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Real-time correlators in Kerr/CFT correspondence
Bin Chen
School of Physics, Peking University
“Taiwan String Group 2011”, Tunghai University December 15, 2010
. . . . . .
.. Related works
BC, Bo Ning and Zhi-bo Xu, JHEP 1002:031,2010;
BC, Chong-sun Chu, JHEP 1005:004,2010;
BC, Jiang Long, JHEP 1006:018,2010; JHEP 1008:065,2010;
BC, Jiang Long and Jia-ju Zhang, Phy.Rev. D82 104017(2010);
BC, Chiang-mei Chen and Bo Ning, arXiv:1010.1379;
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Outline
Brief review of Kerr/CFT correspondence Green’s functions in 2D CFT
Retarded Green’s functions in NHEK/CFT correspondence Hidden conformal symmetry and real-time correlators Hidden conformal symmetry and quasi-normal modes Conclusion and discussions
. . . . . .
.. Black hole
Bekenstein-Hawking entropy:
S = A 4
The entropy is proportional to the area of the horizon, rather than the volume ;
Holographic principle in quantum gravity;
One central issue: how to understand the entropy microscopically?
One of the greatest achievements in string theory: for a class of extremal charged BH, there exists microscopic counting;
Strominger and Vafa (1996)
It relies on string technology: D-branes ...;
Any other ways?
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. AdS/CFT correspondence
Quantum gravity in AdS spacetime is dual to a CFT at AdS boundary;
A concrete realization of holographic principle;
Entropy counting: a well-known 4/3 discrepancy for Sch-AdS black hole;
Madacena’s conjecture: ⇐ physics of D3-branes;
AdS3/CFT2 correspondence: ASG of AdS3;Brown and Henneaux (1986)
Another way to understand AdS/CFT;
It has led to some interesting results in the past few years:
.
.1. Warped AdS/CFT correspondence;.
.2. Kerr/CFT correspondence;. . . . . .
.. Kerr black holes
A Kerr black hole is characterized by the mass M and angular momentum J = aM . It could be described by the metric of the following form
ds2=−∆ ˆ
ρ2(dˆt−a sin2θd ˆϕ)2+sin2θ ˆ ρ2
(
(ˆr2+ a2)d ˆϕ− adˆt)2
+ρˆ2
∆dˆr2+ ˆρ2dθ2, with
∆ = ˆr2− 2M ˆr + a2, ρˆ2= ˆr2+ a2cos2θ, where we have used the unit G =~ = c = 1.
Two horizons: r±= M ±√
M2− a2;
The Hawking temperature, the angular velocity of the horizon and the entropy of the Kerr black hole are
TH =r+− r−
8πM r+, ΩH = a
2M r+, SBH = 2πM r+.
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. A holographic description?
It is asym. flat ...;
One lesson from BH in string theory: let’s focus on near-horizon region;
Universal feature of BH: the property of the horizon geometry determine the entropy, Hawking radiation et.al.;
Another lesson from string theory: start from extremal one;
The first step: consider the near-horizon geometry of an extremal Kerr black hole (NHEK)J.M. Bardeen and G.T. Horowitz (1999)
ds2= 2J Γ (
−r2dt2+dr2
r2 + dθ2+ Λ2(dϕ + rdt)2 )
,
where Γ(θ) = 1+cos2 2θ, Λ(θ) = 1+cos2 sin θ2θ.
For fixed θ, it is a warped AdS3, as a U (1) bundle on AdS2; SL(2, R)R× U(1)L isometry group;
. . . . . .
.. NHEK/CFT correspondence
M.Guica et.al. 0809.4266 Conjecture: the quantum gravity in the NHEK geometry with certain boundary conditions is dual to a 2D CFT;The correspondence was inspired by the properties of the asymptotic symmetry group of NHEK;
Under a certain set of boundary condition, the U (1)L get enhanced into a Virasoro algebra with central charge cL= 12J ;
cR= 12J with a different set of B.C.;Y.Matsuo et.al. (2009)
cRfrom AdS2 quantum gravity;A. Castro and F. Larsen 0908.1121
Open issue: no B.C. gives cL, cRsimultaneously;Y.Matsuo and T.Nishioka 1010.4549
Perfect match of the macroscopic entropy of the black hole with the microscopic (CFT) entropy computed by the Cardy formula.
