Black-Holes in AdS: Hawking-Page Phase Transition

Transcription

1 Black-Holes in AdS: Hawking-Page Phase Transition Guilherme Franzmann December 4, / 14

2 References Thermodynamics of Black Holes in Anti-de Sitter space, S.W. Hawking and Don. N Page (1983); Black Holes in AdS Spacetime, Peng Zhao (2013). 2 / 14

3 References Thermodynamics of Black Holes in Anti-de Sitter space, S.W. Hawking and Don. N Page (1983); Black Holes in AdS Spacetime, Peng Zhao (2013). 2 / 14

4 Outline Introduction; Asymptotically flat case; The AdS and Schwarzschild-AdS metrics; Path Integral, Euclidean formulation and partition function; Hawking-Page phase transition. 3 / 14

5 Outline Introduction; Asymptotically flat case; The AdS and Schwarzschild-AdS metrics; Path Integral, Euclidean formulation and partition function; Hawking-Page phase transition. 3 / 14

6 Outline Introduction; Asymptotically flat case; The AdS and Schwarzschild-AdS metrics; Path Integral, Euclidean formulation and partition function; Hawking-Page phase transition. 3 / 14

7 Outline Introduction; Asymptotically flat case; The AdS and Schwarzschild-AdS metrics; Path Integral, Euclidean formulation and partition function; Hawking-Page phase transition. 3 / 14

8 Outline Introduction; Asymptotically flat case; The AdS and Schwarzschild-AdS metrics; Path Integral, Euclidean formulation and partition function; Hawking-Page phase transition. 3 / 14

9 Introduction AdS space has BH solutions! These solutions have a characteristic temperature and intrinsic entropy that is equal to A/4 (as in the flat case); Difference: there is a minimum temperature for the BH; Implications: positive specific heat and it can be in stable equilibrium with thermal radiation at a fixed temperature; Gravitational potential of AdS acts as a box of finite volume. 4 / 14

10 Introduction AdS space has BH solutions! These solutions have a characteristic temperature and intrinsic entropy that is equal to A/4 (as in the flat case); Difference: there is a minimum temperature for the BH; Implications: positive specific heat and it can be in stable equilibrium with thermal radiation at a fixed temperature; Gravitational potential of AdS acts as a box of finite volume. 4 / 14

11 Introduction AdS space has BH solutions! These solutions have a characteristic temperature and intrinsic entropy that is equal to A/4 (as in the flat case); Difference: there is a minimum temperature for the BH; Implications: positive specific heat and it can be in stable equilibrium with thermal radiation at a fixed temperature; Gravitational potential of AdS acts as a box of finite volume. 4 / 14

12 Introduction AdS space has BH solutions! These solutions have a characteristic temperature and intrinsic entropy that is equal to A/4 (as in the flat case); Difference: there is a minimum temperature for the BH; Implications: positive specific heat and it can be in stable equilibrium with thermal radiation at a fixed temperature; Gravitational potential of AdS acts as a box of finite volume. 4 / 14

13 Introduction AdS space has BH solutions! These solutions have a characteristic temperature and intrinsic entropy that is equal to A/4 (as in the flat case); Difference: there is a minimum temperature for the BH; Implications: positive specific heat and it can be in stable equilibrium with thermal radiation at a fixed temperature; Gravitational potential of AdS acts as a box of finite volume. 4 / 14

14 Asymptotically Flat Case The metric is given by ( ds 2 = 1 2M r ) ( dt M r ) 1 dr 2 + r 2 dω 2 ; (1) The temperature is T H = 1 8πM ; The specific heat is given by E T = M V T = 1 8πT 2 ; Therefore, the BH is unstable if in equilibrium with an infinite heat reservoir. 5 / 14

15 Asymptotically Flat Case The metric is given by ( ds 2 = 1 2M r ) ( dt M r ) 1 dr 2 + r 2 dω 2 ; (1) The temperature is T H = 1 8πM ; The specific heat is given by E T = M V T = 1 8πT 2 ; Therefore, the BH is unstable if in equilibrium with an infinite heat reservoir. 5 / 14

16 Asymptotically Flat Case The metric is given by ( ds 2 = 1 2M r ) ( dt M r ) 1 dr 2 + r 2 dω 2 ; (1) The temperature is T H = 1 8πM ; The specific heat is given by E T = M V T = 1 8πT 2 ; Therefore, the BH is unstable if in equilibrium with an infinite heat reservoir. 5 / 14

17 Asymptotically Flat Case Hawking found that an unphysical box of finite volume and finite heat capacity could stabilize the BH in equilibrium with radiation given that the radiation energy of the box satisfies E rad < M/4; Therefore, a box is nice! However, it is artificial here for this case. This box is naturally provided in AdS! 6 / 14

