Statistical black hole entropy and AdS 2 / CFT 1 Asymptotic symmetries and dual boundary theory Matteo Ciulu 17.5.2017 Universit´ a degli studi di Cagliari
Table of contents 1. Quantum gravity & BH thermodynamics 2. AdS / CFT correspondence 3. Hamiltonian & symmetries 4. 2D Dilaton gravity 5. Boundary theory 1
Quantum gravity & BH thermodynamics
Looking for a quantum gravity theory D = 4 metric degrees of freedom = 10 components - 4 diffeos -4 non-dynamical = 2 d.o.f • Gravity theory in 4 D is not a perturbative renormalizable theory ( [ G ] = − 2 in mass units). • We can interpret thoery of gravity as an Effective field theory : d 4 x √− g {− 2Λ + R + c 1 R 2 + c 2 R µν R µν + c 3 R µνρσ R µνρσ + . . . } 1 � S = 16 π G The theory needs an UV completion • No local observables ; • The graviton is not composite (Weinberg-Witten theorem) • Emergent spacetime . . . 2
BH Thermodynamics • Zero Law : the surface gravity κ is costant over the horizon; • First law :for any stationary black hole with mass M , angular momentum J and charge Q , it turns out to be κ 8 π G δ A + Ω δ J + φδ Q δ M = where Ω is the angular velocity and φ is electrostatic potential. • Second law : The Area A of the event horizon of a black hole never decreases δ A ≥ 0 • Third law :It is impossible to reduce, by any procedure, the surface gravity κ to zero in a finite number of steps. The correspondence between thermodynamic and black hole mechanics is complete if we identify: E → M S → A T → κ • Moreover Bekenstein found: S = η A � G 3
Hawking radiation � c 3 T hawking = κ 8 π GM ∼ 6 × 10 − 8 M 2 π = M ⊙ Can Hawking radiation be observed? • For stellar mass black hole eight orders of magnitude smaller than cosmic microwave background; • More important for primordial black holes; • Analogue of Hawking radiation in condensed matter system. The many derivations of Hawking radiation • Canonical quantization in curved space time (Hawking, 1975); • Path integral derivation (Hartle and Hawking, 1976); • KMS condition (Bisognano and Wichmann,1976); • Gravitational istantons(Gibbons and Hawking,1977); • Tunneling trough the horizon (T.Damour and R.Ruffini,1976; M.K. Parikh and F.Wilczek, 2000); 4
Black hole entropy � 2 A � M S BH ∼ 10 90 S BH = 4 � G 10 6 M ⊙ • No hair theorem(s) : Stationary, asymptotically, flat black hole solutions to general relativity coupled to electromagnetism that are nonsingular outside the event horizon are fully characterized by the parameters of mass, charge and spin. � S = − p n ln p n n Why classical black holes have entropy? • Problem of universality : A great many different models of black hole microphysics yeld the same thermodynamical proprieties; • Loss information paradox : black holes evaporate, emitting Hawking radiation, which contains less information than the one that was originally in the spacetime, therefore information is lost. 5
AdS / CFT correspondence
Holographic principle • (’t Hooft and Susskind) A bulk theory with gravity describing a macroscopic region of space is equivalent to a boundary theory without gravity living on the boundary of that region; • Susskind considered an approximately spherical distribution of matter that is not itself a black hole and that is contained in a closed surface of area A Let us suppose that the mass is induced to collapse to form a black hole, whose horizon area turns out to be smaller than A . The black hole entropy is therefore smaller than A 4 and the generalized second law implies the bound S ≤ A 4 6
AdS/CFT correspondence • The gauge/gravity correspondence ( duality ) is an exact relationship between any theory of quantum gravity in asymptotically AdS d +1 space ( the bulk ) and an ordinary CFT d without gravity ( the boundary ) ; • Each field φ propagating in a (d+1)-dimensional anti-de Sitter spacetime is related, through a one to one correspondence, to an operator O in a d-dimensional conformal field theory defined on the boundary of that space ( GKPW dictionary ). � � � Z grav [ φ i � d d x φ i 0 ( x ) O i ( x ) � � 0 ( x ); ∂ M ] = exp − CFT on ∂ M i This is UV complete!! • The mass of the bulk scalar is related to the scaling dimension of the CFT operator � d 2 ∆ = d m 2 = ∆( d − ∆) , 2 + 4 + m 2 l 2 • Thermal states in CFT are dual to black holes in quantum gravity Z [ φ 0 ; M ] = Z grav [ φ 0 , boundary = M ] 7