This has been generalized to many other cases: Kerr BH in higher dimensions, Kerr-Newmann-AdS-dS, RN,...;
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Superradiant scattering
For a scalar incident scattering an extreme Kerr BH, if 0≤ ω ≤ mΩH where ΩH = 1/2M , then the scattering is superradiant. Zeldovich, Misner 1970s
Superradiance: the classical reflected wave is more energetic than the incident one;
Quantum mechanically, superradiant modes with ω ≤ mΩH
are spontaneously emitted by the BH, even though TH = 0;
Hawking radiation:
Γ = 1
e(ω−mΩH)/TH− 1σabs,
σabs is called greybody factor, or absorption cross section, which modified the spectrum observed at ∞ from that of a blackbody;
. . . . . .
.. Near-NHEK
The scattering brings the Kerr a little away from extremality;
Namely we have to consider near-extremal Kerr;
The near-extremal near-horizon (near-NHEK) metric ds2= 2J Γ
(
−r(r + 2α)dt2+ dr2
r(r + 2α)+ dθ2+ Λ2(dϕ + (r + α)dt)2 )
, where α = 2πTR.
.
.1. If α = 0, it reduces to NHEK;.
.2. Locally, this geometry is diffeomorphic to NHEK;.
.3. However, the coordinate transformation is singular;.
.4. The physics is different: there exists right temperature;.
.5. This is reminiscent of BTZ BH to AdS3, or warped AdS3 BH to global warped AdS3;In the superradiant scattering of near-extreme Kerr(-Newman) black hole, the absorption cross section has been shown to be in perfect match with CFT prediction;(more details later)Bredberg et.al. 0907.3477, Hartman et.al. 0908.3909, Cvetic et.al. 0908.1136, BC and Chong-sun Chu 1001.3208
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Kerr/CFT for generic non-extremal Kerr
Castro et.al. 1004.0996 Kerr/CFT should be true for a generic non-extremal Kerrblack hole, whose near horizon geometry may not be warped AdS;
Conjecture: a generic non-extremal Kerr black hole is dual to a 2D CFT with central charges cL= cR= 12J and
temperatures TL= M2/2πJ and TR=√
M4− J2/2πJ ; Consider the low frequency scalar scattering off a Kerr BH, one can find that there exists a local conformal symmetry acting on the solution space of the wave equation;
.
.1. It is not globally defined;.
.2. It is sufficient to associate a CFT to a Kerr BH;No derivation on cL,R;
Both the entropy counting and the low-frequency scattering amplitude support the conjecture;
. . . . . .
.. Motivation
NHEK is in fact a warped AdS3 spacetime with a warping factor being a function of the angular variable;
Warped AdS/CFT correspondence: Anninos et.al. (2008)
.
.1. Originate from 3D TMG;.
.2. Its discovery is free of string technology;Real-time correlators in warped AdS/CFT correspondence;BC, Bo Ning and Zhi-bo Xu, 0911.0167
.
.1. It turns out that the Minkowski prescription is applicable to compute the real-time correlators;.
.2. The real-time correlators of various fields are in good match with CFT prediction;Question: can we compute the real-time correlators in Kerr/CFT?
YES. We will show in two cases:
.
.1. The frequency is near the superradiant bound;BC and Chong-sun Chu 1001.3208, BC and Jiang Long, 1006.0157.
.2. The very low frequency region;BC and Jiang Long, 1004.5039Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Other questions on hidden conformal symmetry
Holographic descriptions of other generic non-extremal black holes?
What’s the origin of the hidden conformal symmetry?
