EVH black holes, Their AdS 3 throats and EVH/CFT proposal By: M.M. - - PDF document

EVH black holes, Their AdS3 throats and EVH/CFT proposal

By: M.M. Sheikh-Jabbari

Based on: My Recent Work, arXiv:1107.5705 [hep-th] In collaboration with H. Yavartanoo. Istanbul, August 2011

1

Outline

• Motivation and Introduction
• EVH black holes in general dimensions
• 4d EVH black holes of Einstein-Maxwell-dilaton

theories

• EVH black holes have near horizon AdS3

throat

• EVH/CFT correspondence
• Connection to Kerr/CFT
• Summary and outlook

2

Introduction and Motivation

• Black holes can be understood as thermo-

dynamical systems

• Black holes Hawking-radiate
• Formation and evaporation of black holes is

hence not a unitary process, unless

• there exists an underlying stat. mech. sys-

tem, i.e.

• resolution of black hole information loss prob-

lem replies on identification of its microstates.

3

• String theory has been successful in the black

hole microstate counting project of certain supersymmetric BPS black holes.

• The idea is that

the microstates reside on the horizon ⇒ near horizon geometry and not its asymp- totics carries the microstate information.

• Therefore, for BPS black holes the entropy

is ought to be only a function of the charges and independent of the moduli, the attractor mechanism.

• Extremal (but non-BPS) black holes are in

many ways similar to BPS black holes, e.g. attractor mechanism works for them.

• Extremal black holes have zero Hawking tem-

perature, but generically finite Bekenstein- Hawking entropy.

4

• Near horizon geometry of extremal black holes

contain AdS2 throats.

• One may use AdS2/CFT1 for identification
• f microstates of extremal black holes.
• Kerr/CFT or Ext/CFT:

∀ Extremal black holes, ∃ chiral 2d CFT description.

• Kerr/CFT instructs usage of Cardy formula

to relate black hole entropy to the 2d CFT density of states.

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• BUT, AdS2/CFT1 is not well understood

and,

• What is a chiral 2d CFT?

How do we identify it? Does Kerr/CFT have a dynamical content? Can it help with understanding generic non- extremal black holes?

• Black hole microstate counting have been

most successful when based upon AdS3/CFT2, like D1-D5-P system.

• Can we get AdS3 throat as the near hori-

zon limit of a black hole (and not black brane/string)? This is what we explore and its answer is Yes, for EVH black hole.....

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Extremal Vanishing Horizon (EVH) black holes

• Black holes with vanishing T, Ah but with

Ah/T = fixed.

• In the class of n parameter black holes, EVH

are defined by n−2 dim. EVH hypersurface.

• Examples of EVH black holes:

– n = 2: massless BTZ. – n = 3: 5d Kerr with one vanishing angu- lar momentum. – n = 4: two-charge AdS5 black holes of U(1)3 5d gauged SUGRA. – n = 5: three-charge AdS4 black holes of U(1)4 4d gauged SUGRA, and .....

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• Once we specified an EVH point on a given

EVH hypersurface, then Near EVH black holes are determined by two parameters.

• EVH black holes can be supersymmetric or

non-BPS.

• EVH black holes can be asymptotically flat
• r AdS.
• There are examples of stationary and static

EVH black holes.

• NOTE: We have not an example of station-

ary BPS EVH black hole, but no proof that it cannot happen.

8

• Regardless of the details:

Near Horizon geometry of any EVH black hole has a (pinching) AdS3 throat. pinching AdS3 ≡ AdS3/ZK, K → ∞.

• Near horizon limit of Near EVH black hole

contains a (pinching) BTZ geometry.

• We prove above for any EVH solution to

generic 4d (gauged) Einstein-Maxwell-Dilaton theory.

• The above are more general and is presum-

ably true for any non-BPS EVH black hole.

9

4d EVH black holes

Consider the 4d gravity theory

L

= R − 2GAB∂ΦA∂ΦB − fIJ(Φ)F I

µνF J µν

− 1 2√−gǫµναβ ˜ fIJ(Φ)F I µνF J µν + V (Φ). The most general stationary black hole ansatz ds2 = −N2(ρ, θ)dt2 + gρρ(ρ, θ)dρ2 + gθθ(ρ, θ)dθ2 + gφφ(ρ, θ)

• dφ + Nφ(ρ, θ)dt

2,

A = At(ρ, θ)dt + Aρ(ρ, θ)dρ + Aφ(ρ, θ)dφ, Φ = Φ(ρ, θ). In Aθ = 0 gauge. N2 = (ρ − ρ+)(ρ − ρ−)µ(ρ, θ), Nφ = −ω + (ρ − ρ+)η(ρ, θ), gρρ = 1 (ρ − ρ+)(ρ − ρ−)Λ(ρ, θ) ,

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• µ(ρ, θ) and Λ(ρ, θ) do not have zero in (ρ+, ∞).
• Finite horizon angular velocity at the hori-

zon, requires having finite η(ρ+, θ).

