Bin Wang Fudan University Shanghai, China Perturbations around Black Holes.

Bin Wang

Fudan University

Shanghai, China

Perturbations around Black Holes

Outline

Perturbations in Asymptotically flat spacetimes Perturbations in AdS spacetimes

Perturbation behaviors in SAdS, RNAdS etc. BH backgrounds

Testing ground of AdS/CFT, dS/CFT correspondence

QNMs and black hole phase transition Detect extra dimension from the QNMs Conclusions and Outlook

Searching for black holes

Study X-ray binary systems. These systems consist of a visible star in close orbit around an invisible companion star which may be a neutron star or black hole. The companion star pulls gas away from the visible

star.

As this gas forms a flattened disk, it swirls toward the companion. Friction caused by collisions between the particles in the gas heats them to extreme temperatures and they produce X-rays that flicker or vary in intensity within a second.

Many bright X-ray binary sources have been discovered in our galaxy and nearby galaxies. In about ten of these systems, the rapid orbital velocity of the visible star indicates that the unseen companion is a black hole. (The figure at left is an X-ray image of the black hole candidate XTE J1118+480.) The X-rays in these objects are produced by particles very close to the event horizon. In less than a second after they give off their X-rays, they disappear beyond the event horizon.

http://chandra.harvard.edu/resources/glossaryG.html#galaxy

Do black holes have a characteristic “sound”?

Yes.Yes.

During a certain time interval the evolution of initial perturbation is dominated by damped single-frequency oscillation.

Relate to black hole parameters, not on initial perturbation.

IR i

Quasinormal Modes

Why it is called QNM? They are not truly stationary, damped quite rapidly They seem to appear only over a limited time interval, NMs extending

from arbitrary early to late time.

What’s the difference between QNM of BHs and QNM of stars? Stars: fluid making up star carry oscillations, Perturbations exist in

metric and matter quantities over all space of star BH: No matter could sustain such oscillation. Oscillations essentially

involve the spacetime metric outside the horizon.

Wave dynamics in the asymptotically flat space-time

Schematic Picture of the wave evolution:Shape of the wave front (Initial Pulse)Quasi-normal ringingUnique fingerprint to the BH existence

Detection is expected through GW observation

RelaxationK.D.Kokkotas and B.G.Schmidt, gr-qc/9909058

The perturbation equations Introducing small perturbation

In vacuum, the perturbed field equations simply reduce to

These equations are in linear in h

For the spherically symmetric background, the perturbation is forced to be considered with complete angular dependence

The perturbation equations

Different parts of h transform differently under rotations

“S” transform like scalars, represented by scalar spherical harmonics

Vectors and tensors can be constructed from scalar functions


There are two classes of tensor spherical harmonics (polar and axial). The differences are their parity under space inversion .

Function acquires a factor refering to polar perturbation, and axial with a factor

The radial component of perturbation outside the BH satisfy


For axial perturbation:

For polar perturbation:


The perturbation is described by

Incoming wave

transmitted reflected wave wave

Main results of QNM in asymptotically flat spacetimes

ωi always positive damped modes The QNMs in BH are isospectral (same ω for different perturbations eg axial or polar)

This is due to the uniqueness in which BH react to a perturbation

(Not true for relativistic stars)

Damping time ~ M (ωi,n ~ 1/M), shorter for higher-order modes (ωi,n+1 > ωi,n)

Detection of GW emitted from a perturbed BH direct measure of the BH mass

Main results of QNM in asymptotically flat spacetimes

Tail phenomenon of a time-dependent case

Hod PRD66,024001(2002)

V(x,t) is a time-dependent effective curvatue potential which

determines the scattering of the wave by background geometry

QNM in time-dependent background

Vaidya metric

In this coordinate, the scalar perturbation equation is

Where x=r+2m ln(r/2m-1) […]=ln(r/2m -1)-1/(1-2m/r)

Xue, Wang, Abdalla MPLA(02)Shao, Wang, Abdalla, PRD(05)


M with t, ωi

The decay of the

oscillation becomes

slower


M ( ) with t,

the oscillation

period becomes

longer (shorter)

Detectable by ground and space-based instruments

Needs accurate waveforms produced by GR community

Schutz, CQG(96)

Quasi-normal modes in AdS space-time

AdS/CFT correspondence:A large static BH in AdS spacetime corresponds to an

(approximately) thermal state in CFT.

Perturbing the BH corresponds to perturbing this thermal state, and the decay of the perturbation describes the return to thermal equilibrium.