How it acts on other kinds perturbations?
Any other applications ?
. . . . . .
.. Finite temperature AdS/CFT
BH in AdS ∼ finite temperature CFT;
Taking BH as a thermodynamical system, the thermal equil.
in BH system could be compared to thermal equil. of finite temperature CFT;
Perturbations around the BH ∼ corresponding tiny deviations from thermal equilibrium in dual field theory;
Frequencies of quasi-normal modes in BH correspond to the poles of retarded Green’s function;
We can do a little better: real-time correlators, in particular when the dual field theory is a 2D CFT;
The same picture in warped AdS/CFT correspondence;
How about in Kerr/CFT correspondence?
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Prescriptions for retarded Green’s functions
In field theory, one may use the linear response theory to compute the real-time correlators;
Real-time correlators from gravity is subtle Q: Boundary conditions at black hole horizon?
A: Purely ingoing one corresponds to retarded Green’s function;
Analytic continuation? Not clear.
A prescription first suggested by A. Son et.al.(2005);
Its modern version:Gubser et.al.(2008), H.Liu et.al.(2009) GR(ω, ⃗k) = lim
r→∞
Π(r, ω, ⃗k)|ϕR
ϕR(r, ω, ⃗k) ,
where Π is the canonical momentum conjugate to ϕ, taking r as the “time” direction. Now ϕR is the classical solution, which should be purely in-falling at the black hole horizon.
Subtlety: plug in appropriate terms proportional to the power of r to cancel the divergence;
. . . . . .
.. Prescription: continued
In practice, one can get the retarded Green’s function from the asymptotic behavior of the solution;
For example, for a scalar field with the asymptotic behaviour ϕ∼ A(ω,⃗k)r−nA+ B(ω, ⃗k)r−nB,
with nA> nB, the real-time correlator of the scalar field is given by A(ω, ⃗k)/B(ω, ⃗k), up to a constant factor
independent of ω and ⃗k which depends on the normalization of the operator.
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Green’s functions in 2D CFT
Two independent sectors: left-moving one and right-moving one, possibly with different central charges and temperatures;
Consider an operator of conformal dimensions (hL, hR), right charge qR, at temperature (TL, TR) and chemical potential ΩR. Correlators in 2D CFT are very much decided by conformal invariance:J.Cardy (1984)
G(t+, t−) = ⟨Oϕ†(t+, t−)Oϕ(0)⟩
∼ ( πTL
sinh(πTLt+))2hL( πTR
sinh(πTRt−))2hReiqRΩRt−. Left-mover: (in momentum space)
GE(ωL,E)∼ TL2hL−2eiωL,E/2TLΓ(1− 2hL) Γ(1− hL+2πTωL,E
L)Γ(1− hL−2πTωL,EL). (2.1)
. . . . . .
.. Green’s functions in 2D CFT: continued
Right-mover:
GE(ωR,E)∼ TR2hR−2e(iωR,E+qRΩR)/2TRΓ(1− 2hR) Γ(1− hR+ ωR,E2πT−iqRΩR
R )Γ(1− hR−ωR,E2πT−iqRRΩR) . (2.2) The total contribution is the product of the left-mover’s (2.1) and the right-mover’s (2.2):
GE(ωL,E, ωR,E) = GE(ωL,E)GE(ωR,E). (2.3) Cross section: σ = Im(Gret)
σ ∼ TL2hL−1TR2hR−1sinh(ωL 2TL
+ωR− qRΩR 2TR
)
Γ(hL+ i ωL 2πTL
) 2 Γ(hR+ iωR− qRΩR
2πTR )
2. (2.4)
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Superradiant scattering of scalar in near-extreme Kerr
Consider the scalar field Φ of mass µ in the near-NHEK background.
Ansatz: Φ = e−iωt+imϕR(r)S(θ), where ω and m are the quantum numbers;
The angular partS(θ) satisfies the spheroidal harmonic equation:
1 sin θ
d dθ
( sin θ d
dθS )
+ (
Λlm− (m2
4 − Jµ2) sin2θ− m2 sin2θ
) S = 0,
where Λlm is the eigenvalue, which can be computed numerically.
