Black Hole Entropy in LQG: a quick overview and a novel outlook

Transcription

1 Black Hole Entropy in LQG: a quick overview and a novel outlook Helsinki Workshop on Loop Quantum Gravity Helsinki June 2016 Alejandro Perez Centre de Physique Théorique, Marseille, France.

2 Two ingredients to understand BH entropy in LQG (1) Semiclassical inputs: a. BH thermodynamics from a local perspective b. Regularity of the quantum state in the vicinity of the Horizon: Hadamard condition. (2) Quantum geometry: In quantum gravity the BH horizon area is made up from contributions of point-like quanta

3 Two ingredients to understand BH entropy in LQG (1) Semiclassical inputs: a. BH thermodynamics from a local perspective E. Frodden, A. Ghosh, AP Phys.Rev. D87 (2013) no.12, b. Regularity of the quantum state in the vicinity of the horizon: Hadamard condition. (2) Quantum geometry: In quantum gravity the BH horizon area is made up from contributions of point-like quanta

4 Black Hole Thermodynamics A quasilocal perspective `2 << A Introduce a family of local stationary observers ~AMOS H = + t Singularity u a = a k k H W O a = u a r a u b = 1`

5 A thought experiment throwing a test particle from infinity w a Particle s equation of motion `2 << A w a r a w b = qf bc w @ Symmetries of the background L g ab = L g ab = L A a = L A a =0 u a Conserved quantities E w a a qa a a L w a a + qa a a

6 A thought experiment throwing a test particle from infinity w a Conserved quantities E w a a qa a a L w a a + qa a a u a E = Particle at infinity w a a 1 energy L = w a a 1 angular @

7 A thought experiment throwing a test particle from infinity Conserved quantities E w a a qa a a L w a a + qa a a u a E`oc At the local observer w a u a local energy w @

8 = + = = After absorption seen from infinity E = w a a 1 energy L = w a a 1 angular momentum The BH readjusts parameters M = E J = L Q = q u a The area change from 1st law M = apple 8 A + J + Q `2 << A w a apple 8 A = E L q

9 After @t E w a a 1 As seen by the local observer apple 8 A = E L q L = w a a 1 angular momentum At the local observer E`oc w a u a local energy u a = + t u a = a k k E`oc = wa a + w a a k k `2 << A w a E`oc = E L q k k

10 After absorption seen by a local observer apple 8 A = E L q w a E`oc = E L q k k The appropriate local energy notion must be the one such that: H E loc = apple 8 k k A E = E`oc E = apple 8 A E loc = apple 8 A apple apple k k

11 Local first law Main classical result `2 << A w a a = u a r a u b = 1` M = apple 8 A + J + Q, H E = apple 8 A apple apple = 1` + o(`) E = A 8 `

12 Local first law refined field theoretical version J a = T a b b is conserved thus H dv ds T ab a k b = W O J b N b SINGULARITY H H dv ds T ab applevk a {z } a k b = W O k k T ab u a N b k a N a T ab The Raychaudhuri equation d dv = 8 T abk a k b W O H dv ds V d dv = 8 k k apple E, E = apple 8 A

13 Local first law Hamiltonian version J. Engle, Noui, AP, D. Pranzetti Phys.Rev. D82 (2010) Time flow defined by stationary observers at the stretched horizon boundary (L t, )= A 8 G` + b.t.

14 E = A 8 `

15 Two ingredients to understand BH entropy in LQG (1) Semiclassical inputs: a. BH thermodynamics from a local perspective b. Regularity of the quantum state in the vicinity of the horizon: Hadamard condition. (2) Quantum geometry: In quantum gravity the BH horizon area is made up from contributions of point-like quanta

16 The phase space of GR The symplectic potential ( ) = = in connection variable ( IJKL e I ^ e K ) ^! KL = S = = + ( 0jkl e 0 ^ e j ) ^! kl +( 0jkl e j ^ e k ) ^! l0 = ( jkl e j ^ e k ) ^! l0 M S[e,!] = M IJKL e I ^ e K ^ F KL (!) 2 e I IJKL ^ e K ^ F KL (!) + e I ^ e K ^ d! (! KL )= 2 e I IJKL ^ e K ^ F KL (!) d! ( IJKL e I ^ e K ) ^! KL ( IJKL e I ^ e K ) = p

17 The phase space of GR in connection variables S[e,!] = M IJKL e I ^ e K ^ F KL S = = + M 2 e I IJKL ^ e K ^ F KL (!) + e I ^ e K ^ d! (! KL )= 2 e I IJKL ^ e K ^ F KL (!) d! ( IJKL e I ^ e K ) ^! KL ( IJKL e I ^ e K ) ^! KL The symplectic potential ( ) = Number of components: 12 for versus 24 for e I a ( IJKL e I ^ e K ) ^! KL =! IJ a Time gauge e 0 e 2 = = ( 0jkl e 0 ^ e j ) ^! kl +( 0jkl e j ^ e k ) ^! l0 = ( jkl e j ^ e k ) ^! l0 e 1

