Self-Dual Warped AdS
3Black Holes
Bo Ning
with Bin Chen, 1005.4175 School of Physics, Peking University
NCTS
April 22, 2011
Motivation Black Hole Holography Summary
Contents
1
Motivation
2
Self-dual warped AdS
3black hole Geometry
Thermodynamics
3
Holographic description Temperatures
Asymptotic behavior Scalar perturbation
4
Summary
AdS/CFT correspondence
Quantum gravity (string theory/M-theory) in (D + 1)-dimensional anti-de-Sitter spacetime is equivalent to D-dimensional conformal field theory at AdS boundary.
A new way to study the strong coupling problems in field theory: AdS/QCD, AdS/CMT, ...
A tool to study the problems in quantum gravity;
Relies heavily on string theory technology.
Motivation Black Hole Holography Summary
AdS/CFT correspondence
A case not invoking string theory is AdS
3/CFT
2correspondence:
quantum gravity asymptotic to AdS
3is holographically dual to a 2D CFT.
For 3D pure gravity with a negative cosmological constant, the vacuum solution is AdS3, with the isometry groupSL(2; R) SL(2; R);
The analysis of the symmetry of perturbations at asymptotic boundary give rise to a Virasoro algebra with a central term;
The entropy of the BTZ black hole could be read from Cardy formula, which relates the asymptotic density of states in a 2D CFT and the symmetry algebra.
Topological Massive Gravity
Pure 3D gravity: contains no local propagating degrees of freedom, thus hard to explain the microscopic origin of the BTZ black hole entropy.
Topological Massive Gravity: with an additional gravitational Chern-Simons term, resulting a local, massive propagating degree of freedom.
IT MG= 1 16G
Z d3xp
g (R + 2=`2) + 1
ICS
; ( > 0; G > 0) (1) ICS= 12
Z d3xp
g"
@
+ 23
(2) TheAdS3 vacuain TMG are generally perturbatively unstable except at the chiral point` = 1 , where a consistent quantum theory of gravity is conjectured to exist and be dual to a chiral CFT.
Motivation Black Hole Holography Summary
Topological Massive Gravity
TMG possess other vacua for generic values of, namelywarped AdS3, admit isometry groupU(1) SL(2; R). Some of them are stable.
Spacelike:
ds2= `2 (2+ 3)
cosh2d2+ d2+ 42+ 32 (du + sinh d)2
(3)
stable for2> 1 (stretchedcase), where = `=3.
Timelike:
ds2= `2 (2+ 3)
cosh2du2+ d2 42
2+ 3(d + sinh du)2
(4)
Null: solution to TMG only for2= 1
ds2= `2
"
du2
u2 + dx+dx u2
dx u2
2#
(5)
Topological Massive Gravity
Quotients: Just as BTZ black holes are discrete quotients of ordinary AdS3, there are black holes solutions as discrete quotients of warped AdS3.
Spacelike stretched black hole:
ds2
`2 = dt2+ dr2
(2+ 3)(r r+)(r r )+
2r p
r+r (2+ 3) dtd
+ r4
3(2 1)r + (2+ 3)(r++ r ) 4p
r+r (2+ 3)
d2 (6) identifying points along isometry@such that + 2.
Self-dual solution:
ds2 = `2
2+ 3
~x2d~2+ d~x~x22 + 42+ 32 d ~ + ~xd~2 (7)
with identification ~ ~ + 2.
Motivation Black Hole Holography Summary
Warped AdS/CFT correspondence
It is conjectured that the > 1 quantum TMG with asymptotical spacelike stretched AdS3 geometry is holographically dual to a 2D CFT with central charges
cL= 4`
G(2+ 3); cR= (52+ 3)`
G(2+ 3) (8)
Entropy of spacelike stretched black hole could be reproduced through Cardy formula
S = 2`
3 (cLTL+ cRTR) (9) Analysis of theasymptotic symmetrylead to a central extended Virasoro algebra with above central charges.
