The Kerr/CFT correspondence A bird-eye view

Transcription

1 The Kerr/CFT correspondence A bird-eye view Iberian Strings 2012, Bilbao, February 2nd, Geoffrey Compère Universiteit van Amsterdam G. Compère (UvA) 1 / 54

2 Aim of the Kerr/CFT correspondence The Kerr/CFT correspondence proposes that (at least part of) the dynamics of rotating black holes can be identified with the dynamics of (a close cousin of) a conformal field theory (CFT). Rotating black holes CFTs G. Compère (UvA) 2 / 54

3 Aim of the Kerr/CFT correspondence The Kerr/CFT correspondence proposes that (at least part of) the dynamics of rotating black holes can be identified with the dynamics of (a close cousin of) a conformal field theory (CFT). Rotating black holes CFTs G. Compère (UvA) 2 / 54

4 Aim of the Kerr/CFT correspondence The Kerr/CFT correspondence proposes that (at least part of) the dynamics of rotating black holes can be identified with the dynamics of (a close cousin of) a conformal field theory (CFT). Rotating black holes CFTs Metric g µν,...?..., OPEs, Operators Angular momentum J,...? T L, T R Temperatures Symmetries U(1) U(1)? Virasoro Virasoro Entropy? Entropy S BH (M, J,... ) = A 4G +... S CFT = π2 3 (c RT R + c L T L ) Dynamics of probes? Correlators φ = 0,......, O 1 (0)O 2 (x) G. Compère (UvA) 2 / 54

5 Aim of the Kerr/CFT correspondence The Kerr/CFT correspondence proposes that (at least part of) the dynamics of rotating black holes can be identified with the dynamics of (a close cousin of) a conformal field theory (CFT) CFTs?????..., OPEs, Operators T L, T R Temperatures Virasoro Virasoro Entropy S CFT = π2 3 (c RT R + c L T L ) Correlators..., O 1 (0)O 2 (x) G. Compère (UvA) 2 / 54

6 Aim of the Kerr/CFT correspondence The Kerr/CFT correspondence proposes that (at least part of) the dynamics of rotating black holes can be identified with the dynamics of (a close cousin of) a conformal field theory (CFT). Rotating black holes CFTs Metric g µν,...?..., OPEs, Operators Angular momentum J,...? T L, T R Temperatures Symmetries U(1) U(1)? Virasoro Virasoro Entropy? Entropy S BH (M, J,... ) = A 4G +... S CFT = π2 3 (c RT R + c L T L ) Dynamics of probes? Correlators φ = 0,......, O 1 (0)O 2 (x) G. Compère (UvA) 2 / 54

7 Why do we care? List of open physics problems in quantum gravity : (1) Compute the value of the cosmological constant (2) Solve Hawking s paradox (3) Resolve the cosmological singularity (4) Describe the microstates of a realistic black hole The Kerr/CFT correspondence has the potential to help solving (4). G. Compère (UvA) 3 / 54

8 Why do we care? List of open physics problems in quantum gravity : (1) Compute the value of the cosmological constant (2) Solve Hawking s paradox (3) Resolve the cosmological singularity (4) Describe the microstates of a realistic black hole The Kerr/CFT correspondence has the potential to help solving (4). G. Compère (UvA) 3 / 54

9 Why do we care? List of open physics problems in quantum gravity : (4) Describe the microstates of a realistic black hole The Kerr/CFT correspondence has the potential to help solving (4). G. Compère (UvA) 3 / 54

10 What is the main idea? Extreme rotating black holes are IR fixed points (characterized by universal properties) that control the dynamics of non-extreme black holes. G. Compère (UvA) 4 / 54

11 What is the main idea? Extreme rotating black holes are IR fixed points (characterized by universal properties) that control the dynamics of non-extreme black holes. G. Compère (UvA) 4 / 54

12 Diclaimer Is the Kerr/CFT correspondence based on string theory? No! This is an effective (bottom-up) model but one can build UV complete models in string theory. (Current models use non-realistic string compactifications.) G. Compère (UvA) 5 / 54

13 Diclaimer Is the Kerr/CFT correspondence based on string theory? No! This is an effective (bottom-up) model but one can build UV complete models in string theory. (Current models use non-realistic string compactifications.) G. Compère (UvA) 5 / 54

14 Diclaimer Is the Kerr/CFT correspondence based on string theory? No! This is an effective (bottom-up) model but one can build UV complete models in string theory. (Current models use non-realistic string compactifications.) G. Compère (UvA) 5 / 54

15 Diclaimer Is the Kerr/CFT correspondence based on string theory? No! This is an effective (bottom-up) model but one can build UV complete models in string theory. (Current models use non-realistic string compactifications.) G. Compère (UvA) 5 / 54

16 Derivation of Kerr/CFT in three steps ❸ Far from extremality : Hidden conformal symmetry of the Kerr black hole, Castro, Maloney, Strominger, T BH 0 ❷ Close to extremality : Black Hole Superradiance from Kerr/CFT, Bredberg, Hartman, Song, Strominger, T BH = 0 ❶ At extremality : The Kerr/CFT correspondence, Guica, Hartman, Song, Strominger, G. Compère (UvA) 6 / 54

17 Derivation of Kerr/CFT in three steps ❸ Far from extremality : Hidden conformal symmetry of the Kerr black hole, Castro, Maloney, Strominger, T BH 0 ❷ Close to extremality : Black Hole Superradiance from Kerr/CFT, Bredberg, Hartman, Song, Strominger, T BH = 0 ❶ At extremality : The Kerr/CFT correspondence, Guica, Hartman, Song, Strominger, G. Compère (UvA) 6 / 54