18 Asymptotically Flat Case Hawking found that an unphysical box of finite volume and finite heat capacity could stabilize the BH in equilibrium with radiation given that the radiation energy of the box satisfies E rad < M/4; Therefore, a box is nice! However, it is artificial here for this case. This box is naturally provided in AdS! 6 / 14

19 The AdS and Schwarzschild-AdS metrics AdS metric: ds 2 = (1 + r 2 b 2 ) dt 2 + (1 + r 2 ) 1 dr 2 + r 2 dω 2 b 2 ( b t Λ) 3 1/2, t t + 2πb; Schwarzschild-AdS metric: ( dssch AdS 2 = 1 2M + r 2 ) ( r b }{{ 2 dt M r } V (r) + r 2 ) 1 b 2 dr 2 + r 2 dω 2 ; The apparent sing. r = r + solved if τ (Euclid. time) being angular coordinate with period β = 4πb2 r + b 2 + 3r+ 2, giving a natural temperature for the BH. 7 / 14

20 The AdS and Schwarzschild-AdS metrics AdS metric: ds 2 = (1 + r 2 b 2 ) dt 2 + (1 + r 2 ) 1 dr 2 + r 2 dω 2 b 2 ( b t Λ) 3 1/2, t t + 2πb; Schwarzschild-AdS metric: ( dssch AdS 2 = 1 2M + r 2 ) ( r b }{{ 2 dt M r } V (r) + r 2 ) 1 b 2 dr 2 + r 2 dω 2 ; The apparent sing. r = r + solved if τ (Euclid. time) being angular coordinate with period β = 4πb2 r + b 2 + 3r+ 2, giving a natural temperature for the BH. 7 / 14

21 The AdS and Schwarzschild-AdS metrics AdS metric: ds 2 = (1 + r 2 b 2 ) dt 2 + (1 + r 2 ) 1 dr 2 + r 2 dω 2 b 2 ( b t Λ) 3 1/2, t t + 2πb; Schwarzschild-AdS metric: ( dssch AdS 2 = 1 2M + r 2 ) ( r b }{{ 2 dt M r } V (r) + r 2 ) 1 b 2 dr 2 + r 2 dω 2 ; The apparent sing. r = r + solved if τ (Euclid. time) being angular coordinate with period β = 4πb2 r + b 2 + 3r+ 2, giving a natural temperature for the BH. 7 / 14

22 The AdS and Schwarzschild-AdS metrics The locally measured temperature is given by T loc = 1 b ; β 1 2M mp 2 r + r 2 b 2 Since β has a maximum for r + = r 0 = 3 1/2 b, T has a minimum, T 0 = (2π) 1 3 b. (2) 8 / 14

23 The AdS and Schwarzschild-AdS metrics The locally measured temperature is given by T loc = 1 b ; β 1 2M mp 2 r + r 2 b 2 Since β has a maximum for r + = r 0 = 3 1/2 b, T has a minimum, T 0 = (2π) 1 3 b. (2) 8 / 14

24 Path Integral, Euclidean Formulation and Partition Function The partition function in QFT for Euclidean time is Z = Dϕe I [ϕ] ; Setting ϕ 1 = ϕ 2 = ϕ and i(t 2 t 1 ) = τ 2 τ 1 = β, we get Z = ϕ e βh ϕ. Integrating over all ϕ, then Z = Tr exp( βh), (3) where the integral is taken over all fields that are periodic in imaginary time with period β. This is simply the partition function of ϕ at temperature T = β 1 ; The dominant contribution comes from the classical solution, δi = 0. Therefore, Z e I. 9 / 14

25 Path Integral, Euclidean Formulation and Partition Function The partition function in QFT for Euclidean time is Z = Dϕe I [ϕ] ; Setting ϕ 1 = ϕ 2 = ϕ and i(t 2 t 1 ) = τ 2 τ 1 = β, we get Z = ϕ e βh ϕ. Integrating over all ϕ, then Z = Tr exp( βh), (3) where the integral is taken over all fields that are periodic in imaginary time with period β. This is simply the partition function of ϕ at temperature T = β 1 ; The dominant contribution comes from the classical solution, δi = 0. Therefore, Z e I. 9 / 14

26 Path Integral, Euclidean Formulation and Partition Function The partition function in QFT for Euclidean time is Z = Dϕe I [ϕ] ; Setting ϕ 1 = ϕ 2 = ϕ and i(t 2 t 1 ) = τ 2 τ 1 = β, we get Z = ϕ e βh ϕ. Integrating over all ϕ, then Z = Tr exp( βh), (3) where the integral is taken over all fields that are periodic in imaginary time with period β. This is simply the partition function of ϕ at temperature T = β 1 ; The dominant contribution comes from the classical solution, δi = 0. Therefore, Z e I. 9 / 14

