Wiggling Throat of Extremal Black Holes

Transcription

1 Wiggling Throat of Extremal Black Holes Ali Seraj School of Physics Institute for Research in Fundamental Sciences (IPM), Tehran, Iran Recent Trends in String Theory and Related Topics May 2016, IPM based on arxiv: and arxiv: together with G. Compère, K. Hajian, M.M. Sheikh-Jabbari Ali Seraj (IPM) IPM 1 / 21

2 Introduction Ali Seraj (IPM) IPM 2 / 21

3 Motivation Black holes in gravity Attractive nature of gravity Gravitational collapse Formation of horizon Black hole picture from Ali Seraj (IPM) IPM 3 / 21

4 Motivation Black hole Thermodynamics Laws of black hole mechanics [Bardeen et al. (1973)] Black holes are thermodynamic systems [Hawking(1973)] Black hole entropy [Bekenstein(1973)] Ali Seraj (IPM) IPM 4 / 21

5 Motivation Microscopic description of black hole entropy Boltzmann principle The microscopic origin of black hole entropy Ali Seraj (IPM) IPM 5 / 21

6 Motivation Microstates vs. No hair theorem No hair theorems in GR No hair theorems [Robinson(1975), Mazur(1982), Chrusciel et al. (2012)] Ali Seraj (IPM) IPM 6 / 21

7 Motivation Microstates vs. No hair theorem No hair theorems in GR No hair theorems [Robinson(1975), Mazur(1982), Chrusciel et al. (2012)] Microstates? Expensive approaches (String theory, Fuzzballs, Loop Q.Gr, etc.) Ali Seraj (IPM) IPM 6 / 21

8 Motivation Microstates vs. No hair theorem No hair theorems in GR No hair theorems [Robinson(1975), Mazur(1982), Chrusciel et al. (2012)] Microstates? Expensive approaches (String theory, Fuzzballs, Loop Q.Gr, etc.) Minimal resolution (GR compatible): Surface degrees of freedom Ali Seraj (IPM) IPM 6 / 21

9 Motivation Microstates vs. No hair theorem No hair theorems in GR No hair theorems [Robinson(1975), Mazur(1982), Chrusciel et al. (2012)] Microstates? Expensive approaches (String theory, Fuzzballs, Loop Q.Gr, etc.) Minimal resolution (GR compatible): Surface degrees of freedom Black hole thermodynamics is associated to the horizon T H = κ 2π, Area of horizon S = 4G N Ali Seraj (IPM) IPM 6 / 21

10 Motivation Surface degrees of freedom Horizon as an inner boundary Simultaneous breaking of diffeomorphism Surface gravitons as Goldstone modes How to detect these modes? Ali Seraj (IPM) IPM 7 / 21

11 Noether s theorems [E.Noether(1918)] Noether s theorem [E.Noether(1918)] Associated with each continuous symmetry of a theory, there exists a conserved charge (given suitable boundary conditions) Ali Seraj (IPM) IPM 7 / 21

12 Noether s theorems [E.Noether(1918)] Noether s theorem [E.Noether(1918)] Associated with each continuous symmetry of a theory, there exists a conserved charge (given suitable boundary conditions) Noether s Second theorem. Local symmetries imply Bianchi identities as strong equalities Using both above theorems imply global symmetries Conserved charge as a volume integral local symmetries Conserved charge as a surface integral Ali Seraj (IPM) IPM 7 / 21

13 Motivation Microstates at the horizon Using the charges we can distinguish states at the horizon which were previously gauge equivalent. Ali Seraj (IPM) IPM 8 / 21

14 Motivation Microstates at the horizon Using the charges we can distinguish states at the horizon which were previously gauge equivalent. These states have different names in the literature: Boundary gravitons [Brown-Henneaux (1986)] Ali Seraj (IPM) IPM 8 / 21

15 Motivation Microstates at the horizon Using the charges we can distinguish states at the horizon which were previously gauge equivalent. These states have different names in the literature: Boundary gravitons [Brown-Henneaux (1986)] Edge states [Balachandran et al. (1994),Carlip (2005)] Ali Seraj (IPM) IPM 8 / 21