The radial partR satisfies the equation d
dr (
r(r + 2α) d dr
) R(r) =
(
Λlm− m2+ 2J µ2−(ω + m(r + α))2 r(r + 2α)
) R(r).
Key point: Λlm is independent of the frequency;
. . . . . .
.. Scalar retarded Green’s function
Taking into account of the ingoing boundary condition at the horizon, the radial wave function could be written in terms of hypergeometric function. At asymptotic infinity, the radial eigenfunction has the behaviour
R(r) ∼ Ar−12−β+ Br−12+β (2.5) where
β2 = 1
4+ Λlm− 2m2+ 2J µ2. A = N Γ(−2β)Γ(1 − i(m +ωα))
Γ(12 − β − im)Γ(12− β − iωα)(2α)12+β−i2(m+ωα), B = A(β→ −β)
and N is an arbitrary constant.
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Scalar retarded Green’s function
.
.1. Consider a real β > 0, then GR ∼ AB
= (2α)2βΓ(−2β) Γ(2β)
Γ(12+ β− im)Γ(12+ β− iωα) Γ(12− β − im)Γ(12− β − iωα).
.
.2. With the identification:hL= hR=1
2 + β, ωL= m, ω¯R= ω, TL= 1
2π, TR= TR, at the Matsubara frequencies, the expression on scalar retarded Green’s function agrees precisely with CFT predition up to an irrelevant normalization factor which depends only on β, qL and qR
and can be absorbed into the normalization of the operator.
. . . . . .
.. Absorption cross section
The cross section can also be read out from the retarded correlator directly
σ = Im(GR) = (2α)2β
2βπ(Γ(2β))2sinh(π(m+ω α))|Γ(1
2+β−im)Γ(1
2+β−iω α)|2. This agree, up to an irrelevant normalization factor, with CFT
prediction as it should be.
The quasi-normal modes frequencies could be read from the poles of the retarded Green’s function
¯
ωL = −i2πTL(nL+ hL)
¯
ωR = −i2πTR(nR+ hR)
with nL, nRbeing non-negative integers. The left part is not actually the quasi-normal modes since it is related to the quantum number of rotation. The right part gives the contribution.
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Remarks
For other kinds of perturbations with nonzero spin, it’s convenient to use Newman-Penrose formalism;
For near-NHEK geometry, we can derive the Teukolsky’s master equation either directly or taking the scaling limit on the ones in Kerr case;
The radial equation for the superradiant scattering could be solved in terms of hypergeometric function as well;
The real-time correlators could be read directly from the asymptotical behavior in a slightly subtle way, and are in good agreement with CFT prediction;
So are the cross sections;
NHEK itself is dual to the 2D chiral CFT with a temperature in the left-moving sector;
It is thus more natural to take NHEK as a limiting case of near-NHEK.
The scalar scattering off NHEK supports this picture;
. . . . . .
.. Kerr-Newman black hole
BC and Jiang Long, 1004.5039For the Kerr-Newman black hole with mass M , angular momentum J = aM and electric charge Q, its metric takes the following form ds2=−∆
ρ2(dt−a sin2θdϕ)2+ρ2
∆dr2+ρ2dθ2+1
ρ2sin2θ(
adt− (r2+ a2)dϕ)2
, where
∆ = (r2+ a2)− 2Mr + Q2, ρ2 = r2+ a2cos2θ.
The gauge field is A =−Qrρ2(dt− a sin2θdϕ).
There are two horizons r±= M±√
M2− a2− Q2. The entropy: S = π(r2++ a2);
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Charged scalar scattering
Let us consider the complex scalar field with mass µ and charge e scattering with the Kerr-Newman black hole;
The Klein-Gordon equation is
(∇µ+ ieAµ)(∇µ+ ieAµ)Φ− µ2Φ = 0.
