Horizon fluff. A semi-classical approach to (BTZ) black hole microstates

Transcription

1 Horizon fluff A semi-classical approach to (BTZ) black hole microstates with D. Grumiller, M.M. Sheikh-Jabbari and H. Yavartanoo Hamid R. Afshar 11 May Recent Trends in String Theory and Related Topics Institute for Research in Fundamental Sciences (IPM)

2 Near Horizon collaboration Stephane Detournay Daniel Grumiller Wout Merbis Blagoje Oblak Alfredo Perez Shahin Sheikh-Jabbari David Tempo Ricardo Troncoso Hossein Yavartanoo 1

3 Horizon fluff proposal Black hole microstates = horizon fluff: subset of near horizon soft hairs not distinguishable by the observers away from the horizon. Black hole: a state in the Hilbert space of asymptotic symmetries: [L n, L m ] = (n m)l m+n + k 2 n3 δ n+m,0. Soft hairs: states in the Hilbert space of near horizon algebra: [J n, J m ] = k 2 n δ n+m,0. Duality map between asymptotic Hilbert space (BTZ) to the near horizon Hilbert space provides the required degeneracy (entropy). 2

4 Outline 1. Motivation 2. Brown-Henneaux boundary conditions 3. Near horizon boundary conditions 4. Quantization of conical defects 5. Black hole microstates 6. Logarithmic correction to entropy 7. Conclusion 3

5 Motivation

6 Motivation The Bekenstein-Hawking area law for black hole entropy is observer independent and is accessible through semiclassical considerations. S = A/(4G) - q ln A/(4G) + O(1) = ln ( # microstates ) Universality of this result suggests that a statistical description of microstates in the thermodynamic limit does not need full knowledge of the underlying quantum theory. 4

7 Soft hair Diffeomorphic geometries which differ by their boundary behavior can be physically distinct, asymptotic soft hairs. The conserved charges associated with the diffeomorphisms relating them is non-zero and form an infinite dimensional algebra. A black hole spacetime in particular can carry low-energy quantum excitations, near horizon soft hairs, providing a huge degeneracy to their vacuum. Could they amount for microstates of black holes? How to identify microstates among them? 5

8 Brown-Henneaux boundary conditions

9 Bañados geometries All Locally AdS 3 geometries obeying Brown Henneaux b.c.: ds 2 = l 2 dr 2 r 2 r 2( dx + l2 L (x ) r 2 dx )( dx l2 L + (x + ) r 2 dx +) L ± (x ± + 2π) = L ± (x ± ), x ± = t/l ± φ, φ [0, 2π]. Constant family: BTZ black holes: L 0 0 Global AdS 3 : L 0 = 1/4 L + = L = L 0 Conical defects: 1/4 < L 0 < 0 6

10 Brown-Henneaux symmetries Bañados geometries fall into representation of asymptotic (simplectic) symmetry algebra which is two copies of Virasoro at Brown-Henneaux central charge c = 6k. The infinitesimal action of Virasoro symmetries on functions L ± is; δ ɛ± L ± = 2L ± ɛ ± + ɛ ±L ± ɛ ±/2. The corresponding geometries with (L ± ) fall into a one-to-one relation with the coadjoint orbits of these symmetries. They are BTZ black holes, conic spaces and global AdS 3 and their conformal descendants, explicitly: H Vir = H BH H Conic H gads. }{{} H CG 7

11 Near horizon boundary conditions

12 Near horizon boundary conditions All locally AdS 3 geometries with horizon at r = 0 are parametrized by 4 real functions ds 2 = dr 2 l 2 sinh 2 r l [a dt ωdϕ]2 + cosh 2 r [Ω dt + γdϕ]2 l t J ± = ± ϕ ζ ± ; 2ζ ± a ± Ω l and 2J ± γ l ± ω. Constant family: J + = J = ±J 0, L 0 = J 0 2 BTZ black holes: J 0 0 Global AdS 3 : J 0 = i 2 Conical defects: J 0 = iν 2, ν (0, 1) 8

13 Near horizon boundary conditions ds 2 = dr 2 l 2 sinh 2 r l [a dt ωdϕ]2 + cosh 2 r [Ω dt + γdϕ]2 l = dr 2 (ar) 2 dt 2 + γ 2 dϕ 2 + O(r 2 ). ϕ ϕ + 2π Rindler space: Universal near horizon to any non-extremal horizon. 9

14 Near horizon boundary conditions ds 2 = dr 2 l 2 sinh 2 r l [a dt ωdϕ]2 + cosh 2 r [Ω dt + γdϕ]2 l = dr 2 (ar) 2 dt 2 + γ 2 dϕ 2 + O(r 2 ). ϕ ϕ + 2π Rindler space: Universal near horizon to any non-extremal horizon. In the canonical description the Rindler acceleration a is fixed. 9

15 Near horizon boundary conditions ds 2 = dr 2 l 2 sinh 2 r l [a dt ωdϕ]2 + cosh 2 r [Ω dt + γdϕ]2 l = dr 2 (ar) 2 dt 2 + γ 2 dϕ 2 + O(r 2 ). ϕ ϕ + 2π Rindler space: Universal near horizon to any non-extremal horizon. In the canonical description the Rindler acceleration a is fixed. AdS radius l drops out of the near horizon line-element. 9

