Tatsuma Nishioka (Kyoto,IPMU) based on PRD 77:064005,2008 with T. Azeyanagi and T. Takayanagi JHEP 0904:019,2009 with T. Hartman, K. Murata and A. Strominger JHEP 0905:077,2009 with G. Compere and K. Murata
Outline 1 Introduction 2 Entanglement entropy and black hole 3 Extremal black hole/cft correspondence 4 Summary
Introduction What is a black hole? solution of the Einstein equation R µν 1 2 Rg µν + Λg µν = 0 a region of space in which the gravitational field is so powerful that nothing, including electromagnetic radiation (e.g. visible light), can escape its pull after having fallen past its event horizon (from Wikipedia) It seems to exist in our universe!
Hawking temperature Consider a Schwarzschild black hole ( ds 2 = 1 2M ) dt 2 + dr 2 r 1 2M r + r 2 dω 2 2 Near the horizon r = 2M + ρ 2 /8M (and after changing t iτ) ds 2 = ρ2 16M 2 dτ 2 + dρ 2 + (2M) 2 dω 2 2 To avoid a conical singlurality at ρ = 0, we obtain the Hawking temperature τ τ + 1, T H = 1 T H 8πM Then, we can consider the thermodynamics of the black holes!
Black hole horizon and Entropy If a small mass dm is added to the black hole, the entropy increases ds BH = dm T H = d(4πm 2 ) S BH = π(2m) 2 Then the black hole has an entropy given by the Bekenstein-Hawking area law (Bekenstein 73, Hawking 74) S BH = Area(Horizon) 4 This law is appliciable to the other black holes including Kerr and Reissner-Nordstrom black holes
Extremal limit of Reissner-Nordstrom black holes The Reissner-Nordstrom black hole is a solution in Einstein-Maxwell theory The metric is given by ds 2 = f (r)dt 2 + dr 2 f (r) = 1 2M r + Q2 r 2 f (r) + r 2 dω 2 2 There are two horizons r ± as solutions to f (r) = 0 The extremal limit is defined by the situation where outer horizon coincides with inner horizon: r + = r
Extremal limit of Reissner-Nordstrom black holes It means that the near horizon region becomes AdS 2 S 2 : since f (r) (r r + ) 2 ρ 2 ds 2 = ρ 2 dt 2 + dρ2 ρ 2 + r 2 +dω 2 2 In this limit, the Hawking temperature is zero T H (r + r ) = 0 The Bekenstein-Hawking entropy does not vanish even at zero temperature S BH = πr 2 + The finite entropy at zero temperature should be related with the degeneracies of ground states
How to interpret black hole entropy? Naively, Black hole is one particular solution in general relativity Entropy is defined as log(degeneracy) S BH = 0?? We don t know inside the black hole Is there fundamental degrees of freedom inside it? it is mysterious that the entropy is proportional to the area of the black hole, not its volume its (microscopic) origin remains to be fully understood
Relation to condensed matter physics Recently, it is believed that a superconducting phase can be holographically realized by putting a charged scalar field on Reissner-Nordstrome black hole Black hole entropy degeneracy of ground state in CMP It would be important to understand the microscopic configuration in CMP Then, it is worth pursuing the black hole entropy problem!
Counting in string theory String theory compactified to 4 or 5 dimensions provides supersymmetric charged black holes microscopic construction as bound states of D-branes Black hole entropy is the degeneracies of open string (BPS) states between D-branes: S BH (Q) = S open (Q)
Attractor mechanism 4D N = 2 SUGRA is described by special geometry The theory depends on Kahler and complex structure moduli, but the entropy of the black hole is independent of the asymptotic value of moduli Φ Numerical plot ofφ r in AdS4 0.6 0.4 0.2 0.2 5 10 15 20 Α1 Α2 2. Q1 1 2 Q2 2 Φ0 0.346574 r 0.0150842 2.91723 r
AdS 3 /CFT 2 correspondence There exists the BTZ black hole in asymptotically AdS 3 space Brown and Henneaux define the asymptotic symmetry group in asymptotically AdS 3 space (Brown-Henneaux 86) The ASG turned out to be the Virasoro algebras Strominger identified these Virasoro algebras as those in dual CFT 2 in terms of the AdS/CFT correpondence (Strominger 98) S BTZ = S CFT2
OSV conjecture Higher derivative corrections can be easily taken into account in 4D N = 2 SUGRA The structure of the theory is very similar to that of topological string theory Ooguri, Strominger and Vafa rewrote the partition function of the black hole and derived the conjectured relation Z BH = Z top 2 with some relations between 4D SUGRA and topological strings (Ooguri-Strominger-Vafa 04)
Various approaches For specific case, there are several explanations Counting BPS states (SUSY BH) (Strominger-Vafa 96) Attractor mechanism (Extremal) (Ferrrara-Kallosh-Strominger 95, Sen 05 Goldstein-Iizuka-Jena-Trivedi 05) AdS 3 /CFT 2 (BTZ BH) (Strominger 97) Near horizon conformal symmetry (Carlip 98 99) OSV conjecture (SUSY, Higher derivative) (Ooguri-Strominger-Vafa 04) Remarkably, the extremality plays an important role even though the approaches are quite different.