The ansatz Φ = e−iωt+imϕR(r)S(θ);
The angular part is of the form 1
sin θ d dθ
( sin θ d
dθS )
+ (
Λlm− a2(ω2− µ2) sin2θ− m2 sin2θ
) S = 0.
The radial part of the wave function is of the form
∂r(∆∂rR) + VRR = 0 with
VR = −Λlm+ 2amω +H2
∆ − µ2(r2+ a2), H = ω(r2+ a2)− eQr − am.
. . . . . .
.. Low-frequency limit
In the low frequency limit,
ωM << 1, (3.1)
the ω2 term in the angular equation could be neglected.
Note that the low frequency limit (3.1) is very different from the near-extreme case, where only the frequencies near the superradiant bound were studied;
To simplify our discussion, we focus on the massless scalar.
The angular equation is just the Laplacian on the 2-sphere with the separation constants taking values Λlm= l(l + 1).
In the “Near” region,
rω << 1, (3.2)
the radial equation could be simplified even more, and more importantly, could be written in terms of SL(2, R) quadratic Casimir (for neutral scalar);
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Conformal coordinates for non-extremal BH
Let’s introduce the conformal coordinates ω+ =
√r− r+
r− r−e2πTRϕ+2nRt, ω− =
√r− r+
r− r−e2πTLϕ+2nLt, y =
√r+− r−
r− r− eπ(TL+TR)ϕ+(nL+nR)t, Define locally the vector fields
H1 = i∂+ H0 = i
(
ω+∂++1 2y∂y
)
H−1 = i(ω+2∂++ ω+y∂y− y2∂−) which obey the SL(2, R) Lie algebra: [H0, H±1] =∓iH±1; Similarly we can define another set of vector fields ( ˜H0, ˜H±1) with +↔ −;
. . . . . .
.. Casimir
The quadratic Casimir is
H2 = ˜H2 = −H02+1
2(H1H−1+ H−1H1)
= 1
4(y2∂y2− y∂y) + y2∂+∂−. The key point: the neutral scalar Laplacian is just the SL(2, R) Casimir
H˜2R(r) = H2R(r) = l(l + 1)R(r), (3.3) with the following identification:
nR= 0, nL=− 1 4M TR= r+− r−
4πa , TL= (r++ r−)− Q2/M
4πa , (3.4)
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Conjecture on Kerr-Newmann BH
Conjecture: the Kerr-Newman black hole is dual to a CFT with central charges
cL= cR= 12J at finite temperature (TL, TR) given in (3.4).
Entropy: from Cardy’s formula S = π2
3 (cLTL+ cRTR), we get the microscopic entropy
S = π(r+2 + a2)
which is in perfect agreement with the macroscopic Bekenstein-Hawking area law for the entropy of the Kerr-Newman black hole.
. . . . . .
.. Remarks
Actually the conformal coordinates used here has been discussed in the case of BTZ black hole;
The vector fields are only defined locally;
The SL(2, R)× SL(2, R) symmetry is spontaneously broken down to U (1)L× U(1)R subgroup by the periodic identification
ϕ∼ ϕ + 2π.
The relation between conformal and Boyer-Lindquist coordinates is remniscent of the relation between Minkowski and Rindler
coordinates in flat spacetime;
The identification (3.4) reflects the nature of the underlying geometry. It is universal to all kinds of perturbations;
In the Q→ 0 limit, it reduces to the one in the Kerr case.
This provides an effective way to determine (TL, TR) in CFT;
For extremal BHs, a completely new set of conformal coordinates;BC et.al. 1007.4269
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Identification of charged scalar
First law: THδS = δM− ΩδJ − ΦδQ.