18 A canonical transformation to get back the connection variables The symplectic potential ( ) = ( ijk e i ^ e j ) ^! k0 = 1 ( ijk e i ^ e j ) ^ (! k0 ) = 1 ( ijk e i ^ e j ) ^ (! k0 + k (e)) 1 ( ijk e i ^ e j ) ^ k(e)) = 1 ( ijk e i ^ e j ) ^ A k e i ^ e i

19 Une transformation canonique pour retrouver les variables de connexion Immirzi parameter The symplectic potential ( ) = ( ijk e i ^ e j ) ^! k0 = 1 ( ijk e i ^ e j ) ^ (! k0 ) = 1 ( ijk e i ^ e j ) ^ (! k0 + k (e)) 1 ( ijk e i ^ e j ) ^ k(e)) = 1 ( ijk e i ^ e j ) ^ A k e i ^ e i

20 Une transformation canonique pour retrouver les variables de connexion Immirzi parameter The spin connection de i + ijk j ^ e k =0 The symplectic potential ( ) = ( ijk e i ^ e j ) ^! k0 = 1 ( ijk e i ^ e j ) ^ (! k0 ) = 1 ( ijk e i ^ e j ) ^ (! k0 + k (e)) 1 ( ijk e i ^ e j ) ^ k(e)) = 1 ( ijk e i ^ e j ) ^ A k e i ^ e i

21 Une transformation canonique pour retrouver les variables de connexion Immirzi parameter The spin connection de i + ijk j ^ e k =0 d( e i )+ ijk j ^ e k + ijk j ^ e k =0 The symplectic potential ( ) = ( ijk e i ^ e j ) ^! k0 ( ijk e i ^ e j ) ^ k = d(e i ^ e i ) = 1 ( ijk e i ^ e j ) ^ (! k0 ) = 1 ( ijk e i ^ e j ) ^ (! k0 + k (e)) 1 ( ijk e i ^ e j ) ^ k(e)) = 1 ( ijk e i ^ e j ) ^ A k e i ^ e i

22 Une transformation canonique pour retrouver les variables de connexion le paramètre d Immirzi The spin connection de i + ijk j ^ e k =0 d( e i )+ ijk j ^ e k + ijk j ^ e k =0 The symplectic potential ( ) = ( ijk e i ^ e j ) ^! k0 ( ijk e i ^ e j ) ^ k = d(e i ^ e i ) = 1 ( ijk e i ^ e j ) ^ (! k0 ) = 1 ( ijk e i ^ e j ) ^ (! k0 + k (e)) 1 ( ijk e i ^ e j ) ^ k(e)) = 1 ( ijk e i ^ e j ) ^ A k e i ^ e i

23 ( ) = Quantization of area in a nut-shell ( ijk e i ^ e j ) ^! k0 = 1 ( ijk e i ^ e j ) ^ (! k0 ) J. Engle, Noui, AP, D. Pranzetti Phys.Rev. D82 (2010) = 1 ( ijk e i ^ e j ) ^ (! k0 + k (e)) 1 ( ijk e i ^ e j ) ^ k(e)) = 1 ( ijk e i ^ e j ) ^ A k e i ^ e i E 3 = e 1 ^ e 2 In a 2-boundary e 2 {e i a(x),e j b (y)} = ab ij (2) (x, y) e 1 area element ds 2 [e] E i 1 jkl e j ^ e k p Ei E i = p h (2) {E i (x),e j (y)} = ijk E k (2) (x, y)

24 ( ) = Quantization of area in a nut-shell ( ijk e i ^ e j ) ^! k0 = 1 ( ijk e i ^ e j ) ^ (! k0 ) = 1 ( ijk e i ^ e j ) ^ (! k0 + k (e)) 1 ( ijk e i ^ e j ) ^ k(e)) = 1 ( ijk e i ^ e j ) ^ A k e i ^ e i E 3 = e 1 ^ e 2 In a 2-boundary e 2 {e i a(x),e j b (y)} = ab ij (2) (x, y) e 1 area element ds 2 [e] E i 1 jkl e j ^ e k p Ei E i = p h (2) {E i (x),e j (y)} = ijk E k (2) (x, y) area quantum = `2pp j(j + 1)

25 Quantization of area in a nut-shell E(, S) = S i e j ^ e k ijk E(, S) S Tr[ E] S = = Int(S) d(tr[ E]) (Tr[d A ( )E]+Tr[ d A (E)]) Int(S) A. Cattaneo and A.P. to appear somewhere = Int(S) Tr[(d +[A, ])E] {E(, S),E(,S)} = Int(S) Int(S) {Tr[(d +[A, ])E], Tr[(d +[A, ])E]} {E(, S),E(,S)} = E([, ],S)