16 Near horizon boundary conditions ds 2 = dr 2 l 2 sinh 2 r l [a dt ωdϕ]2 + cosh 2 r [Ω dt + γdϕ]2 l = dr 2 (ar) 2 dt 2 + γ 2 dϕ 2 + O(r 2 ). ϕ ϕ + 2π Rindler space: Universal near horizon to any non-extremal horizon. In the canonical description the Rindler acceleration a is fixed. AdS radius l drops out of the near horizon line-element. All solutions have a regular horizon, regardless of the value of γ, ω as long as a/ (2π) is identified with the Unruh temperature. 9

17 Symmetries of the Near-Horizon The most general transformation that preserves this boundary condition and also preserves the the field equation, with δζ = 0, transforms J s as; δ η J = η. u(1) k -algebra [J n, J m ] = k 2 n δ n+m,0. How are these two geometries related? Check how J is transformed under conformal transformaition acting on the metric: δ ɛ J = (ɛj) ɛ /2 L = J 2 + J. L n 6 J n p J p + inj n c p Z 10

18 Quantization of conical defects

19 Wilson lines as primary fields The fields J(φ) are primary fields if the anomalous term ɛ can be ignored that is for black hole sector (J 0 = J 0) with J being large; L = J + J 2 δ ɛ J = (ɛj) ɛ /2 We can construct the primary field W; W(φ) = e 2 φ J δ ɛ W = (ɛw) with the periodicity property: W ± (φ + 2π) = e 4πJ0 W ± (φ) which is a good description for conic spaces (J 0 = J 0). 11

20 Quantization of W fields From δ η J = η /2 we get; δ η± W ± = η ± W ±, η ± = φ η ± iη 0 and using the appropriate mode expansion W(ϕ) = n Z W ν n e i(n±ν)ϕ, W ν n = 6 c Wν n, we get; [J n, W ± m] = iw ± n+m, ( n 0), [J 0, W ± n ] = i c 6 W± n δ n,0. The W operators are like ladder operators of J 0. 12

21 Free field realization of W-fields The interaction terms WWJ and WJJ are suppressed by factors of 1/c. In the large c regime, the algebra can be closed as: [W ±ν n, W ν m ] = c 12 (n ± ν) δ n, mδ(ν ν ), ν, ν (0, 1), This gives a free field realization for W-fields. 13

22 Near horizon soft hairs Bohr-type quantization condition ν = r c, r = 1,, c, c Z To construct the Hilbert space H CG, we use the Fourier modes J n : J cn+r 6W r n, r = νc = 1, 2,, c. The commutation relation of W n s takes a very simple form [J n, J m ] = n 2 δ n, m, We define the vacuum state as global AdS 3 by J n 0 = 0, n 0. 14

23 Virasoro algebra of near horizon In parallel to the black hole sector we can have a Virasoro generator; L r n = 6 c :W r n pw r ( p : ( 2r c 1)2) δ n,0 p Z The generators L n = c r=1 Lr n, can be written in terms of J n modes; L n = 1 c p Z :J nc p J p : 1 24c δ n,0 They satisfy a Virasoro algebra at Brown-Henneaux central charge c. 15

24 Black hole microstates

25 State of a BTZ Duality (L nc = cl n ) 1 J nc p J p = inj n + 6 J n p J p. c c p Z p Z The two Hilbert spaces in H Vir = H BH H CG are related. H CG H BH A given AdS 3 black hole state: BTZ > H BH micro-states = Ψ > H CG L n BTZ = 1 2 (lm ± J)δ n,0. 16

26 Horizon fluffs = Microstates H CG [J m, J n ] = m 2 δ m+n,0 Ψ >= J mi J m2 J m1 0, m i > 0 So Ψ describes a blackhole state if; 1 2 (lm ± J) = L± 0 = 1 c L± 0 = 1 c Mathematically, this reduces to Hardy and Ramanujan combinatorial problem: the number p(n) of ways a positive integer N can be partitioned into non-negative integers in the limit of large N; p(n) 1 4N 3 exp i n i ( ) N 2π, N

27 Entropy Microcanonical entropy as logarithm of the number of states: S 0 = ln p(n + ) + ln p(n ) = 2π cl cl 0 6 where cl ± 0 = c(lm ± J) = i n i = N ±. Reminding L 0 ± = (J 0 ± ) 2, the entropy is; S 0 = πc 3 ( J J ) 0 18

28 Logarithmic correction to entropy

29 Micro-canonical entropy The logarithmic correction: S 0 = πc 3 ( J J ) 0 S = S 0 2 ln S , We should map the J fields onto the microcanonical fields in terms of h, J(φ) = J 0 h (φ) 1 h (φ) 2 h (φ). The microcanonical entropy is obtained through replacing J 0 with J 0 = k 2π 2π 0 dφ J(φ) BTZ. 19

30 Micro-canonical entropy Using the replica trick: ln(h (φ)) 2π 0 = ln J BTZ 0 + J 0 -independent terms. Consequently we find the exact match for the log correction in the mic-canonical ensemble for BTZ black holes; S mic = = S BH 3 2 ln S

31 Conclusion

32 Summary The horizon fluff proposal is a semi-classical proposal for constructing all BTZ microstates. Remarkably, within this semiclassical setting we not only derived the Bekenstein-Hawking entropy but also the logarithmic corrections. We just used semi-classical results about symmetries and some basic Bohr-type quantization on central charge and the dificit angle and the fact that the near horizon and asymptotic energies differ by a factor of 1/c and do not need any detailed UV completed quantum gravity description. 21

33 Questions? 21