We are looking for the conjugate charges δEL and δER such that δS =δEL
TL
+δER TR
. If we identify
δM = ω, δJ = m, δQ = e, ωL=(2M2− Q2)M
J ω, ωR=(2M2− Q2)M
J ω− m,
qL= qR= δQ = e, (3.5)
µL= QM2 J −Q3
2J, µR= QM2
J (3.6)
we have
δEL = ωL− qLµL, δER= ωR− qRµR. (3.7) if the charge e is nonvanishing, it couples to both the left and right chemical potential.
If both e and Q are vanishing, the above identifications reduce to the one in the Kerr case.
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Scalar retarded correlator
The radial wavefunction behaves asymptotically as R(r) ≃ Arh−1+ Br−h
where h is the conformal weight of the scalar field h = 1
2(1 +√
(2l + 1)2− 4e2Q2),
From the Minkowski prescription, the two-point retarded correlator is just
GR ∼ B A
∼ Γ(1− 2h) Γ(2h− 1)
Γ (
h + iωL2πT−qLµL
L
) Γ
(
h + iωR2πT−qRµR
R
)
Γ (
1− h + iωL2πT−qLLµL) Γ
(
1− h + iωR2πT−qRRµR)
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Photons and gravitons in Kerr black hole
Now we focus on the low-frequency limit, instead of the frequencies near the super-radiant bound;
Similar to the near-extreme case, the wave function could be solved exactly in terms of hypergeometric functions;
Due to the existence of hidden conformal symmetry, we apply the Minkowski prescription directly and compute the retarded Green’s functions;
Results: the retarded Green’s functions and the absorption cross sections are both in good agreement with the CFT prediction;
This gives strong support to Kerr/CFT correspondence for generic non-extremal Kerr BH;
. . . . . .
.. Kerr-Newmann-AdS-dS BH
BC and Jiang Long, 1006.0157 The same kind of analysis has been applied to the study of Kerr-Newman-AdS-dS;However, even though the final picture is quite similar to the ones in Kerr and Kerr-Newman case, the details are much subtler;
It turns out that one has to focus on the near-horizon region to find the hidden conformal symmetry for low-frequency scattering;
With the temperature identification from conformal
coordinate transformation, and the charge identification from 1st law of thermodynamics, we recover the macroscopic entropy from Cardy’s formula;
Furthermore, we calculate the superradiant scattering amplitudes and find perfect match with the CFT prediction;
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Remarks
Our treatment is in accordance with the universal behavior of the BH;
One lesson: the hidden conformal symmetry may not from the ambiguity in choosing the matching region. Instead it resides in the near-horizon region;
Similar analysis has been applied to many other kinds of black holes: RN, higher-dim. Kerr, Kerr-Bolt-AdS-dS, ...;
The hidden conformal symmetry has also been discussed in various three-dimensional black holes:
.
.1. Now the hidden conformal symmetry could be seen in the whole spacetime, rather than just the ”Near” region;.
.2. In some cases, the scalar Laplacian could not be just written as the quadratic Casimir, but with small modification;It helps us to find the holographic pictures of the black holes with charges or angular momentum;
In particular, for Kerr-Newman(-AdS-dS) BHs, there exist twofold conformal symmetries, which allows twofold holographic
descriptions;C.M. Chen et.al. 1006.4097, 1010.1379.
. . . . . .
.. Quasi-normal modes
BC and Jiang Long 1009.1010 Quasi-normal Modes as the “Sound” of a BH;Consider the linear perturbations around a BH with physical ingoing purely boundary condition at the horizon;
Complex frequency: “damped” oscillations;
In AdS/CFT correspondence, QNM ∼ tiny deviations Oi of thermal equilibrium in dual field theory;
The frequencies of QNM∼ the poles in the retarded green function of the perturbationsOi in momentum space;
Usually, it is hard to determine the QNM analytically;
However, in many classes of three-dimensional black holes, including BTZ, spacelike warped, null warped and self-dual black holes, QNM of various spin could be determined analytically;
We show that for the 3D black holes mentioned above, their QNM’s could be given elegantly in an algebraic way;
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Lie-induced quadratic Casimir
Another question: how the hidden conformal symmetry acts on the vector and graviton?