Jigsaw puzzle
Two new approaches For extremal black holes, we propose two following approaches: Entanglement entropy (Azeyanagi-TN-Takayanagi 07): Black hole entropy originates from entanglement of two CFTs in the near horizon AdS 2 region Extremal black hole/cft correspondence (Hartman-Murata-TN-Strominger 08, Compere-Murata-TN 08): Black hole entropy is accounted for by the statistical entropy in dual CFT
Outline 1 Introduction 2 Entanglement entropy and black hole 3 Extremal black hole/cft correspondence 4 Summary
AdS 2 from general extremal black holes For any extremal (rotating) black holes constructed in the Einstein-Maxwell-scalar theory with topological terms in four and five dimensions, it is known that the near horizon geometry always have the SO(2, 1) symmetry (Kunduri-Lucietti-Reall, [arxiv:0705.4214], Astefanesei-Goldstein-Jena-Sen-Trivedi, [hep-th/0606244]) it means that the near horizon geometry will be of the form AdS 2 M d 2, where M d 2 is a compact manifold such as S d 2 In fact, the near horizon geometry of the Reissner-Nordstrom black hole is AdS 2 S 2 as we found before
AdS 2 in global coordinate AdS 2 spacetime ds 2 = R 2 dτ 2 + dσ 2 cos 2, σ π τ It has two time-like boundaries at σ = ± π 2 CQM 1 CQM 2 σ The gravity theory on AdS 2 is expected to be dual to two copies of conformal quantum mechanics CQM1 and CQM2 living on the two boundaries via AdS 2 /CFT 1 π π/2 π/2
Black hole entropy as entanglement We would like to claim that CQM1 and CQM2 are actually quantum mechanically entangled with each other To check the above statement we compute the entanglement entropy and see it is non-zero it is equivalent to the BH entropy
Entanglement entropy Divide the space manifold into two parts A and B H tot = H A H B The density matrix ρ A is defined by tracing out the Hilbert space for the subsystem B ρ A = Tr B ρ tot The entanglement entropy is defined by the von Neumann entropy S ent = Tr[ρ A log ρ A ]
Entanglement entropy in QFT The entanglement entropy in (d+1)-dim QFT includes UV divergences Its leading behavior is proportional to the area of the (d-1)-dim boundary A S ent Area( A) a d 1 Very similar to the Bekenstein-Hawking formula of the black-hole entropy B A S BH = Area(horizon) 4G N
Holographic entanglement entropy In the asymptotically AdS space, we can apply the holographic computation of entanglement entropy (Ryu-Takayanagi, [hep-th/0603001], [hep-th/0605073]) B A minimal surface The entropy is given by the formula S ent = Area(γ A) 4G N AdS r
Holographic entanglement entropy in AdS 2 /CFT 1 We regard the devided space A and B as CQM1 and CQM2 respectively Minimal surface is a codimension two surface in the bulk The holographic entanglement entropy becomes S ent = 1 4G (2) N Since the bulk is two dimensional, the minimal surface is a point
Relation to black hole entropy AdS 2 geometry appears as the near horizon geometry of the extremal black holes If we compute the entanglement entropy of this geometry, we have the following relation S ent = S BH because 2D Newton constant in the near horizon geometry is defined as 1 G (2) N = VolM d 2 G (4) N
Outline 1 Introduction 2 Entanglement entropy and black hole 3 Extremal black hole/cft correspondence 4 Summary
The Kerr/CFT correspondence Extremal Kerr black hole in 4D 2D (chiral) CFT (Guica-Hartman-Song-Strominger 08) Key points Extract Virasoro algebra from diffeomorphism Evaluate the central charge c of this Virasoro algebra Define the dual temperature T L Roughly speaking Kerr BH with ang. mom. J state J in CFT
Counting the entropy In 2D CFT, the entropy is easily calculated by the Cardy formula S CFT = π2 3 ct L once we know the central charge c and the temperature T L In the case of the extremal Kerr black hole, it was shown that S CFT = S BH We broaden this correspondence to general case!
Asymptotic symmetry group Under the diffeomorphism ξ, the metric shifts g µν g µν + µ ξ ν + ν ξ µ This changes the asymptotic form of the metric at spatial infinity The asymptoric symmetry group (ASG) of a spacetime is A symmetry which preserves the boundary conditions in the diffeomorphisms ASG is a part of the diffeomorphisms
Concept of ASG The boundary condition determines the family of the geometries L ξ g 0 up to BC ξ : Asymptotic Symmetry Group We require ASG includes the Virasoro algebra (not too strong) Conserved charge is finite (not too week)
How to choose boundary conditions? For the general extremal black holes, the near horizon geometries take the form ds 2 = Γ(θ) [ r 2 dt 2 + dr 2 ] + α(θ)dθ2 + γ(θ)(dφ + krdt) 2 r 2 with the entropy S BH = π 2 π 0 dθ Γ(θ)α(θ)γ(θ) We choose the boundary condition such that the ASG includes the Virasoro algebra the charges is finite
Construct dual CFT Taking the appropriate boundary condition, we obtaine the Virasoro algebra with the central charge c = 3k π 0 dθ Γ(θ)α(θ)γ(θ) The temperature is guessed by many cases as Then, the entropy in CFT is S CFT = π 2 π 0 T L = 1 2πk dθ Γ(θ)α(θ)γ(θ) = S BH This agrees exactly with the black hole entropy!
Outline 1 Introduction 2 Entanglement entropy and black hole 3 Extremal black hole/cft correspondence 4 Summary
Summary Black hole entropy should be accounted for in terms of the microscopic degrees of freedom For supersymmetric and extremal cases, there are several ways of counting Extremality seems to play a crucial role We propose two new approaches to extremal black holes