Strong/Weak duality Aside for certain examples, the corrispondence is well defined and useful only in certain limits. One realization which is understood in great details is: IIB strings on AdS 5 × S 5 = Yang-Mills in 4d with N = 4 supersymmetry The large symmetry group of 5d anti-de Sitter space matches precisely with the group of conformal symmetries of the N = 4 super Yang-Mills theory • gravity side: � √ g ( R + L matter + l 4 s R 4 + . . . ) S IIB ∼ parameters: l s , l p , l AdS • CFT side : SU ( N ) gauge fields + matter fields for supersymmetry. parameters: g YM , N λ = g 2 YM N • The mapping l d − 1 λ ∼ l 4 AdS AdS ∼ N 2 l 4 G N s 8
Hamiltonian & symmetries
Asymptotic symmetries GR is locally diff invariant, but it is not invariant under diff. that reach the boundary: δ ξ ( √− g L ) = � � dA µ ξ µ L M ∂ M symmetries Asymptotic Symmetry Group = trivial symmetries where trivial symmetry is one whose associated vashining conserved charges. Maxwell theory S = − 1 � d 4 x ( F µν F µν + A µ J µ matter ) 4 The action is invariant under trasformations: δ A µ = ∂ µ Λ( x ) , δφ = i Λ( x ) φ For Λ = const . dQ � � d 3 x J 0 d 2 xF tr dt = 0 Q ∼ Matter ∼ ASG = U (1) global Σ ∂ Σ 9
Hamiltonian in GR In Hamiltonian formalism the global charges appears as canonical generators of the asymptotic symmetries of the theory. Let us use as canonical variable h ij ( � x , t ) π ij ( � x , t ) and we parametrize the metric as: ds 2 = − N 2 dt 2 + h ij ( dx i + N i dt )( dx j + N j dr ) Consider the action � √− g ( R − 2Λ) d 4 x + � √ 1 1 ± h ( K − K 0 ) d 3 x , I = 16 π G 8 π G d 3 x √ σ u µ T µν ξ µ � � π ij ˙ � � d 4 x h ij − N H − N i H i I = − M ∂ M where T ij is the boundary (Brown-York) stress tensor : √ T ij = 1 δ I on − shell = 1 � 8 π ( K ij − h ij K ) − (background) − hT ij δ g ij 2 ∂ M wer can read off the Hamiltonian: d 2 x √ σ u µ T µν ξ µ � � � � d 3 x N H + N i H i − H [ ξ ] = Σ ∂ Σ 10
Asymptotic symmetries in GR • The bulk term vanishes on-shell. • The dynamics leaves ξ unspecified. This corresponds to a choice of time evolution: { H [ ξ ] , X } = L ξ X • General spacetimes do not have isometries, so no local conserved quantities. Asymptotic symmetries allow to define global conserved charges. • In General relativity ASG is generated by the conserved charges. { H [ ξ ] , H [ η ] } = H [[ ξ, η ]] + c ( ξ, η ) • In Minkowski spacetime the ASG is Poincar´ e group. • ASG leads to surprise. The isometry group of AdS d is SO ( D − 1 , 2). A natural guess is that the asymptotic simmetry group is the same. This is not true for D ≤ 3 ( Brown and Henneaux ) 11
2D Dilaton gravity
JT model • In 2D dimensions, the curvature tensor has only one independent component, since all nonzero component can be obtained by simmetry R µνρσ = 1 2 R ( g µλ g νρ − g µρ g νλ ) = ⇒ G µν = 0 • For formulating a theory endowed with a not trivial degree of freedom gravity theory coupled with a scalar: � d 2 x η ( R − 2 λ 2 ) . S = which admits BH solutions: ds 2 = − ( λ 2 x 2 − a 2 ) dt 2 + ( λ 2 x 2 − a 2 ) − 1 dx 2 , η = η 0 λ x • The action can be considered a dimensional reduction of an higher dimensional model: 2 ds 2 ( d +2) = ds 2 d d Ω 2 ( κ, d ) (2) + η • Thermodynamics: M = a 2 η 0 λ T = λ a S = 2 πη 0 a 2 2 π 12
Isometries • AdS 2 ( a 2 = 0) is a maximally symmetric space, it admits, therefore, three Killing vectors generating the SO (1 , 2) ∼ SL (2 , R ) group of isometries. χ (1) = 1 ∂ λ ∂ t χ (2) = t ∂ ∂ t − x ∂ ∂ x , 1 λ 4 x 2 ) ∂ ∂ t − 2 λ tx ∂ χ (3) = λ ( t 2 + ∂ x . • For a 2 � = 0 SL (2 , R ) symmetry is realized in a different way δη = L χ η = χ µ ∂ µ η. Symmetries of 2D spacetime are broken by the linear dilaton χ µ = F 0 ǫ µν ∂ ν η, SL (2 , R ) → T • This symmetry breaking pattern will gives rise to a central charge!! 13
Asymptotic symmetries • We define asymptotically AdS 2 if, for x → ∞ g tx ( t ) = γ tx ( t ) λ 2 x 2 + γ xx ( t ) 1 g tt = − λ 2 x 2 + γ tt + o ( x − 2 ) λ 3 x 3 + o ( x − 5 ) λ 4 x 4 + o ( x − 6 ) g xx = • This asymptotic form is preserved by: 2 λ 4 x 2 + α t ( t ) ¨ ǫ ( t ) χ t = ǫ ( t ) + + o ( x − 5 ) , x 4 ǫ ( t ) + α x ( t ) χ x = − x ˙ + o ( x − 2 ) x • The asymptotic behaviour of dilaton compatible with these trasformations is: � λρ ( t ) x + γ φφ ( t ) � + o ( x − 3 ) η = η 0 2 λ x 14
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