18 Derivation of Kerr/CFT in three steps ❸ Far from extremality : Hidden conformal symmetry of the Kerr black hole, Castro, Maloney, Strominger, T BH 0 ❷ Close to extremality : Black Hole Superradiance from Kerr/CFT, Bredberg, Hartman, Song, Strominger, T BH = 0 ❶ At extremality : The Kerr/CFT correspondence, Guica, Hartman, Song, Strominger, G. Compère (UvA) 6 / 54

19 Plan : Derivation of Kerr/CFT in three steps ❸ Far from extremality : Dynamics of probes has broken SL(2, R) SL(2, R) symmetry. Entropy still matches! T BH 0 T BH = 0 ❷ Close to extremality : Dynamics of probes can be obtained from a CFT two-point correlation function. ❶ At extremality : Temperatures T L, T R, Virasoro algebra, Match S BH = S CFT! G. Compère (UvA) 7 / 54

20 Plan : Derivation of Kerr/CFT in three steps ❸ Far from extremality : Dynamics of probes has broken SL(2, R) SL(2, R) symmetry. Entropy still matches! T BH 0 T BH = 0 ❷ Close to extremality : Dynamics of probes can be obtained from a CFT two-point correlation function. ❶ At extremality : Temperatures T L, T R, Virasoro algebra, Match S BH = S CFT! G. Compère (UvA) 7 / 54

21 Plan : Derivation of Kerr/CFT in three steps ❸ Far from extremality : Dynamics of probes has broken SL(2, R) SL(2, R) symmetry. Entropy still matches! T BH 0 T BH = 0 ❷ Close to extremality : Dynamics of probes can be obtained from a CFT two-point correlation function. ❶ At extremality : Temperatures T L, T R, Virasoro algebra, Match S BH = S CFT! G. Compère (UvA) 7 / 54

22 First step of the Kerr/CFT correspondence T BH 0 T BH = 0 ❶ At extremality : Temperatures T L, T R, Virasoro algebra, Match S BH = S CFT! G. Compère (UvA) 8 / 54

23 Extremal black holes are nearly physical Theoretical bounds on the Kerr parameters : J GM 2 1 J GM Observed : J GM 2 > 0.98 J GM 2 > 0.98 J GM 2 > 0.95 no naked singularity [Thorne] from details of matter [GRS (2006)] M 10 15M [MCG (2006)] M M [Cygnus X-1 (2011)] M 14.8M G. Compère (UvA) 9 / 54

24 Extremal black holes are nearly physical Theoretical bounds on the Kerr parameters : J GM 2 1 J GM Observed : J GM 2 > 0.98 J GM 2 > 0.98 J GM 2 > 0.95 no naked singularity [Thorne] from details of matter [GRS (2006)] M 10 15M [MCG (2006)] M M [Cygnus X-1 (2011)] M 14.8M G. Compère (UvA) 9 / 54

25 Two key statements 1 The extremal Kerr black hole can be described by a field theory at the (dimensionless) temperatures T L = 1 2π, T R = 0 2 The axial symmetry can be extended to a Virasoro asymptotic symmetry algebra close to the horizon with central charge c = 12J 6N The black hole entropy can then be reproduced by a CFT counting S BH = A 4G = 2πJ! = π2 3 (c T L + c T R ) = S CFT G. Compère (UvA) 10 / 54

26 Two key statements 1 The extremal Kerr black hole can be described by a field theory at the (dimensionless) temperatures T L = 1 2π, T R = 0 2 The axial symmetry can be extended to a Virasoro asymptotic symmetry algebra close to the horizon with central charge c = 12J 6N The black hole entropy can then be reproduced by a CFT counting S BH = A 4G = 2πJ! = π2 3 (c T L + c T R ) = S CFT G. Compère (UvA) 10 / 54

27 Two key statements 1 The extremal Kerr black hole can be described by a field theory at the (dimensionless) temperatures T L = 1 2π, T R = 0 2 The axial symmetry can be extended to a Virasoro asymptotic symmetry algebra close to the horizon with central charge c = 12J 6N The black hole entropy can then be reproduced by a CFT counting S BH = A 4G = 2πJ! = π2 3 (c T L + c T R ) = S CFT G. Compère (UvA) 10 / 54

28 The Kerr metric [Roy Kerr, 1963] ds 2 = r2 2GMr + a 2 r 2 + a 2 cos 2 θ (dt a sin2 θdφ) 2 + (r 2 + a 2 cos 2 θ)dθ 2 + r2 + a 2 cos 2 θ sin 2 θ r 2 2GMr + a 2 dr2 + r 2 + a 2 cos 2 θ ((r2 + a 2 )dφ adt) 2 where a = J M. Extremality is reached when J = GM2 or a = GM. Then T H = 0, Ω H = 1 2GM, S BH = 2π J, J N 2 Take the near-horizon limit λ 0 : t near = λt, r near = r GM, φ near = φ Ω λ H t G. Compère (UvA) 11 / 54

29 The Kerr metric [Roy Kerr, 1963] ds 2 = r2 2GMr + a 2 r 2 + a 2 cos 2 θ (dt a sin2 θdφ) 2 + (r 2 + a 2 cos 2 θ)dθ 2 + r2 + a 2 cos 2 θ sin 2 θ r 2 2GMr + a 2 dr2 + r 2 + a 2 cos 2 θ ((r2 + a 2 )dφ adt) 2 where a = J M. Extremality is reached when J = GM2 or a = GM. Then T H = 0, Ω H = 1 2GM, S BH = 2π J, J N 2 Take the near-horizon limit λ 0 : t near = λt, r near = r GM, φ near = φ Ω λ H t G. Compère (UvA) 11 / 54