For a tensor, the action of a vector field is induced by Lie-derivative;
For 3D black holes, we show that the equations of various
perturbations could be written as the Lie-induced quadratic Casimir:
(L2+ m2t)T+= 0 where
L2≡ −LV0LV0+1
2(LV1LV−1+LV−1LV1);
is the Casimir commuting with the Lie-derivativesLVi;
In general T+ is an appropriate superposition of tensor components.
Two key relations:
[LX,LY] =L[X,Y ], LaX = aLX
where X, Y are arbitrary vectors and a is an arbitrary constant.
. . . . . .
.. Algebraic construction of QNM
For simplicity, we show our construction of the scalar mode;
We start from the highest weight mode defined by:
LV1Φ(0) = 0, LV0Φ(0)= hRΦ(0)
The equation of motion give the conformal weight of scalar:
hR= 1
2(1 +√
1 + 4m2s).
From Φ(0), we construct an infinite tower of quasi-normal scalar modes Φ(n) as
Φ(n)= (LV−1)nΦ(0), n∈ N.
All the Φ(n) are descendents of the mode Φ(0).
Since the CasimirL2 commutes with LVi, i = 0,±1, Φ(n) satisfy the scalar equation as well.
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Frequencies of QNM
To compute the frequency of the quasi-normal modes, we may expand the scalar as
Φ = e−iωt+ikϕφ,
as ∂tand ∂ϕ are always the Killing vectors of the black holes.
It turns out that the frequencies of the quasi-normal modes take the following form
λ1ωR(n)= λ2k + i(hR+ nR), λ¯1ω(n)L = ¯λ2k + i(hL+ nL) where nL,R are non-negative integers;
λi, ¯λi are parameters in the hidden conformal symmetry of various black holes;
The spectrum of all kinds of quasi-normal modes share the same structure, with the difference being from the conformal weights which are decided by the m2t term.
. . . . . .
.. Remarks
Asymptotically, the QNM’s behave as Φ(n)∼ r−hR.
It is remarkable that for vector and tensor modes, one needs proper combination of components to have the ”GOOD”
equations of motion;
In all the cases, our constructions reproduce the known results consistently;
For the spacelike and null warped black holes, the equations of motion are actually
(L2+ b ¯L2V¯0 + m2t)T+= 0,
which still allows us to construct the quasi-normal modes in the similar way. But now the conformal weight depends not only on the mass but also on the extra quantum numbers.
A similar construction could be applied to extremal black hole, in which case though we have different conformal coordinates;
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
. . . . . .
.. Conclusion and discussions
Scattering off a BH is an important way to check (or set up) BH/CFT correspondence;
The retarded Green’s function could be computed via Minkowski prescription;
Hidden conformal symmetry is very valuable in setting up the Kerr/CFT correspondence for generic non-extremal black hole:
.
.1. It provides a simple way to read the dual temperatures;.
.2. It allows a direct computation of real-time correlators, even though the spacetime is neither AdS nor warped AdS;.
.3. Though it is usually read out from the study of the (neutral) scalar equation of motion, it is actually an intrinsic property of the BH;.
.4. It acts on the tensors via Lie-derivatives;.
.5. It allows us to construct the infinite tower of the QNMs of 3D black hole algebraically;. . . . . .
.. Conclusion and discussions
Kerr/CFT is intriguing, especially for generic black holes:
.
.1. No string technique is needed; (but embedding into string theory may allow us to have a better understanding).
.2. No AdS spacetime;(How to generalize AdS/CFT to Minkowski or dS?).
.3. Unlike extreme or near-extreme ones, there is even no warped AdS spacetime;.
.4. The only clue comes from the hidden conformal symmetry acting on the solution space;.
.5. More study is deserved: central charges, ...;What’s the origin of the hidden conformal symmetry?
Bin Chen, PKU Real-time correlators in Kerr/CFT correspondence
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