27 Path Integral, Euclidean Formulation and Partition Function From statistical mechanics, E = Z, S = β E + logz; β And the free energy (Helmholtz potential) is F = T log Z TI ; All of this can be generalized for curved spacetime considering the geometry as another field in the path integral. For our case, the action is I = 1 d 4 x g(r 2Λ). 16π 10 / 14

28 Path Integral, Euclidean Formulation and Partition Function From statistical mechanics, E = Z, S = β E + logz; β And the free energy (Helmholtz potential) is F = T log Z TI ; All of this can be generalized for curved spacetime considering the geometry as another field in the path integral. For our case, the action is I = 1 d 4 x g(r 2Λ). 16π 10 / 14

29 Path Integral, Euclidean Formulation and Partition Function From statistical mechanics, E = Z, S = β E + logz; β And the free energy (Helmholtz potential) is F = T log Z TI ; All of this can be generalized for curved spacetime considering the geometry as another field in the path integral. For our case, the action is I = 1 d 4 x g(r 2Λ). 16π 10 / 14

30 Hawking-Page phase transition What s is more stable? Thermal radiation or BH s? The idea is to compare the free energy of the Sch-AdS against AdS solution; The solution for EE with cosmological constant provides R = 4Λ. Then, the action becomes I = Λ 8π d 4 x g. (4) 11 / 14

31 Hawking-Page phase transition What s is more stable? Thermal radiation or BH s? The idea is to compare the free energy of the Sch-AdS against AdS solution; The solution for EE with cosmological constant provides R = 4Λ. Then, the action becomes I = Λ 8π d 4 x g. (4) 11 / 14

32 Hawking-Page phase transition What s is more stable? Thermal radiation or BH s? The idea is to compare the free energy of the Sch-AdS against AdS solution; The solution for EE with cosmological constant provides R = 4Λ. Then, the action becomes I = Λ 8π d 4 x g. (4) 11 / 14

33 Hawking-Page phase transition For AdS metric, we have I 1 = Λ β1 K dt r 2 dr dω 2 = Λ 8π 0 0 S 2 6 βk3 ; For the Schwarzschild-AdS metric, I 0 = Λ β0 K dt r 2 dr dω 2 = Λ 8π 0 r + S 2 6 β 0(K 3 r+); 3 (5) Matching the two metrics at r = K, we find a condition for β 1, β K 2 b 2 = β 0 1 2M K + K 2 b / 14

34 Hawking-Page phase transition For AdS metric, we have I 1 = Λ β1 K dt r 2 dr dω 2 = Λ 8π 0 0 S 2 6 βk3 ; For the Schwarzschild-AdS metric, I 0 = Λ β0 K dt r 2 dr dω 2 = Λ 8π 0 r + S 2 6 β 0(K 3 r+); 3 (5) Matching the two metrics at r = K, we find a condition for β 1, β K 2 b 2 = β 0 1 2M K + K 2 b / 14

35 Hawking-Page phase transition For AdS metric, we have I 1 = Λ β1 K dt r 2 dr dω 2 = Λ 8π 0 0 S 2 6 βk3 ; For the Schwarzschild-AdS metric, I 0 = Λ β0 K dt r 2 dr dω 2 = Λ 8π 0 r + S 2 6 β 0(K 3 r+); 3 (5) Matching the two metrics at r = K, we find a condition for β 1, β K 2 b 2 = β 0 1 2M K + K 2 b / 14

36 Hawking-Page phase transition Then, the difference for large K is We end up with: I πr +(b 2 2 r+) 2 b 2 + 3r+ 2 ; r + = b, I = 0; and T H T 1 = 1 πb For r + < b, I > 0 For r + > B, I < 0; While for thermal radiation, the free energy is negative and proportional to T 4 without gravity. With gravity, for T > T 2, it collapses to BH s. We could consider other fields as well, providing different T 2 s for the collapse. 13 / 14

37 Hawking-Page phase transition Then, the difference for large K is We end up with: I πr +(b 2 2 r+) 2 b 2 + 3r+ 2 ; r + = b, I = 0; and T H T 1 = 1 πb For r + < b, I > 0 For r + > B, I < 0; While for thermal radiation, the free energy is negative and proportional to T 4 without gravity. With gravity, for T > T 2, it collapses to BH s. We could consider other fields as well, providing different T 2 s for the collapse. 13 / 14

38 Hawking-Page phase transition Then, the difference for large K is We end up with: I πr +(b 2 2 r+) 2 b 2 + 3r+ 2 ; r + = b, I = 0; and T H T 1 = 1 πb For r + < b, I > 0 For r + > B, I < 0; While for thermal radiation, the free energy is negative and proportional to T 4 without gravity. With gravity, for T > T 2, it collapses to BH s. We could consider other fields as well, providing different T 2 s for the collapse. 13 / 14

39 Hawking-Page phase transition T < T 0 T 0 < T < T 1 T 1 < T < T 2 T > T 2 F AdS (no BH) F > 0 F < 0 F SchAdS (only BH) 14 / 14