bin wang fudan university shanghai, china perturbations around black holes
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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.
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
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
The perturbation equations
For axial perturbation:
For polar perturbation:
The perturbation equations
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)
QNM in time-dependent background
M with t, ωi
The decay of the
oscillation becomes
slower
QNM in time-dependent background
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
Perturbations in the dS spacetimes
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
Perturbations in the dS spacetimes
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
Perturbations in the dS spacetimes
For the dS case
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
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
BAAB XXXXP ')',(
0|)'(),(|0)',( XXconstXXG
0)',()( 22 XXGX
Perturbations in the dS spacetimes
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
Perturbations in the dS spacetimes
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
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)
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
QNM-black hole phase transition
Can QNMs reflect this phase transition?
Martinez etal, PRD(04)
QNM-black hole phase transition
Perturbation equation
MTZ TBH
Above critical value Below critical value
Koutsoumbas et al(06), Shen&Wang(07)
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 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!
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