16 Motivation Microstates at the horizon Using the charges we can distinguish states at the horizon which were previously gauge equivalent. These states have different names in the literature: Boundary gravitons [Brown-Henneaux (1986)] Edge states [Balachandran et al. (1994),Carlip (2005)] Soft gravitons [Strominger et al. ( )] Ali Seraj (IPM) IPM 8 / 21

17 Motivation Microstates at the horizon Using the charges we can distinguish states at the horizon which were previously gauge equivalent. These states have different names in the literature: Boundary gravitons [Brown-Henneaux (1986)] Edge states [Balachandran et al. (1994),Carlip (2005)] Soft gravitons [Strominger et al. ( )] Symplectic symmetries [ G.Compère, K.Hajian, A.S. and M. M.Sheikh-Jabbari, JHEP 1510, 093 (2015).] Ali Seraj (IPM) IPM 8 / 21

18 Motivation Microstates at the horizon Using the charges we can distinguish states at the horizon which were previously gauge equivalent. These states have different names in the literature: Boundary gravitons [Brown-Henneaux (1986)] Edge states [Balachandran et al. (1994),Carlip (2005)] Soft gravitons [Strominger et al. ( )] Symplectic symmetries [ G.Compère, K.Hajian, A.S. and M. M.Sheikh-Jabbari, JHEP 1510, 093 (2015).] Adiabatic modes [Mehrdad Mirbabayi, Marko Simonovi (2016)] Ali Seraj (IPM) IPM 8 / 21

19 Extremal black holes and Near horizon geometry Ali Seraj (IPM) IPM 9 / 21

20 Motivation Extremal Rotating Black Hole Extremal black holes Vanishing Hawking temperature M Minimum energy for given value of angular momenta M = M(J i ) Degenerate Killing horizon Finite entropy T H 0 = S S 0 Extremal black hole Naked Singularity Figure: Parameter space of Kerr black hole Ali Seraj (IPM) IPM 10 / 21

21 Motivation Near Horizon Geometry Approach Ali Seraj (IPM) IPM 11 / 21

22 Motivation Near Horizon Geometry Approach In the extremal case, the near horizon region can be decoupled using the near horizon limit Ali Seraj (IPM) IPM 11 / 21

23 Motivation Near Horizon Geometry Approach In the extremal case, the near horizon region can be decoupled using the near horizon limit Enhancement of symmetries in NHEG SL(2, R) U(1) N Ali Seraj (IPM) IPM 11 / 21

24 Motivation Near Horizon Geometry Approach In the extremal case, the near horizon region can be decoupled using the near horizon limit Enhancement of symmetries in NHEG SL(2, R) U(1) N Near horizon limit picture from: Tom Hartman Ali Seraj (IPM) IPM 11 / 21

25 Motivation Near Horizon Geometry Approach In the extremal case, the near horizon region can be decoupled using the near horizon limit Enhancement of symmetries in NHEG SL(2, R) U(1) N Near horizon limit picture from: Tom Hartman [ ds 2 = Γ(θ) r 2 dt 2 + dr2 d 3 r 2 + ] dθ2 + γ ij (θ)(dϕ i + k i rdt)(dϕ j + k j rdt) i,j=1 Ali Seraj (IPM) IPM 11 / 21

26 Dynamics in the near horizon geometry Kerr/CFT proposal A dynamical duality between asymptotic NHEK geometries and a two dimensional chiral CFT [Strominger et.al. 09] Ali Seraj (IPM) IPM 12 / 21

27 Dynamics in the near horizon geometry Kerr/CFT proposal A dynamical duality between asymptotic NHEK geometries and a two dimensional chiral CFT [Strominger et.al. 09] No Dynamics in NHEK [Reall et al. (2009), Horowitz et al. (2009)] Ali Seraj (IPM) IPM 12 / 21

28 Dynamics in the near horizon geometry Kerr/CFT proposal A dynamical duality between asymptotic NHEK geometries and a two dimensional chiral CFT [Strominger et.al. 09] No Dynamics in NHEK [Reall et al. (2009), Horowitz et al. (2009)] Conclusion. The only states penetrating the near horizon geometry are the soft states Ali Seraj (IPM) IPM 12 / 21