30 The Kerr metric [Roy Kerr, 1963] ds 2 = r2 2GMr + a 2 r 2 + a 2 cos 2 θ (dt a sin2 θdφ) 2 + (r 2 + a 2 cos 2 θ)dθ 2 + r2 + a 2 cos 2 θ sin 2 θ r 2 2GMr + a 2 dr2 + r 2 + a 2 cos 2 θ ((r2 + a 2 )dφ adt) 2 where a = J M. Extremality is reached when J = GM2 or a = GM. Then T H = 0, Ω H = 1 2GM, S BH = 2π J, J N 2 Take the near-horizon limit λ 0 : t near = λt, r near = r GM, φ near = φ Ω λ H t G. Compère (UvA) 11 / 54

31 The Kerr metric [Roy Kerr, 1963] ds 2 = r2 2GMr + a 2 r 2 + a 2 cos 2 θ (dt a sin2 θdφ) 2 + (r 2 + a 2 cos 2 θ)dθ 2 + r2 + a 2 cos 2 θ sin 2 θ r 2 2GMr + a 2 dr2 + r 2 + a 2 cos 2 θ ((r2 + a 2 )dφ adt) 2 where a = J M. Extremality is reached when J = GM2 or a = GM. Then T H = 0, Ω H = 1 2GM, S BH = 2π J, J N 2 Take the near-horizon limit λ 0 : t near = λt, r near = r GM, φ near = φ Ω λ H t G. Compère (UvA) 11 / 54

32 The extremal Kerr causal structure G. Compère (UvA) 12 / 54

33 Near-horizon region of extremal Kerr [Bardeen-Horowitz, 1999] ( ) ds 2 = J(1+cos 2 θ) ( r 2 dt 2 + dr2 r 2 )+dθ2 + 4 sin2 θ (1 + cos 2 (dφ + rdt)2 θ) 2 Decoupling between horizon and flat asymptotics This spacetime is geodesically complete. Enhancement to SL(2, R) U(1) symmetry [Kunduri, Lucietti, Reall] r c r, t t/c t φ reversal invariance ξ = t t r r G. Compère (UvA) 13 / 54

34 Near-horizon region of extremal Kerr [Bardeen-Horowitz, 1999] ( ) ds 2 = J(1+cos 2 θ) ( r 2 dt 2 + dr2 r 2 )+dθ2 + 4 sin2 θ (1 + cos 2 (dφ + rdt)2 θ) 2 Decoupling between horizon and flat asymptotics This spacetime is geodesically complete. Enhancement to SL(2, R) U(1) symmetry [Kunduri, Lucietti, Reall] r c r, t t/c t φ reversal invariance ξ = t t r r G. Compère (UvA) 13 / 54

35 Near-horizon region of extremal Kerr [Bardeen-Horowitz, 1999] ( ) ds 2 = J(1+cos 2 θ) ( r 2 dt 2 + dr2 r 2 )+dθ2 + 4 sin2 θ (1 + cos 2 (dφ + rdt)2 θ) 2 Decoupling between horizon and flat asymptotics This spacetime is geodesically complete. Enhancement to SL(2, R) U(1) symmetry [Kunduri, Lucietti, Reall] r c r, t t/c t φ reversal invariance ξ = t t r r G. Compère (UvA) 13 / 54

36 Near-horizon region of extremal Kerr [Bardeen-Horowitz, 1999] ( ) ds 2 = J(1+cos 2 θ) ( r 2 dt 2 + dr2 r 2 )+dθ2 + 4 sin2 θ (1 + cos 2 (dφ + rdt)2 θ) 2 Decoupling between horizon and flat asymptotics This spacetime is geodesically complete. Enhancement to SL(2, R) U(1) symmetry [Kunduri, Lucietti, Reall] r c r, t t/c t φ reversal invariance ξ = t t r r G. Compère (UvA) 13 / 54

37 Near-horizon region of extremal Kerr [Bardeen-Horowitz, 1999] ( ) ds 2 = J(1+cos 2 θ) ( r 2 dt 2 + dr2 r 2 )+dθ2 + 4 sin2 θ (1 + cos 2 (dφ + rdt)2 θ) 2 Decoupling between horizon and asymptotics Geometry is a warped product AdS 2 warped S 2. [Conjecture] Any solution asymptotic to this geometry is diffeomorphic to it. [Amstel, Horowitz, Marolf, Roberts] G. Compère (UvA) 14 / 54

38 Near-horizon region of extremal Kerr [Bardeen-Horowitz, 1999] ( ) ds 2 = J(1+cos 2 θ) ( r 2 dt 2 + dr2 r 2 )+dθ2 + 4 sin2 θ (1 + cos 2 (dφ + rdt)2 θ) 2 Decoupling between horizon and asymptotics Geometry is a warped product AdS 2 warped S 2. [Conjecture] Any solution asymptotic to this geometry is diffeomorphic to it. [Amstel, Horowitz, Marolf, Roberts] G. Compère (UvA) 14 / 54