• Hawking temperature

T =

  

(N2)′ 4π

• gρρN2

  

ρ=ρ+

prime denotes derivative with respect to ρ and ρ+ is the location of the outer horizon.

• The area of horizon Ah can be expressed as

Ah = 2π

π

• g(0)

θθ (θ)g(0) φφ (θ) dθ

where gθθ(ρ, θ) = g(0)

θθ (θ) + (ρ − ρ+)g(1) θθ (θ) + · · · ,

gφφ(ρ, θ) = g(0)

φφ (θ) + (ρ − ρ+)g(1) φφ (θ) + · · · ,

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• EVH point:

Ah ∼ T ∼ ˜ ǫ

• The above, demanding the regularity of the

geometry, is possible if g(0)

φφ ∼ ˜

ǫ2, ρ+ ∼ ˜ ǫs , ρ+ − ρ− ∼ ˜ ǫv , v ≥ s > 0.

• That is,

horizon is located at ρ = 0.

• Demanding the geometry to be smooth around

ρ = 0 (for generic values of θ) implies N2(ρ = 0) = 0.

• To avoid having a naked singularity and to

keep Ah/T finite d dρN2|ρ=0 = 0.

12

• In summary

N2 = ρµ(ρ, θ) µ is an analytic function with no zeros at ρ > 0.

• Roots of µ are potential loci of EVH black

hole singularity. These roots are hence located at ρ < 0.

• Singularity line

ρ = ρs(θ), µ(ρs, θ) = 0 touches the horizon ρs(θ) = 0 which occurs at some isolated points in θ.

• Away from these isolated points the near

horizon geometry of the EVH black hole is expected to be smooth.

• EVH black holes are different from small

black holes where the horizon and the sin- gularity are basically becoming identical.

13

The most generic 4d EVH black hole metric: ds2 = −ρ ˜ µdt2 + dρ2 ρ2˜ Λ + ρ ˜ gφφ

• dφ + ˜

Nφdt

2 + ˜

gθθdθ2 where

• ˜

µ, ˜ Λ, ˜ gφφ, ˜ Nφ and ˜ gθθ are analytic functions

• f (ρ, θ).
• ˜

µ and ˜ Λ do not have any zero in [0, ∞).

• g−1

ρρ has double roots at the horizon ρ = 0.

• These functions may be determined using

the equations of motion.

• Temperature (surface gravity) must be in-

dependent of the angular coordinate θ: ˜ µ(0, θ)˜ Λ(0, θ) = L2 .

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EVH black holes have AdS3 throats, the Proof

• Under the near horizon limit

ρ = ǫ2r2, ˜ t = ǫt , ˜ φ = ǫ(φ − ˜ ωt), ǫ → 0

• The EVH geometry goes to

ds2 = a(˜ θ)

• −r2d˜

t2 + L2dr2 r2 + b(˜ θ)r2d˜ φ2 + R2d˜ θ2

• where L, R are constants.
• ˜

ω is the angular velocity of the horizon at EVH limit.

• We note that ˜

φ ∈ [0, 2πǫ].

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• How about gauge and scalar fields?

E.o.M. to have only ˜ θ dependent solutions ⇒ All the components of the gauge field strength should vanish.

• E.o.M. for metric, assuming constant poten-

tial V = V0, implies: db d˜ θ = 0 ⇒ b = b0 = const. , d2a d˜ θ2 + 4R2 L2 a − V0R2a2 = 0 .

• For V0 = 0 (ungauged gravity) case

a = a0 sin 2R L ˜ θ , where C = 2R

L a0.

E.o.M for Φ: dΦ d˜ θ = ± √ 3C a ⇒ e

2Φ √ 3 = g0 tan R˜

θ L , where g0 is a constant.

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• In summary, and for V = 0 case

ds2 = R2

AdS3 sin θ

• −r2dτ2 + dr2

r2 + r2dψ2 + 1 4dθ2

• ,

Fµν = 0 , e

2Φ √ 3 = g0 tan θ

2 where

• we have redefined ˜

θ such that θ ∈ [0, π] and,

• ψ ∈ [0, 2πǫ], i.e.

the AdS3 throat in the near horizon limit of the EVH black hole is a pinching AdS3.