Warped AdS/CFT correspondence is essential inKerr/CFT correspondence, in the sense that the warped AdS3 structure appears in the near-horizon geometry of Kerr black hole.
Kerr/CFT correspondence
NHEK: near-horizon geometry of the extreme Kerr black hole ds2 = 2J ()
r2dt2+ drr22 + d2+ ()2(d + rdt)2
(10) with + 2. A slice of NHEK geometry at fixed polar angle is locally aself-dual warped AdS3.
Conjecture: quantum gravity in NHEK is dual to a 2D CFT, with only left-moving temperature.
Near-NHEK: near-horizon geometry of the near-extreme Kerr black hole
ds2= 2J ()
r(r + 4TR)dt2+ dr2
r(r + 4TR)+ d2
+ ()2(d + (r + 2TR)dt)2 (11) Conjecture: near-NHEK is dual to the same CFT with non-vanishing right temperature, since the right-moving sector gets excited.
Motivation Black Hole Holography Summary
Motivation
Is there near-NHEK like solutions in TMG?
If they exist, what’s the holographic description of them?
Relationship with the warped AdS/CFT correspondence?
It turns out that such solutions do exist in TMG:
self-dual warped AdS3 black holes.
Holographic dual to a chiral 2D CFT, with the same left central charge as in warped AdS/CFT, and non-vanishing left- and right-moving temperatures.
Provide another novel support to the conjectured AdS/CFT correspondence.
Motivation
Is there near-NHEK like solutions in TMG?
If they exist, what’s the holographic description of them?
Relationship with the warped AdS/CFT correspondence?
It turns out that such solutions do exist in TMG:
self-dual warped AdS3 black holes.
Holographic dual to a chiral 2D CFT, with the same left central charge as in warped AdS/CFT, and non-vanishing left- and right-moving temperatures.
Provide another novel support to the conjectured AdS/CFT correspondence.
Motivation Black Hole Holography Summary Geometry Thermodynamics
Self-dual warped AdS
3black hole
Self-dual warped AdS3 black hole: non-Einstein black hole solution of TMG, which is asymptotic to spacelike warped AdS3
ds2 = `2
2+ 3
(x x+)(x x ) d2+ 1
(x x+)(x x )dx2 + 42+ 32 (d + (x x++ x
2 ) d)2
; (12)
where 2 [ 1; 1], x 2 [ 1; 1] and + 2.
There seems to be two horizons atx+andx .
The vacuum is chosen to be given byx+= x = 0 and = 1, which is the self-dual warped AdS3.
For2> 1, the solution is free of naked CTCs.
Self-dual warped AdS
3black hole
Self-dual warped AdS3 black hole is related to the self-dual warped AdS3 through coordinate transformation
~ ~ 1~x= tanh
1 4
(x+ x ) ln x xx x+
;
~ = + 12ln
1 (~+)2 1 (~ )2
: (13)
Globally, the maximal analytic extension of the self-dual warped black hole is diffeomorphic to the self-dual warped AdS3.
However, the above coordinate transformations are singular at the boundaryx ! 1, indicating different physics.
The situation here is very similar to the relation between NHEK and near-NHEK, or between AdS2 and AdS2 black hole.
Observer at fixedx measure a Hawking temperature proportional to x+ x . The entropy does not depend on x+ x , but the scattering amplitudes do depend onx+ x .
Motivation Black Hole Holography Summary Geometry Thermodynamics
Self-dual warped AdS
3black hole
Self-dual warped black hole is locally equivalent to spacelike warped AdS3 ds2= `2
2+ 3
cosh2d2+ d2+ 42
2+ 3(du + sinh d)2
(14) through coordinate transformation
= tan 1
"
2p
(x x+)(x x )
2x x+ x sinhx
+ x
2 #
;
= sinh 1
"
2p
(x x+)(x x )
x+ x coshx
+ x
2 #
;
u = + tanh 1
2x x+ x
x+ x cothx
+ x
2
: (15)
Self-dual warped AdS
3black hole
Killing vectors of spacelike warped AdS3: J2 = 2@u
J~1 = 2 sin tanh @ 2 cos @+ 2 sin cosh @u
J~2 = 2 cos tanh @ 2 sin @ 2 cos cosh @u
J~0 = 2@
Self-dual warped AdS3 black hole solution is free of curvature singularity and regular everywhere.