39 Temperature of extremal black holes T H = 0, but... First law suggests the existence of a physical temperature δs = ext = 1 (δm Ω T H δj) H 1 (( M T H J ) ext Ω H )δj = 1 δj T φ This suggests that excitations of extreme rotating black holes lie along φ. One can then argue that there is a 2d QFT description with T L = T φ = 1 2π, T R = 0 G. Compère (UvA) 15 / 54

40 Temperature of extremal black holes T H = 0, but... First law suggests the existence of a physical temperature δs = ext = 1 (δm Ω T H δj) H 1 (( M T H J ) ext Ω H )δj = 1 δj T φ This suggests that excitations of extreme rotating black holes lie along φ. One can then argue that there is a 2d QFT description with T L = T φ = 1 2π, T R = 0 G. Compère (UvA) 15 / 54

41 One more statement to make 1 The extremal Kerr black hole can be described by a field theory at the (dimensionless) temperature T L = 1 2π, T R = 0 2 The axial symmetry can be extended to a Virasoro asymptotic symmetry algebra close to the horizon with central charge c = 12J 6N The black hole entropy can then be reproduced by a CFT counting S BH = A 4G = 2πJ! = π2 3 c T L = S CFT G. Compère (UvA) 16 / 54

42 Interlude : detour to 3d gravity! (Main argument to microscopically derive black hole entropy in string theory!) G. Compère (UvA) 17 / 54

43 Remember : Gravity in AdS 3 Near-horizon geometry of many supersymmetric black holes is asymptotic to ds 2 = l 2 AdS ) ( (1 + r 2 )dt 2 + dr2 1 + r 2 + r2 dφ 2 Semi-classical analysis valid for l AdS G 3. Conformal boundary is a cylinder (t, φ) : x ± = t ± φ with conformal symmetries ξ L = L(x + ) + rl (x + ) r and ξ R = L(x ) r L (x ) r extending SL(2, R) L SL(2, R) R Boundary conditions g µν = O(... ) exist such that charges L n ξ L (L(x + ) = e inx+ ), Ln ξ R ( L(x ) = e inx ), are finite, well-defined and conserved. G. Compère (UvA) 18 / 54

44 Remember : Gravity in AdS 3 Near-horizon geometry of many supersymmetric black holes is asymptotic to ds 2 = l 2 AdS ) ( (1 + r 2 )dt 2 + dr2 1 + r 2 + r2 dφ 2 Semi-classical analysis valid for l AdS G 3. Conformal boundary is a cylinder (t, φ) : x ± = t ± φ with conformal symmetries ξ L = L(x + ) + rl (x + ) r and ξ R = L(x ) r L (x ) r extending SL(2, R) L SL(2, R) R Boundary conditions g µν = O(... ) exist such that charges L n ξ L (L(x + ) = e inx+ ), Ln ξ R ( L(x ) = e inx ), are finite, well-defined and conserved. G. Compère (UvA) 18 / 54

45 Remember : Gravity in AdS 3 Near-horizon geometry of many supersymmetric black holes is asymptotic to ds 2 = l 2 AdS ) ( (1 + r 2 )dt 2 + dr2 1 + r 2 + r2 dφ 2 Semi-classical analysis valid for l AdS G 3. Conformal boundary is a cylinder (t, φ) : x ± = t ± φ with conformal symmetries ξ L = L(x + ) + rl (x + ) r and ξ R = L(x ) r L (x ) r extending SL(2, R) L SL(2, R) R Boundary conditions g µν = O(... ) exist such that charges L n ξ L (L(x + ) = e inx+ ), Ln ξ R ( L(x ) = e inx ), are finite, well-defined and conserved. G. Compère (UvA) 18 / 54

46 Remember : Gravity in AdS 3 Near-horizon geometry of many supersymmetric black holes is asymptotic to ds 2 = l 2 AdS ) ( (1 + r 2 )dt 2 + dr2 1 + r 2 + r2 dφ 2 Semi-classical analysis valid for l AdS G 3. Conformal boundary is a cylinder (t, φ) : x ± = t ± φ with conformal symmetries ξ L = L(x + ) + rl (x + ) r and ξ R = L(x ) r L (x ) r extending SL(2, R) L SL(2, R) R Boundary conditions g µν = O(... ) exist such that charges L n ξ L (L(x + ) = e inx+ ), Ln ξ R ( L(x ) = e inx ), are finite, well-defined and conserved. G. Compère (UvA) 18 / 54

47 [Brown-Henneaux, 1986] Algebra of charges defined via a Dirac bracket : i{l m, L n } = (m n)l m+n + c L 12 m(m2 1)δ m, n i{ L m, L n } = (m n) L m+n + c R 12 m(m2 1)δ m, n i{l m, L n } = 0 Virasoro algebra with central charges c L = c R Einstein = 3l AdS 2G 3 [Banados, Teitelboim, Zanelli, 1993] BTZ black holes have L 0 Einstein = M l AdS J, L0 Einstein = M + l AdS J G. Compère (UvA) 19 / 54

48 [Brown-Henneaux, 1986] Algebra of charges defined via a Dirac bracket : i{l m, L n } = (m n)l m+n + c L 12 m(m2 1)δ m, n i{ L m, L n } = (m n) L m+n + c R 12 m(m2 1)δ m, n i{l m, L n } = 0 Virasoro algebra with central charges c L = c R Einstein = 3l AdS 2G 3 [Banados, Teitelboim, Zanelli, 1993] BTZ black holes have L 0 Einstein = M l AdS J, L0 Einstein = M + l AdS J G. Compère (UvA) 19 / 54

