Bin Wang

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Perturbations around Black Holes

• Bin Wang
• Fudan University
• Shanghai, China

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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

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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.

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• 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 J1118480.) 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.

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Do black holes have a characteristic sound?

• 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.

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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.
• Whats 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.

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Wave dynamics in the asymptotically flat
space-time

• Schematic Picture of the wave evolution
• Shape of the wave front (Initial Pulse)
• Quasi-normal ringing
• Unique fingerprint to the BH existence
• Detection is expected through GW observation
• Relaxation
• K.D.Kokkotas and B.G.Schmidt, gr-qc/9909058

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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

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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

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The perturbation equations

• 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

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The perturbation equations

• For axial perturbation
• For polar perturbation

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The perturbation equations

• The perturbation is described by
• 
  Incoming wave
• transmitted
  reflected wave
• wave

r
3.3r
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Main results of QNM in asymptotically flat
spacetimes

• ?i always positive ? damped modes
• The QNMs in BH are isospectral
• (same ? 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 (?i,n 1/M), shorter for
  higher-order modes (?i,n1 gt ?i,n)
• Detection of GW emitted from a perturbed BH
  ?direct measure of the BH mass

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Main results of QNM in asymptotically flat
spacetimes
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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
• 
  

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QNM in time-dependent background

• Vaidya metric
• In this coordinate, the scalar perturbation
  equation is
• Where xr2m ln(r/2m-1)
• ln(r/2m -1)-1/(1-2m/r)

Xue, Wang, Abdalla MPLA(02) Shao, Wang, Abdalla,
PRD(05)
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QNM in time-dependent background

• M
  with t, ?i
• 
  The decay of the
• 
  oscillation becomes
• 
  slower

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QNM in time-dependent background

• M ( ) with t,
• the oscillation
• period becomes
• longer (shorter)

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Detectable by ground and space-based instruments
Schutz, CQG(96)
Needs accurate waveforms produced by GR community
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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)PRD59,06
  4025(1999)
• G.T.Horowitz and V.E.Hubeny, PRD62,024027(2000)CQ
  G17,1107(2000)
• B.Wang et al, PLB481,79(2000)PRD63,084001(2001)P
  RD63,124004(2001) PRD65,084006(2002)

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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

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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

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QNM in SAdS BHs

• In 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 added, w may have any values.
• Definition 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

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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.

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QNM in SAdS BHs - Results

• For large BH (rgtgtR)
• , r.
• Additional symmetry depend on the BH T
  (Tr/R2)
• For intermediate small BH
• do not scale with the BH T
• r 0,

?
Zhu, Wang, Abdalla, PRD(2001)
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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
• 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

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QNM in RN AdS BHs

• Besides r, R, it has another parameter Q. It
  possesses richer physics to be explored.
• In the extreme case,

Wang, Lin, Abdalla, PLB(2000) Wang, Molina,
Abdalla, PRD(2001) Wang, Lin, Molina, PRD(2004).
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QNM in RN AdS BH

• Consider the massless scalar field obeying
• Using , the radial
  function satisfies
• where

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QNM in RN AdS BH

• Solving the numerical equation

Price et al PRD(1993)
Wang, Lin, Molina, PRD(2004)
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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 , ,

For big Q, it is quicker for the QN ringing to
settle down to thermal equilibrium.
If we perturb an RNAdS BH with high charge, the
surrounding geometry will not ring as much and
as long as that of the BH with small Q.
If we perturb a RNAdS BH with high Q, the
surrounding geometry will not ring as much and
as long as that of BH with small Q
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QNM in RN AdS BH - Results

• QgtQc 0
• QgtQc changes from increasing to decreasing
• Exponential decay
• Q Qmax
• Power-law decay

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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)
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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?
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QNM in BH with nontrivial topology
Wang, Abdalla, Mann, PRD(2003)
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Quasi normal modes in AdS topological Black Holes
QNM depends on curvature coupling spacetime
topology
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Support of (A)dS/CFT from QNM

• AdS/CFT correspondence
• The 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

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QNM in 21 dimensional BTZ BH

• General Solution
• where J is the angular momentum

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QNM in 21 AdS BH

• For the AdS case

Birmingham et al PRL(2002)
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.
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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)

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21-dimensional dS spacetime
The metric of 21-dimensional dS spacetime is
The horizon is obtained from
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Perturbations in the dS spacetimes

• Scalar perturbations is described by the wave
  equation
• Adopting the separation
• The radial wave equation reads

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Perturbations in the dS spacetimes

• Using the Ansatz
• The radial wave equation can be reduced to the
  hypergeometric equation

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Perturbations in the dS spacetimes

• For the dS case

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Perturbations in the dS spacetimes

• 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

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Perturbations in the dS spacetimes

• 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

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Perturbations in the dS spacetimes

• We obtain
• where
• The two point correlator can be got analogously
  to
• hep-th/0106113
• NPB625,295(2002)

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Perturbations in the dS spacetimes

• Using the separation
• The two-point function for QNM is

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Perturbations in the dS spacetimes

• 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)

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QNM way to detect extra dimensions
Maarten et al (04)

• String theory makes the radial prediction
• Spacetime has extra dimensions
• Gravity propagates in higher dimensions.

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QNM way to detect extra dimensions

• QNM behavior
• 4D The late time signal-simple power-law tail
• Black String High frequency signal persists

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QNM way to detect extra dimensions

• Brane-world BH Read Extra Dimension
• Hawking Radiation? -LHC
• QNM? GW Observation?
• (ChenWang PLB07)
• (ShenWang PRD06)
• Black String Stability
• (Thermodynamical ?Dynamical)

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QNM-black hole phase transition

• Topological black hole with scalar hair

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QNM-black hole phase transition
Can QNMs reflect this phase transition?
Martinez etal, PRD(04)
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QNM-black hole phase transition

• Perturbation equation
• MTZ TBH

Above critical value
Below critical value
Koutsoumbas et al(06), ShenWang(07)
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QNM-black hole phase transition

• 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 Modes
• AdS Soliton perturbation equation
• NORMAL Modes

Hawking-Page transition
Surya et al PRL(01)
Question Ricci flat BH and Hawking-Page
phase Transition in GB Gravitydilaton Gravity
Shen Wang(07)
Cai, Kim, Wang(2007)
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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??

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• Thanks!