26 Quantization of area in a nut-shell Immirzi parameter area quantum = `2pp j(j + 1) 1 2

27 Two ingredients to understand the origin of BH entropy (1) Semiclassical inputs: a. BH thermodynamics from a local perspective b. Regularity of the quantum state in the vicinity of the horizon: Hadamard condition. (2) Quantum geometry: In quantum gravity the BH horizon area is made up from contributions of point-like quanta

28 The black hole area spectrum The area gap ba S j 1,j 2 i = " 8 `2p X q j p (j p + 1) # j 1,j 2 i p a min =4 p 3 Isolated Horizon boundary conditions [Ashtekar et al. 2000]

29 INGREDIENTS (1)+(2) [Ghosh, Perez, 2011 PRL] [Frodden, Ghosh, Perez, to appear]

30 The black hole Hamiltonian " `2 bh j p 1,j 2 i= ` The area gap X p q j p (j p + 1) # j 1,j 2 i a min =4 p 3 ` The scale is a fiducial quantity (a regulator) The regulator is natural: York 1983, Hajicek-Israel 1980.

31 Two ingredients to understand the origin of BH entropy (1) Semiclassical inputs: a. BH thermodynamics from a local perspective b. Regularity of the quantum state in the vicinity of the horizon: Hadamard condition. (2) Quantum geometry: In quantum gravity the BH horizon area is made up from contributions of point-like quanta

32 Regularity of the quantum state The temperature of stretched horizon is Unruh temperature T U = 1 2 ` Present ingredient: Quantum IH physical state of boundary Chern-Simons Quantum bulk state = quantum gravity physical state describing BH background (for matter fields in the semiclassical approximation = vacuum regular at horizon) ` arbitrary fixed proper distance to the horizon

33 Entropy Calculation Microcanonical or canonical ensemble Counting geometric excitations S bh = 0 A 4`2p Barbero et al. T = T U = 0 Are geometric excitations all there is? S matter = A 2 + corrections = undertermined constant (UV regularization dependent species problem) = UV cut-o Number of d.o.f. dominated by boundary contribution Ghosh-Noui-AP, PRD 2014 D exp apple A 2

34 An outlook Residual diffeos: extra degrees of freedom at punctures

35 Laurent Freidel and AP. arxiv: work in progress with Freidel and Pranzetti Kac-Moody Charges Q S ( ) := 1 = apple 1 S i e i d A i e i Q p (vye i ) Tangent Diffeos Generator D S ( ) := 1 L e i ^ e i apple D p (v)ˆ= 1 I (vye i )e i apple C p S {Q p ( ),Q p ( )} = 1 8 I C p i d i. q i N Q( i exp(in')) {q i N,q j M} = in 4 ij N+M,0 S

36 An outlook Residual diffeos: extra degrees of freedom at punctures {q i N,q j M} = in 4 ij N+M,0 q i N Q( i exp(in')) X ˆL N =2 X i :ˆq i M ˆq i N M : M2 [ˆL N, ˆL M ]=(N M)ˆL N+M + c 12 N(N 2 1) N+M,0 j c We get Virasoro generators S D N = i(l N LN ) with central charge equal 3

37 An outlook Residual diffeos: extra degrees of freedom at punctures {q i N,q j M} = in 4 ij N+M,0 q i N Q( i exp(in')) X ˆL N =2 X i :ˆq i M ˆq i N M : M2 [ˆL N, ˆL M ]=(N M)ˆL N+M + c 12 N(N 2 1) N+M,0 j c =3 The Cardy formula gives exponential growth of the number of states in S D N = i(l N LN ) of the energy eigenvalues=area

38 Implications for BH entropy HR CFT = LR 0 + L R 0 c R 12 = a R 8 G N R i For the chiral coupling, macroscopic BHs are possible if = i Entropy from the Cardy Formula In the Euclidean, + c 12 = X p2r j p relevant for thermodynamics r cm ( D A exp c m 24 ) = exp A 4G N

39 Outlook A Kac-Moody algebraic structure follows directly from the gravitational symplectic structure when distributional configurations (punctures) are considered. On the completely algebraic level this leads to the construction of Virasoro generators via the Sugawara construction. The central charge of the Virasoro algebra is c = 3. Two dimensional conformal field theoretical degrees of freedom can be associated to punctures. For the moment we only understand this possibility in a chiral example. Can we have both modes? Important implications for BH entropy. Puzzling: =1playsaspecial role in the quantum theory. Is this related to analytic continuation indications? Frodden, Geiller, Noui, AP; Europhys.Lett. 107 (2014) 10005