49 [Brown-Henneaux, 1986] Algebra of charges defined via a Dirac bracket : i{l m, L n } = (m n)l m+n + c L 12 m(m2 1)δ m, n i{ L m, L n } = (m n) L m+n + c R 12 m(m2 1)δ m, n i{l m, L n } = 0 Virasoro algebra with central charges c L = c R Einstein = 3l AdS 2G 3 [Banados, Teitelboim, Zanelli, 1993] BTZ black holes have L 0 Einstein = M l AdS J, L0 Einstein = M + l AdS J [Strominger, 1998] BTZ black holes described by a 2D CFT G. Compère (UvA) 20 / 54

50 [Brown-Henneaux, 1986] Algebra of charges defined via a Dirac bracket {, } i [, ] and L m L m : [L m, L n ] = (m n)l m+n + c L 12 m(m2 1)δ m, n [ Lm, L ] n = (m n) L m+n + c R 12 m(m2 1)δ m, n [ Lm, L ] n = 0 Virasoro algebra with central charges c L = c R Einstein = 3l AdS 2G 3 1 [Banados, Teitelboim, Zanelli, 1993] BTZ black holes have L 0 Einstein = M l AdS J, L0 Einstein = M + l AdS J [Strominger, 1998] BTZ black holes described by a 2D CFT G. Compère (UvA) 21 / 54

51 [Brown-Henneaux, 1986] Algebra of charges defined via a Dirac bracket {, } i [, ] and L m L m : [L m, L n ] = (m n)l m+n + c L 12 m(m2 1)δ m, n [ Lm, L ] n = (m n) L m+n + c R 12 m(m2 1)δ m, n [ Lm, L ] n = 0 Virasoro algebra with central charges c L = c R Einstein = 3l AdS 2G 3 1 [Banados, Teitelboim, Zanelli, 1993] BTZ black holes have L 0 Einstein = 1 (M l AdSJ), L0 Einstein = 1 (M + l AdSJ) [Strominger, 1998] BTZ black holes described by a 2D CFT! cl L S BH = 2π π cr L0 6 Cardy = S CFT G. Compère (UvA) 22 / 54

52 (One can use this reasoning to derive microscopically the entropy of a large class of supersymmetric black hole in string theory. ) End of Interlude : back to Kerr! G. Compère (UvA) 23 / 54

53 Near horizon geometry of extremal Kerr : Asymptotic symmetries Now, we have AdS 2 warped S 2. Idea : conformal symmetries along φ! l n = e inφ φ + inre inφ r Decoupling Boundary conditions : g µν ḡ µν + O(... ), Q t = 0 Representation by covariant charges Brandt, 2001], l n L n [(g µν,... ); L] [Barnich, [Barnich, G.C, 2007] Charges finite. Conserved and integrable after fixing technical details. [Amsel,Marolf,Roberts, 2009] G. Compère (UvA) 24 / 54

54 Near horizon geometry of extremal Kerr : Asymptotic symmetries Now, we have AdS 2 warped S 2. Idea : conformal symmetries along φ! l n = e inφ φ + inre inφ r Decoupling Boundary conditions : g µν ḡ µν + O(... ), Q t = 0 Representation by covariant charges Brandt, 2001], l n L n [(g µν,... ); L] [Barnich, [Barnich, G.C, 2007] Charges finite. Conserved and integrable after fixing technical details. [Amsel,Marolf,Roberts, 2009] G. Compère (UvA) 24 / 54

55 Near horizon geometry of extremal Kerr : Asymptotic symmetries Now, we have AdS 2 warped S 2. Idea : conformal symmetries along φ! l n = e inφ φ + inre inφ r Decoupling Boundary conditions : g µν ḡ µν + O(... ), Q t = 0 Representation by covariant charges Brandt, 2001], l n L n [(g µν,... ); L] [Barnich, [Barnich, G.C, 2007] Charges finite. Conserved and integrable after fixing technical details. [Amsel,Marolf,Roberts, 2009] G. Compère (UvA) 24 / 54

56 Near horizon geometry of extremal Kerr : Asymptotic symmetries Now, we have AdS 2 warped S 2. Idea : conformal symmetries along φ! l n = e inφ φ + inre inφ r Decoupling Boundary conditions : g µν ḡ µν + O(... ), Q t = 0 Representation by covariant charges Brandt, 2001], l n L n [(g µν,... ); L] [Barnich, [Barnich, G.C, 2007] Charges finite. Conserved and integrable after fixing technical details. [Amsel,Marolf,Roberts, 2009] G. Compère (UvA) 24 / 54

57 Charges can be represented via a Dirac bracket, i{l m, L n } = (m n)l m+n + c L 12 m(m2 1)δ m+n,0 and they admit a central term, with central charge c L Kerr = 12J The black hole entropy can then be reproduced by a CFT counting S BH = A 4G = 2πJ! = π2 3 c L T L = S CFT in one chiral sector. Note c L T L. G. Compère (UvA) 25 / 54

58 Charges can be represented via a Dirac bracket, i{l m, L n } = (m n)l m+n + c L 12 m(m2 1)δ m+n,0 and they admit a central term, with central charge c L Kerr = 12J The black hole entropy can then be reproduced by a CFT counting S BH = A 4G = 2πJ! = π2 3 c L T L = S CFT in one chiral sector. Note c L T L. G. Compère (UvA) 25 / 54

