.]

Wavepackets at the horizon

Wavepackets at the horizon

Rate of decay determined by how sharply localised the wave packet isMeasure localisation using Sobolev norms

For a function u(x) defined on Rn, with Fourier transform ˜u(ξ), define

||u(x)||2Hk =

Z

dnξ(1 + |ξ|2)k|˜u(ξ)|2Can extend definition to curved manifolds

The larger k is, the smoother a function with ||u(x)||Hk < ∞ is.Crudely, an outgoing wavepacket localised at the horizon, with

||ψ(x, t)||Hk < ∞ will decay like

|ψ(x, t)| ∼ e−κ(k−12 )

where κ is the surface gravity

What are quasinormal modes? Version 2

What are quasinormal modes? Version 2

[c.f U (t) = eit∆ for Scr¨odinger equation on Rn]

The spectrum of (Dk(A), A)

=(s)

<(s)

k = 1

The spectrum of (Dk(A), A)

=(s)

<(s)

k = 2

The spectrum of (Dk(A), A)

=(s)

<(s)

k = 3

The main theorem

Theorem (Discreteness of QNF [CMW, 2013])

The spectrum of A in the region <(s) > 12 − k

κ consists solely ofisolated eigenvalues of finite multiplicity The eigenfunctions u are smooth

at the horizon and if ψ = estu, we have

gψ − 2

l2ψ = 0

Related work: [Horowitz–Hubeny; Vasy; Bachelot; Gannot; Melrose–S´ a Baretto–Vasy; Dyatlov; S´ a Baretto–Zworski; Bony–H¨ afner; ]

The main theorem

Corollary

Let ψ(x, t) be a smooth solution of the Klein-Gordon equation on anasymptotically AdS black hole Then the Laplace transform

ˆψ(x, s) =

0

e−stψ(x, t)dtextends meromorphically to C, and the location of its poles belong to acountable set ΛQN F which is independent of ψ

The main theorem

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, QNMare honest eigenfunctions of an operator on a Hilbert space

Do not need to restrict to perturbations supported away from thehorizon

Can show that ‘ingoing’ QNF are a subset of these QNF, and theytypically agree

1 Introduction

2 Example: Schwarzschild-AdS

3 Completeness of the quasinormal mode spectrum

4 Conclusions

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

Completeness of the spectrum

Since QNF spectrum is countable, is it true by analogy with Fourierseries that if ψ is a solution of KGE, then

Incompleteness for AdS Schwarzschild

r = 0

Incompleteness for AdS Schwarzschild

r = 0

Incompleteness for AdS Schwarzschild

r = 0

r = 0

Incompleteness for AdS Schwarzschild

r = 0

r = 0

Incompleteness for AdS Schwarzschild

r = 0

r = 0

Incompleteness for AdS Schwarzschild

r = 0

Incompleteness for AdS Schwarzschild

r = 0

ψ ∼ 0 as t → ∞, but ψ 6≡ 0.

QNM should be thought of as eigenvalues of the infinitesimal

generator of the solution operator on Hk× Hk−1 for a regular slicingThe QNF are a discrete, countable set of points in the complex planeThe QNM do not form a complete basis for Hk× Hk−1