• A two parameter family of solutions;

RAdS3, g0 are not fixed by E.o.M.

• NOTE: the near horizon limit and the EVH

black hole limits do not commute.

• θ = 0, π are singular points of the near hori-

zon EVH geometry.

17

Near horizon limit of near EVH black holes

• Near EVH black holes have non-zero but

very small Ah and T with their ratio finite and,

• are defined by two parameters, parameteriz-

ing how we have moved away from the EVH point.

• Their near horizon geometry can be taken

along the same arguments as the EVH case, leading to

ds2 = R2

AdS3 sin θ

• −F(r)dτ 2 + dr2

F(r) + r2(dψ − r+r− r2 dτ)2 + 1 4dθ2

• where

F(r) = (r2 − r2

+)(r2 − r2 −)

r2 .

• i.e.

the same as before but pinching AdS3 is replaced by pinching BTZ.

• The solution has now four parameters:

RAdS3, g0, r±.

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Near horizon structure of the EVH black hole in flat or AdS background:

• The Near horizon region is warped (AdS3 or

BTZ ×I) geometry.

• In the strict near horizon limit the intermedi-

ate and asymptotic regions are cut off from the geometry.

19

Examples of 4d EVH black holes

• KK black holes:

– four parameter family of 4d black holes with electric and magnetic charges Q, P angular momentum J and mass M. – uplifted to 5d Einstein gravity stationary solutions. – may be embedded in the string theory to a rotating D0-D6 brane system Q ∝ N0, P ∝ N6. – have two different branches: ergo-branch PQ < G4J, ergo-free branch PQ > G4J.

20

• KK black hole is usually parameterized by

G4J = √pq(pq + 4m2) 4(p + q) j, M = (p + q) Q2 = q(q2 − 4m2) 4(p + q) , P 2 =p(p2 − 4m2) 4(p + q) .

• non-nakedly-singular black hole solutions have

q ≥ 2m, p ≥ 2m, j ≤ 1.

• Temperature and horizon area of the KK

black holes: T = m π√pq

  pq + 4m2

p + q + 2m

• 1 − j2

  

−1

, Ah = 2π√pq

• 1 − j2

  pq + 4m2

(p + q) + 2m

• 1 − j2

  

• EVH limit

m = ǫµ, 1 − j2 = ǫ2λ2 , ǫ → 0 . m = 0, j = 1, OR PQ = G4J

21

• In the four dimensional (n = 4) parame-

ter space of KK black holes, there exists a two dimensional EVH hypersurface parame- terized by p, q.

• Singularity of EVH KK black hole is generi-

cally sitting behind the horizon.

• Singularity becomes naked only in two points.
• For the EVH KK black hole horizon is gener-

ically far from singularity.

• Horizon of the EVH KK black hole is topo-

logically a two-sphere, but a singular one.

• i.e.

close to the horizon, at constant r, t, metric is more like a cylinder the axis of which is along θ direction and its circle, which has vanishing radius is along φ direction.

22

Horizon geometry as we increase angular momentum (left to right).

• The left figure shows the horizon at J = 0.
• For a fixed P, Q, increasing angular momen-

tum reduces the horizon area and at the crit- ical value G4J = PQ, the EVH point, horizon area vanishes.

• For the near EVH |G4J −PQ| ≪ PQ, horizon

is a thin cylinder.

23

Near horizon of Near EVH KK black holes

• N.H. limit over the KK metric (in its stan-

dard coordinate system) ρ = pq p + q

• r2 − r2

+

• ǫ2 + r2

+ǫ2

t = 2√pq τ ǫ, ǫ → 0 φ = τ + ψ ǫ where r± = 2µ(p + q) ± pqλ 2pq .

• Leads to

ds2 = R2

AdS3| sin θ|

• − F(r)dτ 2 + dr2

F(r) + r2(dψ − r+r− r2 dτ)2 + 1 4dθ2

• F(r) =

(r2−r2

+)(r2−r2 −)

r2

, θ ∈ [0, π],ψ ∈ [0, 2πǫ]. R2

AdS3 = 8PQ = 8G4J = 2(pq)3/2

p + q e

4φ √ 3 = q

p tan2 θ 2

24

Reduction to 3d

• Reduction of 4d KK theory over

ds2 = R2

AdS3| sin θ|

• gabdxadxb + 1

4dθ2

• ,
• Leads to 3d Einstein AdS3 gravity with

Λ = R2

AdS3

• 3d Newton constant

G3 = 2G4 RAdS3 .