59 Question : is this a numerical coincidence? G. Compère (UvA) 26 / 54

60 Extensions of entropy matching [Many authors... ] For any 4d black hole in Einstein gravity + matter fields, the near-horizon region is ) ds 2 = α(θ) (( r 2 dt 2 + dr2 r 2 ) + dθ2 + β(θ)(dφ + k rdt) 2 and one can show S BH = A 4G [Kunduri, Lucietti, Reall, 2007]! = π2 3 c L T L = S CFT with T L = 1 2πk Einstein c L = 3k 2πG A G. Compère (UvA) 27 / 54

61 Extensions of entropy matching [Many authors... ] For any 4d black hole in Einstein gravity + matter fields, the near-horizon region is ) ds 2 = α(θ) (( r 2 dt 2 + dr2 r 2 ) + dθ2 + β(θ)(dφ + k rdt) 2 and one can show S BH = A 4G [Kunduri, Lucietti, Reall, 2007]! = π2 3 c L T L = S CFT with T L = 1 2πk Einstein c L = 3k 2πG A G. Compère (UvA) 27 / 54

62 Question : is this a numerical coincidence? No! G. Compère (UvA) 28 / 54

63 Question : is this a numerical coincidence? No! G. Compère (UvA) 28 / 54

64 Question : Does it depends on the field equations of Einstein gravity? G. Compère (UvA) 29 / 54

65 What about other theories of gravity? Any diffeomorphic covariant Lagrangian L(f (φ)) = f L(φ) can be written as L = d D x L(g ab, R abcd, e R abcd, (e f ) R abcd,..., ψ, a ψ, (a b) ψ,... ) [Iyer Wald 1994] This encompasses all α corrections. G. Compère (UvA) 30 / 54

66 What about other theories of gravity? Any diffeomorphic covariant Lagrangian L(f (φ)) = f L(φ) can be written as L = d D x L(g ab, R abcd, e R abcd, (e f ) R abcd,..., ψ, a ψ, (a b) ψ,... ) [Iyer Wald 1994] This encompasses all α corrections. G. Compère (UvA) 30 / 54

67 Extensions of entropy matching What happens with higher curvature corrections? Near-horizon geometry still ds 2 = α(θ) ) (( r 2 dt 2 + dr2 r 2 ) + dθ2 + β(θ)(dφ + k rdt) 2 but now the black hole entropy is δl S BH = 2π ɛ δr ab ɛ cd vol(s) A abcd 4G where δl δr abcd S L L e +... R abcd e R abcd Obtained from the black hole thermodynamics T BH δs BH = δm ΩδJ [Wald 1993 ; Iyer, Wald 1994] G. Compère (UvA) 31 / 54

68 Extensions of entropy matching What happens with higher curvature corrections? Near-horizon geometry still ds 2 = α(θ) ) (( r 2 dt 2 + dr2 r 2 ) + dθ2 + β(θ)(dφ + k rdt) 2 but now the black hole entropy is δl S BH = 2π ɛ δr ab ɛ cd vol(s) A abcd 4G where δl δr abcd S L L e +... R abcd e R abcd Obtained from the black hole thermodynamics T BH δs BH = δm ΩδJ [Wald 1993 ; Iyer, Wald 1994] G. Compère (UvA) 31 / 54

69 Extensions of entropy matching What happens with higher curvature corrections? Near-horizon geometry still ds 2 = α(θ) ) (( r 2 dt 2 + dr2 r 2 ) + dθ2 + β(θ)(dφ + k rdt) 2 but now the black hole entropy is δl S BH = 2π ɛ δr ab ɛ cd vol(s) A abcd 4G where δl δr abcd S L L e +... R abcd e R abcd Obtained from the black hole thermodynamics T BH δs BH = δm ΩδJ [Wald 1993 ; Iyer, Wald 1994] G. Compère (UvA) 31 / 54

70 Extensions of entropy matching [Azeyanagi, G.C., Ogawa, Tachikawa, Terashima, 2009] The temperature can be derived from free quantum fields in the curved geometry T L does not depend on gravitational field equations T L = 1 2πk Now, need some tricks Any Diff-invariant theory can be recasted in second order form using field redefinitions and auxiliary fields. The geometry has SL(2, R) U(1) and t φ symmetry Using asymptotic charge technology, we find δl c L = 12k ɛ S δr ab ɛ cd vol(s) + 0 abcd (checked up to 30 derivatives of the Riemann) G. Compère (UvA) 32 / 54

71 Extensions of entropy matching [Azeyanagi, G.C., Ogawa, Tachikawa, Terashima, 2009] The temperature can be derived from free quantum fields in the curved geometry T L does not depend on gravitational field equations T L = 1 2πk Now, need some tricks Any Diff-invariant theory can be recasted in second order form using field redefinitions and auxiliary fields. The geometry has SL(2, R) U(1) and t φ symmetry Using asymptotic charge technology, we find δl c L = 12k ɛ S δr ab ɛ cd vol(s) + 0 abcd (checked up to 30 derivatives of the Riemann) G. Compère (UvA) 32 / 54

72 Extensions of entropy matching [Azeyanagi, G.C., Ogawa, Tachikawa, Terashima, 2009] The temperature can be derived from free quantum fields in the curved geometry T L does not depend on gravitational field equations T L = 1 2πk Now, need some tricks Any Diff-invariant theory can be recasted in second order form using field redefinitions and auxiliary fields. The geometry has SL(2, R) U(1) and t φ symmetry Using asymptotic charge technology, we find δl c L = 12k ɛ S δr ab ɛ cd vol(s) + 0 abcd (checked up to 30 derivatives of the Riemann) G. Compère (UvA) 32 / 54