Strictly speaking, it does not belong to the category of regular black holes, which require the existence of a geometric or causal singularity shielded by an event horizon.
On the other hand, similar to black holes, this solution have regular thermodynamic behavior, satisfying the first law of thermodynamics.
Motivation Black Hole Holography Summary Geometry Thermodynamics
Thermodynamics
Conserved charges:
MADT = 0 ; JADT = (2 1)`
6G(2+ 3) (16)
Entropy: with the Chern-Simons contribution S = SE+ SCS= 2`
3G(2+ 3) (17)
Hawking temperatureTH and the angular velocity of the horizonh, TH= x+ x
4` ; h= x+ x
2` (18)
The first law
dMADT = THdS + hdJADT (19)
is satisfied for a variation of the black hole parameter.
Thermodynamics
Remarks:
ADT mass and angular momentum do not depend onx+ andx . EntropyS does not depend on x+ andx either.
Observers at fixedx measure a Hawking temperature proportional to x+ x .
Similar to the case of near-NHEK.
Even with a different choice of vacuum, the first law of thermodynamics still holds.
Motivation Black Hole Holography Summary Temperatures Asymptotic behavior Scalar perturbation
Temperatures in dual CFT
Since the self-dual warped black hole metric looks much similar to the near-NHEK geometry, it is sensible to define a quantum vacuum in analogy to the Frolov-Thorne vacuum of Kerr.
The construction of the vacuum begins by expanding the quantum fields in eigenmodes of the asymptotic energy! and angular momentum k. For a scalar field,
=X
!;k;l
!kle( i!+ik)Rl(x) (20)
After tracing over the region inside the horizon, the vacuum is a diagonal density matrix in the energy-angular momentum eigenbasis with a Boltzmann weighting factor
exp
~! kH
TH
(21)
Temperatures in dual CFT
In AdS/CFT dictionary,black hole in AdSis dual tofinite temperature CFT.
Taking black hole as a thermodynamical system, the thermal equilibrium in black hole system could be compared to thermal equilibrium of finite temperature CFT.
2D CFT contains two independent sectors: left-moving one and right-moving one, possibly with different central charges and temperatures.
Motivation Black Hole Holography Summary Temperatures Asymptotic behavior Scalar perturbation
Temperatures in dual CFT
The Boltzman factor should be identified with that in CFT exp
~! kH
TH
= exp n
L
TL
nR
TR
(22)
The left and right chargesnL; nRassociated to@ and@ are
nL k ; nR ! (23)
This defines the left and right temperatures:
TL= 2`; TR= x+ x
4` (24)
Temperatures in dual CFT
The right temperaturedenotes the deviation from extremality, originates essentially similar to the case that Rindler observers detect radiations in the Minkowski vacuum (as discussed for AdS2 black hole).
The left temperaturearises from the periodical identification of points in spacelike warped AdS3 along@. Expressing@ in terms of the spacelike warped AdS3 coordinates,
@= 2J2= ` TLJ2 (25) the temperature of the dual 2D CFT can be read from the coefficient of the shift. This periodical identification makes no contribution to the right temperature.
Motivation Black Hole Holography Summary Temperatures Asymptotic behavior Scalar perturbation
Asymptotic behavior
To acquire the central charge of the dual CFT through the asymptotic symmetry analysis, we impose the following consistent boundary conditions
h = O(1) hx= O(1=x3) h= O(x ) hx = hx hxx= O(1=x3) hx= O(1=x ) h = h hx= hx h= O(1)
!