29 Phase Space of Near Horizon Extremal Geometries Ali Seraj (IPM) IPM 13 / 21

30 NHEG Phase Space Field Configurations Start from the background metric ḡ µν : [ ds 2 = Γ(θ) r 2 dt 2 + dr2 d 3 r 2 + ] dθ2 + γ ij (θ)(dϕ i + k i rdt)(dϕ j + k j rdt) i,j=1 Ali Seraj (IPM) IPM 14 / 21

31 NHEG Phase Space Field Configurations Start from the background metric ḡ µν : [ ds 2 = Γ(θ) r 2 dt 2 + dr2 d 3 r 2 + ] dθ2 + γ ij (θ)(dϕ i + k i rdt)(dϕ j + k j rdt) i,j=1 Soft modes are given by g (ɛ) µν = ḡ µν + (µ ξ ν) Ali Seraj (IPM) IPM 14 / 21

32 NHEG Phase Space Field Configurations Start from the background metric ḡ µν : [ ds 2 = Γ(θ) r 2 dt 2 + dr2 d 3 r 2 + ] dθ2 + γ ij (θ)(dϕ i + k i rdt)(dϕ j + k j rdt) i,j=1 Soft modes are given by g (ɛ) µν = ḡ µν + (µ ξ ν) By some physical arguments, we fix ξ to be ξ[ɛ( ϕ)] = ɛ(ϕ i )k i ϕ i (k i ϕ iɛ) ( 1 r t + r r ) Ali Seraj (IPM) IPM 14 / 21

33 NHEG Phase Space Field Configurations Start from the background metric ḡ µν : [ ds 2 = Γ(θ) r 2 dt 2 + dr2 d 3 r 2 + ] dθ2 + γ ij (θ)(dϕ i + k i rdt)(dϕ j + k j rdt) i,j=1 Soft modes are given by g (ɛ) µν = ḡ µν + (µ ξ ν) By some physical arguments, we fix ξ to be ξ[ɛ( ϕ)] = ɛ(ϕ i )k i ϕ i (k i ϕ iɛ) ( 1 r t + r r ) g (ɛ) µν is smooth at poles Ali Seraj (IPM) IPM 14 / 21

34 NHEG Phase Space Field Configurations Generic geometries in the phase space are obtained by exponentiating the infinitesimal transformations Ali Seraj (IPM) IPM 15 / 21

35 NHEG Phase Space Field Configurations Generic geometries in the phase space are obtained by exponentiating the infinitesimal transformations ϕ i = ϕ i + k i F ( ϕ), r = re Ψ( ϕ), t = t eψ( ϕ) 1, r where e Ψ 1 + k F. Therefore ḡ µν g µν [Ψ(ϕ i )] Ali Seraj (IPM) IPM 15 / 21

36 NHEG Phase Space Field Configurations Generic geometries in the phase space are obtained by exponentiating the infinitesimal transformations ϕ i = ϕ i + k i F ( ϕ), r = re Ψ( ϕ), t = t eψ( ϕ) 1, r where e Ψ 1 + k F. Therefore ḡ µν g µν [Ψ(ϕ i )] NHEG phase space: M = { } g µν [Ψ(ϕ i )], Ψ(ϕ i ) periodic Ali Seraj (IPM) IPM 15 / 21

37 Covariant phase space description of Gravity Ali Seraj (IPM) IPM 16 / 21

38 Covariant phase space description of Gravity Using the covariant phase space method, one can make M a phase space M Γ = (M, Ω) Ali Seraj (IPM) IPM 16 / 21

39 Covariant phase space description of Gravity Using the covariant phase space method, one can make M a phase space M Γ = (M, Ω) Symplectic structure Ω is constructed covariantly through Ω(δ 1 φ, δ 2 φ) = ω(φ, δ 1 φ, δ 2 φ) where ω is constructed from the second variation of Lagrangian Σ Ali Seraj (IPM) IPM 16 / 21