73 Extensions of entropy matching Therefore, we get S BH! = π2 3 c L T L +0 = S CFT G. Compère (UvA) 33 / 54

74 Question : Does it depends on the field equations of Einstein gravity? No! It is a property of extreme horizons and diffeomorphism-invariance! G. Compère (UvA) 34 / 54

75 Question : Does it depends on the field equations of Einstein gravity? No! It is a property of extreme horizons and diffeomorphism-invariance! G. Compère (UvA) 34 / 54

76 To be a CFT or not to be a CFT? Universal matching S BH = π2 3 c LT L for all rotating extremal black holes. No SL(2, R) L SL(2, R) R invariant ground state Only one Virasoro algebra appears in the asymptotic symmetry group. No other solutions than NHEK. Only one chiral sector. The central charge depends on the parameters of the black hole G. Compère (UvA) 35 / 54

77 Question : What is the nature of what looks like a chiral sector of a CFT? This is an open question. There are several embeddings in string/m-theory. [Song, Strominger ; G.C., Song, Virmani ; Guica, El-Showk ;...] None is conclusive. G. Compère (UvA) 36 / 54

78 Question : What is the nature of what looks like a chiral sector of a CFT? This is an open question. There are several embeddings in string/m-theory. [Song, Strominger ; G.C., Song, Virmani ; Guica, El-Showk ;...] None is conclusive. G. Compère (UvA) 36 / 54

79 Second step of the Kerr/CFT correspondence T BH 0 T BH = 0 ❷ Close to extremality : Dynamics of probes can be obtained from a CFT two-point correlation function. ❶ At extremality : Temperatures T L, T R, Virasoro algebra, Match S BH = S CFT! G. Compère (UvA) 37 / 54

80 Scattering close to extremality A near-extremal rotating black hole is defined from the condition M T H 1, τ H r + r r + 1. We will look at the scattering of a probe field Φ = 0, Φ = e iωt e imφ Θ(θ)R(r) close to the superradiant bound ω mω H + q e Φ e G. Compère (UvA) 38 / 54

81 Scattering close to extremality A near-extremal rotating black hole is defined from the condition M T H 1, τ H r + r r + 1. We will look at the scattering of a probe field Φ = 0, Φ = e iωt e imφ Θ(θ)R(r) close to the superradiant bound ω mω H + q e Φ e G. Compère (UvA) 38 / 54

82 Scattering process Exterior region τ H «x Matching region τ H «x «1 Near-horizon region x «1 x r r + r +, τ H = r + r r + 1 G. Compère (UvA) 39 / 54

83 Superradiant scattering Define the absorption probability σ abs (ω, m, l; M, J) de abs/dt de in /dt. [Computed by Press and Teukolsky in the 70s] What we really want to match with a CFT computation is Result : σ near abs = de abs/dt Φ(x = x B ) 2, τ H x B 1. σabs near (ω, m, l; M, J) = Gamma functions Feature : Superradiance (σ abs < 0) occurs when ω < mω H + q e Φ e Spontaneous emission also occurs : extremal Kerr-Neumann black holes spontaneously decay. G. Compère (UvA) 40 / 54

84 Superradiant scattering Define the absorption probability σ abs (ω, m, l; M, J) de abs/dt de in /dt. [Computed by Press and Teukolsky in the 70s] What we really want to match with a CFT computation is Result : σ near abs = de abs/dt Φ(x = x B ) 2, τ H x B 1. σabs near (ω, m, l; M, J) = Gamma functions Feature : Superradiance (σ abs < 0) occurs when ω < mω H + q e Φ e Spontaneous emission also occurs : extremal Kerr-Neumann black holes spontaneously decay. G. Compère (UvA) 40 / 54

85 Dual CFT picture The process is controlled by the two-point function G(t +, t ) = O (t +, t )O(0), where t ± are the coordinates of the CFT. From the so-called Fermi s golden rule, the absorption probability σ abs is given by σ abs dt + dt e iω Rt iω L t + [G(t + iɛ, t iɛ) G(t + + iɛ, t + iɛ)] G. Compère (UvA) 41 / 54

86 Dual CFT picture : Matching More precisely, the form of the thermal two-point function is dictated by conformal invariance to be ( ) G(t +, t πt 2hL ( ) L sinh (πt L t + ) πt R sinh (πt R t ) ) 2hR e iq Lµ L t + +iq R µ R t at temperatures (T L, T R ) and chemical potentials (µ L, µ R ) when the operator has conformal dimensions (h L, h R ) and charges (q L, q R ). Matching the gravity result around the Kerr black hole gives h L = h L (m, l, aω) T L = 1 2π, ω L = m q L = e, µ L = Q3 2J h R = h R (m, l, aω) T R = T ext ω R q R µ R T R q R = m R µ R = Ω H = ω mω H 2πT H 2r + ω G. Compère (UvA) 42 / 54

87 Dual CFT picture : Matching More precisely, the form of the thermal two-point function is dictated by conformal invariance to be ( ) G(t +, t πt 2hL ( ) L sinh (πt L t + ) πt R sinh (πt R t ) ) 2hR e iq Lµ L t + +iq R µ R t at temperatures (T L, T R ) and chemical potentials (µ L, µ R ) when the operator has conformal dimensions (h L, h R ) and charges (q L, q R ). Matching the gravity result around the Kerr black hole gives h L = h L (m, l, aω) T L = 1 2π, ω L = m q L = e, µ L = Q3 2J h R = h R (m, l, aω) T R = T ext ω R q R µ R T R q R = m R µ R = Ω H = ω mω H 2πT H 2r + ω G. Compère (UvA) 42 / 54