• The Bekenstein-Hawking entropy of the pinch-

ing BTZ solution: S3d = 2πǫr+RAdS3 4G3 = R2

AdS3

G4 · πǫr+ 4 .

• The entropy of the original 4d near EVH KK

black hole: S4d = Ah 4G4 = 2π√pq λǫ 4G4

• pq

p + q + 2µ λ

• = R2

AdS3

G4 · πǫr+ 4 ,

25

• Hawking temperatures of the 4d near EVH

KK black hole vs. that of pinching BTZ: T4d = ǫ 2√pq TBTZ , TBTZ = r2

+ − r2 −

2πr+ . The prefactor

ǫ 2√pq is expected N.H. scaling.

• Uplift to 5d:

ds2

5

= 2

p

qR2

AdS3

• cos2 ϑ
• −r2dτ 2 + dr2

r2 + r2dψ2

• +
• cos2 ϑdϑ2 + tan2 ϑ dχ2

, ϑ = θ/2, 0 ≤ ϑ ≤ π/2.

• This is the 5d geometry obtained as the near

horizon limit of EVH 5d Kerr.

• As expected the process of uplifting does not

change temperature or entropy.

26

Static charged EVH black holes in 4d U(1)4 SUGRA

• The solution is specified by five parameters,

four charges qI and µ: ds2 = −H−1/2fdt2 + H1/2

• dr2

f + r2dΩ2

2

• ,

AI = ˜ qI qI

• 1

HI − 1

• ,

XI = H1/4 HI , where HI = 1 + qI

r , H = H1H2H3H4,

f = 1 − µ r + 4r2 L2 H, ˜ qI =

• qI(qI + µ)
• The three charge black holes are EVH:

q1 = 0, µ = 4q2q3q4 L2

• This family has both supersymmetric and

non-supersymmetric EVH solutions.

• 11d uplift of these solutions are stationary

(not static).

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• In 11d the geometry corresponds to three

stacks of intersecting rotating (giant) spher- ical M5-branes.

• Near horizon EVH limit is singular in 4d,

while the limit is well-defined over the 11d uplift.

• The near horizon geometry of supersymmet-

ric EVH geometry develops an AdS3 throat without the pinching,

• while for the non-supersymmetric case we
• btain a pinching AdS3 throat.

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• In both cases the “circular” part of AdS3,

comes from the seven dimensional part of the 11d solution and is not a part of the

• riginal asymptotic AdS4 geometry.
• The EVH solutions interpolate between the

AdS3 throat on the horizon and AdS4 in the asymptotic region.

• Entropy of the near EVH geometry, for both

BPS and non-BPS case, before taking the near horizon limit is equal to that of the 3d BTZ geometry obtained after the limit.

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EVH/CFT correspondence

Gravity theory on the near horizon limit of EVH black holes is governed by a 2d CFT. Evidence in support of the proposal

• Appearance of AdS3 throat in the near hori-

zon geometry. Caveat:pinching AdS3, rather than AdS3

• Entropy of the original and the BTZ in near

horizon geometry are equal.

• Near horizon limit is a decoupling limit.

30

Near horizon limit is a decoupling limit 1 √−g∂µ

√−ggµν∂νΨ

• = 0

where g is the EVH black hole metric, before taking the near horizon limit. Ψ = e−i(ωt+kφ)Flkω(ρ)Ylkω(θ). Ylkω satisfies Heun’s equation, which is a gener- alized form of Legendre equation. With ρ = r2 , Flkω = Rlkω(r) r3/2 , radial equation takes a Schrodinger-type eq: R′′

lkω − U(r)Rlkω = 0 ,

with

U(r) = −4ω2r2 − 4(p + q)ω2+ + 4ql(l + 1) + 3/4 − 6pqω2 r2 − 4ωp3/2q3/2ω(k + √pqω) (p + q)r4

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Behaviour of the potential as we vary ω, and for ω(k + √pqω) > 0. As we decrease ω the hight of the maximum

• f the potential increases and in the low energy

ω → 0 limit the potential develops an infinite barrier, signaling the decoupling.

32

Behavior of the potential

• around r = 0, where the horizon of the EVH

black hole resides and

• in low energy, when the energy and angu-

lar momentum of the probing particle ω, k is scaling to zero is governed by the terms in the potential in red.