40 Covariant phase space description of Gravity Using the covariant phase space method, one can make M a phase space M Γ = (M, Ω) Symplectic structure Ω is constructed covariantly through Ω(δ 1 φ, δ 2 φ) = ω(φ, δ 1 φ, δ 2 φ) where ω is constructed from the second variation of Lagrangian Σ Symmetries. ξ such that φ + L ξ φ M, φ M Conserved Charges are defined through δq ξ = ω(φ, δφ, L ξ φ) Σ Ali Seraj (IPM) IPM 16 / 21

41 NHEG Phase Space Symmetry Algebra Vectors ξ[ɛ( ϕ)] are also symmetries of the phase space Fourier expansion ξ = n c n ξ n [ξ n, ξ m ] = i k ( n m) ξ n+ m d = 4 Witt algebra d = 5 See figure Infinitely many Virasoro sub algebras k Q n vs. k Q n U(1) factors Ali Seraj (IPM) IPM 17 / 21

42 NHEG Phase Space Algebra of Charges: The NHEG Algebra There is a well defined symplectic structure on this phase space Ali Seraj (IPM) IPM 18 / 21

43 NHEG Phase Space Algebra of Charges: The NHEG Algebra There is a well defined symplectic structure on this phase space Algebra of charges is a central extension of the algebra of symmetry vectors Ali Seraj (IPM) IPM 18 / 21

44 NHEG Phase Space Algebra of Charges: The NHEG Algebra There is a well defined symplectic structure on this phase space Algebra of charges is a central extension of the algebra of symmetry vectors Quantized Algebra after { } 1 i [ ], H n L n, [L m, L n ] = k ( m n)l m+ n + S 2π ( k m) 3 δ m+ n,0 Ali Seraj (IPM) IPM 18 / 21

45 NHEG Phase Space Algebra of Charges: The NHEG Algebra There is a well defined symplectic structure on this phase space Algebra of charges is a central extension of the algebra of symmetry vectors Quantized Algebra after { } 1 i [ ], H n L n, [L m, L n ] = k ( m n)l m+ n + S 2π ( k m) 3 δ m+ n,0 In d = 4 dimensions, it is a chiral Virasoro algebra In d 5 we have an extended Virasoso algebra Ali Seraj (IPM) IPM 18 / 21

46 NHEG Phase Space Charges Charge H n [Ψ] : Generator of ξ n over the field configuration g µν [Ψ(ϕ i )] H n = ɛ T [Ψ]e i n ϕ Charges are Fourier modes of a Liouville-type stress tensor T [Ψ] = 1 ((Ψ ) 2 2Ψ + 2e 2Ψ) 16πG Ψ is a primary field of weight one δ ɛ Ψ = ɛ Ψ + ɛ, T [Ψ] is a quasi-primary of weight two δ ɛ T = ɛ T + 2ɛ T 1 8πG ɛ However Ψ = Ψ(ϕ i ), Ψ = k Ψ Ali Seraj (IPM) IPM 19 / 21

47 Summary and Outlook Summary We constructed the classical phase space of extermal black holes We obtrained the symmetry algebra (The NHEG algebra) We obtained the exact form of charges on the phase space Outlook Look for a field theory with the same charges (it should be much similar to Liouville theory) Is there a notion of modular invariance here? Look for a Cardy like formula for counting of states? Is the black hole entropy reproduced? Ali Seraj (IPM) IPM 20 / 21

48 Thank you for your attention Ali Seraj (IPM) IPM 21 / 21

49 Details Infinitesimal phase space transformation χ[ɛ( ϕ)] = ɛ(ϕ i )k i ϕ i (k i ϕ iɛ) ( b r t + r r ) (1) Phase space field configurations [ ( dr ) 2 ] ds 2 = Γ(θ) (σ dψ) 2 + r dψ + dθ 2 + γ ij (d ϕ i + k i σ)(d ϕ j + k j σ) where τ = t + 1 r and σ = e Ψ rdτ + dr r, ϕi = ϕ i + k i (F Ψ). (2) Ali Seraj (IPM) IPM 21 / 21