88 Features CFT description forms a representation under the symmetries Virasoro L Current L (Virasoro R Current R ) while only Virasoro L belongs to the asymptotic symmetry group of the near-horizon geometry. Real and complex weights h L (m, l, aω) = h R (m, l, aω) R or ir Imaginary conformal weights are related to Schwinger pair production. Rotating extremal black holes decay no stable ground state Many features to be better understood! G. Compère (UvA) 43 / 54

89 Features CFT description forms a representation under the symmetries Virasoro L Current L (Virasoro R Current R ) while only Virasoro L belongs to the asymptotic symmetry group of the near-horizon geometry. Real and complex weights h L (m, l, aω) = h R (m, l, aω) R or ir Imaginary conformal weights are related to Schwinger pair production. Rotating extremal black holes decay no stable ground state Many features to be better understood! G. Compère (UvA) 43 / 54

90 Features CFT description forms a representation under the symmetries Virasoro L Current L (Virasoro R Current R ) while only Virasoro L belongs to the asymptotic symmetry group of the near-horizon geometry. Real and complex weights h L (m, l, aω) = h R (m, l, aω) R or ir Imaginary conformal weights are related to Schwinger pair production. Rotating extremal black holes decay no stable ground state Many features to be better understood! G. Compère (UvA) 43 / 54

91 Third and last step of the Kerr/CFT correspondence ❸ Far from extremality : Dynamics of probes has broken SL(2, R) SL(2, R) symmetry. Entropy still matches! T BH 0 T BH = 0 ❷ Close to extremality : Dynamics of probes can be obtained from a CFT two-point correlation function. ❶ At extremality : Temperatures T L, T R, Virasoro algebra, Match S BH = S CFT! G. Compère (UvA) 44 / 54

92 Conformal invariance away from extremality Away from extremality, there is no decoupled near-horizon geometry with conformal invariance. Yet, conformal invariance can be present in the solution space of fields propagating on the geometry and control the scattering amplitudes and the asymptotic growth of states. G. Compère (UvA) 45 / 54

93 Scattering Process Let s look again at a scalar probe Φ obeying Φ = 0. We have Φ = e iωt+imφ Θ(θ)R(r) We look only in the low-energy regime, M ω 1 G. Compère (UvA) 46 / 54

94 Scattering Process Far region M «r Matching region M «r «ω -1 Near region r «ω -1 In the near region r ω 1, Θ(θ) = Y lm (θ), R(r) = hypergeometric G. Compère (UvA) 47 / 54

95 SL(2, R) L SL(2, R) R symmetry Trick : define new coordinates related to (t, r, φ, θ) as (w +, w, y, θ) where w + = w = y = T R = T H Ω H, r r+ e 2πT Rφ r r r r+ e 2πT Lφ t 2M r r r+ r e π(t L+T R )φ t 4M r r T L = 1 2π ( ) 1 + M2 J. J G. Compère (UvA) 48 / 54

96 SL(2, R) L SL(2, R) R symmetry The radial part of the wave equation can then be written as ( H 2 Φ H ) 2 (H 1H 1 + H 1 H 1 ) Φ = l(l + 1)Φ where Φ = e iωt R(r) and H 1 = i +, H 0 = i(w y y) H 1 = i(w w + y y y 2 ) obey the SL(2, R) L algebra. There is also a second set of generators SL(2, R) R such that H 2 Φ = l(l + 1)Φ. G. Compère (UvA) 49 / 54

97 Identification breaks symmetry to U(1) L U(1) R At fixed r, the change of coordinates is w + = e 2πT Rφ w = e 2πT Lφ t 2M where T R = T H Ω H, T L = 1 2π ( ) 1 + M2 J J The identification φ φ + 2π breaks SL(2, R) L SL(2, R) R U(1) L U(1) R. G. Compère (UvA) 50 / 54

98 T L, T R can be interpreted as temperatures Assume that (w +, w ) plane describes a SL(2, R) SL(2, R) invariant CFT vacuum. We define the coordinates t + and t as Then the identification can be written as w + e t + = e 2πT Rφ w e t = e 2πT Lφ t 2M φ φ + 2π t + t + + 4π 2 T R, t t 4π 2 T L Tracing over the region outside the strip leads to a density matrix with temperatures T L, T R. G. Compère (UvA) 51 / 54

99 Entropy matching Remarkably, the entropy of the Kerr black hole S BH = 2π ( ) M 2 + M G 4 J 2 agrees with the Cardy formula with S CFT = π2 3 (c LT L + c R T R ) c L = c R = 12J. G. Compère (UvA) 52 / 54

100 Conclusions Remarkable agreement between black hole entropy and Cardy formula S BH = π2 3 (c LT L + c R T R ) where c L computed for extremal black holes, c R = c L, and T L, T R computed independently for extremal black holes and non-extremal black holes. Two regimes where the dynamics of probes is governed by conformal symmetry : at near-extremality close to superradiant bound away from extremality at low energy Open question : what is the nature of this CFT? Open question : Are there other regimes where the CFT description is valid? G. Compère (UvA) 53 / 54

101 For further reading : check the reviews Cargèse Lectures on the Kerr/CFT correspondence, Bredberg, Keeler, Lysov, Strominger, The Kerr/CFT correspondence, a bird-eye view, G.C., 1202.XXXX G. Compère (UvA) 54 / 54