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

EVH black holes, Their AdS 3 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 AdS 3 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 AdS 2 throats. • One may use AdS2/CFT1 for identification of 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. 5

• 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..... 6

Extremal Vanishing Horizon (EVH) black holes • Black holes with vanishing T , A h but with A h /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 ..... 7

• 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 or 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) AdS 3 throat. pinching AdS 3 ≡ AdS 3 /Z K , 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 R − 2 G AB ∂ Φ A ∂ Φ B − f IJ (Φ) F I µν F J µν L = 1 f IJ (Φ) F I µν F J µν + V (Φ) . 2 √− gǫ µναβ ˜ − The most general stationary black hole ansatz − N 2 ( ρ, θ ) dt 2 + g ρρ ( ρ, θ ) dρ 2 ds 2 = � 2 , g θθ ( ρ, θ ) dθ 2 + g φφ ( ρ, θ ) � dφ + N φ ( ρ, θ ) dt + A = A t ( ρ, θ ) dt + A ρ ( ρ, θ ) dρ + A φ ( ρ, θ ) dφ, Φ = Φ( ρ, θ ) . In A θ = 0 gauge. N 2 = ( ρ − ρ + )( ρ − ρ − ) µ ( ρ, θ ) , N φ = − ω + ( ρ − ρ + ) η ( ρ, θ ) , 1 g ρρ = ( ρ − ρ + )( ρ − ρ − )Λ( ρ, θ ) , 10

• 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   ( N 2 ) ′ T =   �   g ρρ N 2 4 π ρ = ρ + prime denotes derivative with respect to ρ and ρ + is the location of the outer horizon. • The area of horizon A h can be expressed as � π � g (0) θθ ( θ ) g (0) A h = 2 π φφ ( θ ) dθ 0 where g (0) θθ ( θ ) + ( ρ − ρ + ) g (1) g θθ ( ρ, θ ) = θθ ( θ ) + · · · , g (0) φφ ( θ ) + ( ρ − ρ + ) g (1) g φφ ( ρ, θ ) = φφ ( θ ) + · · · , 11

• EVH point: A h ∼ T ∼ ˜ ǫ • The above, demanding the regularity of the geometry, is possible if g (0) ǫ 2 , φφ ∼ ˜ ǫ s , ǫ v , ρ + ∼ ˜ ρ + − ρ − ∼ ˜ 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 N 2 ( ρ = 0) = 0 . • To avoid having a naked singularity and to keep A h /T finite d dρN 2 | ρ =0 � = 0 . 12

• In summary N 2 = 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 = ρ s ( θ ) , µ ( ρ s , θ ) = 0 touches the horizon ρ s ( 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: µdt 2 + dρ 2 � 2 + ˜ ds 2 = − ρ ˜ g θθ dθ 2 � N φ dt dφ + ˜ Λ + ρ ˜ g φφ ρ 2 ˜ where N φ and ˜ µ, ˜ ˜ • ˜ Λ , ˜ g θθ are analytic functions g φφ , of ( 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 : Λ(0 , θ ) = L 2 . µ (0 , θ )˜ ˜ 14

� EVH black holes have AdS 3 throats, the Proof • Under the near horizon limit ρ = ǫ 2 r 2 , ˜ ˜ t = ǫt , φ = ǫ ( φ − ˜ ωt ) , ǫ → 0 • The EVH geometry goes to t 2 + L 2 dr 2 � � ds 2 = a (˜ φ 2 + R 2 d ˜ − r 2 d ˜ θ ) r 2 d ˜ θ 2 r 2 + b (˜ θ ) where L, R are constants. • ˜ ω is the angular velocity of the horizon at EVH limit. • We note that ˜ φ ∈ [0 , 2 πǫ ]. 15

• 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 = V 0 , implies: db θ = 0 b = b 0 = const. , ⇒ d ˜ d 2 a θ 2 + 4 R 2 L 2 a − V 0 R 2 a 2 = 0 . d ˜ • For V 0 = 0 (ungauged gravity) case a = a 0 sin 2 R ˜ θ , L where C = 2 R L a 0 . E.o.M for Φ: √ 3 = g 0 tan R ˜ 2Φ d Φ 3 C θ √ θ = ± ⇒ e L , d ˜ a where g 0 is a constant. 16

• In summary, and for V = 0 case − r 2 dτ 2 + dr 2 � � r 2 + r 2 dψ 2 + 1 ds 2 = R 2 4 dθ 2 AdS 3 sin θ , 2Φ 3 = g 0 tan θ √ F µν = 0 , e 2 where • we have redefined ˜ θ such that θ ∈ [0 , π ] and, • ψ ∈ [0 , 2 πǫ ], i.e. the AdS 3 throat in the near horizon limit of the EVH black hole is a pinching AdS 3 . • A two parameter family of solutions; R AdS 3 , g 0 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 A h 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 − F ( r ) dτ 2 + dr 2 � � F ( r ) + r 2 ( dψ − r + r − r 2 dτ ) 2 + 1 ds 2 = R 2 4 dθ 2 AdS 3 sin θ where ( r 2 − r 2 + )( r 2 − r 2 − ) F ( r ) = . r 2 • i.e. the same as before but pinching AdS 3 is replaced by pinching BTZ. • The solution has now four parameters: R AdS 3 , g 0 , r ± . 18

Near horizon structure of the EVH black hole in flat or AdS background: • The Near horizon region is warped (AdS 3 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 ∝ N 0 , P ∝ N 6 . – have two different branches: ergo-branch PQ < G 4 J, ergo-free branch PQ > G 4 J . 20