The quasinormal frequencies of AdS BH have direct interpretation in terms of the dual CFT

J.S.F.Chan and R.B.Mann, PRD55,7546(1997);PRD59,064025(1999)G.T.Horowitz and V.E.Hubeny, PRD62,024027(2000);CQG17,1107(2000)B.Wang et al, PLB481,79(2000);PRD63,084001(2001);PRD63,124004(2001);

PRD65,084006(2002)

QNM in Schwarzschild AdS BHs Horowitz et al PRD(99)

D-dimensional SAdS BH metric:

R is the AdS radius, is related to the BH mass

is the area of a unit d-2 sphere. The Hawking temperature is

QNM in SAdS BHs The minimally coupled scalar wave equation

If we consider modes

where Y denotes the spherical harmonics on

The wave equations reads

QNM in SAdS BHs In the absence of the BHIn the absence of the BH, r* has only a finite

range and solutions exist for only a discrete set of real w.

Once BH is addedOnce BH is added, w may have any values. Definition of QNM in AdS BHsDefinition of QNM in AdS BHs: QNMs are defined to be modes with only ingoing waves

near the horizon. Exists for only a discrete set of complex w

We want modes with behavior near the horizon

QNM in SAdS BHs It is convenient to set and work

with the ingoing Eddington coordinates.

Radial wave equation reads

We wish to find the complex values of w such that Eq. has a solution with only ingoing modes near the horizon and vanishing at infinity.

QNM in SAdS BHs - Results For large BH (r+>>R) , r+. Additional symmetry: depend on the BH T (T~r+/R^2)

For intermediate & small BH

do not scale with the BH T

r+ 0,

∝

QNM in SAdS BHs - Results

SBH has only one dimensionful parameter-T must be multiples of this T Small SAdS BH do not behave like SBHs Decay at very late time SBH: power law tail SAdS BH: exponential decay Reason:Reason: The boundary conditions at infinity are changed. Physically, the late time behavior of the field is

affected by waves bouncing off the potential at large r

QNM in RN AdS BHs

Besides r+, R, it has another parameter Q. It possesses richer physics to be explored.

In the extreme case,

QNM in RN AdS BH

Consider the massless scalar field obeying

Using , the radial function satisfies

where

QNM in RN AdS BH

Solving the numerical equation

Price et al PRD(1993)

Wang, Lin, Molina, PRD(2004)

QNM in RN AdS BH - Results With additional parameter Q, neither nor

linearly depend on r+ as found in SAdS BH. For not big Q: Q , ,

If we perturb a RNAdS BH with high Q, the

surrounding geometrywill not ring as much and as

long as that of BH with small Q

QNM in RN AdS BH - Results Q>Qc: 0

Q>Qc: changes from increasing to decreasing

Exponential decay

Q Qmax

Power-law decay

QNM in RN AdS BH - Results

Higher modes:Asymptotically flat spacetime

const., while with large With some (not clear yet) correspondence between classical

and quantum states, assuming this constant just the right one to make LQG give the correct result for the BH entropy.

Whether such kind of coincidence holds for other spacetimes? In AdS space ?

For the same value of the charge, both real and imaginary part of QN frequencies increases with the overtone number n.

Hod. PRL(98)

QNM in RN AdS BH - Results

Higher modes:For the large black hole regime the frequencies become

evenly spaced for high overtone number n. For lowly charged RNAdS black hole, choosing bigger

values of the charge, the real part in the spacing expression becomes smaller, while the imaginary part becomes bigger.

Call for further Understanding

from CFT?

QNM in BH with nontrivial topology

Wang, Abdalla, Mann, PRD(2003)

Quasi normal modes in AdS topological Black Holes

QNM depends on curvature coupling & spacetime topology

Support of (A)dS/CFT from QNM AdS/CFT correspondenceThe decay of small perturbations of a BH at

equilibrium is described by the QNMs.

For a small perturbation, the relaxation process is completely determined by the poles, in the momentum representation, of the retarded correlation function of the perturbation.

？QNMs in AdS BH Linear response theory in FTFT

QNM in 2+1 dimensional BTZ BH

General Solution

where J is the angular momentum

QNM in 2+1 AdS BH For the AdS case

Exact agreement: QNM frequencies & location of the poles of the retarded correlation function of the corresponding perturbations in the dual CFT

A Quantitative test of the AdS/CFT

correspondence.