(26)
whereh is the deviation of the full metric from the vacuum.
These boundary conditions differ from both the ones inspacelike warped AdS3 and theKerr black holes, since the allowed deviationshandh are of the same order as the leading terms.
Asymptotic behavior
Theasymptotic symmetry group, preserving the above boundary conditions, contains one copy of the conformal group of the circle generated by
= () @ (27)
Since + 2, it is convenient to define n() = ein and
n= (n). They admit the following commutators
i [m; n] = (m n) m+n (28) and0 generates theU(1) rotational isometry. That is, the U(1)
isometry is enhanced to a Virasoro algebra.
The conserved charge associated with an asymptotic Killing vector represent the asymptotic symmetries algebra via a covariant Poisson bracket, up to acentral term
ifQm; Qng = (m n)Qm+n+ c12Lm(m2 2)m+n;0; (29) where the central charge
cL = 4`
G(2+ 3) (30)
Motivation Black Hole Holography Summary Temperatures Asymptotic behavior Scalar perturbation
Asymptotic behavior
ThecLis exactly the value of the left central charge conjectured in warped AdS/CFT correspondence for spacelike warped AdS3.
The entropy of the self-dual warped black hole can be reproduced from the Cardy formula
S = 2`
3G(2+ 3)= 2`
3 4`
G(2+ 3) 2` 2`
3 cLTL (31)
Conjecture: our self-dual warped black hole solution is dual to a chiral 2D CFT with the temperatures and central charges
TL= 2`; TR= x+ x
4` ; cL = 4`
G(2+ 3) (32) Further supported by the real-time correlator of scalar perturbations.
Scalar perturbation
Consider a scalar field with mass m in black hole background
= e i!+ikR(x) (33)
The radial wave functionR(x) satisfies the equation d
dx
(x x+)(x x ) ddx
R(x)
2+ 3 42
k2 2 + `2
2+ 3m2
! +k(x x++x2 )2 (x x+)(x x )
!
R(x) = 0 (34)
Choose ingoing boundary condition at the horizon for calculating the retarded Green’s function, the solution is
R(x) = N
x x+
x x
2i k+ 2!
x+ x x+ x
x x
12
· F
1
2 i k; 12 i 2!
x+ x ; 1 i
k
+ 2!
x+ x
; x xx x+
(35)
Motivation Black Hole Holography Summary Temperatures Asymptotic behavior Scalar perturbation
Scalar perturbation
At asymptotical infinity, the radial eigenfunction has the behavior
R(x) Ax 12 + Bx 12+ (36)
where
A = N (x+ x )12+ ( 2) 1 i
k+x+2!x
12 ik 1
2 ix+2!x (37)
B = A ( ! ) (38)
The real-time retarded Green’s function could be computed using a prescription in terms of the boundary values of the bulk fields. Consider a real > 0 without loss of generality, the retarded correlator is given by
GR AB = (x+ x )2 ( 2) (2)
12 + ik 1
2+ ix+2!x
12 ik 1
2 ix+2!x (39)
Scalar perturbation
Throwing a scalar at the black hole is dual to exciting the CFT by acting with an operatorO. For an operator of dimensions (hL; hR) at temperature (TL; TR), the momentum-space Euclidean Green’s function is determined by conformal invariance
GE(!L;E; !R;E) TL2hL 1TR2hR 1ei!L;E2TLei!L;E2TR (hL+ !2TL;EL) (hL !L;E
2TL)
· (hR+ !R;E
2TR) (hR !R;E
2TR) (40) the Euclidean correlatorGE(!L;E; !R;E) is related to the value of the retarded correlatorGR(!L; !R) by
GE(!L;E; !R;E) = GR(i!L;E; i!R;E) (41) Comparing the arguments of the Gamma functions among (39) and (41), we find preciseagreementunder the following identification
hL= hR= 12+ ; !L= k=`; !R= !=`; TL= 2`; TR= x+ x 4`
Motivation Black Hole Holography Summary