• It has a maximum if

ω(k + √pqω) > 0

• and the potential at the maximum goes to

infinity in the near horizon limit and hence the decoupling.

33

Resolution of pinching AdS3 orbifold and EVH/CFT

• A system of vanishing entropy does not have

a good stat. mech. description.

• Neither, the gravity description is trustable.

Resolution: make the entropy finite with accom- panying the near horizon limit with the scaling G4 = ǫℓ2 , ℓ, RAdS3 = fixed . [The above idea is from arXiv:1011.1897]. Recalling the 3d reduction: G3 = 2G4 RAdS3 = 2ℓ2 RAdS3 ǫ .

34

• For AdS3/ZK orbifold, the Brown-Henneaux

argument leads to cK = 3RAdS3 2G3 · 1 K NOTE: A simple way to see the above is to recall that B.H. central charge is a result

• f integration over a “circle” of a conserved
• ne-form so integration over [0, 2π/K] leads

to an extra 1/K factor. [Martinec-McElgin, hep-th/0106171].

• For our case K = 1/ǫ, and central charge of

the proposed dual 2d CFT is c =3RAdS3 2G3 ǫ = 3 2

RAdS3

ℓ

2

.

• Next item in our EVH/CFT dictionary is to

identify L0, ¯ L0 of 2d CFT.....

35

• L0, ¯

L0 in terms of near EVH parameters, r±: L0 = c 24

r+ + r−

RAdS3

2

, ¯ L0 = c 24

r+ − r−

RAdS3

2

,

• NOTE: EVH geometry corresponds to the

ground state of the 2d CFT.

• For other quantum perturbations on EVH

geometry: recall the “decoupling” discus- sions: kφ + ωt = 1 ǫ [kψ + τ(k + 2√pqω)] .

• Natural choice is to identify

coefficient of ψ by J≡ L0 − ¯ L0 and coefficient of τ by ∆≡ L0 + ¯ L0: L0 = √pqω + k , ¯ L0 = √pqω .

• NOTE: Decoupling condition

ω(√pqω + k) > 0 turns to unitarity condition of the 2d CFT.

36

Concluding Remarks and Outlook

———————– Our two main results:

⊛ The near horizon limit of any EVH black hole

has an AdS3 throat.

• For non-BPS EVH cases this AdS3 is a pinch-

ing AdS3: AdS3/ZK, Z → ∞.

• BPS EVH cases pinching is not there.
• near EVH

⇒ pinching BTZ.

⊛ The EVH/CFT correspondence:

gravity on near horizon EVH geometry is dual to or de- scribed by a 2d CFT.

37

Evidence for EVH/CFT

• Appearance of AdS3 throat.
• Near horizon as decoupling limit.
• Resolution of pinching orbifold:triple scaling

limit Ah, T, G4 → 0 , Ah/T and Ah/G4 held fixed .

⊛The above implies:

2d CFT with central charge c on cylinder R×S1 is dual to 2d CFT with central charge cK on R × S1/ZK in the K → ∞ limit.

38

⊛ Connection between EVH/CFT and Kerr/CFT:

• DLCQ, a possible connection between the

two: The EVH/CFT in the DLCQ descrip- tion reproduces Kerr/CFT.

⊛ EVH/CFT and Schwarzchild type black holes?

• How far from the EVH point (hypersurface)

the validity of EVH/CFT can be extended?!

• In the strict “triple decoupling” limit we can-

not probe beyond the AdS3 throat into the intermediate or asymptotic flat regions

⊛ Embedding EVH black holes in string theory?

• EVH KK black holes: D0-D6 brane at the

tip of Taub-NUT.

39

⊛ EVH black holes in d > 4? ⊛ EVH black holes vs. EVH black rings.

• In the near horizon limit EVH black hole hori-

zon is cylindrical.

• We can have EVH black ring and EVH black

holes of the same quantum numbers.

• horizon topology change from a hole to a

ring, can happen through the EVH point.

⊛ 2d description of the topology change?!

• EVH black hole and EVH black ring respec-

tively correspond to vacuum of 2d CFT, the massless BTZ and to vacuum of DLCQ of 2d CFT, the null self-dual orbifold.

40

⊛ Other extensions to EVH black holes.

• EVHn black holes, for which

Ah, T → 0 Ah/T n = finite n = 1 is the EVH case we studied here.

• For n > 1, again based on the generic prop-

erty of n + 1 dimensional CFT’s for which S ∝ c T n with c being the central charge, one would expect to obtain an AdSn+2 throat in the near horizon geometry.

Thank You For Your Attention

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