[Birmingham et al PRL(2002)]

Perturbations in the dS spacetimes

We live in a flat world with possibly a positive cosmological constant

Supernova observation, COBE satellite

Holographic duality: dS/CFT conjecture A.Strominger, hep-th/0106113

Motivation: Quantitative test of the dS/CFT conjecture E.Abdalla, B.Wang et al, PLB (2002)

2+1-dimensional dS spacetime

22

2212

2

2

22

2

2

2

22 )

2()

4()

4( dt

r

Jdrdr

r

J

l

rMdt

r

J

l

rMds

The metric of 2+1-dimensional dS spacetime is:

The horizon is obtained from

04 2

2

2

2

r

J

l

rM


Scalar perturbations is described by the wave equation

Adopting the separation

The radial wave equation reads

0)(1 2

ggg

imti eerRrt )(),,(

Rg

Rmr

J

l

rMm

rrdr

dR

g

r

rdr

d

g rrrrrr

222

22

22

2 1])(

1[)(

1


Using the Ansatz

The radial wave equation can be reduced to the hypergeometric equation

)()1()( zFzzzR

0])1([)1(2

2

abFdz

dFzbac

dz

Fdzz


For the dS case


Investigate the quasinormal modes from the CFT side:

For a thermodynamical system the relaxation process of a small perturbation is determined by the poles, in the momentum representation, of the retarded correlation function of the perturbation


Define an invariant P(X,X’)associated to two points X and X’ in dS space

The Hadamard two-point function is defined as

Which obeys

BAAB XXXXP ')',(

0|)'(),(|0)',( XXconstXXG

0)',()( 22 XXGX


We obtain

where

The two point correlator can be got analogously to

hep-th/0106113;

NPB625,295(2002)

)2/)1(,2/3,,(Re)( PhhFPG

2211 lh

**2

2)'(''lim rr

rG

l

rrddtdtd


Using the separation:

The two-point function for QNM is

imti eerRrt )(),,(

)2

2/2/2/()

2

2/2/2/(

)2

2/2/2/()

2

2/2/2/()'(

]2

))((sinh

2

))((sinh2[

)''''exp(''

'

22

T

limh

T

limh

T

limh

T

limh

l

tilrir

l

tilrirtiimtiim

ddtdtd

mm

h


The poles of such a correlator corresponds exactly to the QNM obtained from the wave equation in the bulk.

These results provide a quantitative test of the dS/CFT correspondence

This work has been extended to four-dimensional

dS spacetimes Abdalla et al PRD(02)

QNM – way to detect extra dimensions

String theory makes the radial prediction:Spacetime has extra dimensions

Gravity propagates in higher dimensions.

Maarten et al (04)

QNM – way to detect extra dimensions QNM behavior:

4D: The late time signal-simple power-law tail

Black String: High frequency signal persists

QNM – way to detect extra dimensions

Brane-world BH – Read Extra Dimension: Hawking Radiation? -LHC QNM? –GW Observation?(Chen&Wang PLB07)(Shen&Wang PRD06)

Black String Stability (Thermodynamical =?Dynamical)

QNM-black hole phase transition

Topological black hole with scalar hair


Can QNMs reflect this phase transition?

Martinez etal, PRD(04)


Perturbation equation

MTZ TBH

Above critical value Below critical value

Koutsoumbas et al(06), Shen&Wang(07)


ADS BLACK HOLES WITH RICCI FLAT HORIZONS ON THE ADS SOLITON BACKGROUND

AdS BH with Ricci flat horizon AdS soliton

Flat AdS BH perturbation equation

DECAY ModesAdS Soliton perturbation equationNORMAL Modes

Hawking-Page transition

Surya et al PRL(01)

Shen & Wang(07)

Question: Ricci flat BH and Hawking-Page phase

Transition in GB Gravity&dilaton Gravity Cai, Kim, Wang(2007)

Conclusions and Outlook Importance of the study in order to foresee

gravitational waves accurate QNM waveforms are needed

QNM in different stationary BHs QNM in time-dependent spacetimes QNM around colliding BHs

Testing ground of Relation between AdS space and Conformal Field Theory Relation between dS space and Conformal Field Theory

Possible way to detect extra-dimensions Possible way to test BHs’ phase transition More??

Thanks!

Bin Wang Fudan University Shanghai, China Perturbations around Black Holes.

Documents

galaxy slide

polar perturbation

visible star

outlook slide

small perturbation

black hole parameters

neutron star

space of star bh